Uncertainties have a Meaning: Quantitative Interpretation of the Relationship between ... · Uncertainties have a Meaning: Quantitative Interpretation of the Relationship between

Uncertainties have a Meaning:

Quantitative Interpretation of the Relationship

between Subsurface Flow and

Geological Data Quality

J. Florian Wellmann

Diplom (M.Sc. equiv.) in Geology

This thesis is presented for the degree of Doctor of Philosophy

of The University of Western Australia

School of Earth and Environment

2011

Abstract

Long-term predictions of subsurface flow are important for societal issues such as ground-

water flow, renewable and non-renewable energy resources, nuclear waste disposal and

CO2 sequestration. In complex realistic settings, numerical simulations are dependent on

the distribution of below-ground properties defined by geological models, constructed from

observed data. The quality of the data therefore directly influences the predictions of sub-

surface flow. To date, no framework exists that allows a direct evaluation of effects of data

quality on these flow fields. This thesis presents a first comprehensive method to analyze,

visualize, quantify, and couple geological data uncertainty to flow field predictions.

Methods are introduced to simulate and evaluate uncertainties in complex 3-D struc-

tural geological models. Based on probability distributions assigned to the underlying

data, realizations of the geological model are created with an automated implicit mod-

eling technique for complex 3-D spatial settings. The concept of information entropy is

applied to visualize and analyze uncertainties in the resulting geological models. Infor-

mation entropy values are a measure of the minimum number of geological units that can

exist at any point in the domain. In addition, measures of mean model entropy can be

used to derive quality estimates of the discretization required for geological modeling.

Techniques are described that significantly simplify the integration of complex geologi-

cal modeling into flow simulation, allowing an automatic update of flow and temperature

outputs when data in the geological model are added or changed. The new scripting meth-

ods enable an integration of geological modeling and flow simulations into one automated

workflow: based on a set of geological data, the geological model is constructed, then

mapped on a pre-defined grid structure, flow parameters are assigned to the grid cells

according to the geological unit, and the input file for the flow simulation is generated.

The flow simulation is then performed and results can be post-processed, analyzed and

visualized. The workflows utility is demonstrated with applications to testing the effects

of geologic scenarios and to determining mesh discretization.

In analogy to information entropy for interpretation of uncertainty in geological data, the

thermal entropy production is introduced to interpret uncertainty in fluid flow simulations.

Application to simple scenarios of conductive and convective heat transport shows that

local and global entropy productions provide a measure of the criticality and the mode

of heat transfer in a hydrothermal system. The measure is also related to the available

work. This valuable information for a whole system is encapsulated within a scalar value

allowing a simple comparison of a vast range of different flow realizations.

Finally, this thesis looks at the combination of all methods into one framework to identify

the influence of geological data uncertainties on a hydrothermal system. The methodology

is applied to a geothermal resource study in the North Perth Basin, Western Australia.

The structural geological setting consists of a deep (>12 km) sedimentary basin offset

by several faults. Coupled fluid and heat flow simulation shows that convection occurs

in the permeable layers within the basin, strongly affecting the temperature field. The

uncertainty of the geological model can neatly be encapsulated by the information entropy

measure. The influence of geological data quality on flow realizations is demonstrated along

vertical profiles and cross-sections. Although both, the geological model and the simulated

flow fields are clearly affected by geological data quality, regions of highest uncertainty do

not always coincide. However, the global measures of mean information entropy and

the spread of average thermal entropy productions indicate that the uncertainties are

correlated.

The results of the case study clearly show that it is possible to visualize, analyze, and

quantify the effect of geological data uncertainties on geological models and hydrother-

mal systems. The combination of stochastic geological modeling and coupled fluid and

heat flow simulations provides a detailed insight into uncertainties in the predicted flow

fields. Applying information entropy and thermal entropy production as global measures

to classify the uncertainty of the geological models and the hydrothermal fields enables a

new way forward to a system based uncertainty analysis in complex geological settings.

Finally, both measures can be interpreted quantitatively and therefore give uncertainties

a meaning.

Contents

1. Introduction 1

1.1. Relevance and context of work . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Geological models, flow simulations and uncertainties . . . . . . . . . . . . . 3

1.2.1. Structural geological models and associated uncertainties . . . . . . 3

1.2.2. Geological models as basis for subsequent flow modeling . . . . . . . 4

1.2.3. Influence of structural uncertainties on flow predictions . . . . . . . 4

1.3. Structure of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4. Own publications in the context of this thesis . . . . . . . . . . . . . . . . . 8

1.4.1. Publications directly relevant to the thesis . . . . . . . . . . . . . . . 8

1.4.2. Additional publications . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Brief Overview of Theory 11

2.1. Geological modeling with the potential-field method . . . . . . . . . . . . . 11

2.1.1. Implicit potential-field method and geostatistical interpolation . . . 13

2.2. Coupled simulations of fluid and heat flow . . . . . . . . . . . . . . . . . . . 14

2.2.1. Fluid flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2. Heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3. Combining conductive and advective heat flow equations . . . . . . . 18

2.2.4. Numerical solution of the flow equations . . . . . . . . . . . . . . . . 18

3. Uncertainty Simulation of Structural Geological Models 21

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1. Structural modeling methods . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2. Uncertainties in structural modeling . . . . . . . . . . . . . . . . . . 24

3.2. Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1. Method overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2. Construction of the initial geological model . . . . . . . . . . . . . . 28

3.2.3. Probability distributions for input data . . . . . . . . . . . . . . . . 28

3.2.4. Simulation of different input data sets . . . . . . . . . . . . . . . . . 30

3.2.5. Analysis and visualization . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1. Simple graben model . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2. Doming structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4. Information Entropy as a Measure of Uncertainty 43

ii Contents

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45


4.2.1. Visualizing uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2. Information entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.3. Entropy as a measure of fuzziness . . . . . . . . . . . . . . . . . . . 49

4.2.4. Total model entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.5. Application to uncertainties in geological models . . . . . . . . . . . 50

4.3. Geological modeling and uncertainty simulation . . . . . . . . . . . . . . . . 50

4.3.1. Type of geological modeling considered here . . . . . . . . . . . . . . 50

4.3.2. Uncertainty simulation for geological models . . . . . . . . . . . . . 51

4.4. Application of information entropy to visualize and analyze uncertainties . 52

4.4.1. Geological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.2. Uncertainty simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.3. Model 1: Visualization of model uncertainties . . . . . . . . . . . . . 53

4.4.4. Model 2: Uncertainty reduction with additional data . . . . . . . . . 54

4.4.5. Model 3: Geological hypothesis testing . . . . . . . . . . . . . . . . . 55

4.4.6. Model 4 and 5: Evaluate uncertainty reduction with additional data 56

4.5. Potential applications beyond visualization . . . . . . . . . . . . . . . . . . 58

4.5.1. Using mean entropy and fuzziness to compare models . . . . . . . . 58

4.5.2. Determination of representative cell sizes . . . . . . . . . . . . . . . 59

4.5.3. Entropies as convergence criteria for uncertainty simulation . . . . . 59

4.6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5. Controlling Flow Simulations with flexible Scripting Libraries 63

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.1. PySHEMAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.2. PyTOUGH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.3. Availability and licensing . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.1. Simplified model set-up with PyTOUGH . . . . . . . . . . . . . . . 72

5.3.2. Grid refinement study . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.3. Determining the onset of convection in a 3-D box . . . . . . . . . . . 75

5.3.4. Automatic determination of convection onset . . . . . . . . . . . . . 77

5.3.5. Using scripting in conjunction with other approaches . . . . . . . . . 78

5.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6. Link between Geological Modeling and Flow Simulation 83

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


6.2.1. From geological data to simulated flow field . . . . . . . . . . . . . . 87

6.2.2. Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . 91

Contents iii

6.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3.1. Example model in the North Perth Basin . . . . . . . . . . . . . . . 93

6.3.2. Testing of different cell discretization schemes . . . . . . . . . . . . . 95

6.3.3. Testing of different geological scenarios . . . . . . . . . . . . . . . . . 101

6.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7. Entropy Characterization of Hydrothermal Flows 109

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2. Entropy production in a thermo-hydraulic system . . . . . . . . . . . . . . . 111

7.3. Analysis of entropy production in conductive flow . . . . . . . . . . . . . . . 113

7.3.1. Basic considerations for the conductive case . . . . . . . . . . . . . . 113

7.3.2. Entropy production in a transient conductive system . . . . . . . . . 113

7.4. Analysis of entropy production in a convective system . . . . . . . . . . . . 114

7.4.1. Thermal entropy production and advective heat transport . . . . . . 114

7.4.2. Visualization of convective flow with entropy production . . . . . . . 116

7.4.3. Entropy production during the onset of convection . . . . . . . . . . 117

7.4.4. Relationship between Nusselt number and entropy production . . . . 119

7.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8. Effect of Geological Data Quality on Geothermal Flow Fields 123

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125


8.2.1. Generation of multiple flow realizations . . . . . . . . . . . . . . . . 125

8.2.2. Visualization and analysis of uncertainties . . . . . . . . . . . . . . . 126

8.2.3. Workflow of uncertainty analysis . . . . . . . . . . . . . . . . . . . . 127

8.3. Case study: North Perth Basin, Western Australia . . . . . . . . . . . . . . 129

8.3.1. Regional context, model scenarios and data quality . . . . . . . . . . 129

8.3.2. Analysis of uncertainties in the geological model . . . . . . . . . . . 131

8.3.3. Analysis of uncertainties in the flow fields . . . . . . . . . . . . . . . 137

8.3.4. Flow uncertainties for different geological scenarios . . . . . . . . . . 142

8.3.5. Entropy production as a measure of the system state variability . . . 144

8.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9. Key Findings and Further Outlook 149

9.1. Key findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.2. Further outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.2.1. System classification with thermodynamic measures . . . . . . . . . 150

9.2.2. Integrated geological and hydrothermal simulation . . . . . . . . . . 151

Bibliography 154

A. Improved Geothermal Resource Estimation 167

iv Contents

List of Figures

1.1. Graphical abstract: the overarching research question . . . . . . . . . . . . 2

1.2. Evolution of scientific paradigms towards more data-intensive studies . . . . 3

1.3. Example of structural influence on flow fields . . . . . . . . . . . . . . . . . 5

1.4. Organization of research questions addressed in this thesis . . . . . . . . . . 7

3.1. Simple and complex structural modeling settings . . . . . . . . . . . . . . . 23

3.2. Uncertainty types in structural geological modeling . . . . . . . . . . . . . . 25

3.3. Workflow for geological uncertainty simulation . . . . . . . . . . . . . . . . 27

3.4. Probability distributions for geological data . . . . . . . . . . . . . . . . . . 29

3.5. Simple graben model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6. Uncertainty visualization for Example model 1 . . . . . . . . . . . . . . . . 36

3.7. Map-based statistical analysis for elevation surface structures . . . . . . . . 37

3.8. Full 3-D dome structure model . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.9. Visualization of uncertainties in the dome model . . . . . . . . . . . . . . . 38

4.1. Shannon information entropy for 1 bit system . . . . . . . . . . . . . . . . . 47

4.2. Information entropy to visualize uncertainties in a spatial context . . . . . . 48

4.3. Geological model for information entropy calculations . . . . . . . . . . . . 53

4.4. Visualization of uncertainties for Model 1 . . . . . . . . . . . . . . . . . . . 54

4.5. Reduction of uncertainties with additional data in Model 2 . . . . . . . . . 55

4.6. Results of a geological hypothesis test, Model 3 . . . . . . . . . . . . . . . . 56

4.7. Testing the effect of additional drillhole data . . . . . . . . . . . . . . . . . 57

4.8. Mean entropy and unit fuzziness for the different models . . . . . . . . . . . 58

4.9. Evaluation of unit fuzziness for different numbers of cells . . . . . . . . . . . 59

4.10. Several fuzziness realizations for one geological unit . . . . . . . . . . . . . . 60

5.1. Conceptual model of the first example model . . . . . . . . . . . . . . . . . 72

5.2. Visualization of the simulated fluid and heat flow fields . . . . . . . . . . . . 73

5.3. Vertical temperature profiles at the center of the model . . . . . . . . . . . 75

5.4. Analysis of the onset of convection in a permeable layer . . . . . . . . . . . 76

5.5. Advanced experiments with PySHEMAT . . . . . . . . . . . . . . . . . . . 78

5.6. Study of the effect of varying permeability contrasts . . . . . . . . . . . . . 79

6.1. Workflow from geological data to simulated flow fields . . . . . . . . . . . . 86

6.2. Geological data considered in the workflow . . . . . . . . . . . . . . . . . . 88

6.3. Rectilinear cartesian grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

vi List of Figures

6.4. Model set-up of the geothermal simulation, Dandaragan Trough . . . . . . . 94

6.5. Cell discretization study: discretization types and simulation time . . . . . 96

6.6. Section through geological models for different discretizations . . . . . . . . 97

6.7. Section view of simulated temperatures for different discretizations . . . . . 98

6.8. Velocities for different discretizations . . . . . . . . . . . . . . . . . . . . . . 99

6.9. 3-D view of temperature field for different discretizations . . . . . . . . . . . 100

6.10. Modification of data points for the first geological scenario test . . . . . . . 102

6.11. Comparison of temperature fields for different geological scenarios . . . . . 103

6.12. Additional data points for the second geological scenario . . . . . . . . . . . 104

6.13. Simulated temperatures for second geological scenario . . . . . . . . . . . . 105

7.1. Entropy production during equilibration of a conductive system . . . . . . . 114

7.2. Entropy production during equilibration in a convective system . . . . . . . 115

7.3. Entropy production and temperature during onset of convection . . . . . . 118

7.4. Specific entropy production during convective equilibration . . . . . . . . . 119

7.5. Average entropy production during the onset of convection . . . . . . . . . 120

7.6. Relationship between average entropy production and Nusselt number . . . 121

8.1. Workflow for combining uncertainty analyses . . . . . . . . . . . . . . . . . 128

8.2. Geological model scenarios for uncertainty simulation . . . . . . . . . . . . 130

8.3. Probability visualization for Yarragedee formation . . . . . . . . . . . . . . 132

8.4. Interpretation of information entropy in a 1-D profile . . . . . . . . . . . . . 133

8.5. Information entropy in 1-D profiles for different scenarios . . . . . . . . . . 135

8.6. Visualization of information entropy for scenario 1C . . . . . . . . . . . . . 136

8.7. Total information entropy for different scenarios . . . . . . . . . . . . . . . . 137

8.8. Example of flow fields in a 1-D profile . . . . . . . . . . . . . . . . . . . . . 138

8.9. Temperature and coefficient of variability for scenario 2C in a cross-section 140

8.10. Mean and standard deviation of absolute velocities in a cross-section . . . . 141

8.11. Comparison of flow field and geological uncertainties in 1-D . . . . . . . . . 143

8.12. Entropy production during model equilibration for scenario 2C . . . . . . . 145

8.13. Comparison of finite values of entropy production . . . . . . . . . . . . . . . 146

8.14. Spread of entropy production and geological information entropy . . . . . . 147

9.1. Geologically based flow calibration . . . . . . . . . . . . . . . . . . . . . . . 152

9.2. Probabilistic estimation of heat in place . . . . . . . . . . . . . . . . . . . . 153

List of Tables

3.1. Standard deviations for the graben model . . . . . . . . . . . . . . . . . . . 35

3.2. Standard deviations for the dome model . . . . . . . . . . . . . . . . . . . . 38

4.1. Standard deviations for observation points . . . . . . . . . . . . . . . . . . . 54

6.1. Material properties used for the fluid and heat flow simulations . . . . . . . 96

8.1. Standard deviations for the case study . . . . . . . . . . . . . . . . . . . . . 131

viii List of Tables

Acknowledgements

I am deeply thankful to my main supervisor Klaus Regenauer-Lieb for a great scientific

guidance throughout the course of the PhD, for stimulating discussions, millions of ideas,

and unforgettable working retreats that somehow always ended up in snorkeling and surfing

sessions. Many thanks also go to Christa Lieb for all the patience she had when I was

invading at the weekend. Peter Cawood was a great inspiration as my second supervisor

for the first half of the PhD. I am very thankful that Lynn Reid took over the role of second

supervisor after Peter left UWA. Lynn was a wonderful help through the final stages of

the work, a source of constructive criticism but also of happiness and joy. And thanks to

her, I never got cold feet. Apart from my supervisors, I am very grateful that I had Frank

Horowitz as a mentor for so many questions of science, programming and — everything;

and especially for the fact that he always had an answer to the even most tricky question:

“42” (and here it is!). And talking of good humor: I am very thankful to Mary Gee for all

those stressful, and still so funny days of guiding hundreds of students through their first

steps in the field. An extra thank you also goes to Christine Riordan and Jo Francis for

guiding me through administrative tasks.

I am thankful for financial support through an Australian International Postgraduate

Research Scholarship (IPRS) and a Top-Up Scholarship by Green Rock Energy Ltd.

All my colleges and friends at the Western Australian Geothermal Centre of Excellence

and the University of Western Australia were of great importance for me during the years

of the study. Thank you all so much for all the great moments, wonderful barbies, great

discussions... so many of the things we experienced together will stay on my mind forever.

Very special thanks go to Oli for being such a great office mate and friend and to Soazig

for all the motivating coffee breaks. Thank you all, it was such a great time!

Very special thanks go to my family who has been of such a wonderful continuous

support for me for all my life, and especially during my PhD, even from a distance. Every

paper was celebrated, every little success was welcomed and admired. I am so happy to

have you as my constant support.

Finally, and most importantly, I am deeply grateful to my lovely wife Eva, for all her

kind help and understanding, and for getting along with my confusion. She was of such a

wonderful and selfless support that I wonder how I could ever give it back. But one thing

I know for sure: the days that she had to go climbing alone are over!

x List of Tables

1. Introduction

1.1. Relevance and context of work

Accurate predictions of subsurface flow systems are of major relevance for our society,

from aspects of groundwater flow to renewable and non-renewable energy resources, nu-

clear waste disposal and CO2 sequestration. In a recently published book, the National

Research Council of the US identified the detailed understanding of subsurface flow, and

specifically the requirement of precise models for long-term predictions of flow systems,

as one of the most important “Research Questions for a Changing Planet” (US National

Research Council, 2008). A variety of numerical methods have been developed to analyze

flow in complex realistic settings (e.g. Bundschuh and Arriaga, 2010). One fundamental

component of these simulations is the distribution of properties (e.g. porosity, density,

permeability). The spatial distribution of these properties in the subsurface is commonly

derived from geological models.

Geological models are representations of the structural setting in the subsurface and

contribute a first-order indication of the distribution of rock types and their properties

below ground. It is, therefore, reasonable to assume that uncertainties in the subsurface

structure will have a direct influence on the predicted flow fields. For example, recent

hydrogeological studies suggest that uncertainties in the conceptual model—of which the

geological model is a major part—are probably the most important source of uncertainty

for long-term flow predictions in realistic settings (Nilsson et al., 2007; Troldborg et al.,

2007; Refsgaard et al., 2011). Still, these types of uncertainties are rarely considered in

modern, full 3-D analyses of complex geological settings. The reason for this neglect is

two-fold: (i) there is no simple way to analyze and visualize uncertainties in complex 3-D

structural geological models, and (ii) no general method is available for an automated

processing of complex structural models into flow simulations, as a simple and straight-

forward way to test different geological scenarios in flow simulations.

Following the well-known dogma that “all models are wrong, but some are useful”1 (Box

and Draper, 1987), a crucial question is, therefore, how accurate simulated flow fields are

with respect to uncertainties in the geological model (fig. 1.1). The fundamental focus of

the work presented in this thesis is to derive quantitative methods that enable a comparison

of uncertainties in the geological model and in the flow fields in a meaningful way.

1The original quote in Box’ book on page 74 is: “Remember that all models are wrong; the practical

question is how wrong they have to be to not be useful.”

2 1. Introduction

The approach taken here is to combine stochastic uncertainty simulation for structural

geological models with coupled hydrothermal simulations. The development of techniques

for a combination of these, usually separate, areas will be a significant contribution to

knowledge. Furthermore, quantitative measures have to be determined that enable a quan-

titative interpretation of uncertainties in the geological model and the flow simulations for

a whole model. A successful quantitative uncertainty analysis of flow fields and geological

models with respect to uncertainties in the geological data set is an important step towards

a comprehensive analysis of uncertainties in subsurface flow. The results are, therefore,

directly applicable to important questions of subsurface flow, from CO2-sequestration to

geothermal exploration.

Figure 1.1.: The overarching research question of this thesis: how accurate are predictions of coupledhydrothermal flow processes with respect to uncertainties in the structural geological model?

The evaluation of data uncertainties in the context of integrated stochastic simulation

and modeling as taken in this thesis is a timely approach as it defines methods for data-

intensive scientific studies. From the early theoretical and empirical studies of physical

laws in the 17th century, to networks of scientific observatories and large-scale simulations

on supercomputers (Bell et al., 2009), scientific studies became more data-intense (fig.

1.2). This development is sometimes referred to as the “4th paradigm of science” (Hey

et al., 2009), related to the realization that a whole range of novel principles is required

to integrate the large amount of available data into coherent evaluations in a sensible way.

Data-intensive scientific studies are already performed in other branches of sciences, for

1. Introduction 3

example for long-term climate predictions and astronomical studies. The work of this

thesis provides important contributions with the definition of methods to combine raw

geological observations and coupled multiphysics process simulations, and with innovative

methods to analyze uncertainties.

21 centuryth

17 centuryth

low

hig

h

Time

Data

inte

nsity

Simulation

Empiricism

Theory

DataIntensiveScience

Figure 1.2.: Evolution of scientific paradigms towards more data-intensive studies

1.2. Geological models, flow simulations and uncertainties

The typical workflow for a flow simulation in a realistic setting consists of several steps.

Firstly, a digital geological model is established based on available geophysical and ge-

ological data and regional knowledge. This model is then processed into a format that

enables the numerical solution to the flow equations, on a discretized grid geometry. Usu-

ally, relevant rock properties are assigned to geological units and these properties are then

assigned to elements or grid cells with a specific lithology. Further important steps are

the definition of boundary conditions and specific simulation parameters for simulation

time. The discrete version of the geological model, together with the defined boundary

conditions, is sometimes denoted as the conceptual model of the flow problem (Bundschuh

and Arriaga, 2010). All these steps can contain uncertainties. In the context of this work,

uncertainties in the geological data and their influence on flow simulations is evaluated.

1.2.1. Structural geological models and associated uncertainties

A detailed understanding of the subsurface has always been essential for resource eval-

uations. The first systematic understanding of the spatial structure of the subsurface

began with the creation of the first geological map by William Smith in the 18th century

(Winchester and Vannoi, 2001). Since those days, geological mapping has become one

4 1. Introduction

of the most important methods in geology and is still extensively used today. The first

method in geological mapping was to infer deep structures from surface observations. But

in the search for ever deeper resources, and the requirement to understand more detail,

more techniques were developed to get a better insight into the structural setting below

ground. Today, a range of different methods are applied to infer the subsurface geometry.

These range from direct geological observations (in outcrops at the surface or in drillholes)

to interpretations of indirect datasets (e.g. seismics, potential-field data, well logs). All

available information is then combined in a geological model, which can be considered as

the logical extension of a geological map into the third dimension.

One aspect remained common, from the first geological maps to the highly complex

digital geological models created today: representations of the subsurface always con-

tain uncertainties. These uncertainties are due to a variety of reasons. They can be

broadly classified into the categories of (1) imprecision and measurement error, (2) in-

herent stochasticity, and (3) imprecise knowledge (Cox, 1982; Mann, 1993; Bardossy and

Fodor, 2001). Specific uncertainties depend greatly on the scale and type of data used to

create the geological model. In the context of structural modeling as considered in this

work, the most significant uncertainties are related to the uncertainties in the raw data, in

the interpolation and extrapolation of structures between known data points, and impre-

cise knowledge of structural existence (for further details, see section 3.1.2). Even though

it is widely accepted that uncertainties are omnipresent in geological models, to date, no

general approach exists to evaluate, analyze and visualize these uncertainties (Bardossy

and Fodor, 2004).

1.2.2. Geological models as basis for subsequent flow modeling

A common purpose of creating geological models is to use them to determine property

distributions for process simulations (Caumon et al., 2009). Simulations of physical pro-

cesses are commonly applied for subsurface studies, for example to understand contaminant

transport in groundwater, the distribution of oil in a reservoir or the processes involved in

geothermal systems. The physical processes can be described with equations. Depending

on the studied system, these include any type of thermal, hydraulic, chemical or mechanical

processes, or coupled equations for more complex problems. For any reasonably complex

realistic scenario, these equations cannot be solved analytically. However, many numerical

methods have been developed to solve the coupled problems numerically (Huyakorn and

Pinder, 1987; Holzbecher, 1998; Bundschuh and Arriaga, 2010).

1.2.3. Influence of structural uncertainties on flow predictions

As structural geological models are a main foundation for complex and realistic flow sim-

ulations, it can be assumed that uncertainties in the structural model have an influence

on predicted flow fields. This influence can be specifically substantial in coupled hydro-

thermal processes due to their non-linear behavior. An illustrative example is the onset

of convection in a porous medium heated from below. The theory of convection in porous

1. Introduction 5

media is well established (e.g. Nield and Bejan, 2006). An important point is the onset

of convection, as at this point (bifurcation) the flow behavior changes completely. This

is a typical non-linear effect, where small changes in some properties lead to a significant

change in the system. For a simple, homogeneous porous medium with a fixed temperature

difference between top and bottom, the point of onset of convection can be determined

analytically with a linear stability analysis (e.g. Turcotte and Schubert, 2002). A simple

example showing this important effect is presented in figure 1.3. The figure shows a part

of a simulated subsurface flow field in a vertical section for two scenarios with a slightly

different structure: in scenario 2, the permeable layer is slightly thicker than in scenario 1.

All other simulation parameters are exactly the same. This little difference in the struc-

tural setting leads to a completely different flow behavior and a very different subsurface

temperature distribution.

Impermeable

Permeablelayer, thicknessin order ofseveral 100 m

Impermeable

Permeable

Conductive Convective

Scenario 1 Scenario 2

Increasedthickness ofpermeable

layer

Convective heattransport sets indue to increase

in layerthickness

Hot

Cold

Hot

Cold

Extracted partsfrom larger model

Figure 1.3.: Example of structural influence on flow fields: the scenarios exemplify a vertical sectionthrough the subsurface, consisting of one permeable layer with a thickness in the order of severalhundred meters overlain by an impermeable layer. In scenario 1, the heat flow through the layeris purely conductive. For scenario 2, the thickness of the permeable layer is slightly increased. Allother parameters and settings are the same as before. This little structural difference leads to theonset of convection, an effect that completely changes the flow behavior and associated temperaturedistribution.

The example is a very specific and simple case. In a more realistic setting the inter-

action between structure and flow fields can be expected to be even more complex. For

example, recent studies by Nilsson et al. (2007) and Refsgaard et al. (2011) have shown

that in a realistic setting, different geological models can lead to significant differences in

the predicted flow fields. In a detailed study of the influence of geological uncertainties

on reservoir predictions, Suzuki et al. (2008) have shown how geological uncertainties in

6 1. Introduction

the reservoir volume effect the simulation of oil production, for several geological hypothe-

ses (i.e. structural models). These previous studies and the theoretical considerations

clearly show that the determination of the influence of structural geological uncertainties

on predicted flow fields is a relevant research question.

1.3. Structure of this thesis

In line with the regulations of The University of Western Australia, this thesis is organized

as a series of scientific publications. The publications are each investigating separate re-

search questions that lead up to the general hypothesis that the influence of uncertainties

in geological data on subsurface flow fields can be quantified. An overview of the organiza-

tion and the link between chapters is given in figure 1.4. After a brief introduction to the

relevant theory (chapter 2), these research questions are presented in different chapters:

➤ Chapter 3 addresses the problem of uncertainty evaluation in structural geological

models with a stochastic simulation method. This chapter was published in Tectono-

physics (Wellmann et al., 2010).

➤ The focus of chapter 4 is to evaluate if information entropy is applicable as a mean-

ingful measure to quantify uncertainties in structural geological models. This chapter

was published in Tectonophysics (Wellmann and Regenauer-Lieb, 2011).

➤ In chapter 5, an approach is presented that greatly simplifies the input file gener-

ation for flow simulations. This method is the basis for a completely integrated

workflow from geological input data to simulated hydrothermal flow fields (chapter

6). Chapter 5 has been accepted for publication after minor revisions in Computers

and Geosciences (Wellmann et al., 2011a). Chapter 6 is an extended version of an

extended conference abstract of the GeoProc 2011 (Wellmann et al., 2011b).

➤ In chapter 7 it is evaluated how thermal entropy production can be applied as a

thermodynamic measure to evaluate uncertainties in simulated flow fields. In sym-

metry to chapter 4, a main focus is to derive a meaningful quantity to classify an

entire flow system.

➤ The chapters before provide all required components to determine the influence of

geological data quality on uncertainties in geological models and subsurface flow

simulations. These aspects are combined in chapter 8 and applied to a case study of

a geothermal resource model in the North Perth Basin, Western Australia.

In addition to the research related to the quantification of uncertainties, methods were

developed for an optimization of geothermal resource evaluations based on geothermal

simulations. Two extended conference abstracts focused on this topic are presented in the

appendix.

1. Introduction 7

Principles for combininggeological modeling and flow

simulation

Information entropyas a measure of

geological uncertainty

Uncertainty simulationfor geological models

Automated scriptingcontrol for flow

simulations

Geological Modeling Flow Simulation Engine

Geological Uncertainties Integrated Simulation

Advanced geothermalresource assessment

Entropy production forhydrothermal flowcharacterization

Case study: Quantitativeinterpretation of uncertainties

Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7

Figure 1.4.: Organization of research questions addressed in this thesis: the chapters in this thesisfollow two main streams. In one stream, uncertainties in structural geological models are evaluated.In the other stream, methods are developed to combine geological modeling and flow simulations. Ina symmetrical way, the last chapters of each stream develop system-based measures to characterizea complete system: information entropy (chapter 4) and thermal entropy production (chapter 7). Inchapter 8, all methods are combined for an uncertainty analysis in a case study. A practical use ofthe developed methods is described in the Appendix.

8 1. Introduction

As the text of the following chapters is in the format of scientific publications, each

chapter contains a short introduction, a materials and methods section and an own dis-

cussion. Due to the presentation as publications, some overlap between the chapters can

unfortunately not be avoided.

1.4. Own publications in the context of this thesis

The work of chapters 3 to 6 of this thesis has been peer-reviewed, published in inter-

national scientific journals, or is in revision. The extended abstracts, peer-reviewed and

published in conference proceedings, contain aspects of this work that are directly rele-

vant to geothermal exploration. This work that was mainly presented at conferences as a

suitable venue to address industry and researchers in the field of geothermal energy. As

an illustration of this work, two extended abstracts are included in the appendix.

1.4.1. Publications directly relevant to the thesis

Wellmann, J. Florian, Adrian Croucher, Klaus Regenauer-Lieb, Simplifying and ex-

tending subsurface fluid and heat flow simulations using Python scripting libraries. Com-

puters and Geosciences, in revision.

Wellmann, J. Florian, Adrian Croucher, Lynn B. Reid, Klaus Regenauer-Lieb, From

outcrop to flow fields: combining geological modeling and coupled THMC simulations. In

review, GeoProc Conference 2011, Perth, WA, Australia, July 6-9, 2011.

Wellmann, J. Florian, Klaus Regenauer-Lieb, Uncertainties have a meaning: Infor-

mation entropy as a quality measure for 3-D geological models. Tectonophysics, In Press,

Accepted Manuscript, Available online 8 May 2011.

Wellmann, J. Florian, Franklin G. Horowitz, Eva Schill, Klaus Regenauer-Lieb, To-

wards incorporating uncertainty of structural data in 3D geological inversion. Tectono-

physics, Volume 490, Issues 3-4, 30 July 2010, Pages 141-151.

1.4.2. Additional publications

Wellmann, J. Florian, Lynn B. Reid, Franklin G. Horowitz, Klaus Regenauer-Lieb,

Geothermal Resource Assessment: Combining Uncertainty Evaluation and Geothermal

Simulation. Proceedings, AAPG Hedberg Conference, Napa Valley, California, USA,

March 14-18, 2011.

Wellmann, J. Florian, Franklin. G. Horowitz, Ludovic P. Ricard, Klaus Regenauer-

Lieb, Estimates of sustainable pumping in Hot Sedimentary Aquifers: Theoretical consid-

erations, numerical simulations and their application to resource mapping. Proceedings,

AGEC Conference, Adelaide, SA, Australia November 16-19, 2010.

Wellmann, J. Florian, Franklin G. Horowitz, Klaus Regenauer-Lieb,. Influence of

Geological Structures on Fluid and Heat Flow Fields. Proceedings, Thirty-Fifth Workshop

on Geothermal Reservoir Engineering, Stanford University, Stanford, California, February

1-3, 2010.

1. Introduction 9

Wellmann, J. Florian, Adrian Croucher, Franklin G. Horowitz, Klaus Regenauer-

Lieb,. Integrated Workflow enables Geological Sensitivity Analysis in Geothermal Simu-

lations. Proceedings, NZ Geothermal Workshop, 16-18 November, 2009

Wellmann, J. Florian, Franklin G. Horowitz, Klaus Regenauer-Lieb, Concept of an

Integrated Workflow for Geothermal Exploration in Hot Sedimentary Aquifers. Proceed-

ings, Australian Geothermal Energy Conference, Brisbane, 10-13 November, 2009

10 1. Introduction

2. Brief Overview of Theory

Overview

The theory parts in the main chapters of the thesis are intentionally brief, as they are

written in the format of scientific publications and only the most relevant theoretical

aspects are explained there. Therefore, a brief overview of the fundamental methods that

are applied throughout the thesis is provided here.

2.1. Geological modeling with the potential-field method

In this work, geological modeling, or geomodeling, is considered as the construction of dig-

ital structural representations of the subsurface (subsequently called geological models).

The purpose of geological models is to integrate all available information about the sub-

surface with relevant geological concepts to understand the structures below ground (e.g.

Caumon et al., 2009). The structural setting is itself the starting point for further analyses,

from subsurface flow to the development of mineral systems, to engineering questions.

Several methods exist for computer assisted geological modeling and the appropriate

method has to be chosen for each specific problem. Many methods are available to model

simple structural geological settings that can be described as “layer-cakes” (sometimes

called 2.5-D settings, see section 3.1.1 and figure 3.1), where the subsurface is a system

of sub-parallel, mostly sedimentary layers. This assumption is useful in the shallow sub-

surface in sedimentary basins, but not valid for larger-scale settings that almost always

contain more complex structures like fault networks, folding or doming structures (see

figure 3.1).

To model complex structural settings in the subsurface, two main approaches can be

distinguished: explicit, surface-based methods and implicit volumetric approaches. The

former methods are based on a concept of constructing a structural model with explicitly

interpolating surfaces that match the data for faults and boundary surfaces between geo-

logical units in the range of specified constraints (see Mallet, 1992, 1997). In a next step,

surface intersections are constructed, for example between a geological surface and a fault.

Finally, the volume of a geological unit is defined by its enclosing surfaces. The workflow

is described in detail by Caumon et al. (2009).

Explicit modeling methods are very flexible and can model a wide range of different

structures (e.g. Sprague et al., 2006; Bistacchi et al., 2008; Zanchi et al., 2009). They are

12 2. Brief Overview of Theory

widely applied in the petroleum and mineral industry and specifically useful to incorporate

large amounts of data, for example from 3-D seismic surveys and well logs. However,

explicit modeling techniques do not allow a direct model update when the input data set

is changed, even if most of the workflow can be automated (Kaufmann and Martin, 2008).

Implicit modeling techniques directly interpolate the structural model based on all avail-

able data, honoring several geological constraints (Caumon et al., 2009). The basic concept

is usually to interpolate a 3-D function that defines the geology in an abstract way. This

function can be described as a geological potential field (see Lajaunie et al., 1997; Calcagno

et al., 2008, and below). In this approach, boundaries between geological surfaces can be

considered as isosurfaces of the potential field. Several implicit modeling methods have

been developed to date. Mallet (2004) developed an approach based on paleogeographic

coordinates and depositional time, called GeoChron. In the GeoChron definition, a ge-

ological structure is mapped into a depositional space-time where subsequent geological

surfaces are interpreted as isosurfaces of depositional time. Frank et al. (2007) developed

an implicit technique for the reconstruction of complex geological structures from point

clouds. Another implicit potential-field method is implemented in the software LeapFrog

(Carr et al., 2001; Cowan et al., 2002; Milicich et al., 2010), based on an interpolation of

radial basis functions. Generally speaking, implicit modeling methods are recognized as

an important step forward in geological modeling (see discussion in Caumon et al., 2009).

For the purpose of this work, an implicit potential field method based on dual-kriging

(Lajaunie et al., 1997; Calcagno et al., 2008), implemented in the software GeoModeller

is applied. This method has been used to create complex 3-D models for a wide range

of problems, from large-scale modeling in mountain belts (Martelet et al., 2004; Maxelon

and Mancktelow, 2005; Maxelon et al., 2009) to lithospheric studies of magmatic intrusions

(Joly et al., 2008, 2009), to mineral exploration in cratons (McInerny et al., 2004b). This

methodology was considered appropriate for the analyses in this thesis because of the

following specifications:

➤ The method has been shown to create geologically realistic models based on a limited

amount of subsurface data (Putz et al., 2006);

➤ It is possible to automatically reconstruct a geological model when the input data

are moderately changed (McInerny et al., 2004a);

➤ The method can handle full 3-D geological settings of complex fault networks with

reverse faults, as well as overturned folding and dome structures (sec. 3.1.1);

➤ It is possible to access the interpolated model directly with an external program

interface, without requiring the graphical user interface. This is important for the

integrated workflow developed here (chapter 6).

The mathematical background for the potential-field method is briefly described below.


2.1.1. Implicit potential-field method and geostatistical interpolation

The theory of the potential-field method and the geostatistical interpolation are described

in detail in Lajaunie et al. (1997). An overview is also given in Chiles et al. (2004)

and Calcagno et al. (2008). These authors also describe the application of the method

for multiple interacting potential-fields to model erosional or onlapping structures and

complex fault networks.

The basic concept of the method is that a scalar potential-field U can be used to describe

subsequent layering of geological structures. Lajaunie et al. (1997) proposed that this

assumption is not only applicable to sedimentary systems where a layering structure is

the norm, but also to other rock types. Each surface between two geological units is an

isosurface of this potential field with a specific value uk. The value of the potential-field

for every surface is initially unknown and has to be interpolated from the available data.

Two types of data are considered (see Lajaunie et al., 1997):

(1) points with location vector ~p at the surface between two geological units:

U(~p) = uk ; (2.1)

(2) orientation measurements of the structure, for example the primary orientation in a

sedimentary layering, here interpreted as related to the gradients of the potential field:

∂U(~p)

∂x. (2.2)

The potential-field can be estimated with a cokriging method. Considering a total number

of surface points M and gradient values N , the estimator U⋆ is of the form (see Lajaunie

et al., 1997; Calcagno et al., 2008):

U⋆(~p) − U⋆(~p0) =

M∑

i,j

µα (U(~pi) − U(~pj)) +

N∑

β=1

νβ∂U

∂xβ(~pβ) (2.3)

In this equation, the increments of of the potential field U between a point ~p and a reference

point ~p0 are estimated from the knowledge that the field difference between two points

i, j ∈ M, i 6= j of the same surface, U(~pi) − U(~pj), should vanish, and that the gradients

of the field should equal the gradients from the observation data. Variables µα and νβ

denote weights that are part of the cokriging solution.

In order to solve this cokriging equation, a random function model is applied where the

potential field U is assumed to be a random function with a polynomial drift and a sta-

tionary covariance (Chiles and Delfiner, 1999). The covariance function itself is estimated

from the orientation measurements (Aug, 2004). Once the potential field is estimated,

the values uk of the potential field belonging to the geological surfaces can be determined

(eq. 2.1). Per definition, different isosurfaces of the scalar potential field can not intersect

each other. Hence, the succession of different geological units is always guaranteed, for


example the layering of sediments in a basin. Erosional and onlapping structures are mod-

eled with several intersecting potential fields, and faults are considered as discontinuities

in one field (Calcagno et al., 2008).

An interesting detail of this approach is that the interpolation uncertainty can be deter-

mined from the cokriging standard deviations. In the standard procedure implemented in

the modeling software, a dual form of the cokriging is applied because it greatly reduces

computation time. But Aug (2004) and Chiles et al. (2004) have shown that the standard

form can be used to analyze and visualize interpolation uncertainties. The method applied

in this thesis is to evaluate the uncertainties in the geological model due to uncertainties

in the input data. In the mathematical context described here, these are uncertainties in

the support point positions ~p and gradients ∂U(~p)∂x

(see also section 3.1.2 and figure 3.2).

2.2. Coupled simulations of fluid and heat flow

The flows of fluids and heat in the subsurface can be derived from principles of conservation

of energy and mass and the response to pressure and temperature differences. The typical

assumptions are presented below, together with considerations of scale and applicability.

Finally, the equations that are used in the numerical solver will be described, with a

short description of the numerical method itself. The derivation is following the excellent

textbooks of DeMarsily (1986), Phillips (2009), and Bundschuh and Arriaga (2010).

2.2.1. Fluid flow

Mass conservation

The principle of mass conservation states that the total mass is constant in a closed system.

If we consider an arbitrary volume V , a flux through the surface S therefore has to be

balanced buy a change in mass within the volume. The mass change can be associated

with a change in density or a source or sink. If we consider a porous medium where the

medium itself is incompressible and therefore its mass does not change with time, we can

write:∫

V

∂(φρf )

∂tdV +

∮

A

ρf ~qf · ~ndA =

∫

V

ρfSf dV . (2.4)

The first term represent changes in fluid mass in the volume V due to changes in density ρf

or porosity φ. The second term describes the fluid flux ~qf (in all subsequent equations, the

subscript f denotes fluid flux) through a surface A and the third term represents sources

or sinks of fluid mass within the volume, with a specific source strength per unit volume

Sf .

For the case that the volume is taken small enough (V ≈ dV ), equation (2.4) can be

written in the differential form of fluid mass conservation. Here, the continuity equation

in Eulerian coordinates is

∂(φρf )

∂t+ ∇ · (ρf ~qf ) = ρfSf . (2.5)


Simpler forms of the continuity equation can be derived for specific cases. For the steady

state case that fluid density and porosity don’t change with time:

∇ · (ρf ~qf ) = ρfSf . (2.6)

If, furthermore, the fluid is considered as being incompressible (constant density) and

no fluid sources or sinks exist, the conservation of mass can be written in its simplest

steady-state form as

∇ · ~qf = 0 . (2.7)

However, for an accurate description of non-isothermal compressible flow, the general form

(2.5) has to be applied.

Relationship between pressure and flow in porous media: Darcy’s Law

After considering the basic statement of conservation of fluid above, the next step is to

derive an understanding about the forces driving fluid flow in the subsurface. This is a

complex problem on a pore scale in a natural heterogeneous rock, as fluid flows through

a complex network of connected pores and fractures. Even though fluid flow can be

described on this scale using Naiver-Stokes equations, it is impossible to do this for large-

scale (> cm–m) models, mainly because it is not possible to obtain such huge amounts of

detailed data. The general approach to overcome this problem is to apply a continuum

approach (eg. Bear, 2007) assuming that, given a minimum scale, the rock properties

average out. Following the definition of Phillips (2009), the average length scale should

be chosen in a way that the properties of the medium vary smoothly, and that, for a

macroscopic study with length scale L and an internal macroscopic length scale l0:

l0 ≪ lAV ≪ L . (2.8)

The first experiments about the relationship between pressure and flow in porous me-

dia were performed by Darcy (1856), investigating the flow for the fountains of Dijion,

France. He realized the linear relationship between pressure difference and discharge rate

Qf through a porous media with area A:

Qf ∝ ∆P

∆xA . (2.9)

The constant of proportionality is k, the permeability of the medium. Furthermore, the

dynamic viscosity µf of the fluid has to be considered, and with Q/A = u:

u = − k

µf

∆P

∆x. (2.10)

This relationship is commonly known as “Darcy’s Law”, where u is the discharge per unit

area, sometimes also called transport or Darcy velocity. It is related to the mean interstitial


velocity vi by the porosity:

u = φvi . (2.11)

In a realistic 3-D system, considering hydrostatic pressure ρ0~g dz acting in the z-direction,

Darcy’s Law can be written as:

~u = − 1

µfk · (∇P − ρf~g) , (2.12)

where k is the permeability tensor. For many hydrogeological studies, it is convenient to

write Darcy’s Law as a function of hydraulic head h and hydraulic conductivity K (as it

was originally derived by Darcy):

~u = −K · ∇h − h + z

ρfK · ∇ρf ; (2.13)

where K =ρf g

µfk and the hydraulic head (or piezometric head) relates the energy of an

incompressible fluid to the height of a static fluid column above a reference height z0

with h = Pρf g

+ z0. The additional second term is required for cases of variable density

fluids, as, for example, the case in coupled hydrothermal simulations. Even if convenient

for hydrogeological analysis, this form of Darcy’s Law is not as useful for hydrothermal

simulations as the hydraulic conductivity K is itself a function of density and viscosity,

whereas permeability is an intrinsic property of the rock itself (e.g. Bundschuh and Arriaga,

2010).

Darcy’s Law is only valid for laminar viscous flows. The upper limit for the validity

under these conditions can be estimated from the Reynolds number Re = ~ud/ν, where d is

a length scale related to grain size and ν the kinematic fluid viscosity. Bear (2007) states

that there is broad evidence that Darcy’s Law is valid in most natural groundwater systems.

For laminar flow with higher Reynolds numbers, for example in karstic formations or near

wells and springs (Bear, 2007), non-linear motion equations can be applied (Forchheimer,

1901). However, these situations are not considered here.

From Darcy’s Law and the conservation of mass to the general fluid flow

equation

The general groundwater flow equation can now be obtained denoting the fluid flux ~qf

as the Darcy velocity ~u and combining the continuity equation (2.5) with Darcy’s Law

(eq. 2.12):

∂

∂t(φρf ) = −∇ · (ρf~u) + ρfSf

∂

∂t(φρf ) = ∇ ·

[

ρf

µfk · (∇P − ρf~g)

]

+ ρfSf (2.14)

This is one of the fundamental equations of fluid flow in porous media and is analogous

to the equation of heat diffusion described below.


2.2.2. Heat flow

Heat flow and thermal energy conservation

The principle statement of energy conservation is analogous to the statement of mass

conservation:∫

V

ρrcprdT

dtdV +

∮

A

~qh · ~n dA =

∫

V

ρrcprSh dV (2.15)

The first term describes temporal changes of heat in the volume, the second term the heat

flux ~qh (subscript h for heat) entering and leaving a domain enclosed by the surface A and

the third term sources or sinks with the specific heat source strength Sh in the domain. If

as before the divergence theorem is applied and we consider a small sub-volume, we derive

the continuity equation for heat transport:

∂

∂t(cpρrT ) = ∇ · ~qh + ρrcprSh (2.16)

Heat conduction in a medium: Fourier’s Law

Heat can be transported in essentially three ways: as conductive, advective and radiative

heat transport. For thermo-hydraulic process simulations, conductive and advective trans-

port have to be considered. The law of heat conduction is based on experimental studies,

similar to Darcy’s Law described above. Fourier, a French mathematician of the 19th

century, discovered the linear relationship between temperature difference and conductive

heat flow ~qc, expressed as Fourier’s law in 3-D as (e.g. Bundschuh and Arriaga, 2010):

~qc = −λ · ∇T , (2.17)

where λ is the tensor of thermal conductivity.

From Fourier’s Law and the conservation of thermal energy to the conductive

heat flow equation

Again in analogy to the groundwater flow equation, the equation for conductive heat trans-

port can be obtained from Fourier’s Law of heat conduction (2.17) and the conservation

of thermal energy (2.16):

∂

∂t(ρrcprT ) = ∇ · [λ · ∇T ] + ρrcprSh (2.18)

For the specific case that the rock is homogeneous and isotropic and rock density ρr and

heat capacity cpr do not change in time, the equation can be simplified:

∂T

∂t= κ∇2T + Sh , (2.19)

where κ is the thermal diffusivity. This equation is a typical diffusion equation and, for

example, equivalent to the equation for chemical diffusion.


Advective heat transport

A second important heat transport mechanism in geological systems is advective heat

transport. It occurs due to heat transported with fluid movement. The specific heat

content of a volume of fluid is the product of fluid density with specific enthalpy hf . The

advective heat transport qa is therefore:

~qa = ρfhf~u (2.20)

The specific enthalpy is related to the specific heat at constant pressure:

cp =

(

∂h

∂T

)

P

(2.21)

Under the assumption that heat transport occurs locally under small pressure changes we

derive the commonly applied equation for the advective transport of heat:

~qa = ρf cpf ~uT (2.22)

2.2.3. Combining conductive and advective heat flow equations

The equations for conductive (2.17) and advective (2.22) heat transport can now be com-

bined with the continuity equation for heat (2.16) and the consideration of heat sources

and sinks. The total heat flux ~qh in equation (2.16) is now the sum of advective and

conductive heat transport:

~qh = ~qa + ~qc = ρfcpf ~uT −∇ · T (2.23)

Equation (2.16) becomes:

∂

∂tcprρrT = ∇ · [λ · ∇T − ρf cpf ~uT ] + ρrSh (2.24)

2.2.4. Numerical solution of the flow equations

The equations stated before can only be solved analytically for simple cases. For the real-

istic modeling of complex 3-D heterogeneous systems, and considering feed-back mecha-

nisms and temperature dependency of properties, numerical methods have to be applied.

A range of different methods have been developed that are suitable to solve the differen-

tial equations numerically. Most common are Finite Difference (FD), Finite Volume (FV)

and Finite Element (FE) methods. A description of each of these solution strategies is

beyond the scope of this introduction, but many text books are available that describe

their implementation (e.g. Holzbecher, 1998; Konikow and Mercer, 1988; Bundschuh and

Arriaga, 2010). Also, a variety of software codes exists for each of these methods. Many

different codes are compared by O’Sullivan et al. (2001) for geothermal simulations and a

recent overview is also provided in the book of Bundschuh and Arriaga (2010).


General considerations for a suitable flow simulation code in the context of

this thesis

This work addresses the question of how uncertainties in geological data affect simulated

fluid and heat flow fields. The choice of the simulation code to address this question was

taken based on the following considerations:

➤ The simulation code has to simulate coupled thermo-hydraulic processes as expressed

in equations (2.14) and (2.24);

➤ It should be able to handle automatically created discrete formats of the geological

model (see section 6.2.1);

➤ For a realistic application, it should be able to handle a large number of cells for a

detailed consideration of geological structures;

➤ In order to include the simulation code into an automated framework as in the work

of this thesis, a code is required that can be executed on the command line and

without the requirement of a graphical user interface.

The main reason for the last two points is that the problem stated in this thesis is addressed

with a stochastic approach. Here a range of possible geological data sets are created that

are all used as an input for subsequent flow simulations, resulting in a large number of

different models. Considering additionally that the models should be used for realistic 3-D

models in the scale of several kilometers, a code is required that can easily deal with over

1 million cells and ideally can be executed on a supercomputer.

Several simulation codes fulfill these requirements. For the work presented here, the

FD code SHEMAT (Simulator for HEat and MAss Transport) was applied (see Clauser

and Bartels, 2003). Because it is a FD code, SHEMAT can handle several million cells

without significant computational requirements. It has been widely applied to geothermal

problems and studies in mineral systems (e.g. Kuhn and Gessner, 2006; Kuhn et al., 2006;

Gessner et al., 2009; Mottaghy and Rath, 2006).

A FE code was not considered as useful at this stage because it is not yet possible to

automatically create meshes for FE simulations from complex full 3-D geological models,

even though this might be possible in the near future (see sec. 6.2.1). In addition to SHE-

MAT, the methods described here were implemented and tested with the code TOUGH2, a

widely used code to simulate coupled multi-phase subsurface flow problems, from geother-

mal systems to CO2-sequestration. First tests of feasibility showed that TOUGH2 can

be used to address the problems stated in the thesis as well. However, the simulation

of flow fields with more than a million cells becomes computationally demanding with

TOUGH2. It should, still, be possible to combine the methods developed in this thesis

with a parallelized version of TOUGH2 in future work.


3. Uncertainty Simulation of

Structural Geological Models


simulation





simulations





Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7



Chapter 3

22 3. Uncertainty Simulation of Structural Geological Models

Overview The next two chapters address the evaluation, visualization and quantifica-

tion of structural uncertainties in complex geological models. In this chapter, a method is

described for a stochastic simulation of geological models. The method is directly based

on discrete geological observations of surface contacts and orientation measurements. A

probability distribution is then assigned to all data points. From the data and the as-

signed distributions, several realizations of probable input data sets are generated. For

each of these data sets, a geological model is automatically computed, applying a standard

geological modeling technique based on a potential-field approach. The whole procedure

is completely automated. This method is simple and straightforward, and can still be

used even for complex geological models like overturned folds or complex fault systems.

In the context of this thesis, the method is the important first technical development that

is required to evaluate the effect of geological data uncertainties on simulated flow fields.

The text of this chapter was published in Tectonophysics (Wellmann et al., 2010).


3.1. Introduction

Three dimensional models of the subsurface structure (geological models) require quality

estimation for interpretation and further use. To date, no comprehensive 3-D approach

exists to asses uncertainties in a geological model (Turner, 2006). Here, we present a

statistical simulation method aiming to assess and communicate the accuracy of complex

3-D geological models based on uncertainties in the input data. As input data, we denote

here data typically used for structural modeling, i.e. contact points for formation and fault

surfaces and orientation measurements.

Our uncertainty evaluation differentiates itself from previous approaches (e.g. Thore

et al., 2002; Tacher et al., 2006; Bistacchi et al., 2008; Suzuki et al., 2008) since we do

not consider a probabilistic measure of the quality of the raw data in the final model.

Rather, we directly simulate the effect of data uncertainty in full 3-D. This complements

and extends typical geostatistic procedures (e.g. Chiles and Delfiner, 1999; Deutsch, 2002)

and is a first step towards 3-D geological inversion. A complementary approach to combine

geophysical and geological inversion based on the definition of misfit functions has been

developed by Jessell et al. (2010).

complexrelationships

reverse faults(a) (b)

(c) (d)

normal faults

every contact defined only once in drillholes contacts partly more than once

more than once and partly overturned

dome structure

envelope of dome structure at more than one point

overturned folds

Figure 3.1.: Simple and complex geological settings in a modeling sense; (a) normal faulting in alayer-cake model, e.g. a basin setting; no complications for elevation surface modeling methods (b)reverse faulting, e.g. an inverted basin; due to the reverse faulting, formations are doubled at somelocations (c) overturned fold; again, the folded formation exists at more than one (x,y) point inspace; the same applies for (d) a salt dome or magmatic intrusion.

Our technique is based on a recent geological modeling method (Calcagno et al., 2008;

Guillen et al., 2008). Their method allows model construction directly from input data,

once all parameters are defined. It also incorporates geological laws and is capable of

dealing with complicated geological settings like domes and overturned folds (fig. 3.1).

3.1.1. Structural modeling methods

3-D structural modeling usually aims to create a digital model representing the geometry of

structures in the subsurface, from a microscale to a crustal scale. The modeling is typically


based on different types of data like boreholes, geological maps or seismic reflection surveys.

There are several approaches to the construction, modeling and representation of geo-

logical objects. These approaches mostly differ in the details of their interpolation of data

and in their ability to represent complex structures in 3-D. Several common approaches

represent geological structures as elevation surfaces (see fig. 3.1a). Here, one point on

a surface is associated to a specific reference (u, v), for example a geographic position.

The coordinates of the point are calculated with functions (usually polynomial) for each

coordinate direction, e.g. x = f1(u, v), y = f2(u, v), z = f3(u, v). In these cases, the

surface can be imagined as a plane that can be completely projected onto the flat refer-

ence plane. These interpolation methods are sometimes referred to as 2.5-D methods (e.g.

MacEachren et al., 2005; Wu et al., 2005; Bistacchi et al., 2008; Caumon et al., 2009).

However, the term 2.5-D can be ambiguous as it is also used to describe methods where

the geology is defined in a section (e.g. a X-Z-section) and with extrusion perpendicular to

it (in Y-direction) as done for section balancing (e.g. Galera et al., 2003; Moretti, 2008) or

geophysical modeling (e.g. Sander and Cawthorn, 1996; Malengreau et al., 1999; Truffert

et al., 2001).

Geostatistical approaches can be applied (e.g. Goovaerts, 1997; Chiles and Delfiner,

1999), allowing for more geologically reasonable results. A drawback of these techniques

is their difficulty in handling some complex geological structures such as reverse fault-

ing (fig. 3.1b), overturned folding (fig. 3.1c), or the complexities of doming structures

(fig. 3.1d). For such structures, approaches that can handle structures independent of

their orientation are required. One common approach is the construction of triangulated

irregular nets (TINs). The geological structure is interpolated explicitly between defined

points and it is possible to model any type of structure. These surface methods are imple-

mented in a variety of software packages (e.g. Turner, 2006; Howard et al., 2009; Kessler

et al., 2009; Wycisk et al., 2009). Similar approaches have been developed for direct

volume construction using Voronoı cells (Courrioux et al., 2001). However, for complex

structures, closely spaced sections and a high data density are required.

Implicit function techniques overcome some of these limitations. In such techniques,

geological surfaces are represented as isovalues of certain functions defined everywhere in

space. The functions are constructed from the locations and attitudes of measured data

points, along with other external constraints (e.g. Lajaunie et al., 1997; Mallet, 1992; Frank

et al., 2007). These methods are commonly used in various geological modeling packages.

Specifically, for our uncertainty simulations, we apply the implicit function approach that

is based on an interpolation of a harmonic potential field (Lajaunie et al., 1997; Calcagno

et al., 2008). This approach enables the fast construction of a model directly using position

and orientation data. This is described in more detail below (sec. 3.2.2).

3.1.2. Uncertainties in structural modeling

Almost all types of geological data are subject to several sources of uncertainty (e.g. Mann,

1993; Davis, 2002). These include measurement inaccuracies, sampling limitations, insuf-


ficient sample numbers, imperfect concepts and hypotheses, the need for simplifications,

heterogeneity, inherent randomness and many others. These different types of uncertain-

ties can be broadly separated into three categories (Cox, 1982; Mann, 1993; Bardossy and

Fodor, 2001): (1) imprecision and measurement error, (2) stochasticity, and (3) imprecise

knowledge.

We adapt the classification of Mann (1993) to the case of structural modeling. Typical

examples are presented in figure 3.2:

Type 1 (error, bias, and imprecision): uncertainty in all types of raw data that are used

for modeling, like the position of a formation boundary or the orientation of a

structure (fig. 3.2a).

Type 2 (stochasticity, and inherent randomness): this commonly shows up as the un-

certainty in interpolation between (and extrapolation from) known data points

(fig. 3.2b).

Type 3 (imprecise knowledge): applies to incomplete and imprecise knowledge of struc-

tural existence, general conceptual ambiguities and the need for generalizations

(fig. 3.2c).

With fault

Uncertainty at points

Accurate position Uncertainty away from points

Without fault

(c) Type 3: Problem of uncomplete knowledge of structures in subsurface

Fault

(b) Type 2: Uncertainty of interpolation and extrapolation away from known points

(a) Type 1: Ambiguity of structure based on uncertainties in raw data

Contact location uncertainty

Actual position of formation surface Possible ambiguous interpretations

Contact location uncertainty

Figure 3.2.: Adapting the classifications of Mann (1993) to the uncertainties in structural modeling; (a)Interpretation of a geological formation boundary based on ill-defined input data points (i.e. wherethe contact position itself is uncertain) and resulting uncertainty in the interpreted boundary (b)Uncertainty of interpolation between and extrapolation away from known data points (c) Incompleteknowledge of structures in the subsurface, e.g. does a fault exist or not.


In the current literature, a comprehensive treatment of all types of uncertainties in 3-D

structural geological modeling is still lacking. Type 1 uncertainties can be handled with

geostatistical methods for simple structures (Chiles and Delfiner, 1999). Even when the

gross uncertainties have been reduced with seismic data, the remaining uncertainty affects

the resulting models (e.g. Thore et al., 2002; Glinsky et al., 2005). But to date, there is

no method to evaluate the influence of different data uncertainties, including orientation

measurements, in complex geological settings.

Uncertainties of Type 2 have been addressed with a variety of statistical and geostatis-

tical methods (Chiles and Delfiner, 1999; Deutsch, 2002; Davis, 2002, and others). Un-

certainties of Type 3 are more difficult and sometimes impossible to evaluate (e.g. Mann,

1993; Bardossy and Fodor, 2001).

Generally, all three types of uncertainties can be addressed by the formulation of a

suitable inverse problem. With the method presented here, we are concentrating on Type

1 uncertainties, based on error, bias and imprecision. But the method is equally suitable

to analyze special cases of Type 3 uncertainties.

3.2. Materials and methods

3.2.1. Method overview

We define geological uncertainties in the following section as uncertainties based on the

position, orientation, and interpretation of contextual geological information and data.

We also explicitly investigate the influence of raw data uncertainties on the model quality

while assuming that the applied interpolation is correct (sec. 3.4).

Our procedure to simulate uncertainty in a model is simple and straightforward. It can

generally be separated into five steps (fig. 3.3):

Step 1: Construction of an initial 3-D geological model. This model is considered as the

best possible model that can be constructed using all available input data (for-

mation and fault positions and orientation measurements). All relevant modeling

parameters are defined in this model (e.g. model extent, geographic projection,

digital elevation model) as well as the fault network and the stratigraphy.

Step 2: Identification of the input data quality and assignment of probability distribu-

tions. This can be based on either direct observations or statistical inference

and educated guess. Both data positioning and orientation uncertainties can be

considered.

Step 3: Simulation of several input data sets based on the raw data in the initial model

and the assigned probability distributions of the previous steps.

Step 4: Construction of multiple geological model representations with the simulated input

data sets.

Step 5: Visualization, processing and analysis of results.


Step 1: Construct initial model

Step 2: Define uncertainties

Step 3:Create different input data sets

Step 4: Model multiple realisations

Step 5: Analyse and visualise uncertainties

Boundary position Orientation measurements

Figure 3.3.: Simplified work-flow for the five steps of our approach: (1) Initial model construction fromformation boundary position and orientation measurements and (2) defined probability distributionsare used to (3) simulate different data sets for (4) multiple model realizations. These are thenprocessed to (5) analyze and visualize model uncertainties.


All important steps are described in detail below.

3.2.2. Construction of the initial geological model

The initial model is the starting point for our analysis. We apply an implicit potential-

field approach for the geological modeling (Lajaunie et al., 1997). The interpolation it-

self is based on universal cokriging (Chiles et al., 2004). It is implemented in the soft-

ware GeoModeller/ Editeur Geologique developed by BRGM and Intrepid Geophysics

(http://www.geomodeller.com). In two recent publications, Calcagno et al. (2008) and

Guillen et al. (2008) describe all relevant features. The method has been applied in com-

plex geological settings (e.g. Martelet et al., 2004; Maxelon and Mancktelow, 2005; Joly

et al., 2008) and produces reasonable geological models based on limited input data (Putz

et al., 2006).

The main advantage of this approach is that geological models are created based directly

on the geological contact and orientation data. Once the model is set-up and structural

and stratigraphic relationships are defined (Calcagno et al., 2008). It is thus possible to

regard a geological model M solely as a computed function C of the input data set ~k as

M = C(k1, k2, . . . , kn) = C(~k) (3.1)

The components ki of the input data set can be any one coordinate x, y or z of a

boundary point P(x, y, z) or the location or attitude parameters (azimuth ϕ and dip θ) of

a structural measurement O(x, y, z, ϕ, θ) for both formations and faults.

The ability to directly model the effect of a changed or augmented input data set, even

in complex geological settings while honoring defined stratigraphic relationships and fault

systems, makes this approach an ideal tool for our uncertainty simulation.

3.2.3. Probability distributions for input data

The next step is identifying uncertainties and assigning probabilities to the model input

data set ~k. Geological and geophysical data used for modeling are never absolutely accurate

(e.g. Davis, 2002). Uncertainties range from the position at the surface to the estimation

of dip angles at depth. All types of uncertainty have a specific effect on a 3-D model

depending on model scale and complexity. An uncertainty of 10 meters may be irrelevant

in a model at regional scale but significant for a high-resolution local-scale model.

Every geologist is well aware of the raw data problems encountered when constructing

maps and cross-sections. Geophysicists know of similar problems, for example when pro-

cessing seismic data. It may be possible to derive a probability distribution directly from

repeated measurements, for example for variation of dip and strike in orientation measure-

ments. For example, Bistacchi et al. (2008) present some considerations about uncertainty

quantification related to faults and folds when observed in the field. Other aspects of un-

certainties in raw data have also been analyzed, e.g. for seismic processing (Thore et al.,

2002; Glinsky et al., 2005) and for the problem of geological complexity (Tacher et al.,


2006). Still, a general quantitative treatment of uncertainties in the context of structural

modeling is not available to date. Therefore, probability distributions sometimes have to

be assigned based on heuristic knowledge and educated guess.

In our uncertainty simulation, we consider the positions of bounding points and orien-

tations of bedding or other planar elements in formations and faults. These points and

orientations can be derived from any type of data (outcrop observations, seismic data,

boreholes).

In many geological situations, it is reasonable to express the uncertainty as a normal

distribution with a standard deviation σ around an expected value µ (e.g. Davis, 2002).

A typical case is that the contact between two formations in a drillhole or outcrop is not

easily recognized (fig. 3.4a).

In our approach, we usually assume the expected value of the probability distribution

to be the data value of ki of the initial model. The probability distribution p(ξi) for a

value ξi drawn from this population is then

p(ξi) =1

σ√

2πe−

(ξi−ki)2

2σ2 (3.2)

ki

range a

p( )xi

xi

s

ki

k+bi

a

b

(a) (b)

gradual contact direct contact missing

uncertainty aboutcorrect position

(c)

p( )xi

p( )xi

ki

xi

xi

Typical Probability Distributions

Figure 3.4.: Examples of problems for the direct determination of a formation boundary and possibleprobability distributions, e.g. in a drillhole: (a) Contact is gradual: normal distribution (b) Contactitself is missing or its position is uncertain: continuous uniform distribution (c) Two (or more) discretepositions for a contact are possible: discrete probability distribution

But even if a normal distribution is commonly used, other statistical distributions may

also be reasonable in geological circumstances. A continuous uniform distribution should

be used where all points in a finite interval are equally probable. This might be the case

when a contact between two formations is not outcroping or when a segment is missing in

a drillhole (see fig. 3.4b), expressing the notion that there is no reason to assume that a


boundary located at one point is more likely than at another. The probability distribution

for one point ki with an uncertainty range of a in both directions can then be described

as

p(ξi) =

{

12a

for ki − a ≤ ξi ≤ ki + a

0 for all other cases(3.3)

Another typical case is that some discrete points are possible positions for one data

point. In this case, a probability αi can be assigned to every possible discrete value such

that∑

i

αi = 1 (3.4)

An example is the determination of a geological contact out of a wireline well-log (e.g.

gamma-ray log). Often, an uncertainty exists of which peak in the log might correspond

to the exact contact even if every peak itself is well defined (see fig. 3.4c). If, for example,

there are two possible positions for one parameter, with a probability of α1 ∈ [0, 1] for

the first position (which can be the position in the base model, ki) and a probability of

α2 = 1−α1 for the second position (with an offset of b to ki), we can describe the discrete

probability distribution as

p(ξi) =

α1 for ξi = ki

α2 for ξi = ki + b

0 for all other cases

(3.5)

In other cases, a combination of the above described probability distributions might be

required, for example a normal distribution for two different mean values at two possible,

but inexact, positions along the well-log. Another distribution which is applicable for some

types of geological data is the lognormal distribution (e.g. Davis, 2002).

Special care has to be taken for the implementation of dip and azimuth uncertainties.

For simple cases with small angular changes not too close to 0◦ and 90◦, a statistical

distribution like the ones discussed above can be used. In more complex cases, a spherical

distribution should be applied (Fisher, 1953; Davis, 2002).

3.2.4. Simulation of different input data sets

After the initial model is set up and uncertainties of orientation and contact data are

evaluated, we can start the simulation of new input data sets. This is performed in a

straight-forward way:

Step 1: We use our initially created base model Mbase and its data set ~kbase as starting

point for our simulation (see sec. 3.2.2). This base input data set consists of surface

and boundary positions of formations or faults, and the position and attitude of

structural measurements. All or a subset of these values can be changed. All

modeling parameters (e.g. model space, formations, etc.) are kept during the

simulation.


Step 2: For every simulation run ν of a total number of simulations n, we create a new

model data set ~kν where every data point ki,ν of the set ~kν represents a random

sampling from its associated probability distribution

ki,ν = p(ξi) (3.6)

Step 3: We repeat step 2 to create n new datasets. The number of created datasets

depends on the required level of significance for the analysis and the available

computation power (see below).

Step 4: We compute a geological model Mν for every simulated dataset:

Mν = C(

~kν

)

(3.7)

Step 5: After creating these different models, we can compare model results or process the

models for visualization.

The simulation workflow is implemented in a python module to process the original

input data structure and create new input data sets that can directly be recomputed. The

workflow itself is also suitable for computational parallelization. The complete uncertainty

simulation process would then only take slightly longer than a normal model run.

The n realizations of the model all represent possible subsurface models given the initial

model and assigned data probability distributions. We can now use statistical methods to

analyze and display the model uncertainties.

3.2.5. Analysis and visualization

The choice of methods to analyze and visualize the simulated uncertainties depends on the

complexity of the geological setting, the model configuration, and the further use of the

model. We can distinguish between an overall visualization of the model uncertainties,

the analysis of uncertainty of a specific model feature and the processing of all model

realizations to further simulation and inversion tools.

Different approaches to visualize location uncertainties have been proposed and dis-

cussed in literature (see MacEachren et al., 2005, for an extensive review). These ap-

proaches are mainly developed for map representations. Similarly, Viard et al. (2007)

have developed techniques to represent uncertainties in stratigraphic grids, specifically

suited for geological and petrophysical parameters. Still, even if displayed in 3-D, these

analyzes are only suitable for elevation surface/ 2.5-D situations.

Most of these visualization concepts can be extended for 3-D analyzes, for example

maximum likelihood, multiple realizations and multinominal probability fields (Goodchild

et al., 1994a). We present here the application of standard approaches and propose specific

adaptations to 3-D setting.


A useful method is to display the results of all model realizations simultaneously (e.g.

Goodchild et al., 1994a; MacEachren et al., 2005). For example, it can be convenient to

plot all simulated formation top surfaces in one cross-section or map. This delivers a good

first insight into the effect of uncertainty on the model. A similar possibility is to plot all

model realizations along a virtual drillhole at a location.

Simple analyzes of the uncertainties can be based on these visualizations. For example,

we can evaluate the min/max surfaces of formations and plot them into cross-sections,

map-views and along virtual drillholes. We can extend this into three dimensions and

calculate the min/max envelope surfaces of all realizations.

Further analyzes and visualizations are possible using statistical evaluations. One suit-

able analysis is the determination of the median for a modeled structure, e.g. a formation

top, out of all realizations. Assuming a normal distribution of a simulated surface, it is

furthermore possible to calculate a mean surface and the standard deviation at every point

in space.

In simple cases where the structure can be treated as an elevation surface (fig. 3.1a),

it is convenient to visualize these analyzes in a map view (e.g. MacEachren et al., 2005).

But if the structural setting of the model is more complex, this is no longer possible and

we need special procedures to analyze and visualize uncertainties. One suitable method is

the use of indicator functions as the basis for further analyzes. The indicator function of

a formation F is a subset of the whole model space defined as

IF (~x) =

{

1 for ~x ∈ F

0 for ~x /∈ F(3.8)

for all points ~x in the model domain. For practical reasons, we can quantize every model

realization onto a grid and calculate the indicator function for every grid cell G. For the n

simulated models, we thus obtain n indicator fields IFν (G) for every modeled formation.

This formal definition allows for a wide range of further analyzes. For example, the minimal

and maximal extent of a structure in its discretized grid form can be expressed as indicator

fields IFminand IFmax and directly be calculated as

IFmin(G) =

∏

k∈n

IFk(G) (3.9)

and

IFmax(G) = 1 −∏

k∈n

(1 − IFk(G)) (3.10)

Where (1 − IFk(G)) is the complement of the indicator function.

We can also use the indicator functions to derive an indicator probability function

PF (G) ∈ [0, 1] for every formation in space. This function contains probability estimates

for the formation represented by the indicator at every location and can be derived from


the mean value of the local indicator functions as

PF (G) =∑

k∈n

IFk

n(3.11)

This definition is similar to the multinominal probability field definition of Goodchild

et al. (1994a). The result is a scalar field in three dimensions and we can use it to derive a

contour of a probability threshold for one formation, e.g. the 95% probability to encounter

a formation. This can be visualized as an isosurface in 3-D or as 2-D isolines in a section.

We can use the probability indicator field for a more detailed analysis of accuracy. The

gradient of this field, ∇PF , is a vector field pointing out the main direction and rate of

change at every point in space. If we take the absolute value of this vector field

AF = ‖∇PF ‖ (3.12)

we obtain an estimate in full 3-D space about the total rate of change in the probability

field. Coming back to our goal, i.e. the identification of accuracy in the model, we can

interpret the derived scalar field AF as:

➤ High values ⇒ steep gradient in the indicator probability field ⇒ high probability

to encounter the boundary of formation F

➤ Low values ⇒ shallow gradient in the indicator probability field ⇒ low probability

to encounter the boundary of formation F

If we apply this method to all formations, it gives us a direct indication of the overall

accuracy in the whole model domain.

Another possible evaluation of uncertainties in more complex cases is to analyze a specific

feature of the model, for example the volume of a modeled formation. One way to calculate

the total volume of a formation is again based on the indicator functions defined above

(eq. 3.8). We can estimate the total volume of a formation as the sum over all grid cells

multiplied by the grid cell volume:

VFν ≈∑

g∈G

{IFν (g)V (g)} (3.13)

Taking the volume of all simulated models for one formation, we can perform further

statistical evaluations, plot volume histograms for all realizations or use these results

directly in business models that rely on an estimate of this property, for example ore

tonnage.

3.3. Results

We apply our method to two generic 3-D geological models. The first model is a typ-

ical example of a graben structure with tabular sedimentary formations cut by normal

faults. This is a simple setting (in a modeling sense) and all analyzes and visualizations


can be performed on an elevation surface basis (see fig. 3.1a). The second example is a

dome structure, and is more challenging in both modeling and visualization as complex

relationships between structures exist (fig. 3.1d).

Both examples are common geological structures. Even if simplified for the purposes of

this work, they are ideal to demonstrate the relevant steps in our uncertainty evaluation,

possible applications, and challenges of analyzing and visualizing the results.

For a detailed description of all aspects concerning the modeling itself, please refer to

Calcagno et al. (2008).

3.3.1. Simple graben model

Model setup

With our first example, we want to demonstrate the application of our uncertainty eval-

uation in a simple 3-D geological model. The geological setting is a graben structure

with non-planar normal faults cutting and displacing sub-horizontal sedimentary forma-

tions (fig. 3.5). In terms of modeling, this is an elevation surface setting that could be

treated with many typical modeling methods (see sec. 3.1.1). All data are defined in two

cross-sections (insets in fig. 3.5) which could, for example, be derived from seismic cross-

sections. The contact points are at the top surface of a formation or fault, structural

symbols indicate orientation of faults and geological surfaces at these positions. General

modeling settings are that the formation surfaces of the sedimentary pile are sub-parallel.

The model stratigraphy consists of four onlapping formations, Formation 1 (oldest) to

Formation 4 (cover layer, transparent in fig. 3.5). Both faults are infinite in extent and

affect all formations. The model covers an area of 2000 m in East-West direction, 1500 m

in North-South and has a depth of 1000 m from the surface.

Sedimentary Pile(top formation transparent)

Fault East Fault West

E-W: 2000 m

N-S: 1

500 m

z:

10

00

m

Formation 1

Formation 3

Formation 2

Model 1: simple graben setting

Figure 3.5.: Evaluation Model 1: Simple graben setting, sedimentary sequence cut by two faults; dataare defined in the cross-sections (insets).


Simulation parameters and settings

After the initial model is constructed, we can introduce uncertainties for the input data

(contact points and orientations) in the cross-sections. In this simple case, we want to

consider two types of uncertainties:

1. The depth of the formation surface: assuming that we obtained the basic information

for our cross-section from a 2-D seismic line, it is reasonable to infer an error in the

exact depth of a formation (e.g. due to an inaccurate time-depth conversion). This

error is similar for all data points of one formation and we assume that it increases

with depth.

2. The lateral extension of the formations: we can assume that the dip of the fault

surfaces defined in the 2-D section is subject to an error perpendicular to the section

as we do not have any other information about it.

We could also include further uncertainties, specifically the position of the faults at the

surface and the position of the orientation measurements. But we intentionally kept this

example simple to highlight the interaction of the above described uncertainties.

In a real geological model, the imprecision that would be assigned to the data can be

identified directly from the source (see sec. 3.2.3) or estimated with an educated guess,

ideally supported by additional information (like the general geological setting, etc.). In

this model, we assign a normal distribution (eq. 3.2) to the data points. As an estimate

for the mean value of the distribution, we take the data value of the initial model. The

assigned standard deviations for the distribution are noted in table 3.1. For these example

models, we perform 20 simulation runs.

Formation Data type Direction Stdev

Formation 1 Surface contact point Depth (z) 30 mFormation 2 Surface contact point Depth (z) 20 mFormation 3 Surface contact point Depth (z) 10 m

Formation 1 Orientation Dip angle 5◦

Fault E Orientation Dip direction 5◦

Fault W Orientation Dip direction 10◦

Table 3.1.: Estimations of the standard deviation assigned to the data in the graben model.

Analysis and visualization of results

A first good insight into the effect of uncertainties on the model can be obtained if we plot

all surface intersections on one cross-section (fig. 3.6a). As expected, the surfaces of all

simulated models vary most for the bottom formation (Formation 1) which had the largest

standard deviation. Another representation that provides a quick and easily interpreted

visualization is a histogram view of surface occurrences in a virtual well (fig. 3.6b). We

determine the depth value of all simulated surfaces at one location and plot these in


histogram view. This can, for example, be used to communicate model uncertainties at a

proposed drilling location.

We can extend the analysis of our results into a map view. This is possible in this case

since we have a geological setting which can be represented with elevation surfaces and

at every geographical point, there is only one surface point at depth (see fig. 3.1a). If we

analyze all simulated surfaces of one formation statistically, assuming a normal distribution

of results, we can determine the standard deviation of the simulated surfaces in a map view.

Figure 3.7a shows an example of this method for Formation 3, projected onto the mean

of all simulated surfaces. We can see that the areas of highest uncertainties correspond to

the intersections of the formation surface with faults in the model as, of course, we would

expect in this simple case. Still, it is important to note that the standard deviations exceed

the input standard deviations (10 m for Formation 3). This can also be visualized in a

histogram of local standard deviations (fig. 3.7b) and is mainly due to the high fluctuations

around the faults. The analysis shows how different uncertainties interact in the model:

the uncertainty in the resulting model is not simply equal to the standard deviation (or

any other probability distribution) in the input data.

a) Multiple models in one sectionleft centre right

b) Formation histograms along virtual wells

all simulated surfaces for Formation 3

virtualwells

Figure 3.6.: Visualization of results for Evaluation Model 1, simple visualization: (a) Visualization ofmultiple models within one cross-section; (b) Histogram of formation depth along virtual wells.

3.3.2. Doming structures

Model setup

With the second model, we apply our approach to a more complex situation, both in the

geological setting and in the interpretation of the results. We model a doming structure

(for example a salt dome or a magmatic structure, called Dome unit in the following) that

is cutting through a sequence of sub-horizontal formations (for example a sedimentary

sequence). This setting is more complex than the previous one as the boundary of the

doming structure is overturned and may be a multivalued function at each location, i.e. the

formation boundary is present more than once along a vertical line (see fig. 3.1d for a


a) Analysis of all simulated surfaces for Formation 3:Map of standard deviations projected on mean surface

Standard deviation increases around faults Standard Deviation [m]

No

rma

lise

d C

ou

nts

b) Histogram of standard deviations forall simulated surfaces of Formation 3

Most values around inputstandard deviation of 10m

But distribution is skewedtowards higher values dueto intersection with fault

Figure 3.7.: Map-based statistical analysis for elevation surface structures: (a) contour plot of standarddeviations of all simulated surfaces for top of Formation 3, projected onto the mean of these surfaces;The standard deviation increases significantly around the faults above the input value of 10 m; This isalso visible in (b) a histogram of standard deviations at all map positions: the distribution is skewedtowards higher values.

schematic view and fig. 3.8 for a representation of the example model). We now have

the case of a full three-dimensional geological setting. This model extends 1000 m in the

East-West, the North-South, and also in the depth direction. Figure 3.8 shows a 3-D

visualization of this model and the two orthogonal cross-sections where input data points

and orientations are defined.

Model 2: dome structure

E-W: 1000 m

N-S

: 100

0 m

z:

10

00

m

Dome unitSedimentary Pile (visualised incross-sections where datapoints are defined)

Figure 3.8.: Evaluation Model 2: Complex full three-dimensional dome structure; data are defined intwo orthogonal cross-sections (insets).

Simulation parameters and settings

In this example, we want to evaluate the application of our method to the complex doming

structure. We are thus only changing the parameters affecting the Dome unit. Also, we


assign the same probability distributions to all points and orientation measurements (see

tab. 3.2). Again, we perform 20 simulation runs.

Formation Data type Direction Stdev

Dome unit Surface contact point Depth (z) 20 m

Dome unit Orientation Dip angle 5◦

Table 3.2.: Estimations of the standard deviation assigned to the data in the dome model

Analysis and visualization of results

Processing the results is not as simple as in the first model since we are now dealing with a

full 3-D setting. We can still plot all modeled realization surfaces as lines within a section

but the representation and analysis in a depth-to-surface plot (as in fig. 3.7a) or similar

map views are no longer feasible.

We thus use here the indicator function methods introduced above (sec. 3.2.5). In this

case, we export all simulated models into a regular (voxel) grid with cell dimensions of 20

x 20 x 20 m. We then apply equation (4.5) to all simulations of the Dome unit and obtain

20 discrete indicator functions. Taking the sum of all functions at each cell (eq. 4.6), we

derive an indicator probability estimate for the Dome unit (based on these 20 simulations)

as a scalar function in 3-D, describing the probability that the dome exists at this grid

cell.

We can visualize these estimates in cross-sections or as isosurfaces of probability in

3-D. Figure 3.9a shows the surface contour of the minimal extent and maximal extent

of all simulated domes. In the cross-section, isolines of different probability values are

plotted. As both visualizations are based on the same scalar function, they are completely

self-consistent.

a) Extent of dome unit b) Probability to encounter unit boundary

minimal extent

maximal extent very low probability

very high probability (isosurface for mag. > 5.5)

Figure 3.9.: Visualization of results for Evaluation Model 2, complex structures: (a) Representationof minimal and maximal extent of the dome unit for all simulated models; the analysis is performedusing indicator functions, the indicator sum is a measure of the probability to encounter the unit ina cell (b) Gradient magnitude of the indicator field in (a), calculated with eq. 3.12; this provides ameasure of the probability to encounter the unit boundary in full 3-D


The analysis above provides an estimate of the probability to have a unit at any one

point in space. Still, in many cases, it is important to know where the boundary of a unit

is. Therefore, we apply equation (3.12) to derive the absolute value of the gradient of this

field. This provides a measure of the probability to encounter a boundary (fig. 3.9b) with

high values indicating high probability.

A further possibility would be to analyze a specific model feature, for example the

different volumes for all simulated dome models, using equation (3.13).

3.4. Discussion

We have shown that it is possible to quantitatively evaluate uncertainties in 3-D geological

modeling that are introduced by imprecision in different types of input data. The main

feature of our analysis is that we perform all simulations based on statistical manipulations

(i.e. sampling from the distributions) of the input data. These can be orientation measure-

ments and formation or fault contact points. Such data usually are the basis for all types

of structural geological modeling, from observations in the field, to those in a drillhole or

those indirectly interpreted in a seismic profile. Assigning uncertainties to these data types

is thus close to geological thinking. The same applies to the complete workflow which is

simple and straight-forward, starting from an initial model, creating several model repre-

sentations and finally comparing them. We think that this is an important step to enable

a 3-D uncertainty estimation and interpretation of geological models for the non-expert in

an intuitive way.

The approach is simple, intuitive and still applicable in both structurally simple and

complex geological models. As our simulation is based on the full 3-D potential-field

interpolation method (Lajaunie et al., 1997; Calcagno et al., 2008), we are not limited to

structural settings where only one value can be defined at every location as in the case

of elevation surface/ 2.5-D representations (see fig. 3.1a and the approaches of Thore

et al., 2002; Turner, 2006; Bistacchi et al., 2008; Suzuki et al., 2008). In the two presented

examples, we have shown how we can apply the method to a simple graben-type setting and

to a complicated dome structure. These models are kept simple to show the application

of the method. Still, the uncertainty simulation can be applied to all cases where the

implicit potential-field approach can be used for modeling. Complex modeling examples

have been presented in the literature (e.g. Maxelon and Mancktelow, 2005; Putz et al.,

2006; Calcagno et al., 2008; Joly et al., 2008). The advantage of our method is that it is

not limited to a specific geological setting and that the workflow itself is always performed

in the same way.

The main difference in the application of our approach for simple and complex settings

is the visualization and analysis of the results. Simple visualization techniques, like repre-

sentation of all simulated surfaces in one section or map view and occurrence histogram

of a surface intersection in a drillhole can directly be applied in many cases (see example

in fig. 3.6). A statistical analysis of the simulation results can be applied, for example to

estimate the maximum and minimum extent or mean and standard deviation of a surface


at depth (fig. 3.7).

In complex settings, the results of the uncertainty simulation can be visualized with the

use of indicator functions in full 3-D. The dome model (sec. 3.3.2) is an example where this

method is used to visualize the maximum change of models in the simulation run, high-

lighting the areas of highest uncertainties. These visualization techniques can be extended

to include all formations of a model and to derive further measures, like the minimum

and maximum extent (fig. 3.9a) or the probability to encounter the boundary (fig. 3.9b)

of a modeled formation. They can also directly be used to represent the uncertainties

in the volume estimation for one formation (eq. 3.13). This is of great interest in many

exploration situations. The use of indicator functions is a powerful method to analyze and

visualize uncertainties in complex settings.

Limitations of our method are mainly related to the geological modeling technique it-

self. The modeling can become slow when very large datasets are used. In these cases, it

could be reasonable to subdivide the model range into smaller areas for the uncertainty

simulation. Still, since our approach is ideal for parallel implementation, the whole uncer-

tainty estimation will, when optimized and performed on a suitable computer, only take

minimally longer than one model run.

As our method is primarily targeting uncertainties in the input data (i.e. uncertainties of

Type I, fig. 3.2a), the analysis is mainly representative of areas where data are available. In-

between and away from data points an additional uncertainty exists, the problem of correct

interpolation and extrapolation (Type II, fig. 3.2b). This uncertainty is strongly related

to the modeling interpolation technique. For the implicit potential-field interpolation that

we apply, this uncertainty has been evaluated by Aug (2004) and Chiles et al. (2004).

Uncertainties due to incomplete geological knowledge (Type III, fig. 3.2c) are very hard

to assess but could partly be integrated into our simulation, e.g. the existence of a fault

could be included on a probabilistic basis. Thus, even if our method is currently mainly

focused on Type 1 uncertainty – the imprecision of the raw data – it can be combined and

extended to other types for a comprehensive evaluation of model uncertainty.

Apart from the applications of our method described above, the presented examples

also suggest that it is important to analyze uncertainties introduced by the input data

and their influence on each other. The analysis of results for Model 1 in figure. 3.7a shows

how the uncertainty of the dip direction of the faults and the formation tops interact. The

resulting uncertainty for the formation tops exceeds the assumed input standard deviation

significantly in this simple example (fig. 3.7b). We thus conclude that it is not sufficient

to assign a simple statistical measure to a simulated surface, derived from input data

distributions, but that it is important to consider all input data uncertainties and their

interaction. Our method provides a possibility to achieve this.

In addition to the visualization of uncertainties, the simulated models can also be used

directly as an input for further simulation and inversion techniques. It is common practice

to use a 3-D structural model as an input for geophysical simulations, like stress-field

analyzes or fluid and heat flow studies. And even though it is accepted that the geological


basis model itself is a major source of uncertainty in these simulations (e.g. Subbey et al.,

2004; Caumon et al., 2009), it is usually not evaluated. With our method we can directly

use a variety of different model realizations as input. We are thus respecting the possible

uncertainties in the geological model for further simulations. The geophysical simulations

can then be performed in the light of the quality of the geological model. For example,

we plan to use the created range of models as a direct input for geothermal simulations.

This type of combination of structural modeling with other types of complex data (e.g.

fluid flow simulations) has been identified by Caumon et al. (2009) as an important di-

rection of current research. For example, Suzuki et al. (2008) proposed a method to use

reservoir production data to identify possible structural interpretations. Our approach

is similar to this method on a conceptual level. But whereas Suzuki et al. (2008) and

other approaches (typically implemented in oil exploration and production software) con-

centrate on structural modeling derived from seismic data, our approach is suitable to all

types of input data, including orientation measurements. Also, our method is not limited

to typical sedimentary settings as encountered in oil exploration but suitable for complex

3-D settings in any geological environment. And as we perform all simulations directly

based on parameterized structural measurements, the effect of every single measurement

can be analyzed. Our structural uncertainty measure is also quite distinct from other

lithological uncertainty measures where the focus is classically laid on inversion of prop-

erties assigned to grid cells (e.g. Guillen et al., 2008) with respect to assigned properties

based on geophysical observations (e.g. magnetics and gravity) but does not incorporate

the parameterized uncertainty of the structural measurements themselves. We call this

classical method geophysically based inversion whereas ours would be more correctly la-

belled as geologically based inversion. Both techniques complement each other and can be

used in tandem. In this sense, our method opens-up the way to a unified geological and

geophysical data-driven ensemble modeling and inversion.

Our uncertainty simulation scripts will be distributed free of charge for all scientific

purposes. Please contact the corresponding author for a copy of the program and further

information. An evaluation license for the software GeoModeller is available on the website

www.geomodeller.com.


4. Information Entropy as a Measure

of Uncertainty in Geological

Models


simulation





simulations





Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7




Chapter 4

44 4. Information Entropy as a Measure of Uncertainty

Overview The work presented in this chapter is a systematic extension of the stochastic

uncertainty simulation approach of the last chapter. It addresses the problem that no

general approach to quantify uncertainties in geological modeling is available. It is here

proposed to use information entropy as an objective measure to compare and evaluate

model and observational results. Information entropy was introduced in the 50’s and is

based on a solid mathematical framework.

In this chapter, information entropy is applied to visualize uncertainties in a suite of

simple geological models. A quantitative analysis suggests that a global measure of entropy

for the whole model and single geological units within the model can be used to evaluate

how uncertainties are reduced with additional data. This quantitative aspect that describes

uncertainties in the whole model with a single scalar value will further be used in the last

chapter of the thesis, as measure to compare geological uncertainties and uncertainties in

the simulated flow fields (chapter 8)

The text of this chapter is in press in Tectonophysics (Wellmann and Regenauer-Lieb,

2011).


4.1. Introduction

Visualizing and analyzing uncertainties in 3-D structural geological models are widely rec-

ognized as important issues (e.g. Jones et al., 2004; Bond et al., 2007; Caumon et al.,

2009; Jessell et al., 2010). We propose here an information entropy method as a quan-

titative measure for the quality of a geological model and its geological sub-units, based

on the Shannon Entropy model (Shannon, 1948). Goodchild et al. (1994b) already used

this method to visualize uncertainties in map applications in a 2-D context for fuzzy sets

resulting from poorly constrained data. We extend it here into the third dimension and

combine it with an uncertainty simulation approach that is applicable to complex 3-D

geological settings (Wellmann et al., 2010). In the context presented here, we propose

that information entropy is a sound method to visualize uncertainties in any complex 3-D

setting. But due to its underlying concept of information, we expect that it can provide

significant insights into the used and missing information to constrain a model.

In the following, we will briefly review the concept of information entropy and will show

how this measure can be used to evaluate uncertainty at one position within a model

domain. We will then apply this measure to visualize a 3-D uncertainty field, calculated

with a simulation method for geological uncertainty and discuss the feasibility of this

method and the possible extension beyond visualization.


4.2.1. Visualizing uncertainty

A variety of different methods exists to evaluate the quality of a geological model. These

range from simple uncertainty measures, determined from data uncertainties, over geosta-

tistical evaluations (e.g. Chiles and Delfiner, 1999) to model simulations (Suzuki et al.,

2008). A challenge is, in many cases, the communication of the model quality. Several

authors have developed methods to visualize uncertainties in a geographic reference frame

(e.g. MacEachren et al., 2005) or in a stratigraphic grid (Viard et al., 2007). The problem

with most of these visualizations is that they are restricted to specific geological settings

(i.e. most can not handle complex three-dimensional settings like domes or overturned

folds), that many estimations are subjective and ’ad-hoc’ approaches (Jones et al., 2004)

or only suitable to represent specific types of uncertainties (see Bond et al., 2007, and

references therein for a discussion of several approaches).

Our aim of visualizing the uncertainty in a structural geological model is to find a

measure that predicts the accuracy of the model at every location in the model space,

for any type of complex geological model. To achieve this, we subdivide the whole model

space into a regular raster with equal cell sizes (also referred to as voxels). We can then

evaluate the accuracy of every voxel, everywhere within a unit – rather than only the

accuracy of a polygon/ surface boundary (see Leung et al., 1993, for a discussion of raster

vs. polygon estimation).


Within each of the raster cells, we want to find a measure of uncertainty fulfilling the

following criteria:

1. The value should be 0 when no uncertainty exists, i.e. there is only one possible

outcome (i.e. geological unit) i with probability pi = 1;

2. The value should be maximal when all n possible outcomes are equally likely: pi =1n∀ i ∈ n. Or, formulated for the case of a geological model: there is absolutely no

reason to expect a specific geological unit at this location;

3. If more outcomes are equally likely, the measure should have a higher value (“mono-

tonicity”);

4. The measure should fulfill the criteria of expansibility, i.e. the value should not

change when an additional outcome with probability 0 is added;

5. The measure should be independent of the order of results (“symmetry”).

The information entropy, first defined by Shannon (1948), fulfills all of these properties

(see Klir et al., 1988; Yager, 1995, for further details). We apply it here as a method to

visualize uncertainties and discuss some potential applications beyond pure visualization.

4.2.2. Information entropy

The concept of information entropy was first defined by Shannon (1948) in a study per-

formed to identify the amount of information required to transmit English text. The

underlying idea was that, given the probabilities of letters occurring in the English alpha-

bet, it is possible to derive a measure describing the missing information to determine

the full text of a partially transmitted message, where information is understood as the

information required to identify the message, not the information of the message itself.

Based on several theoretical considerations, Shannon derived the following equation to

classify a measure of the missing information, often referred to as information entropy:

H = −N

∑

i

pi log pi (4.1)

The information entropy H is defined as the sum of all products of probabilities p for

each possible outcome i of N total possible outcomes with its logarithm. The minimum

value is 0, because log 1 = 0 and limx→0(x log x) = 0 (possible to prove with L’Hopital’s

theorem, see Ben-Naim, 2008). The logarithm can be taken with any base, depending on

the applied unit of information. We will use the logarithm with base “2” in the following

examples and discussion, as it relates to the information unit of one bit (see below).

Information entropy of a 1 bit system with two possible outcomes

As an example, we will briefly examine the information entropy of a very simple system

with only two exclusive outcomes A and B, with P (A) and P (B) = P (A)′. We can, for


example, consider the example of a coin flip where the outcome is either head or number

(1-bit system with two possible outcomes). The information entropy quantifies the amount

of missing information to classify this system.

0.0 0.2 0.4 0.6 0.8 1.0Probability of first component (p1 )

0.0

0.2

0.4

0.6

0.8

1.0In

form

ati

on E

ntr

opy H

Information Entropy for 1-bit system

Figure 4.1.: Shannon entropy for 1 bit system (two possible outcomes) as a function of the probabilityof the first outcome. When the probability of one outcome is close to 0 or 1, the entropy is low;whereas when the probability of both outcomes is 0.5, the entropy is maximal and equal to 1 if theinformation unit is 1 bit and the logarithm to base 2 is taken.

If the coin is fair, i.e. outcome of head and tail is equal, then the entropy is highest.

Whereas, when the coin is unfair, the entropy is lower, because one outcome is more

probable than the other. In the extreme case (an extremely unfair coin) where the outcome

is always the same, the information entropy is 0, because the outcome is known. Using

equation (4.1), this simple example is presented in figure 4.1.

Information entropy in an evolutionary spatial context

In a spatial or modeling context, we can interpret the information entropy of a model

subregion (e.g. a cell) as the amount of missing information with respect to the discrete

properties of the cell. In the following, we consider the membership to a specific geological

unit as this property. This is possible, as we consider geological units as exclusive events,

i.e. a cell belongs either to the geological unit 1, or unit 2, ... M .

For each discrete subregion, we can describe the information entropy as

H(~x, t) = −M∑

m=1

pm(~x, t) log pm(~x, t) (4.2)

Where ~x denotes the location of the subregion and M the number of possible (exclusive)


members the subregion can contain. t could be physical time or any other parameter

describing the evolution of a model.

This measure can then be used to visualize the uncertainty of cells in a straight-forward

way. Let us, for example, consider a map with discrete subdivisions into a regular grid

(fig. 4.2a) where each cell can contain one of three possible members and where we can de-

rive probabilities for each possible outcome in each cell (fig. 4.2b). Applying equation (4.2)

for each cell with position ~x, we obtain a map of information entropies (fig. 4.2c). We can

Figure 4.2.: Application of the concept of information entropy to visualize uncertainties in a spatialcontext: (a) Subdivision of a map of uncertain members into a regular grid; (b) Probabilities ofpossible outcomes for each member in each cell; (c) Map of information entropies for each cell,where the entropy is 0 when no uncertainty exists (i.e. one member has the probability 1) and theentropy is highest when all members are equally probable.

see that the entropy measure fulfills the criteria described above (sec. 4.2.1), specifically:

➤ The entropy is 0 for cells where one member has the probability 1 and all others are

0 (e.g. cell A).

➤ When two members are equally probable, the entropy is highest when both members

have the same probability (cells B and C).

➤ Finally, when three members are equally probable, the entropy is higher than for the

case that only two members are equally probable (cells B and D).

The map of information entropies provides a direct overview of uncertainties associated


with each subdivision (cell) of the map. We propose therefore that the cell-based measure

of information entropy is an effective measure to visualize uncertainties in a spatial context.

A similar method has been applied by Leung et al. (1993) to visualize the entropy of

multinominal probability fields for uncertainty maps. An important difference between

their approach and the method presented here is that they used a normalized version of

the measure which does not fulfill the criterion of monotonicity, stated in section 4.2.1.

We regard this criterion as relevant for the type of quantitative interpretation presented

in this work. But if, for example, it is relevant to visualize the state of uncertainty for

each individual cell, with 1 denoting that the cell is in its maximum possible state of

uncertainty, the criterion of monotonicity does not any more apply and the normalized

version could be used.

The extension of the information entropy method to a full 3-D structural geological

model is straight-forward when we apply it to a cell-based discretization of the model,

shown in the examples below.

4.2.3. Entropy as a measure of fuzziness

Related to the question of uncertainty in a subregion of the model, as described above, is

the question how well each single member is defined in the whole model region. Or, in the

context of geological models: how precise is the occurrence of a specific unit of the geo

logical model known?

A measure, related again to the information entropy, can be derived on basis of the

fuzzy set theory (Zadeh, 1965). In this theory, a set is not necessarily clearly defined

but may contain some form of indefiniteness, for example imprecise boundaries, usually

referred to as fuzziness (Yager, 1995). The idea to apply information entropy as a measure

of fuzziness was established by De Luca and Termini (1972), based on similar criteria as

described above. Applied to a fuzzy set, where f ∈ [0, 1] is a measure of fuzziness of each

part of the set, the most important properties are:

➤ The measure should be 0 if, and only if, f is either 0 or 1 everywhere;

➤ The measure has its maximum value when f is 0.5 everywhere.

These conditions are, again, met by the Shannon Entropy function. Denoting f as a

probability pm of one outcome m ∈ M , we can quantify the fuzziness as the entropy Hm,

normalized by the total number of cells N :

Hm(t) = − 1

N· (4.3)

N∑

~x=1

[pm(~x, t) log pm(~x, t) + (1 − pm(~x, t)) log(1 − pm(~x, t))]

As for the example of cell entropy, this method has been applied by Leung et al. (1993)

to describe uncertainties of members in a surface map.


4.2.4. Total model entropy

As an extension of the two concepts described above, a total information entropy HT for

a whole model space can be calculated as:

HT (t) = − 1

N

N∑

~x=1

H(~x, t)

= − 1

N

N∑

~x=1

M∑

m=1

pm(~x, t) log pm(~x, t) (4.4)

The total model entropy is equal to zero when all subparts ~x in the model are precisely

associated to one member and it is maximal when, in all subparts of the model, the

probability of all members is exactly 1/M .

4.2.5. Application to uncertainties in geological models

For the case of t = 0, the above described concept can directly be applied to geological

models. If we consider the possible outcomes, or members m ∈ M as possible geological

units in a model, a subdivision of the model space into a discrete number of equally sized

cells with a position ~x, and the probability that a geological unit exists in a cell as pm(~x),

we can apply the entropy measures to:

➤ Calculate cell entropies and use this measure to visualize uncertainties;

➤ Evaluate the uncertainty of a whole geological unit as the related fuzziness;

➤ Use the total model entropy as a way to quantify the uncertainty of the whole model

with a single number.

The entropy measures can be applied to a variety of uncertainties in the context of

geological models if the problem can be classified into discrete exclusive outcomes and

probabilities at each cell can be assigned or estimated. In the following, we are applying

the concepts to uncertainty simulations of structural geological models.

4.3. Geological modeling and uncertainty simulation

4.3.1. Type of geological modeling considered here

We apply here the information entropy method to visualize and analyze uncertainties in

structural geological models. These models are commonly used to show and analyze the

structural setting of the subsurface (i.e. the distribution of geological units and faults

or other structural features), based on a variety of different data sources. These sources

can be direct observations of the geology, for example in surface outcrops or from drill-

core, or interpretations from indirect measurements, for example seismics, gravity and

magnetic measurements or wireline logs in wells. The distribution of geological units,

surfaces and contacts between units and offsets on faults is modeled from interpolations


and extrapolations of these data sources, based on geological constraints, for a defined

model region. A variety of methods exist to construct geological models (see, for example,

Chiles and Delfiner, 1999; Caumon et al., 2009, for a more general introduction to modeling

and concepts).

4.3.2. Uncertainty simulation for geological models

All geological models are subject to several kinds of uncertainty, ranging from conceptual

uncertainty and incomplete knowledge to uncertainties associated with the model con-

struction methods and imprecision in the input data itself (e.g. Mann, 1993). We apply

here an approach to simulate uncertainties in geological models that are due to impreci-

sion in the input data with a stochastic approach. The method is described in detail in

Wellmann et al. (2010) and we will provide only a brief summary here.

The method consists essentially of five steps:

1. Construction of an initial representation of the subsurface structures (i.e. the geo-

logical model), as the best possible model that can be constructed based on all

available input data. For the model construction, we use an implicit potential-field

modeling approach (Lajaunie et al., 1997), implemented in the software GeoModeller

(for further details see Calcagno et al., 2008; Guillen et al., 2008). Main advantages

of using this approach are that it incorporates a meaningful geological reasoning and

that model reconstruction is automatically possible when input data are moderately

changed.

2. Assignment of probability distributions to the input data (data positions and orien-

tation measurements). Input data can be considered as discretized geological obser-

vations and a variety of different probability distributions are possible, depending

on the type of uncertainty.

3. Based on the initial geological model and defined probability distributions, n different

input data sets are generated with a stochastic approach.

4. The different input data sets are automatically recomputed to generate n possible

geological model representations.

5. The geological models are exported into a useful format and processed to analyze

and visualize uncertainties.

To apply the entropy concepts described above, we require a subdivision of the whole

model space into a regular sub-domain (i.e. cell) structure and an estimation of probability

pF for each geological unit to occur in any one cell at position ~x.

Firstly, we are determining an indicator function for each geological unit F . This func-

tion is a subset of the whole model space and defined as:

IF (~x) =

{

1 for ~x ∈ F

0 for ~x /∈ F(4.5)


For the n geological models, we obtain n indicator fields for each geological unit F and

can use these functions to estimate an indicator probability function

PF (~x) =∑

k∈n

IFk(~x)

n(4.6)

We can now use these indicator probability functions to evaluate the entropy measures

described in section 4.2.2.

4.4. Application of information entropy to visualize and analyze

uncertainties

As a test of feasibility, we apply the techniques described above (sec 4.2.2 to 4.2.4) to

visualize uncertainties in a simple geological model. We will also apply the techniques to

identify and visualize the differences of uncertainties when a model data base is extended

with additional information and different geological hypotheses are tested.

4.4.1. Geological model

We are considering here a very simple geological setting with three geological units, sep-

arated by sub-parallel surfaces, dipping to the East (fig. 4.3a). The model is constructed

with an extent of 5 km in East-West and North-South direction and to a depth of 2 km .

As model input data, we consider here observations of unit boundaries and one orientation

measurement at the land surface (fig. 4.3b). In the first example, we visualize uncertainties

at depth originating from uncertainties in these data. For the second model, we use ad-

ditional (uncertain) data for the geological units at depth (visualized in the cross-section,

fig. 4.3c) and analyze the change in subsurface uncertainty. Also in the cross-section, we

define surface data and orientation of a fault (red dots and line) that we use for a geological

hypothesis test in Model 3. Further data is added for Models 4 and 5, to simulate the

effect of additional drillhole data on the model uncertainty (fig. 4.3d).

4.4.2. Uncertainty simulation

We apply the method described in section 4.3.2 to simulate the effect of uncertainties

in the input data on the geological model. For each of the example models, 50 possible

realizations are computed and stored. We assume here that all input data points and

measurements are subject to some uncertainty and assign a normal distribution to all

values. Mean value of the distribution is the initial (best guess) value. The standard

deviations are listed in table 4.1. The models are exported into a regular mesh with 1

million cells and each cell has the dimensions 50 x 50 x 20 meters. Indicator functions for

all units are determined with equation (4.5).


(a) Geological Model (b) Top surface data

(c) Cross-section

Reverse Fault(Model 3)

Additional datafor Model 2

Modelled surface of Unit 1

Modelled surface of Unit 2

De

pth

2 k

m30

Position ofCross-section

Top Unit 2

Top Unit 1

Strike/ Dip

E-W 5 km

E-W 5 kmN-S

5 km

De

pth

2 k

m

Drillhole dataused for Model 4

Data used for Model 5E-W 5 km

De

pth

2 k

m

(d) Hypothetical drillholes

N-S5 km

E-W 5 km

Figure 4.3.: Geological model used as the basis for uncertainty simulations and information entropycalculations: (a) 3-D view of geological model containing three units and two natural geologicalcontact surfaces, e.g. between sedimentary units; (b) Surface (geology) map of the data used formodeling: observations of unit boundaries (dots) and measurement of strike and dip; (c) E-W cross-section with additional data used for Model 2 and 3: one more observation point per surface andone orientation measurement; Red dots and line: data used to constrain the fault in Model 3; (d)Position of hypothetical drillholes for Models 4 and 5. Additionally, the data used for model two(points and orientation) are also used for these models.

4.4.3. Model 1: Visualization of model uncertainties

With the first example model, we want to evaluate the effect of uncertain model input

data at the surface to modeled structures in the deeper parts of the model. We are only

using the data defined in the surface map (fig. 4.3b) and assign associated uncertainties

to observation point positions and orientation measurement (tab. 4.1).

Results of the uncertainty simulation for the first model are presented in figure 4.4. One

common possibility is to plot estimated probabilities, calculated here as indicator function

probabilities with equation (4.6), for all geological units separately (fig. 4.4a). Although

this is a convenient way to represent uncertainties of one unit, it is not useful to visualize

overall model uncertainties. Applying equation (4.2) for each cell, we obtain a measure

of uncertainty combining all unit probabilities (fig. 4.4b). We can now identify areas

where the geological unit is accurately known (with respect to the considered raw data

uncertainties), where H = 0. Where H > 0 (center figure), at least some uncertainties

exist. Where H > 1 (right figure), all three units are possible and where H = log2 3 ≈ 1.58,

all three geological units are equally probable and the uncertainty is maximal.


Section Unit Data type Direction Stdev

Surface Unit 1 Contact Position 50 mSurface Unit 2 Contact Position 50 mSurface Unit 1 Orientation Dip angle 10◦

Section Unit 1 Contact Depth (z) 50 mSection Unit 2 Contact Depth (z) 25 mSection Unit 1 Orientation Dip angle 10◦

Table 4.1.: Standard deviations of observation point position and orientation measurements in thefollowing examples

This example shows that information entropy can be used to visualize uncertainties of

the whole model and, additionally, provides a quantitative measure for the uncertainties.

(b) Information entropy ( )H

(a) Unit probabilities

0 0.4 0.8 1.2 1.58

0 10.5 0 10.5 0 10.5

H > 0 H > 1

Unit 2Unit 1 Unit 3

H = 0: unit precisely known H > 0: at least some uncertainty All three units probable

Figure 4.4.: Visualization of uncertainties for Model 1: (a) Estimated probabilities to encounter onespecific geological unit, useful to interpret uncertainties of one single unit only; (b) Informationentropy H of the model cells, providing a measure for the model uncertainty at each cell position;Where H > 0 (center figure), uncertainty exists about the unit at this position; where H > 1 (rightfigure), all units could occur and where H is maximal (≈ 1.58), all three units are approximatelyequally probable and the uncertainty is maximal.

4.4.4. Model 2: Uncertainty reduction with additional data

In example Model 1, we could observe that large model uncertainties at depth exist due

to only small measurement uncertainties at the surface. We will now evaluate how, and

where, additional data at depth will reduce uncertainties in the model. Additionally to

the data points used in Model 1, we are now using data defined in the cross-section, also


considered uncertain (fig. 4.3c).

As expected, additional data at depth clearly reduces the model information entropy

(fig. 4.5a). In order to identify areas where uncertainties are reduced, and by how much

the additional data reduces the information entropy, we can calculate entropy differences

at each cell. We can then plot areas where the entropy in the new model is lower (fig. 4.5b,

left figure, green colors) and areas where the entropy is increased (fig. 4.5b, right figure, red

colors). These difference plots provide a detailed insight into where the model uncertainties

are different for two simulated scenarios.

Figure 4.5.: Reduction of uncertainties with additional data in Model 2: (a) Visualization of informationentropy (right: only for H > 0) indicates that uncertainties are significantly reduced in many partsof the model and are, generally, also lower; (b) Entropy differences between Model 1 and Model 2show directly that uncertainties are reduced (left figure, green colors) and only slightly increased insome parts (right figure, red colors).

4.4.5. Model 3: Geological hypothesis testing

In Model 2, we visualized the decrease of model uncertainties with additional data. We now

want to evaluate how a different geological hypothesis would change the model entropy.

We are testing the hypothesis that a reverse fault exists somewhere between the data

points at the surface and the points at depth. Fault position and behavior are controlled

by the additional red points in the cross-section (fig. 4.3c). A 3-D representation of the


geological model is shown in figure 4.6a, with a reverse fault (red surface) offsetting the

geological units.

Looking at the information entropy of this model (fig. 4.6b), we note that there are high

values of information entropy around the position of the fault. This is also clearly visible

in the difference plots (fig. 4.6c), where the entropy is reduced in some parts of the model

(left figure, green colors), but greatly increased around the fault (right figure, red colors).

Visualizing the changes with entropy differences provides a direct insight into uncertainty

changes for hypothesis testing.

(a) 3-D representation of model

Reverse fault

(b) Information Entropy ( > 0)H

0 0.8 1.58

-1.58 -1.06 -0.53 0 0.53 1.06 1.58

High Entropyaround fault

(c) Entropy difference to Model 2

Additional fault reducesentropy in some parts butincreases it in others

Reduced information entropy Increased information entropy

Figure 4.6.: Results of a geological hypothesis test, Model 3: (a) 3-D representation of one possiblerealization with a reverse fault offsetting the geological units; (b) Information entropy (H > 0) showsthat uncertainties are specifically high around the fault; (c) Comparing Model 3 to Model 2 withdifference plots, we observe that uncertainties are reduced in some parts of the model (left, greencolors), but greatly increased in other parts (right, red colors).

4.4.6. Model 4 and 5: Evaluate uncertainty reduction with additional data

A common question for many applications of geological models is where, and how, ad-

ditional data would help constrain and optimize the geological model. This is a very

complex question and has to be considered in detail for each case. We want to show here


an application of the information entropy measures to identify how additional informa-

tion, for example derived from new drillholes, would change uncertainties in the model.

We are testing the effect of two hypothetical new drillholes positioned along the cross-

section (fig. 4.3d). In the first case, the drillhole is placed between the known data points

(Model 4) and in the other case, near the Eastern end of the model where the geological

units can be expected to be at a deeper position (Model 5). Positions of the unit contacts

in the drillhole are estimated and, again, subject to uncertainty.

We are now mainly interested in the entropy differences between the two drillhole scenar-

ios and Model 2 (containing the data at depth, but not the fault). Visualizing differences

of information entropy between Model 4 and Model 2 (fig. 4.7a) we observe no great re-

duction in information entropy and model uncertainty. But when the drillhole is placed in

the extrapolated parts of the model in the far East, as simulated in Model 5, information

entropy is significantly reduced in these areas (fig. 4.7b). Information entropy is here used

as a measure to identify and visualize the value of additional data.

Figure 4.7.: Testing the effect of additional drillhole data: (a) For additional data between the knownpoints at depth and the surface, simulated in Model 4, there is no significant reduction in entropy;(b) But when the drillhole is placed in the far East (Model 5), entropy is visibly reduced in theextrapolated parts of the model.


4.5. Potential applications beyond visualization

4.5.1. Using mean entropy and fuzziness to compare models

In the examples presented above, we used visualizations of cell information entropies to

illustrate model uncertainties. Furthermore, difference plots between simulation scenarios

could be used to highlight where, in the model space, information entropy is reduced

and where increased, from model to model. As other measures to compare entropies

between different scenarios, we can calculate the mean entropies for the models (eq. 4.4)

and fuzziness entropies for the single units (eq. 4.3). We obtain single numbers describing,

to some extent, the uncertainties of the geological models.

If we plot the values for mean entropy and unit fuzziness for the example models above

(fig. 4.8), we observe the same behavior in the model differences, now quantified with

one value for each unit and model: starting from the model where only surface data are

used (Model 1), entropies are significantly reduced when additional data at depth are

considered (Model 2). For the hypothesis of a fault between the data points (Model 3),

entropies are slightly increased. Finally, testing the effect of additional information in

two new drillholes, at different positions, the drillhole between the data at depth and

the surface (as in Model 2) only decreases entropies slightly (comparing Model 4 and

Model 2), whereas a drillhole in the far East (Model 5) clearly leads to the lowest entropies.

Model 5 contains, over all, the lowest uncertainties. These values are in accordance to the

qualitative considerations presented before, comparing the different models.

Figure 4.8.: Mean entropy and unit fuzziness for the different models; There is an clear reduction inmean entropy and fuzziness from Model 1 to Model 2. Also, the additional drillhole data used forModel 5 reduces the entropy, and therefore the uncertainties in the model, further.


4.5.2. Determination of representative cell sizes

From the example models shown above, and also from the simple sketch presented in

figure 4.2, it is evident that the spatial discretization of the model has an important

influence on the information entropy. If we discretize the model into cells that are too

large, details of the spatial distribution of uncertainties will be masked. On the other

hand, if cell sizes are very small, we will capture many details but increase computation

time.

We are using here the fuzziness entropies of the geological units as a way to determine

a useful cell size, based on one scenario, that can be expected to be applicable to other,

similar, scenarios. The geological model is subsequently subdivided into smaller cell sizes

and the total number of cells is increased. When we analyze the entropies of the unit

fuzziness for all these spatial subdivision schemes, we observe that, until approximately a

total number of one million cells, entropies of unit fuzziness are fluctuating. Increasing the

number of cells above one million does not seem to have a significant effect in this case,

suggesting that the discretization into a regular grid with a total of one million cells (as

used in the example models above) is here sufficient to capture the model uncertainties.

Similar estimations could be useful for other geological modes containing uncertainties.

Figure 4.9.: Evaluation of unit fuzziness for different numbers of cells; Above 106 cells the estimationsdo not change significantly indicating that the chosen discretization with this number of cells seemsto capture all relevant model fluctuations.

4.5.3. Entropies as convergence criteria for uncertainty simulation

Related to the question of spatial discretization is the question how many simulation

runs for the geological uncertainty simulation have to be performed to obtain a sound


statistical estimate. This question is, again, not easy to answer for complex models where

uncertainties between a variety of data points can interact. One possibility is to use the

measure of unit fuzziness and examine how much it is changing for subsequent numbers

of model simulations.

In figure 4.10, we show the development of the unit fuzziness for one geological unit in one

model, for subsets of the total number of 50 simulations. The different lines show different

random orders of simulation results in the subsets. For less than 20 simulation runs,

estimations of fuzziness are greatly fluctuating. But above approximately 30 simulation

runs, the unit fuzziness seems to stabilize.

Figure 4.10.: Fuzziness of one geo logical unit for randomly chosen subsets of all 50 simulated real-izations; fluctuations in the estimation of fuzziness in this model are very high for a low number ofsimulations (<20) but reduce for n > 30.

4.6. Discussion

The examples show that information entropy can be used to visualize uncertainties in com-

plex 3-D geological models and, additionally, gives uncertainties a meaning. Compared to

visualizations of probabilistic estimations, the advantage of information entropy is that it

combines probabilities from multiple members into one number, for every sub-region of

the model. It is based on a sound theory and we consider the properties of the measure,

mainly that the entropy is 0 when no uncertainty exists and maximal for highest uncer-

tainty, as appropriate to represent the quality of geological models. The examples show

further that, beyond pure visualization, the measure can be interpreted in a quantitative

way. This is useful to separate areas of uncertainties. In the first example, we used it

to highlight areas where all three units could occur (fig. 4.4b). The quantitative aspect


can also be applied to compare different models. We have shown that it is easily possible

to separate areas where a new model step reduces and where it increases uncertainties.

The value of the entropy difference can again be related to the gain in information. All

examples show that information entropy is a powerful method to visualize and analyze

uncertainties for complex, 3-D geological models.

The related measures of entropies for the fuzziness of a geological unit and the mean

entropy of the whole model are, furthermore, very useful to describe overall uncertainties

with a single number. This has been shown to be useful for tracking the model evolution,

when additional data are incorporated. But many further applications can be envisaged.

In the examples, we have shown that these measures can be applied to evaluate a represen-

tative cell size for model discretization (representative volumes) and convergence criteria

for the geological uncertainty simulation that we have applied for the examples. The

measures could, for example, further be implemented to classify model suites in an en-

semble modeling framework or in sensitivity analyses. Generally, we conclude that the

possibility to quantify uncertainties with these measures has a potential for many related

applications.

In the presented examples, we combined uncertainty simulation of geological models

with novel interpretation, analysis and visualization techniques. It can directly be applied

to similar approaches, for example the combination of geological modeling with potential

field inversions of magnetics and gravity as presented by Jessell et al. (2010) and Lindsay

et al. (2010). Also, other recent developments in geological modeling apply an implicit

modeling concept (Durand-Riard et al., 2010; Cherpeau et al., 2010). We presume that

our methods, the visualization and analysis techniques using entropy methods, and the

uncertainty simulation for geological model, can be useful for these approaches and we are

expecting more studies in this contemporary research area in the near future.

We applied information entropy here in the context of structural geological models.

Many other properties could be analyzed in similar ways, for example, sedimentary facies.

It could also be applied to a variety of other probabilistic estimations, for visualization

and quantification of uncertainties. For example, it could be applied to probabilistic

estimations of material properties, based on potential- field inversion (e.g. Guillen et al.,

2008) or estimations of interpolation uncertainties (e.g. Aug, 2004). We propose that the

method can be useful in many cases where estimations of probability for multiple, spatially

overlapping outcomes have to be compared and visualized.

Apart from applying the presented method to other probabilistic estimations, it could

furthermore be applied to physical processes where time has a meaning. Our examples in-

cluded the aspect of pseudo-evolutionary models strictly only for the purpose of geological

hypothesis testing. We showed examples of the same geological model with and without

a reverse fault (fig. 4.6). We emphasize here, that this discussion was still in the context

of a static interpretation where the evolution parameter t did not take on the meaning of

explicit time. Explicit time couples the evolution to an underlying physical process. In the

case of 4-D models with slow-time evolution, new (Gibbs-type) probabilities arise. These


probabilities can be described by maximum information entropy (MaxEnt), as shown by

Dewar (2005). This differentiation is important. The goal of a static exercise is to identify

a global and local measure where the modeler seeks to identify a minimum of information

entropy (MinEnt) through the exercise of gaining additional data. This method can be

extremely useful for assessing the investment risk to reduce uncertainty for the placement

of e.g. additional seismic sections or drillholes.

5. Controlling Flow Simulations with

flexible Scripting Libraries


simulation





simulations





Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7



simulations

Chapter 5

64 5. Controlling Flow Simulations with flexible Scripting Libraries

Overview This chapter presents the first part of the integrated workflow combining

geological modeling and flow simulations. The second part of the workflow is presented in

chapter 6.

Numerical simulations of subsurface fluid and heat flow are commonly controlled man-

ually via input files or from graphical user interfaces (GUIs). Manual editing of input files

is often tedious and error-prone, while GUIs typically limit the full capability of the sim-

ulator. Neither approach lends itself to automation, which is desirable for more complex

simulations.

An alternative approach is here proposed based on the use of scripting. To this end we

have developed Python libraries for scripting subsurface simulations using the SHEMAT

and TOUGH2 simulators. For many problems the entire modeling process including grid

generation, model setup, execution, post-processing and analysis of results can be carried

out from a single Python script.

Through example problems we demonstrate some of the potential power of the scripting

approach, which does not only make model setup simpler and less error-prone, but also

facilitates more complex simulations involving, for example, multiple model runs with

varying parameters (e.g. permeabilities, heat inputs, and the level of grid refinement). It

is also possible to apply the developed methods for extending the functionality of graphical

user interfaces.

A manuscript with the text of this chapter has been submitted to Computers and

Geosciences and has been accepted for publication pending minor revisions (Wellmann

et al., 2011a).


5.1. Introduction

Numerical simulations are commonly performed to study fluid and heat flows in subsurface

environments, using a wide variety of simulation codes (see, e.g. O’Sullivan et al., 2009,

for an overview). Most of these codes are operated from the command line, with one or

more input files containing the problem specification and simulation settings. Creating

these input files manually (particularly if they have a fixed-width plain text layout) is

often tedious and error-prone. Graphical user interfaces (GUIs) are available for many

fluid and heat flow codes, offering graphical post-processing of results and often also some

pre-processing capabilities to assist with the preparation of input files. However, these

interfaces often limit the full potential of the simulation code, and offer little flexibility

in the ways input files can be created. Neither manual input preparation nor GUIs lend

themselves readily to more complex simulations requiring, for example, numerous model

runs carried out in batches with automatic input preparation.

We therefore propose an alternative approach to the set-up and control of subsurface

simulations, based on the use of scripting. Scripts are simple programs, written in a high-

level scripting language, which can automate much of the pre-processing, control and post-

processing of simulations, eliminating the direct preparation of input files while retaining

full access to all of the simulation code’s capabilities. All simulation parameters, including

complex mesh structures, rock properties, boundary conditions and simulator options,

can be controlled by the script. Scripting offers great flexibility and almost unlimited

automation potential in the creation of model input – indeed, as we will show, it can

facilitate more sophisticated simulations that would be difficult or impossible using either

manual or GUI approaches. Furthermore, existing scripts may be adapted and re-used,

saving considerable time when setting up new simulations.

In this publication, we present a scripting approach for two commonly used coupled fluid

and heat flow simulators: SHEMAT (Clauser and Bartels, 2003) and TOUGH2 (Pruess

et al., 1999; Pruess, 2004). Both simulation codes have been applied to a wide variety of

flow problems. However, setting up the input files for these simulators can be daunting,

and the file structures are very different. Pre-processors (with graphical user interfaces)

exist for both codes, but they have, again, their own logic and limitations. Therefore, using

the full capabilities of these codes, specifically when it comes to complex mesh structures

and process automation, is restricted to expert users – and usually to one code.

In this contribution, we describe the technical details of our scripting approach for

SHEMAT and TOUGH2, and then show examples of how this can be used to set up

model input, run simulations and post-process results, with possibilities for automation

beyond those available through traditional manual or GUI approaches.

5.2. Methods

We have based our approach on Python, a scripting language that has become increas-

ingly useful for computational science and engineering (Langtangen, 2008). Python is a


versatile, powerful and easy-to- learn language with a broad user community. It is freely

available under an open-source license and runs on all major computing platforms. Unlike

traditional coding languages such as Fortran or C/C++, Python scripts do not need to

be compiled, which simplifies their development. Python features clean syntax, modern

and flexible object-oriented data structures, and support for efficient numerical comput-

ing. Extension libraries are available for a very wide range of tasks including scientific

computation.

A somewhat analogous Python scripting approach was described by Paterson (2009)

for computational fluid dynamics (CFD) simulations using the OpenFOAM simulator. In

modeling subsurface environments, no similar scripting approaches have been reported,

although Audigane et al. (2011) recently presented a “workflow tool” for TOUGH2 simu-

lations of carbon dioxide capture and storage (CCS), consisting of a set of Fortran subrou-

tines for pre- and post-processing. However, this tool was of relatively limited scope, being

designed mainly for translating meshes generated using the Petrel geo-modeling software

into TOUGH2 meshes, assigning rock properties and boundary conditions, and producing

graphical output, rather than providing complete control of the simulation. In addition,

Fortran is not well suited for high-level scripting, lacking many of the features listed above

of a modern scripting language like Python.

Making a simulation code like SHEMAT or TOUGH2 operable via Python scripts essen-

tially involves developing a new extension library for interacting with the simulator, which

can then be called from any Python script. Below we describe the libraries, PySHEMAT

and PyTOUGH,which we have developed for these two simulators.

5.2.1. PySHEMAT

SHEMAT (Simulator of HEat and MAss Transport) is a coupled fluid, heat and reac-

tive transport simulation code (Clauser and Bartels, 2003). It is widely applied in low-

temperature geothermal and mineral systems simulation (e.g. Bartels et al., 2002; Kuhn

et al., 2006; Mottaghy and Rath, 2006; Gessner, 2009; Ruhaak et al., 2010). Fluid and heat

flow simulations with SHEMAT can be performed from the command line (on Windows

and Linux/Unix) using an ASCII input file. The input file is structured in a Keyword-

Variable notation (see Clauser and Bartels, 2003, sec. 2.4.3 for a detailed description).

Cell lithology/geology, flow properties and boundary conditions are not directly assigned

to cells but stored in 1-D arrays where the first element corresponds to the lower-left cell

of the model and cells are iterated from left to right and bottom to top of the model.

The grid is specified by the total number of cells and the cell sizes in each coordinate

direction. Results of the simulation are stored in exactly the same file format, enabling

the use of the same Python code for input and output files. Simulations are limited to rec-

tilinear (orthogonal) mesh structures, although, recently a method has been developed to

enable simulation with non-orthogonal meshes, using a coordinate transformation method

(Ruhaak et al., 2010).


Methods for input file generation (Pre-processing)

The PySHEMAT library takes an object-oriented approach, representing a SHEMAT input

file as an object, with all relevant variables stored in the object and specific methods of

the object providing access to them. The first step is therefore either to load an existing

SHEMAT .nml input file or to create an empty file from a template into a Python object:

S1 = Shemat_file ( filename = ”model . nml”)

Model variables can easily be accessed and stored in temporary python arrays for changes.

Different methods are used to read single variables (e.g. NDIM, the number of cells in

x-direction) and array variables (e.g. TEMP, the cell temperatures):

ndim = S1 . get ( ”NDIM”)

temp = S1 . get_array ( ”TEMP”)

The variables can now be manipulated with standard Python methods and stored back in

the object with the according methods, e.g. for the variables above:

S1 . set ( ”NDIM”)

S1 . set_array ( ”TEMP”)

For a full list of variables, see Clauser and Bartels (2003), table 2.7 - 2.15.

The commands above can be considered as“low-level”commands, useful for direct access

of the variables in the SHEMAT object. A number of “high-level” commands are also

available for more complex common tasks. In many cases, a 3-D array is more useful to

read or manipulate the data, instead of the 1-D array structure used in the input file. A

variable can directly be obtained as a data[x][y][z] structure with:

data = S1 . get_array_as_xyz_structure( ”TEMP”)

or converted from a 1-D array:

temp_array = S1 . array_to_xyz_structure( temp , idim , jdim , kdim )

where idim, jdim, kdim are the dimensions of the model. Other methods have been

written to allow a change in the mesh structure, the number of cells in each direction or

the positions of the mesh boundaries.

With all parameters adjusted and stored back in the object, a new input file for the

SHEMAT simulation can be created:

S1 . write_file ( filename )

Additional methods are provided to perform tasks related to running the simulation (not

related to preparation of the input file). With

create_nml_dir ( )

a new directory is created for the simulation run, with automatic continuous numbering.

The control file, essential for the SHEMAT simulation, can be created with

create_shemat_control_file( filename ) .


Finally, the simulation can directly be executed (if SHEMAT is installed on the system)

with

execute_shemat ( )

Several “batch-methods”have also been implemented to simplify and automate common

tasks. For example, when a new model is created, a variety of parameters and settings

have to be adjusted, including, for example, the number of cells, boundary conditions

and initial values for parameters like pressure and temperature. The relevant model set-

up steps are integrated into a single function call, simplifying and automating the whole

process. Essential arguments are the cell spacings in each direction, passed as arrays

dx,dy,dz (cell sizes can change along each axis direction). Several other settings can be

provided as optional keywords, e.g.:

S1_new = create_empty_model(

dx=dx , dy=dy , dz=dz ,

title=”Convection example ” ,

seitet=”NFLO” ,

baset=”TEMP” ,

topt=”TEMP” ,

base_temperature=40,

compute_heat = True ,

compute_fluid = True ,

coupled_fluid_heat = True ,

initialize_temp_grad = True ,

initialize_heads = True ,

set_heads = 1000 ,

nml_filename = ”conv ex 1 ”)

This function provides a simple and direct way to create an empty SHEMAT model from

scratch, even for complex rectilinear mesh geometries. The model can then be further

adapted with the methods described above. These functions and object methods provide

a wide variety of options for automatic model generation.

Methods for output analysis and visualization (post- processing)

As pointed out before, the SHEMAT output file is in the same format as the input file,

but with the file ending .nlo. It can, therefore, be loaded into the same object structure,

e.g.:

S1_out = Shemat_file ( filename = ”model . n lo ”)

The methods that were used for input file manipulation are directly applicable to process

and analyze the results. Additional functions can be used for more detailed analyses and

plot generation, based on the free Python plotting library Matplotlib.


Simulated values at any point (x, y, z) in the model can directly be assessed with a

simple function call, e.g.:

value = S1_out . get_value_xyz ( ”TEMP” ,x , y , z )

A more complex method is available to create isohypse maps, for example the depth of

the 100◦C temperature isosurface in the model (interpolated between the known values at

the cell centers), e.g.:

temp_100_2D = S1_out . get_isohypse_data( ”TEMP” ,100)

In many cases, a summary map for one property or for a calculated mean value is re-

quired, for example to produce a map view of the temperatures at the top or the mean

temperatures of a specified geological unit. Using PySHEMAT we can, for example, cal-

culate mean temperatures of the geological unit 2 with

mean_temp = S1_out . calc_mean_formation_value (2 , ”TEMP”)

A simple 2-D map can be created with another function and directly saved as a figure:

S1_out . create_2D_property_plot( mean_temp ,

interpolation=’ b i l i n e a r ’ ,

title = ’Mean temperatures un it 2 ’ ,

colorbar = True ,

colorbar_label = ’ Temperature ’ ,

xscale = ’ k i l omete r ’ ,

yscale = ’ k i l omete r ’ ,

xlabel = ’E−W [km] ’ ,

ylabel = ’N−S [km] ’ ,

savefig = True ,

filename = ”mean temp . png”)

This plotting function can, for example, also be used to create a map of calculated isosur-

faces.

Further postprocessing options are implemented in the module. These include methods

to export data into a grid format for Geographic Information Systems and into a voxet

format for 3-D visualizations, to create slice plots through the model or histograms of

simulated values. The results of transient simulations, stored in a different file format, are

also accessible with a further set of Python modules

5.2.2. PyTOUGH

TOUGH2 is a general-purpose simulator for modeling multi-phase, multi- component sub-

surface fluid and heat flow (Pruess, 1991). The PyTOUGH library has been developed

specifically for use with AUTOUGH2, the University of Auckland version of TOUGH2.

This version uses a more flexible way of handling equation of state (EOS) modules, and

introduces a range of additional generator types for simulating various configurations of

geothermal wells, but is otherwise very similar to standard TOUGH2.


The TOUGH2 simulation parameters, generator data and model grid are all stored in

a single data input file. This is a text file with a fixed- format layout that is particular to

TOUGH2 (and a source of many subtle errors for the inexperienced user). As TOUGH2

uses a finite-volume formulation, the model grid is described in the data file only in terms

of a set of grid block volumes and the distances and areas associated with the connections

between the blocks. There is no need to specify the locations of the blocks in space. This

gives TOUGH2 the flexibility to simulate one-, two- or three-dimensional problems with

equal ease.

However, pre- and post-processing (e.g. grid generation and visualization of results)

do generally require reference to a coordinate system, so for these purposes additional

geometrical information (e.g. the locations of the block vertices) is needed. AUTOUGH2

is typically used in conjunction with a separate ‘Mulgraph geometry’ file (named for the

in-house visualization software developed in the 1990s for AUTOUGH2, see O’Sullivan and

Bullivant, 1995), which specifies the model grid as an irregular unstructured horizontal

mesh projected vertically down through a series of layers.

Methods for input file generation (pre-processing)

PyTOUGH also takes an object-oriented approach and provides Python classes for both

Mulgraph geometry files and TOUGH2 data files (mulgrid and t2data classes respec-

tively), so the data in these files can be entirely constructed and manipulated in the

computer memory, as well as read from or written to disk. The mulgrid class provides

methods for creating simple irregular rectangular 3-D grids, transforming grids (by rota-

tion or translation), checking grids for errors, evaluating and optimizing mesh quality, and

adding, editing or deleting individual grid nodes, columns or layers.

The t2data class contains properties for all TOUGH2 simulation parameters and gen-

erator data, as well as a grid property which contains the finite-volume description of the

grid. A separate t2grid class is defined for this type of grid. Typically, the finite-volume

grid is either read directly from disk, or created from the mulgrid geometry using the

t2grid.fromgeo() method. For example:

dat=t2data ( ’ model . dat ’ )

reads the TOUGH2 data file ’model.dat’ from disk into the object dat. Its finite-

volume grid can then be accessed using the property dat.grid. Alternatively:

geo=mulgrid ( ) . rectangular (dx , dy , dz )

dat . grid=t2grid ( ) . fromgeo ( geo )

creates a new rectangular geometry object geo from the grid size lists dx, dy and dz

and uses it to create the finite-volume grid of the t2data object dat.

The t2grid class contains Python dictionary properties for accessing the TOUGH2

grid’s blocks, connections and rock types, each of which is also represented by its own

PyTOUGH class. Hence, for example, the permeability in block ‘AR213’ of the grid in a

t2data object dat can be accessed simply by


dat . grid . block [ ’AR213 ’ ] . rocktype . permeability

At the end of the simulation, TOUGH2 writes the values of the primary thermodynamic

variables in each block to a ‘save’ file, which has the same format as the files used for

specifying initial conditions (to enable simple restarting). PyTOUGH provides a t2incon

class to represent initial conditions files, so these can also be created from scripts or copied

from the results of a previous run.

The t2data class has a run() method, which causes TOUGH2 to run a simulation with

the corresponding data file. Hence, the entire workflow from pre-processing through sim-

ulation to post-processing (see below) can potentially be encapsulated in a single Python

script, if desired. As well as enabling automation and the simulation of more complex

problems, scripting the workflow also eliminates input data formatting errors.

Methods for output analysis and visualization (post- processing)

TOUGH2 writes all simulation results to a ‘listing’ file, which contains results tables for

elements (i.e. blocks), connections and generators. This file is again a text file with a fixed-

format layout that is particular to TOUGH2, and varies depending on the EOS module

used for the simulation.

PyTOUGH provides a t2listing class for representing TOUGH2 listing files. Reading

a listing file ‘model.listing’ into a t2listing object lst can be carried out simply using

lst=t2listing ( ’ model . l i s t i n g ’ )

This class has methods for navigating through the listing file in time and easily accessing

all data in the results tables. For example, the heat flow from block ‘AR213’ to block

‘AR214’ in the t2listing object lst can be accessed using:

lst . connection [ ’AR213 ’ , ’AR214 ’ ] [ ’ Heat f low ’ ]

Alternatively, rows and columns of the results tables can be extracted. For example,

lst.element[’Temperature’] gives a one-dimensional array of all temperature results

over the model grid, and lst.element[’AR213’] gives a dictionary of all results (indexed

by name) at block ‘AR213’. The t2listing class also has a history() method which

returns time histories of specified quantities for given blocks, connections or generators.

PyTOUGH can produce simple two-dimensional results plots via the Python library

Matplotlib. The mulgrid class has methods for generating plots of any array quantity over

grid layers, vertical slices or along arbitrary lines. Typically, such an array is extracted

from the element table of a t2listing object, but any derived quantity may also be

plotted.

In addition, PyTOUGH can export TOUGH2 results to Visualization Tool Kit (VTK)

XML files for three-dimensional visualization using Paraview, Mayavi or similar software.

These files are produced using a Python interface to the VTK library and use base-64

encoding for writing the data, decreasing file sizes and enabling faster animation of tran-

sient results. Connection data (e.g. mass and heat flows) are converted to approximate

block-average flux vectors to enable flow visualization.


5.2.3. Availability and licensing

The PySHEMAT and PyTOUGH libraries are free software, released under the GNU

General Public License, both to make them accessible and to allow users to contribute to

them. For more information, and the scripts of the examples below, contact the authors.

5.3. Results

We apply the scripting approach to some example problems, to demonstrate some of its

capabilities. We first use our methods to set up relatively simple models, then show the

power of the scripting approach beyond model set-up and into more complicated research

questions. Each example is simulated with either SHEMAT or TOUGH2, but as the

scripting methods developed for both simulators are similar, each problem could be easily

adapted to run with either simulation code.

5.3.1. Simplified model set-up with PyTOUGH

In the first example, we apply the PyTOUGH library to set up a simple flow simulation

with TOUGH2. We want to evaluate the fluid and heat flow fields in a confined aquifer

above an area of elevated heat flow. The conceptual model is a 2-D layer structure with

a lateral extent of 15 km and a height of 1.5 km. The area of elevated heat flow is 5

km wide and located at the bottom center of the model (fig. 5.1). The heat flow in this

area is applied according to a Gaussian distribution in space with peak value 0.3 W/m2

and standard deviation 750 m. The high-permeability aquifer (with a permeability of

10−13 m2) is bound by confining beds with a thickness of 250 m each. In order to save

computation time, we create a fine mesh for the central part of the model with a width

of 5 km consisting of 51 cells, and a mesh that is gradually coarser towards the lateral

boundaries, consisting of 30 cells at each side. In the vertical direction, the spacing is

regular, with a total of 30 cells. Boundary conditions are no-flow elsewhere and with fixed

temperature and pressure at the top (see fig. 5.1).

Figure 5.1.: Conceptual model of the first example model to evaluate the convective fluid and heatflow fields above an area of elevated basal heat flow.


Once the cell spacings dx, dy, dz are defined (the linear increase towards the side can

be calculated with an arithmetic series), the model geometry is created with

geo=mulgrid ( ) . rectangular (dx , dy , dz , atmos_type=0,convention=2,

origin=[0 ,0 ,

height ] )

The next step is to define the area of elevated heat flux in the center of the model. Heat

flux (Neumann) boundary conditions are assigned as generators in TOUGH2. Therefore,

we first need to determine the blocks at the bottom center. The central columns can be

determined with

cols=[col f o r col in geo . columnlist i f side_width <= col . centre

[ 0 ] <=

side_width+centre_width ]

and the lowest layer in the model is determined with

layer=geo . layerlist [−1]

7500 8750 1000062505000

9620 Temperature [ C]o

7500 8750 1000062505000

0

-1500

Ele

va

tio

n [

m]

E-W [m]

Mass flow (absolute value) [kg/s] 5*10-6

5*10-7

Ele

va

tio

n [

m]

0

-1500

Figure 5.2.: Visualization of the simulated fluid and heat flow fields with a VTK-viewer (Paraview).The zone of elevated heat flow leads to convection in the central part of the model

The heat flux in these columns, following a Gaussian distribution, can be calculated by

qmax , qsd=0.3 , 750 .

x0=0.5∗(cols [ 0 ] . centre [0 ]+ cols [ −1 ] . centre [ 0 ] )

dxs=np . array ( [ col . centre [0]−x0 f o r col in cols ] ) /qsd

qcol=qmax∗np . exp (−0.5∗ dxs∗dxs )


Generators can now easily be created for these heat inputs with:

f o r col , q in zip ( cols , qcol ) :

blkname=geo . block_name ( layer . name , col . name )

genname=’ q ’+col . name

dat . add_generator ( t2generator ( name=genname , block=blkname , type=’

HEAT’ ,gx=q

∗col . area ) )

This simple scripted assignment of generators for the variable heat flux boundary con-

dition over a defined area in the model is significantly easier than manual assignment

directly in the input file.

Essential parameters for the simulation runtime control are assigned with

dat . parameter [ ’ max timesteps ’ ]=150

dat . parameter [ ’ p r i n t i n t e r v a l ’ ]=50

dat . parameter [ ’ t s top ’ ]=1. e14

dat . parameter [ ’ d e r i v a t i v e i n c r emen t ’ ]=1. e−10

An initial simulation run revealed a long equilibration time, because of the low per-

meability in the confining layers. This is a common situation but time-consuming to

address using standard approaches. However, using the scripting approach, we can write

a loop iterating over progressively reduced confining permeabilities, running the model

and assigning the calculated results as new initial conditions, until the desired confining

permeability of 10−16 m2 is reached:

f o r k_confining in [ 1 . e−13, 1 . e−14, 1 . e−15, 5 . e−16, 2 . e−16, 1 . e

−16] :

dat . grid . rocktype [ ’ confn ’ ] . permeability=[k_confining ]∗3

dat . write ( ’ setup . dat ’ )

dat . run ( )

t2incon ( ’ setup . save ’ ) . write ( ’ setup . incon ’ )

This iterative adjustment of the permeability structure leads to a significantly reduced

simulation time and more stable results.

The simulated fluid and heat flow fields can be exported into VTK format for visualiza-

tion (see fig. 5.2). This clearly shows the convection zone above the area of elevated heat

flow, with decreasing fluid velocities towards the lateral boundaries of the model.

5.3.2. Grid refinement study

A common problem for coupled fluid and heat flow simulations is the determination of a

suitable cell size. The optimal cell size is generally taken as the largest one that can resolve

all important features of the problem. Larger cell sizes will give model results that are

dependent on cell size, while smaller ones introduce unnecessary computational overheads.


For the example the problem above, a grid refinement study can easily be carried out,

since the grid is generated using commands in a script. Creating results for a finer grid

is a simple matter of changing a few parameters in the script. The whole grid refinement

study can be scripted as a loop.

Figure 5.3 (produced using the Matplotlib Python plotting library as part of the script)

shows vertical temperature profiles at the center of the model for a range of different cell

sizes. It can be seen that increasing the grid resolution from 71x20 cells up to 221x60

cells alters the temperature profiles noticeably, but further refinement to 331x90 cells does

not produce appreciable further improvement. It can be concluded that 221x60 cells is an

appropriate resolution for this problem.

Figure 5.3.: Computed vertical temperature profiles at the center of the model for different levels ofgrid refinement.

5.3.3. Determining the onset of convection in a 3-D box

The aim of this study is to determine the basal heat flux that is required to initiate con-

vection in a permeable layer, with numerical methods. The heat flux is stepwise increased

and, after the simulation, the heat transfer mode is evaluated. As before, all relevant

steps are included into one Python script. The model set-up is similar to that of the

previous problem, except that we now consider a 3-D problem with closed lateral no-flow

boundaries. The grid has dimensions of 2.5 x 2.5 x 1 km and a resolution of 100 m in the

x- and y-directions and 20 m in z- direction. We are, again, considering a 3-layer system

with a confined permeable layer (permeability: 10−12 m2) between low-permeability layers

at the top and bottom (10−18 m2). A uniform heat flux is applied over the entire lower

boundary. Coupled fluid and heat flow is simulated with SHEMAT.


m2)

fluid simulation, of the heat transfer, all heat flux for a whole

Heat transfer across a layer can be described with the dimensionless Nusselt (Nu) number

(e.g. Holzbecher, 1998). This number is defined as the ratio of total heat transport q to

conductive heat transport qc only:

Nu ≡ q

qc≥ 1 (5.1)

The Nusselt number is equal to 1 for conductive heat transfer and larger than 1 if advective

transport occurs. We can, therefore, use it here as a metric to evaluate the onset of

convection.

A numerical estimation of the Nusselt number for a confined permeable layer has been

proposed by Holzbecher (1998) as a comparison of the mean thermal gradients across

the (impermeable) layer boundary, where only conduction occurs, to the thermal gradient

across the layer:

Nu ≈1L

∫

L∂T∂z

dxTmax−Tmin

H

(5.2)

The thermal gradient across the layer can be estimated using the temperatures of cells

across the layer, or with the cells bounding the layer.

Figure 5.4.: Analysis of the onset of convection in a permeable layer as a function of basal heat flux:Below a heat flux value of approximately 0.05 W/m2, no convection occurs, and the Nu-number is 1;above 0.05 W/m2, convection commences, resulting in an increased heat flux. Typical temperaturefields for the different flow behaviors are presented with the inset figures. The red area highlightsthe starting range for the search algorithm presented in section 5.3.4 which located the onset ofconvection at a heat flux of q = 0.053 W/m2.


This experiment is performed for an already existing input file. All relevant aspects of

the experiment – adjusting the basal heat flow boundary condition, performing the simu-

lation run and evaluating the Nusselt number for the simulated flow fields – are combined

into one function call (determine_Nu). As well as calculating the Nusselt number, a slice

plot through the center of the model is automatically created for visual inspection of the

flow fields. With a simple for-loop in Python, the input heat fluxes from 0.005 W/m2

to 0.2 W/m2 are simulated (“parameter-sweep”), the Nusselt numbers are evaluated (for

upper and lower boundary) and stored in a variable:

f o r i in range (40) :

basal_heat_flux = 0.005 ∗ float (i+1)

( Nu_bottom , Nu_top ) = determine_Nu ( basal_heat_flux , plot=

False ,

top_and_bottom=True )

Nus_top . append ( Nu_top )

Nus_bottom . append ( Nu_bottom )

basal_heat_fluxes . append ( basal_heat_flux )

The evaluated Nusselt numbers as a function of basal heat flux are presented in fig-

ure 5.4. For a heat flux of less than 0.05 W/m2, conductive heat transport dominates: the

Nusselt number is equal to 1 (slight deviations are due to inaccuracies in the numerical

simulation and Nu-evaluation). For higher heat fluxes, convection commences and the

Nusselt number increases above 1. This behavior is also visible in the 3-D representations

of the temperature field (small insets in fig. 5.4).

5.3.4. Automatic determination of convection onset with a root-finding

algorithm

An interesting extension of the previous example would be to determine the required

heat flux for the onset of convection automatically. We will test this here with the root-

finding algorithm bisect that is a part of the Python module SciPy (www.scipy.org). The

root-finding algorithm can be applied to the function determine_Nu used in the previous

example, with a linear shift and a defined threshold level:

de f Nu_minus_1 ( basal_heat_flux ) :

threshold=0.025

return determine_Nu ( basal_heat_flux )−(1+threshold )

This function can then be called with the root-find algorithm, defining a starting range

between 0.04 and 0.08 W/m2:

(x0 , r ) = bisect ( Nu_minus_1 , 0 . 04 , 0 . 08 , full_output = True )

When the keyword full_output is True, a detailed information about the results of the

root-find search is provided, including the number of iterations required to find the root.

Tolerance level and maximum iteration can be defined with additional keywords.


With this simple implementation, the onset of convection was automatically determined

at a heat flux value of 0.053 W/m2 after 6 iterations for a tolerance level of 0.001. This is

in accordance with the visual determination from the previous example (fig. 5.4).

5.3.5. Using scripting in conjunction with other approaches

The above simulations contain relatively simple geometric configurations of parallel layers.

It is possible to define more complex geometries with the scripting methods. It is also

possible to create a more heterogeneous input model with a graphical user interface and

then apply our scripting methods to extend the functionality of the GUI – an approach

taken with this example. We consider here a heterogeneous 3-D geological structure with

a high permeability matrix containing irregular bodies of lower permeability. These could,

for example, represent low-permeability clay layers or other internal sedimentary structures

(fig. 5.5a). This model was created using Processing SHEMAT (Clauser and Bartels, 2003).

Simulation results for a high permeability contrast (10−20 m2 for the low-permeability

bodies) and a basal heat flux of 0.1 W/m2 are presented in figure 5.5b.

Figure 5.5.: Using an existing model (created with a graphical user interface) for advanced experimentswith PySHEMAT: (a) visualization of model structure, permeability of dark green bodies is changedin experiment; (b) results of flow simulation (low-permeability bodies: 10−20 m2 and q = 0.1 W/m2):temperature is plotted in the cross-sections, gray vectors and yellow streamlines represent fluid flow.

We now want to determine the effect of different permeability contrasts between matrix

and internal bodies on the flow system. More specifically, we will determine the required

heat flux for the onset of convection in the system, for a range of permeabilities in the

bodies. We can readily apply the function Nu_minus_1 described before, and the combi-

nation with the root-finding algorithm bisect. With a few lines of code, the permeability

range is defined, the specified permeability assigned to the internal bodies (with geology

identifier 3 in the function update_property_from_dict), the simulation is executed and

the required heat flux leading to the onset of convection is determined:

f o r perm in logspace ( min_perm , max_perm , n ) :

basename2 = ’ onse t nu lowres 6 ’

S1 = Shemat_file ( basename2 + ’ o r i b a s e . nml ’ )

S1 . update_property_from_dict( ”PERM” , {3 : perm})


S1 . write_file ( basename2 + ’ o r i . nml ’ )

(x0 , r ) = bisect ( Nu_minus_1 , 0 . 04 , 0 . 08 , full_output = True )

The full code for the problem also contains the determination of the maximum temperature

and the Nusselt number for a basal heat flux of 0.1 W/m2. For a decreasing permeability

in the internal bodies from 10−12 m2 to approximately 10−14 m2, the critical heat flux that

leads to the onset of convection increases (fig. 5.6a). But for lower permeabilities, there

is no significant further change. This behavior is also reflected in the heat transfer across

the layer (fig. 5.6b) and the maximum temperature in the system (fig. 5.6c). This result

suggests that the system behavior is completely controlled by the higher permeability

part, when the permeability of the internal bodies is below 10−14 m2, even though the

low permeability bodies take up more than 25 percent of the layer volume. This type of

analysis is not easily possible with the standard user interface or the input file itself.

Figure 5.6.: Study of the effect of varying permeability contrasts between matrix and internal bodies onthe system behavior: (a) basal heat flux required for the onset of convection: when the permeabilityof the internal bodies is below 10−14 m2, no higher heat flux is required; (b) maximum temperaturesin the model and (c) the heat transfer across the layer for a basal heat flux of 0.1 W/(m K) showthe same behavior; (d) Representation of flow vectors (arrows) and magnitude (colored sections) fortwo different permeabilities of the internal bodies.

5.4. Discussion

The above examples show that our scripting methods PySHEMAT and PyTOUGH not

only make simulation setup simpler, but, indeed, provide a powerful extension to tradi-

tional manual or GUI approaches. More complex sets of simulations with varying pa-

rameters (e.g. the confining permeabilities in section 5.3.1, the level of grid refinement

in section 5.3.2, the heat inputs in sections 5.3.3 and 5.3.5 and the permeabilities of the

internal bodies in section 5.3.5) that would be tedious and time-consuming to carry out


manually or via a GUI can be done easily using a script. Because our scripting libraries

contain complete representations of their respective simulators’ input and output files,

and allow scripts to run the simulators themselves, many simulations can be carried out

entirely from scripts; however it is also possible to combine scripting with manual or GUI

approaches, or use scripting with existing models, where this is useful.

The scripting approach eliminates the need to interact directly with simulation input

files, making creation and configuration of these files less error-prone than it is with manual

methods (although it is of course still possible to make errors in writing a script). Scripts

can also save considerable time, as they can be re-used and adapted for new problems, or

simply re-run when, for example, updated input data become available.

Basing our approach on the Python scripting language brings additional benefits, as it

is possible to take advantage of the extensive libraries available for scientific and other

computation. Complex calculations can often be carried out with a one-line call to a

library function, such as the parameter searches in sections 5.3.3 and 5.3.5.

We have shown here only a few examples of the possible applications of the PySHEMAT

and PyTOUGH scripting libraries. All capabilities of the simulators are available via

these libraries, so for example high- temperature multi-phase TOUGH2 simulations on

unstructured grids can be carried out using scripts. And because each of the scripting

libraries is provided in the form of a ‘toolbox’ of functions, which can be combined in

almost limitless ways, the flexibility and potential of the scripting approach is much greater

than that of manual or GUI approaches.

For example, the graphical output capabilities of our libraries allow the user to create

customized mini-GUIs tailored to specific problems (or to make use of other dedicated

3-D visualization software). Simple parameter estimation by inverse modeling is possible

by combining PySHEMAT or PyTOUGH with available Python libraries for non-linear

optimization. Populating a model grid with rock properties from a geological model could

be automated with a link to geological modeling software. Additionally, ensembles of

models with stochastically-generated rock properties sampled from a distribution can be

automatically created and run to perform sensitivity analysis. grids. For the purpose of

this work, these grid types were chosen as a matter of convenience for the TOUGH2 case

and as a restriction implied by the numerical code, for SHEMAT. As block geometries are

not explicitly defined with TOUGH2, more complex mesh geometries are in principle pos-

sible – but can be tedious to implement. Several pre-processing tools exist to create more

complex meshes, for example Mulgraph, mentioned already before, or WinGridder (Pan,

2008). The possibility to deal with more complex mesh structures is already implemented

in PyTOUGH for extruded meshes: meshes can be defined in 2-D and then extruded into

the third dimension. SHEMAT itself can only handle rectilinear mesh structures but in

a recent development, Ruhaak et al. (2008) presented a method to enable the simulation

with non-orthogonal meshes, using a coordinate transformation method. The scripting

methods presented here are not limited to rectilinear meshes only.

Although we have here demonstrated the scripting approach only for the SHEMAT and


TOUGH2 simulators, it is broadly applicable to other subsurface modeling simulators,

and indeed to other types of simulators as well. The creation of similar libraries for other

simulators would multiply the possibilities of the approach even further. For example,

converting models from one simulator to another (as has been done recently by Audigane

et al., 2011; Borgia et al., 2011) would be simplified. Interesting hybrid simulations would

also become much more easily possible, using more than one simulator and a Python script

to control their interaction – for example, combining TOUGH2 and a rock mechanics

simulator to model subsidence in a geothermal field.


6. Link between Geological Modeling

and Flow Simulation


simulation





simulations





Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7



simulation

Chapter 6

84 6. Link between Geological Modeling and Flow Simulation

Overview This chapter presents the integration of geological modeling and flow simu-

lations into one framework. This integration is an essential aspect in order to enable the

stochastic method to evaluate the influence of uncertainties in geological data on simulated

flow fields. The first part of this chapter briefly outlines the standard workflow from geo-

logical data to flow simulation, highlighting the common “bottle-necks” in this procedure.

Next, it is described how these limitations are overcome with a combination of already

available programs and several newly written programs, all integrated into the scripting

framework of Python.

The hypothesis underpinning this chapter is that these novel methods greatly simplify

the inclusion of realistic full 3-D geological models in coupled flow simulations. It is

tested with two common problems: (i) the set-up of different conceptual models, based on

different possible geological scenarios, and (ii) determining a useful discretization of the

geological model.

The text in this chapter is an extended version of a peer-reviewed extended conference

abstract (Wellmann et al., 2011b).


6.1. Introduction

Subsurface flow and transport processes are commonly evaluated with numerical meth-

ods. An important element of numerical flow simulations is the distribution of relevant

rock properties in space, for example permeability, porosity or thermal conductivity. It

is common practice to determine this property distribution from the subsurface geology.

Therefore, the usual workflow is to create subsurface structural models, here referred to as

geological models, from geological information (outcrop observations, drillhole measure-

ments, interpreted geophysical data, etc.), then to export this model into a mesh structure

that is suitable for the flow simulation, to assign flow properties according to the geology,

define boundary conditions and initial settings and, finally, perform the simulation (e.g.

Franke et al., 1987; Bear, 2007; Bundschuh and Arriaga, 2010).

As models are by definition only an approximation of reality they always contain uncer-

tainties. In order to address these uncertainties, many methods for parameter optimization

and calibration have been developed (e.g. Finsterle, 2004; Poeter and Anderson, 2005; Do-

herty, 1994). However, geological uncertainties are usually not, or only partly, considered

(e.g. Refsgaard et al., 2006; Troldborg et al., 2007; Suzuki et al., 2008). This can be a prob-

lem because geological models always contain uncertainties (Mann, 1993; Bardossy and

Fodor, 2004; Wellmann et al., 2010). In fact, recent studies have shown that uncertainties

in the geological model can be the most significant source of uncertainties for long-term

flow predictions (Nilsson et al., 2007; Refsgaard et al., 2011).

An important reason for this neglect of geological uncertainties is that the workflow from

geological data to flow simulation requires manual interaction. We hypothesize here that

an automation of the whole workflow will greatly simplify testing of different geological

scenarios and provide a novel way to evaluate the effect of geological uncertainties on sim-

ulated flow fields. We will describe our methods to integrate all steps into one automated

workflow below. As a test of feasibility, the novel techniques are applied to a sensitivity

analysis of different possible geological scenarios in a realistic geothermal setting in the

North Perth Basin, Western Australia. Furthermore, we will evaluate an optimized grid

discretization scheme based on geological complexity.


The common procedure to create a subsurface flow simulation in a realistic setting can

be subdivided into several steps (e.g. Bundschuh and Arriaga, 2010). These steps are

visualized in figure 6.1:

Step 1: Based on a geological data set, a geological model is constructed as a representation

of all relevant structures below ground.

Step 2: The continuous geological model is transformed into a discrete version, suitable

for the numerical solver.


Figure 6.1.: Typical modeling steps from original geological data to simulated fluid and heat flow fieldsand the typical “bottle-neck” steps: the geological modeling and mesh generation are difficult toautomate (red arrows). We are addressing these steps with our approach to enable an automaticupdate of the simulated flow fields when geological input points are changed (blue points and dottedlines). The input file generation for the flow simulation itself is less difficult to automate (greenarrow).


Step 3: The distribution of relevant rock properties is determined from the discrete version

of the geological model and boundary conditions are defined.

Step 4: Further specific simulation settings and temporal discretization (time steps) are

assigned and the simulation is performed.

The color of the arrows in figure 6.1 indicates the complexity associated with an automation

of these steps: red arrows denote steps that are difficult to automate, the green arrow for

Step 4 indicates that this step is relatively easy to automate. Methods based on the

automation of Step 3 and 4 are already widely implemented in parameter estimation and

sensitivity methods (Doherty, 1994; Finsterle, 2004; Poeter and Anderson, 2005). However,

the aim of the approach here is to evaluate directly how a change in the geological input

data (from gray to blue points in figure 6.1) affects the flow field and this is only possible

with a complete automation of the workflow. In the following, we describe our approach

to automate all of these steps.

6.2.1. From geological data to simulated flow field

Step 1: Geological modeling

The first step is usually to create a structural representation of the subsurface, a geological

model, as a starting point for the fluid and heat flow simulation (fig. 6.1, Step 1). The

complexity of the model can range from very simple plate models to highly complex

interpolations of a variety of input data in full 3-D. Many different and mostly commercial

tools exist to create subsurface models (e.g. Mallet, 1992; Turner, 2006; Calcagno et al.,

2008). They differ mainly in the way they interpolate between data points (e.g. surface or

volume methods) and the flexibility of input data types they can deal with (seismics, well-

logs, structural measurements, etc.). All these approaches have different advantages and

disadvantages but ultimately the aim to create a representation of the subsurface structure.

For the purpose of this work, an important distinction between different methods is that

an automation is only possible with some approaches. With explicit modeling methods,

manual steps are required for most cases to create a valid 3-D geological model based on

a set of input data (Caumon et al., 2009), especially in complex geological settings with

fault networks or overturned folds. Alternatively, implicit methods enable an automatic

model reconstruction for cases where the input data set is moderately changed (Carr et al.,

2001; McInerny et al., 2004a; Caumon et al., 2007).

We apply here an implicit geological modeling method based on a potential-field ap-

proach (Lajaunie et al., 1997) that is implemented in a commercial program (GeoMod-

eller) to automate the first step. The geological setting is treated as a potential-field

where a geological boundary corresponds to an isosurface of this potential-field. The field

is interpolated from discrete geological observations with a cokriging technique. With this

method, it is possible to recompute the geological model automatically, once all other

model settings and parameters are defined (see Calcagno et al., 2008, for a detailed de-

scription of the modeling method). A further benefit of this method is that it is possible


to consider observations of surface contacts and orientation measurements directly. A

contact point is defined with its coordinates (x, y, z) in space. Orientation measurements

additionally contain the information about the azimuth φ and the inclination θ of the

measured plane (see fig. 6.2). In the following, we denote any of these parameters as kj ,

parametrized geological values:

kj ∈ (~x, φ, θ) (6.1)

Unit 1

Unit 2P1(~x)

P2(~x)

O(~x, φ, θ)

x

yz

Figure 6.2.: Types of geological raw data used for geological modeling and their parameterization:(a) contact points (P1 and P2) between two distinct geological units, ki ∈ (~x) where ~x are thecoordinates of the point and (b) orientation measurement (O) within one unit, ki ∈ (~x, φ, θ) where,additionally, φ is the azimuth and θ the inclination of a measured plane, anywhere within the unit(not only along the surface).

We consider here the geological model symbolically as a computation function

M(~x) = f1(K,αk, βℓ) (6.2)

of the set K of parametrized geological observations kj ∈ K, specific parameters αk of the

interpolation function (e.g. range, sill and nugget effect of the variogram) and βℓ, the space

partitioning in which different potential-fields are valid, defined by fault compartments and

model stratigraphy (Calcagno et al., 2008). The subscript of the function f1 corresponds

to Step 1 in figure 6.1. The domain of possible values of M are the geological units, or

lithologies Li, defined by the volumetric space between interpolated geological boundaries.

Step 2: From subsurface model to simulation grid

The geological model can be considered as a continuous function in 3-D space. On the

other hand, the continuous differential equations that describe the studied flow processes

are typically solved numerically for discrete cells and elements (e.g. Huyakorn and Pinder,

1987; Holzbecher, 1998; Ingebritsen and Sanford, 1998). The next typical step in the

workflow is therefore to create a discretized version of the geological model, according to

the mesh structure that is required for the numerical solver. Solvers that are based on

finite difference (FD) methods require a rectilinear (cartesian) mesh format (see fig. 6.3),

whereas finite volume (FV) and finite element (FE) methods allow a more flexible mesh

structure.

After the grid structure is defined, lithologies are mapped onto this mesh from the

continuous geological model. As we apply here a FD simulation method, we first define a


DD

D

x ,x , ...

x

1

2

n

Dy ,y , ..

.y

1

2

m

D

D

DD

Dz

,z

, ...

z1

2o

Figure 6.3.: Rectilinear cartesian grid, typically used for FD simulations and applied here: grid re-finement in areas of interest is possible but not as flexible as with a completely unstructured grid(modified after Wikipedia, 2006).

rectilinear mesh. The grid structure can be described as an array of cells with defined width

in every direction (see fig. 6.3), for example as ∆x1, . . . ∆xm for m cells in x-direction. The

position of cell centers can then be calculated as

xm = x0 +

m−1∑

µ=1

∆xµ + ∆xm/2 , (6.3)

where x0 is the position of the mesh origin. Cell centers in y and z-direction are determined

accordingly. The lithological unit according to each cell can then be mapped from the

continuous geological model. We assign here the value of the lithological unit at the cell

center ~xc to the whole grid cell G:

L(G) = f2(M(~xc)) . (6.4)

Step 3: Definition of the conceptual model

After the subsurface model is discretized into a mesh format, relevant rock properties are

assigned to each grid cell. The required properties depend on the type of simulated process.

These properties include for example permeability and porosity for flow simulations and

thermal conductivity and heat capacity for thermal simulations. It is common practice to

assign properties according to a rock type or lithological unit, according to measurements of

the property at rock samples, evaluations of wire-line logs in drillholes or the mineralogical

rock composition (e.g. Clauser and Bartels, 2003; Clauser, 2006). If we denote the relevant

set of properties per grid cell as P and we associate each lithology in the model with a

defined set of properties, we can write:

P(G) = f3(L(G)) (6.5)


Where the function f2 can denote a direct association of one property set for each lithology

for the simplest cases. In more complex cases, a property distribution could be assigned,

for example to respect spatial trends with geostatistical methods (e.g. Deutsch and Journel,

1998; Isaaks and Srivastava, 1989; Chiles and Delfiner, 1999).

Another important step at this stage is to assign boundary conditions Bq and initial

values that are suitable to solve the considered problem. These conditions have a significant

influence on the simulated flow fields (Bundschuh and Arriaga, 2010; Refsgaard et al.,

2011) and they can be difficult to estimate. For the purpose of this work, we consider the

boundary conditions not to be directly related to the geological model and therefore as

unaffected by changes in the structural geological model. For example, surficial recharge

conditions are not affected by internal structural changes. However, it is possible that

boundary conditions could be related to the subsurface structure, for example when inflow

at depth exists. These considerations have to be evaluated for each specific system.

Step 4: Process simulation

The process simulation can be performed based on the conceptual model, defined in the

last step, and further simulation settings that have to be assigned according to the applied

simulation method. These settings include time step sizes and parameters of the numerical

solvers, designated γr. A wide variety of numerical solvers is available, both commercially

and as open-source codes, to study subsurface problems (see e.g. O’Sullivan et al., 2001;

Bundschuh and Arriaga, 2010, for an overview of solvers for coupled thermo-hydraulic

simulations). We apply here SHEMAT (Simulator for HEat and MAss Transport, Clauser

and Bartels, 2003) to solve a coupled hydrothermal problem. SHEMAT is a FD simulation

code and has been applied to study geothermal and mineral systems (e.g. Kuhn and

Gessner, 2006; Gessner et al., 2009). SHEMAT can directly be used to simulate flow fields

on the basis of defined conceptual model and simulation settings, stored in an input file.

We can therefore consider the simulation step as a function that calculates the flow fields

Fs based on the properties defined in the mesh Pp, boundary conditions and initial values

Bq and simulation settings γr as:

Fs = f4(Pp,Bq, γr) (6.6)

Combination of all previous steps

The aim is now to obtain the updated flow fields Fs when a part of the geological input

data set K is changed. Under the considerations described above, and further assuming

that the settings for the geological interpolation αk remain the same, the space partitioning

βℓ doesn’t change and boundary conditions Bq and simulation settings γr are independent

of the geological model, all previous steps denoted as functions f1–f4 (eq. 6.2 – 6.6) can

be combined into one function that enables a direct determination of the flow fields as a


function of the changed input data set, here denoted as the “workflow function” fW :

FW = f5(K) (6.7)

6.2.2. Numerical implementation

In order to derive a complete automation of all previous steps to allow a direct re-

computation of flow fields with a changed input data set (eq. 6.6), we integrated existing

codes for geological modeling and coupled process simulation into one framework. A set

of modules in the scripting language Python (www.python.org) was developed to control

and link the external programs, to access the initial data set, and to process and ana-

lyze results. Python is widely used in scientific computations (e.g. Langtangen, 2008),

because it is a flexible, object-oriented language and a wide variety of extension modules

are available for numerical methods, data analysis and visualization. Furthermore, Python

interpreters are available for almost every operating system. Our developed methods were

tested and executed on Linux, MacOS and Windows, with the exception of the geological

modeling itself which is to date limited to Windows.

The Python module controlling the geological modeling enables access to the whole

geological data set, including the definition of stratigraphy, faults and geological modeling

settings. The functionality of the model interpolation itself can be accessed externally

with a program interface to the commercial geomodeling software. A Python module has

been defined to integrate this interface into the workflow.

The method to automate the set-up of the input files for the numerical simulation clearly

depends on the simulation code. We have here integrated the simulation code SHEMAT

(Clauser and Bartels, 2003), but other codes could be included, as well (see 6.4. SHEMAT

can be executed from the command line and all settings and parameters are passed in

an ASCII input file. We developed a module to read, manipulate and create these input

files. Execution of the program itself can also be performed from within the modules. The

results of the simulation time steps are saved in ASCII files that are structurally similar

to the input files and can also be accessed with the Python modules. More details about

the modules to control SHEMAT are described in chapter 4.

Integration into one workflow

With the developed scripts, it is straightforward to combine all relevant steps of the work-

flow (fig. 6.1) into one controlling script, adapted to investigate a specific problem. An in-

put file for a coupled fluid-heat flow simulation, with a simple mesh structure and boundary

conditions, and the property distribution derived from a geological model can be created

with one function call:

geomodeller2shemat(<filename of geomodel>, <directory of

geomodel>,

nx , ny , nz ,

property_file = <path to property file>,

www.python.org


nml_filename = <name of new nml>,

spacing = ’ r egu l a r ’ ,

compute_heat = True ,

compute_fluid = True ,

coupled_fluid_heat = True ,

baset=”WSD” ,

basal_heat_flux = 0.06 ,

topt=”TEMP” ,

top_temperature=20,

execute_shemat = True )

This function already integrates many essential parts of the workflow. The mesh structure

is defined as a regular mesh with nx, ny, nz cells in each coordinate direction. In the

property_file, relevant properties for the simulation (porosity, permeability, etc.) are

assigned to the geological units in the geomodel. Boundary conditions in this example

are a basal heat flux of 0.06 W/m2 and a fixed temperature of 20◦C at the top. Peculiar

identifiers like WSD are used in accordance to the notation in the SHEMAT input file, (see

Clauser and Bartels, 2003). Further adaptions of the input file of the flow simulation are

accessible through the Python modules PySHEMAT, see section 5.2.1.

Another illustrative example is the automation of the complete process depicted in

figure 6.1: changing the position of one data point in the geological data set, re-computing

the geological model, updating property distribution in the flow input file, performing the

flow simulation and exporting data from the simulated flow fields. This could, for example,

be required to adjust for a changed seismic interpretation or a re-evaluation of drillhole

data. Shifting data points for one formation by 50 meters in z-direction, then recomputing

the model, updating the flow simulation and determining the value for temperature at a

location can all be done with a few lines of code:

G1 = Geomodel ( ’ geomodel . xml ’ )

pts = G1 . get_formation_point_data( G1 . section_dict [ ’ Sect ion1 ’ ] )

f o r point in pts :

i f point . find ( Data . get [ ’Name ’ ] == ’ Formation1 ’ :

G1 . change_formation_point_pos( point , add_z_coord = 50)

G1 . recompute ( )

S1 = Shemat_file ( ’ f lowsim . nml ’ )

S1 . update_from_geomodel( ’ geomodel . xml ’ , ’ p r op e r t i e s . csv ’ )

execute_shemat (S1 )

S2 = Shemat_file ( ’ f lowsim . n lo ’ )

p r i n t (S2 . get_data_at_xyz ( ’TEMP’ ,500 ,500 ,−1000) )

The advantage of the organization in scripting modules is that parameters can easily

be changed and adapted. Also, the methods lend themselves easily to automation and

the modules can be combined with a the wide range of powerful modules for numerical


methods, scientific analysis and visualization that are available for Python. The mod-

ular scripting approach has been implemented to enable the study of a wide range of

scientific problems that involve geological modeling and flow simulations. We will test it

with two simple examples that might be possible, but very time-consuming with standard

approaches.

6.3. Results

The possibilities that are enabled with the methods to automate the workflow from ge-

ological data to simulated flow fields will be tested to address two problems, considered

here as important aspects for cases where a realistic geological model is used as the basis

for a flow simulation:

➤ The influence of the discretization of the geological model on the flow simulation;

➤ The sensitivity of simulated fluid and heat flow fields to different geological scenarios.

To address the first problem, we mainly make use of the fact that the second step in the

workflow (fig. 6.1), the processing of the model to the simulation grid, is automated with

our methods. With the second example, we test the complete automation of all four steps,

from initial geological observations to simulated flow fields.

6.3.1. Example model in the North Perth Basin

We are applying our methods to a large-scale hydrothermal simulation. The model is a

geothermal resource scale study for a part of the Dandaragan Trough, Western Australia

(fig. 6.4a). The Dandaragan Trough is part of the Perth Basin, a deep elongated sed-

imentary basin, extending 1000 km along the Western margin of Australia. The Perth

Basin is bound to the East by the Yilgarn Craton, with a major offset along the Darling

Fault (Mory and Iasky, 1996; Song and Cawood, 2000). The Perth Basin formed in several

stages from the Permian up to today, with the main tectonic event occurring during and

after the break-up of Gondwana that separated Australia from Greater India in the Early

Cretaceous (e.g. Mory and Iasky, 1996). The Basin is filled with sediments that were

mainly deposited before the final break-up of Gondwana and is structurally subdivided.

The Dandaragan Trough, forming the main feature in the model researched here, is an

elongated structure along the Eastern margin of the basin, between the Urella Fault in

the East and the Eneabba Fault System and the Cadda Terrace in the West (see Mory

and Iasky, 1996, fig. 6). The Dandaragan Trough is more than 12 km deep and contains

sedimentary sequences of Permian to Mesozoic age.

The North Perth Basin drew recent attention as a potential geothermal resource area

and several companies obtained licenses for geothermal exploration in the region (Ghori,

2008). A main focus of exploration has been on the fault structures in the Western part

of the onshore Basin, mainly because of measurements of elevated heat flow (Chopra and

Holgate, 2005). The Dandaragan Trough, researched in this study, is believed to have a


Basement

Geraldton

Project area(green: submodel)

Jurien

Perth

DarlingFault

(a) Location of modelarea

(c) Discretized geological model

(d) Simulated temperature field

Faults

Sediments

Temperature isosurfaces(50-250 C)

o

Convection cells

250 x 25 x 200 cells(total: 1,250,000)

(b) Extent of geological model andsimulated sub-model

63 km EW

70 k

m N

S

15 k

m

Submodel

Wells with estimatedtemperature gradient

Faults

Figure 6.4.: Simulation of a geothermal resource area: (a) location of the model area in the NorthPerth Basin; (b) Surface map for the area of the geological model, showing faults (red lines), seismiccross-sections (green lines) and the area used for the geothermal simulation (shaded area), shownin: (c) the discrete version of the geological model in a rectilinear mesh with 1,250,000 cells as basisfor the temperature simulation; (d) results of the hydro-thermal simulation showing the effect ofconvection on the temperature field.


low to medium (2 - 2.5◦C) geothermal gradient (see Mory and Iasky, 1996). On the other

hand, preliminary studies on the permeability of the sediments, and considerations of the

extensive thickness of permeable layers, suggest that free convection might exist in some

of the sedimentary sequences (Horowitz et al., 2008; Sheldon et al., 2009). Advective heat

transport, associated with the convection, is a very effective heat transport mechanism and

can strongly influence the local temperature field, providing patterns of warmer and colder

regions compared to the average geothermal gradient, as for example shown by Bachler

(2003), Kohl et al. (2003), Cacace et al. (2010), or Garibaldi et al. (2010). These authors

also discussed the influence of the subsurface structure on the wavelength of convection.

We will here evaluate how uncertainties in the geological model influence the predicted

flow fields for the simulated geothermal resource area.

Geological model and its uncertainties

A full 3-D geological model was constructed for the area, mainly based on previously

published data and interpreted cross-sections from Mory and Iasky (1996). The authors

of this study discuss the uncertainties in their interpolations and maps. For example, they

state that, generally, structural uncertainty greatly increases with depth due to a lack of

data and partially poor seismic data. Especially in the the Dandaragan Trough modeled

here, additional faults might exist in the basin that were not visible in the seismic section

(Mory and Iasky, 1996). Starting from an initial base model, we will create several possible

geological models that are all feasible in the range of these uncertainties.

The initial model was constructed directly honoring the interpreted seismic sections and

the geological map of Mory and Iasky (1996), as the“best guess”geological model. Relevant

material properties (thermal conductivity, porosity and permeability) were compiled from

data sets published by Chopra and Holgate (2005) and Hot Dry Rocks Ltd (2009). A

summary is presented in table 6.1. The geological model was then discretized into a

regular mesh structure with 250 x 25 x 200 grid cells (fig. 6.4b). Boundary conditions are

constant base and surface for heat, and no-flow at lateral boundaries. Flow is constrained

towards the East and to depth by an impermeable formation in the East, and continuous

boundary conditions are applied in the other directions.

The simulated temperature field (for a base temperature of 300◦C as lower boundary

condition, and 20◦C at the surface) is presented in figure 6.4c. The influence of free

convection is dominating the temperature field in the parts of the model with the thickest

sedimentary succession.

6.3.2. Testing of different cell discretization schemes

The determination of a suitable discretization of the geological model for a process simu-

lation is a known problem (Bundschuh and Arriaga, 2010; Mehl and Hill. Mary C., 2010;

Graf and Degener, 2011). The optimal cell size is always a balance between simulation

accuracy and computation time. Typically, cell size criteria are determined from the view-

point of numerical stability (e.g. Huyakorn and Pinder, 1987). However, an additional


Unit Name Unit ID Permeability Porosity Thermal Cond.[m2] [ ] [W m−1 K−1]

Basement 1 1 · 10−20 0.01 2.39Permian 2 1 · 10−17 0.05 3.00Woodada-Kockatea 3 1 · 10−16 0.05 2.79Lesueur 4 1 · 10−15 0.10 3.56Eneabba 5 5 · 10−16 0.05 2.62Cattamarra 6 8 · 10−15 0.10 3.73Cadda 7 1 · 10−17 0.05 3.80Yarragadee 8 1 · 10−14 0.20 3.54Cretaceous 9 5 · 10−14 0.25 1.42

Table 6.1.: Material properties assigned to the units in the geological model. Values are sourced froma variety of publications. Anisotropy of permeability is assigned to all units (10 times higher lateralthen vertical permeability). The unit IDs correspond to those shown in figure 6.6.

(a) Model discretization

1.00E+04

1.00E+05

1.00E+06

1.00E+07

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Model

Nu

mb

er

of

ce

lls

0

50

100

150

200

250

Ce

lls

ize

z-d

ire

cti

on

[m]Total number of cells

Cell height [m]

(b) Model discretization

0.01

0.1

1

10

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Model

CP

Uti

me

(ho

urs

)

CPU time [h]

Large symbols: modelsused for visualization below

Figure 6.5.: (a) Different discretizations used here to evaluate the mesh dependency: Total number ofcells of the 19 models used in this study and corresponding cell height. The larger symbols representthose models for which results are presented in the following figures; (b) Simulation time for thedifferent discretizations, ranging from several minutes to more than 10 hours of CPU time, showingthe clear influence of model discretization on simulation time.

important consideration is how fine the spatial discretization has to be in order to capture

all geological structures that are relevant for the flow field. This is not easy to deter-

mine with standard methods for heterogeneous and complex subsurface structures and

rarely done because creating different mesh structures from the geological model (fig. 6.1,

Step 2) requires manual interaction. We will now apply our methods to perform a cell

discretization study for geological complexity.

The temperature field in figure 6.4c was simulated on a regular mesh with 1.25 million

cells and the simulation lasted approximately 6 hours on a 24 core Linux computer. For the

cell size study, 19 different mesh discretizations were defined with a total number of cells

ranging from approximately 50,000 to 4,000,000 cells. The discretization is determined in

a way that the cells have a five times larger horizontal than vertical extent. As expected,

the simulation time increases significantly for the high resolution models (fig. 6.5b). Cross-

sections through the discretized geological models for the different meshes (fig. 6.6) show

the different levels of refinement for the different cell sizes.


Unit ID

Model 1: 50,400 cells



Model 15: 1,538,636 cells


Figure 6.6.: Cross-section through the discretized geological model for different discretization steps.The refinement is clearly visible around the faults.


Temperature [ C]o






Figure 6.7.: Simulated temperatures in a cross-section through the model for different cell refinements.It is obvious that the simulated temperatures differ significantly from Model 1 to Model 10. Mostdetails of the very high resolution Model 19 are already present in Model 15.


Velocity in z-direction [m/s]






Figure 6.8.: Velocities in z- direction in cross-sections through the models with different discretizationshow a similar behavior as the temperature fields: clear differences exist between the first models,the last two models are similar, specifically in the deeper parts.


Temperature isosurfaces

50 Co

270 Co




Figure 6.9.: 3-D representation of simulated temperature fields for a model with very low resolution(Model 1), intermediate resolution (Model 10) and very high resolution (Model 19); The overallstructure of the convection system is already present in the medium resolution model, although thetemperature fields differ clearly in details around the convection cells.


In figure 6.7, simulated temperature fields are plotted in cross-sections through the

simulated 3-D models. It is clearly visible that the temperature field changes from low-

to mid-resolution models (Model 1 to Model 10). However, from visual inspection no

significant change is visible between mid- to high-resolution models, although more details

are clearly visible in the highest resolution case (Model 19). A similar trend is visible in

the plots of velocity in z-direction (fig. 6.8) and in the 3-D plots of temperature isosurfaces

(fig. 6.9). Even though these evaluations are only qualitative, they provide an indication

about the required level or discretization from the viewpoint of geologic complexity and

solution stability. For the purpose of the model and the analysis performed here, the

results indicate that a mid- to medium-high resolution model of approximately 1.5 million

cells captures most of the details of the flow field, with a reasonable computation time.

The typical considerations of numerical stability (e.g. Huyakorn and Pinder, 1987) have

to be considered additionally.

6.3.3. Testing of different geological scenarios

As described above, the geological model is mainly based on the publication by Mory and

Iasky (1996). The authors state that two main structural uncertainties exist in this part

of the basin:

➤ The exact depth of the sedimentary layers is unknown, as no drillhole exists that

penetrates the whole sequence in the deepest areas of the basin;

➤ Additional faults might exist in the basin that were not recognized due to a poor

seismic coverage of the deeper layers.

In the following, we will test if these two uncertainties affect the simulated fluid and heat

flow fields.

Scenario 1: Testing the effect of unknown layer thickness at depth

We will first test how the uncertain position of layers at depth affects the simulated

temperature field. This test is very simple to perform once the whole workflow from

geological data to simulated flow field is set-up with the original model. The position

of the geological observation can be adjusted with the Python modules described above

(sec. 6.2.2) or, if more convenient, interactively within the graphical user interface of the

modeling software. It is noteworthy that changes are directly performed on the geological

data as opposed to a simulated surface. That is, the position of points that define a

geological surface, and the position and value of orientation measurements are adapted in

the original data (fig. 6.10). The relationship to the uncertainty of the original data is

thus always preserved. All steps from the geological model construction to the simulated

flow fields are then performed within the scripting framework without any further manual

interaction.


Figure 6.10.: Modifying data points to test the effect of the uncertain thickness of the sedimentarylayers: (a) data set of the original model, surface observation points are the raw geological informa-tion, the lines represent the already modeled structure; (b) changed input data points for a thickersedimentary sequence, and (c) for a thinner sedimentary sequence.


In figure 6.11, the simulated temperature fields for the original model (a), the thicker

sedimentary sequence (b) and the thinner sedimentary sequence (c) are plotted in a 2-

D section in the center of the model. The flow fields are simulated for the whole 3-D

model. The changed sedimentary thickness has an influence on the temperature field,

mainly in the area of the main convection system: the convection cell seems to be of a

wider lateral extent in the case of the thicker sedimentary sequence. This is in accordance

to theoretical considerations as the dominant wavelength of convection changes with layer

thickness (Nield and Bejan, 2006). However, the position of the upwelling zone does not

change significantly. This indicates that the upwelling position might be controlled by the

structure itself, that is the shape of the permeable units, rather than the thickness of the

sediments.

15750 315000 47250 63000E-W Section [m]

15750 315000 47250 63000

15750 315000 47250 63000

1000

-2000

-5000

-8000

-11000

Depth

[m

]

1000

-2000

-5000

-8000

-11000

Depth

[m

]

1000

-2000

-5000

-8000

-11000

Depth

[m

]

(b) Thicker sedimentary sequence

(c) Thinner sedimentary sequence

(a) Original model (Temperature in cross-section)

Differences in area ofconvection

Temperature [ C]o

20 300160

Figure 6.11.: Simulated temperature fields for the scenarios of uncertain sediment thickness in thebasin, visualized in a cross-section in the centre of the model: (a) temperature field of the originalmodel; (b) temperature field for the case of a thicker sedimentary sequence, and (c) a thinnersedimentary sequence; Differences in the temperature field to the original model (a) are concentratedin the area of the main convection cell.


Scenario 2: Influence of additional faults

The last example indicated that the thickness of the sedimentary layers does not affect

the flow greatly but implies that structure was important. We will now evaluate the

influence of additional faults in the basin. This scenario is achieved through the addition

of a few data points using the graphical user interface of the geological modeling software

(fig. 6.12). Only a few additional data points have to be added to the model and the

interaction between faults and sedimentary layers has to be defined in order to obtain the

different geological scenario.

Figure 6.12.: Additional data points (now: surface orientation and surface contact) to define twoadditional normal faults in the basin. The geological model (here represented as lines) is thenautomatically reconstructed, honoring the faults.

The simulated flow fields, again, represented with the temperatures in the cross-section

(fig. 6.13a), now clearly show an influence of the change in the geological structure. The

upwelling zone is broader and slightly shifted, compared to the original model (fig. 6.11a).

Also, a stronger upwelling exists in the deeper part of the basin. The differences are even

more evident in a representation of the flow field. The intensity of the vertical flow for the

fault scenario is presented in figure 6.13b, and for the original model in figure 6.13c (red

colors: upward flow, blue colors: downward flow). The influence of the fault structure on

the flow field is clearly visible with upwelling in the center and down-welling along the

faults. This differs from the flow field in the original model (fig. 6.13c) where a stronger

upwelling dominates in the center, with an additional upwelling along the Eastern fault

and a broad downwelling in the Western area of the model.

6.4. Discussion

The results of the case study show that our developed methods greatly simplify the eval-

uation of sensitivities of hydrothermal flow fields with respect to geological uncertainties.

We successfully integrated all steps, from primary geological data, to geological modeling,

to flow simulation and the analysis of results, into one automated framework. No further

manual interaction is required to update the results of flow simulations when the geolog-

ical input data are changed moderately. This opens-up the way to a whole new range

of sensitivity testing and uncertainty evaluations of simulated flow fields with respect to

the original geological data. We presented two studies that are not easily possible with

standard methods. The first was an automatic cell discretization study, evaluating the


15750 315000 47250 63000

1000

-2000

-5000

-8000

-11000

De

pth

[m

]

(a) Temperature field with additional faults Character of temperature fielddifferent to original model

(b) Fault scenario, vertical flow

(c) Original model

Vertical flow

upflowdownflow

Detailed view offlow field

Upflow in center, downflowconcentrated along faults

Stronger upflow in centerand also along E faultdownflow over wide area in W

green lines:geology surfaces

Figure 6.13.: Simulated temperature field for scenario 2 with additional faults in the central part ofthe basin. The temperature field (a) is now very different to the original model. The difference iseven more visible when the flow vector fields are compared: in the original model (b), the up-flowin the central part is broader, down-flow is distributed over a wide lateral extent; in the case of theadditional faults (c), the fault structure has a clear influence on the down-welling (blue areas).


required cell discretization to respect geological heterogeneities. The second study was

a scenario evaluation, testing how different geological scenarios, all possible in the range

of geological uncertainties, influence the simulated flow fields. In the standard approach

for flow simulations, at least some manual interaction is required to perform these types

of analyses (e.g. Suzuki et al., 2008). This is no longer the case with our methods: once

the geological model is linked with the simulation code and all settings are defined, the

flow field is automatically recomputed when the raw geological information is changed.

A similar method based on the same geological modeling technique for the simulation of

conductive heat transport only is already available (Gibson et al., 2007). However, our

methods provide techniques to evaluate the influence of the structural geological model

and its uncertainties on coupled fluid and heat flow simulations.

The methods presented here address structural geological uncertainties but rock prop-

erties within the units were kept constant. Additional uncertainty exists about the hetero-

geneities within a structural domain, for example due to uncertainties in the sedimentary

deposition on the reservoir scale (e.g. Feyen and Caers, 2006). These types of uncertainties

are commonly evaluated with geostatistical methods (Isaaks and Srivastava, 1989; Caers,

2001; Deutsch, 2002) and many types of codes are available for uncertainty simulations

of those systems (e.g. Deutsch and Journel, 1998; Deutsch and Tran, 2002). However,

recent studies suggest that geological uncertainties can have a significant influence on pre-

dicted flow fields, even in the case of relatively shallow groundwater models (e.g. Nilsson

et al., 2007; Troldborg et al., 2007; Refsgaard et al., 2011). The question if the structural

geological uncertainty is more important than the heterogeneity within one unit clearly

depends on the considered scale of the problem (Højberg and Refsgaard, 2005). In fact,

both types should be considered for a complete uncertainty evaluation. This has, to date,

only been done with manually constructed geological scenarios (Troldborg et al., 2007;

Suzuki et al., 2008), but not with inclusion of the complete geological modeling step. We

envisage that our approach provides an important step forward for a consideration of both

types of uncertainties, enabling a sensitivity analysis that provides a clear indication about

the relevant type of uncertainty for each specific case.

In the models presented here, we applied a regular mesh for the discrete geological model,

as a matter of simplicity. More complex rectilinear mesh structures could be envisaged to

reduce the total number of cells in the model. It could be, for example, possible to refine

across faults and to reduce lateral resolution in the gently folded areas. Also, it would be

possible to reduce the cell-size in z-direction in the conductive parts of the model. Any of

these adaptions are possible with the Python modules.

The cell discretization study presented in this paper is different to standard adaptive

mesh approaches that are implemented in some flow simulation solvers (Mansell et al.,

2002; Watson et al., 2005). Adaptive mesh methods refine the mesh in a specific cell to

ensure a more accurate and stable temporal process simulation, that is an existing cell is

refined, and all new cells within this refined cell obtain the same properties as the “parent”

cell, without consideration of geological “refinements” in this area. The cell discretization


study presented in this work addresses the refinement required to honor the geological

structure whereas other methods are mainly concerned about numerical stability of the

solution. For an accurate realistic simulation, both should be considered.

In the workflow presented in this thesis, we performed simulations with the simulation

code SHEMAT. However, due to the modular definition of all parts in the workflow and the

integration into single Python scripting modules, different flow simulation codes can easily

be integrated. First experiments using using a similar scripting approach, PyTOUGH

(chapter 5) to control pre- and post-processing of the multi-phase flow simulation code

TOUGH2 (Pruess, 2004) were very promising.

The results of the cell size study were here only evaluated qualitatively with a visual

comparison of temperature and velocity fields for the different discretization schemes.

Although this method already provides an insight into the model sensitivity with respect

to geological complexity, a more quantitative analysis could be applied. For example,

model validation techniques at observation points could be used to test which level of

discretization is required for a correct model prediction. However, comparisons of values

at single observation points are of limited use for a sensitivity classification of a whole

complex system. We therefore intend to evaluate the use of thermodynamic measures to

classify flow fields in the near future (chapter 7 of this thesis.)


7. Entropy Characterization of

Hydrothermal Flows


simulation





simulations




Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7


Chapter 7


110 7. Entropy Characterization of Hydrothermal Flows

Overview The main aim of this thesis is to analyze how uncertainties in geological

data influence the accuracy of thermo-hydraulic predictions for long-term and large-scale

geothermal problems. As the stochastic method for geological models, described in chapter

3, will be applied to generate a suite of possible realizations, and all of those will be pro-

cessed to a flow simulation, a measure is required that will, finally, enable the comparison

and analysis of the entire flow system for a whole range of simulated flow fields.

Typical methods for the comparison of stochastic simulation results are aimed at com-

paring simulated values to observations. In this chapter, a different approach is taken. It

will be evaluated if the average specific entropy production, as a measure of dissipative

processes within a system, can be applied as a measure of the thermodynamic state of a

whole system.

The motivation is here similar to the considerations that led to the application of infor-

mation entropy as a measure of the geological uncertainties in chapter 4. The symmetry

between these two chapters is therefore no coincidence. A difference is that the entropy

production described here is evaluated for one flow simulation result whereas the informa-

tion entropy describes a set of geological models. Both aspects will be combined in the

final chapter.


7.1. Introduction

Uncertainties in simulations of subsurface processes are commonly evaluated with stochas-

tic simulations (e.g. Doherty, 1994; Subbey et al., 2004; Meixner et al., 2010; Riva et al.,

2010; Vogt et al., 2010a). Instead of one specific result, a variety of probable realiza-

tions is generated, within the range of input data or parameter uncertainty. As a large

quantity of simulation results are generated with these methods, effective measures are

required to identify and classify the results. The aim of the work presented here is to eval-

uate whether a thermodynamic measure can be applied to classify simulated flow fields in

coupled hydrothermal systems.

A variety of methods has already been developed to evaluate results of stochastic sim-

ulations. The main scope of such analyses is to evaluate how accurately the simulation

can predict a set of observables, for example temperatures at observation points (e.g. Vogt

et al., 2010b; Meixner et al., 2010), or the production history in an oil reservoir (Suzuki

et al., 2008). Even, though, these methods are well suited for typical problems of cali-

bration and production forecast, they do not provide a measure of the state of the whole

system.

In the work presented here, it will be evaluated if a thermodynamic measure, specific

thermal entropy production, can be applied to characterize the system state. In the

classical sense, thermodynamic measures can be applied to predict the response of a system

with macroscopic measures, without having to know all the detailed processes within the

system. A simple and typical example is the “Ideal Gas Law”, describing the relationship

between pressure P , temperature T and volume V in an ideal gas:

PV = nkBT , (7.1)

where n is the number of molecules in the gas and the proportionality factor kB the

Bolzmann’s constant. Based on the kinetic theory of gases, this formula can be used to

evaluate how, for example, a volume change affects temperature – without having to know

the kinetic energy of every single molecule in the gas. On the appropriate scale and for a

specific question, thermodynamic measures are useful to describe systems without having

to know exactly the details inside the system itself.

Based on these considerations, it will here be evaluated if thermal entropy production

is useful as a measure of the thermodynamic state of a hydrothermal flow system.

7.2. Entropy production in a thermo-hydraulic system

Entropy production is related to dissipative heat processes within a system. The entropy

of a diabatic system changes if heat is supplied or removed from the system. The change

of entropy, the entropy production, is defined as the ratio between the change in heat Q


and the temperature T (e.g. Callen, 1985):

S ≡ δQ

T(7.2)

The second law of thermodynamics states in the traditional (non-statistical) form that

entropy in a closed system is either constant or increases and therefore:

S ≥ 0 (7.3)

Entropy is produced due to reversible and irreversible processes (see Regenauer-Lieb et al.,

2010, and references therein). Here, only the entropy production for slow fluid flow in a

permeable matrix is considered, using an approach initially developed for climate systems.

For a thermo-hydraulic system that exchanges heat with its surroundings, the entropy

production S for the system and its surrounding can be described as:

S =

∫

V

1

T

[

∂(ρcpT )

∂t+ ∇ · (ρcpT ~vf ) + p∇ · ~v

]

dV +

∮

A

1

T~qh · ~n dA (7.4)

The volume integral on the right side describes entropy production within the system due

to viscous dissipation. The surface integral represents entropy production due to thermal

dissipation.

If we only consider thermal dissipation, and assume that the internal system is in steady

state (in a statistical sense), the entropy production is reduced to the heat that is supplied

through the boundary by the heat flux ~q (Ozawa et al., 2003):

S =

∮

A

1

T~qh · ~n dA (7.5)

Only the conductive heat transport is considered here as relevant to entropy production.

This is following the argument of Ozawa et al. (2003) that advective heat transport is,

in principle, a reversible process and does therefore not contribute to viscous dissipation.

However, advective heat transport implicitly induces entropy production as it can lead

locally to very large temperature gradients, and therefore increasing conductive processes.

As the entropy production, defined in the description of equation (7.5) depends on the

size of the subsystem through the integration over the surface, it can be scaled by the

mass V ρ of the system to obtain the specific entropy production:

s =S

V ρ(7.6)

As a measure of the whole entropy production in a larger system, the average specific

entropy production can be calculated:

〈s〉 =1

V

∫

V

s dV (7.7)


7.3. Application of entropy production to analyze a transient

conductive heat flow field

7.3.1. Basic considerations for the conductive case

As a first example, the average specific entropy production for a system in a transient

conductive state will be evaluated. In the following considerations, conductive heat fluxes

are aligned with coordinate axes. The fluxes are per definition (eq. 7.5) oriented towards

the system. As an example, denoting qx1 as the flux into the cell from the left side, qx2

from the right side, the total contribution in x-direction is ~q ·~x = qx = qx1 −qx2, where ~x is

the unit vector in x-direction. From these considerations directly follows that the entropy

production of a system in conductive steady state is zero as all heat fluxes into and out

of the system are completely balanced. For example, considering a simple system with a

vertical heat flux only, no heat flux in or out of the cell exists in x-direction: qx1 = qx2 = 0,

and the same applies to the y-direction. The heat flux in z-direction is the same into and

out of the cell qz1 = qz2. Therefore, entropy production is zero.

7.3.2. Entropy production in a transient conductive system

We will now consider a system that initiates from a conductive steady state but then

experiences a change in the boundary conditions. The system is a conductive porous

medium with a thickness of 2500 m. Temperature is fixed at the top (10◦C) and the

system is homogeneous and isotropic, with a thermal conductivity of λ = 2.9 W K−1m−1)

and a thermal diffusivity of κ = 10−6m−2. Lateral no flux boundary conditions apply.

The system is initially in a conductive steady state with a temperature at the base of

60◦C. Then, temperature at the base is instantaneously increased to a higher value of

90◦C. The high temperature at the base will lead to transient effects in the system until

a new steady state is reached. This equilibration time span is simulated here with 50

logarithmically spaced time steps for a total time of 106 years. With the parameters of

the transport problem considered here, the system equilibration can be evaluated from the

characteristic time scale τ of diffusive heat transport (e.g. Turcotte and Schubert, 2002):

τ =1

2κl2 ≈ 50, 000 years. (7.8)

It can therefore be expected that the system reaches the steady state in the time scale of

the simulation.

In figure 7.1a, vertical profiles of temperatures and entropy production for different times

after the temperature increase are presented. The temperature profiles reflect the sudden

temperature increase at the base and the subsequent propagation of the temperature front

towards the top of the system. The profile of specific entropy production shows that the

entropy production is maximal in the region of the system where the temperature front

propagates. The peak itself is decreasing over time as the temperature front becomes

broader.


(a) Vertical profiles of T and s

25 50 75�2500�2000�1500�1000�5000

0.0 0.4 0.8 1.21e�9

1545421512393210020399543

s

z

T

(b) < s > during model equilibration

101 102 103 104 105 1060.0

0.5

1.0

1.5

2.0

2.51e�10

T

τ

<s

>Figure 7.1.: Entropy production during the equilibration phase in a conductive system starting from

a steady state in two different scenarios: (a) vertical profiles of temperature T and specific entropyproduction s for different time steps; (b) diagram of the temporal development of average entropyproduction < s > over time. Red dots indicate the time steps for the vertical profiles in (a) and thedotted line denotes the characteristic time scale τ .

The temporal development of the average specific entropy production over time (fig. 7.1b)

shows that entropy production is initially very high in the system and then subsequently

decreases back to zero when the system reaches the new steady state. The high increase

at the beginning is due to the high temperature contrast at the base of the system. The

new equilibrium state is reached after approximately 105 years. This is in the order of the

characteristic time scale τ of the system, indicated with the vertical dotted line in figure

7.1b.

This simple example showed that the average entropy production can be applied to

evaluate the internal thermodynamic state of a conductive system during the equilibration

phase. The time scale for equilibration and the decrease of the value to zero are in

accordance to theoretical considerations. We will now evaluate the application of the

measure to visualize and analyze more complex systems.

7.4. Analysis of entropy production in a convective system

7.4.1. Relationship between thermal entropy production and advective heat

transport

As a second example, we will now examine how entropy production within the system is

affected by advective heat transport with a simple convective system heated from below

(fig. 7.2). Heat is transported with the fluid in the upwelling and downwelling parts

of the convection cell, leading to the typical temperature profile of a convection system

(background picture in fig. 7.2).

We consider now the processes in a small sub-part of the system where colder fluid is

transported downwards. The advecting fluid disturbs the temperature field and leads to


SH < 0

qc

SC > 0

Figure 7.2.: Entropy production in a convective system: the advective heat transport leads to a higherentropy production. For example, if a cold fluid particle is transported downwards in a convectioncell, it will be transported into a warmer region, leading to an increase in entropy production.

a temperature gradient between adjacent sub-volumes in the system. This temperature

gradient causes a conductive heat flow qx between from the hotter to the colder volume,

with temperatures TH and TC . According to equation (7.2), this heat flow leads to an

entropy change in both systems:

SH =−qx

TH< 0 and SC =

qx

TC> 0 (7.9)

It is interesting to note that the entropy is decreased in the hotter sub-volume but increased

in the colder system. However, this is not a violation of the second law of thermodynamics

(eq. 7.3) because each sub-volume is not a closed system. Also, considering the two sub-

volumes, the average entropy of this small subsystem is increased:

⟨

SHC

⟩

= SH + SC

=−qx

TH+

qx

TC

=qx (TH − TC)

THTC> 0 (7.10)

This simple consideration indicates that entropy production is non-zero in a convective

system because temperature disturbance due to advective heat transport leads to an in-

crease in entropy. Furthermore, it can be expected that entropy production increases with

more vigorous convection. A measure commonly applied to determine the heat transport

through a system is the non-dimensional Nusselt number, the ratio between the total heat

flow to conductive heat flow:

Nu =qT

qc(7.11)

For the case of pure conduction, qT = qc and the Nusselt number is 1. In a convective


system, the Nusselt number is greater than 1 and increases with more vigorous convection

(Nield and Bejan, 2006). Regenauer-Lieb et al. (2010) showed that the Nusselt number can

be related to the thermal dissipation in a system. Specifically for the case considered here,

the Nusselt number can be expected to be proportional to the thermal entropy production

in the system

Nu ∝ S , (7.12)

suggesting that higher entropy production can be related to a higher heat transfer rate

through the system which, for the case of a convective system, is associated with higher

fluid velocities (e.g. Nield and Bejan, 2006).

7.4.2. Visualization of convective flow with entropy production

We will now apply the entropy production analysis to a simple hydro-thermal system:

a homogeneous porous medium heated from below. The dimensions of the system are

exactly the same as before (sec. 7.3.2). In the conductive case above, the permeability of

the porous medium was very low (k = 10−18m2). Instead of raising the temperature as

before, we will now increase the permeability of the system. At a certain permeability,

the flow field will reach an unstable state and convection will set in. This change from a

conductive to a convective state is well known from linear instability theory and commonly

determined with a Rayleigh number analysis (Nield and Bejan, 2006): convection sets in

when the Rayleigh number exceeds a critical value of 4π2:

Ra =ρ0gβkH∆T

µκ> 4π2 , (7.13)

where k is the permeability, H the layer thickness, ∆T the temperature difference over the

layer, ρ0 the fluid density, β the coefficient of thermal expansion, κ the thermal diffusivity

of the porous medium and µ dynamic fluid viscosity. This condition is determined for

the Oberbeck-Boussinesq assumption that only density changes with temperature and

all other parameters are kept constant (e.g. Turcotte and Schubert, 2002; Holzbecher,

1998). However, Nield and Bejan (2006) state that the Rayleigh number criteria is still

approximately correct when mean values for the other temperature dependent parameters

are applied. Applying the parameters of this system and standard values for viscosity

and thermal expansivity, the onset of convection should occur for permeability values

k > 1.25 · 10−13 m2.

As a first test, we will simulate the behavior in a system with a permeability of k =

5 · 10−13 m2 over a time period of 22500 years. Convection leads to significant lateral

temperature variations (fig. 7.2). A vertical 1-D profile through the system is no longer

sufficient and we will evaluate the temperature and entropy production in vertical slices in

the (x-z)-plane through the center of the system. A slice through the system for different

time steps is presented in figure 7.3. The left columns show the temperature field in the

model. Starting from a conductive state (Fig. 7.3a), convection sets in (fig. 7.3d). For

the last two time steps presented here, the temperature field does not any more change


significantly (fig. 7.3g and 7.3j). Visualization of the specific entropy production in the

2-D slices, center columns, shows a similar behavior. However, even if the first time step

shown here does not reveal any deviation from the conductive state in the temperature field

(fig. 7.3a), small instabilities are already visible in the entropy production view. Entropy

production for later time steps then reveals the dissipative processes leading to entropy

production, as expected from the considerations before: entropy production increases in

the downwelling areas and directly above the convection upwelling zones. Similar to the

temperature field, entropy production for the two last time steps (fig. 7.3h and 7.3k) shows

a similar pattern.

The figures of the third column show the cubic transform of the specific entropy pro-

duction. This transformation is here applied because the internal structure of the entropy

production in the central slices is not well visible due to high peak values in the system.

Clearly, the cube transformed representations reveal more details of the internal struc-

ture, specifically for areas in the center of the convection cell. Also, neutral areas without

significant entropy production emerge within the convection cell (fig. 7.3i and 7.3l) and

differences between negative and positive entropy production areas are now clearly visible.

The cube root transform also has a theoretical justification because it is the cube root of

the Nusselt number (Nield and Bejan, 2006). As the Nusselt number is proportional to

the entropy production (eq. 7.12), the cube root representation also provides a physical

sense. However, this is not evaluated here in more detail.

7.4.3. Entropy production during the onset of convection

The visual inspection of temperature field and entropy production in figure 7.3 reveals

that the system state changes during the initial time steps until it reaches a convective

equilibrium state. In analogy to the study of the conductive system presented before, we

want to evaluate if the average entropy measure can be used to determine the state of the

system from conductive to convective equilibrium state. Similar to before, the average

specific entropy production is calculated for every time step. The average specific entropy

production curve for the onset of convection in the porous system with a permeability of

5 · 10−13 m2 as used in the example above, is presented in figure 7.4. The system starts

from a conductive steady state with no entropy production. When convection sets in,

entropy production reaches a maximum and then decreases and converges to a finite value

larger than zero, in accordance to what was expected from the theoretical considerations

above.

The time steps of the slice plots are highlighted with red dots in figure 7.4. The average

entropy production exhibits the same behavior as observed in the visualization of entropy

in slices through the system (fig. 7.3): entropy production changes initially, but there is no

significant change in the system between the last represented time steps (16250 to 22500

years).

This example shows that the average entropy production provides an insight into the

global behavior of the system between two equilibrium states. We will now evaluate how


(a) T (1250a)

0 5 10 15 20 250

10

20

30

40

12 18 24 30 36 42 48 54 60

Cells

Cells

(b) 〈s〉 (1250a)

0 5 10 15 20 250

10

20

30

40

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.01e 8

Specific entropy production

Cells

Cells

(c) 3

√

〈s〉(1250a)

0 5 10 15 20 250

10

20

30

40

0.0020 0.0016 0.0012 0.0008 0.00040.0000 0.0004 0.0008 0.0012 0.0016 0.0020

sdotcbrt

Cells

Cells

(d) T (3750a)

0 5 10 15 20 250

10

20

30

40

12 18 24 30 36 42 48 54 60

Cells

Cells

(e) 〈s〉 (3750a)

0 5 10 15 20 250

10

20

30

40

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.01e 8


Cells

Cells

(f) 3

√

〈s〉(3750a)

0 5 10 15 20 250

10

20

30

40

0.0020 0.0016 0.0012 0.0008 0.00040.0000 0.0004 0.0008 0.0012 0.0016 0.0020

sdotcbrt

Cells

Cells

(g) T (16250a)

0 5 10 15 20 250

10

20

30

40

12 18 24 30 36 42 48 54 60

Cells

Cells

(h) 〈s〉 (16250a)

0 5 10 15 20 250

10

20

30

40

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.01e 8


Cells

Cells

(i) 3

√

〈s〉(16250a)

0 5 10 15 20 250

10

20

30

40

0.0020 0.0016 0.0012 0.0008 0.00040.0000 0.0004 0.0008 0.0012 0.0016 0.0020

sdotcbrt

Cells

Cells

(j) T (22500a)

0 5 10 15 20 250

10

20

30

40

12 18 24 30 36 42 48 54 60

Cells

Cells

(k) 〈s〉 (22500a)

0 5 10 15 20 250

10

20

30

40

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.01e 8


Cells

Cells

(l) 3

√

〈s〉(22500a)

0 5 10 15 20 250

10

20

30

40

0.0020 0.0016 0.0012 0.0008 0.00040.0000 0.0004 0.0008 0.0012 0.0016 0.0020

sdotcbrt

Cells

Cells

Temperature ◦C0 5 10 15 20 250

10

20

30

40

12 18 24 30 36 42 48 54 60

Spec. entropy production0 5 10 15 20 250

10

20

30

40

�1.0�0.8�0.6�0.4�0.2 0.0 0.2 0.4 0.6 0.8 1.01e�8


Cuberoot transform0 5 10 15 20 250

10

20

30

40

0.0020�0.0016�0.0012�0.0008�0.00040.0000 0.0004 0.0008 0.0012 0.0016 0.0020

sdotcbrt

Figure 7.3.: Visualization of temperature and entropy production in a z-x plane through the systemfor different time steps during the onset of convection. The figures on the right side show the cubetransform of the specific entropy production, applied here to highlight the internal structure.


0 5000 10000 15000 20000 25000�0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

3.51e�11

1250

3750

15000 22500

Time [a]

<s

>

Figure 7.4.: Specific entropy production during the onset of convection in a porous medium with apermeability of k = 5 ·10−13 m2. From an initial conductive steady-state with no entropy production,the system goes through a phase of high entropy production until it reaches a convective equilibriumstate with a finite non-zero entropy production. The time steps used for the slice plots are markedwith red dots.

the behavior changes with different system properties. As evaluated before, the onset of

convection in the system can be expected for permeabilities larger than approximately

1.25 · 10−13 m2. In the following experiment, we will evaluate the entropy production in

the same system for a range of different permeabilities, from 10−13 m2 to 10−11 m2. All

other parameters and settings are kept constant.

Graphs for the average specific entropy production during the onset of convection in

these models are presented in figure 7.5. The specific entropy production for a permeability

of 10−13 m2 remains zero, indicating that the system stays in a conductive steady state.

For higher values, convection sets in and the same pattern is observed as before, with an

increase of entropy production during the onset of convection, leveling out to a constant

finite value when the system reaches the convective equilibrium state. Due to a higher

heat transfer in the system, the onset of convection occurs at earlier times for systems

with higher permeabilities. For very high permeabilities, flow velocities become too high

for the grid resolution considered here.

7.4.4. Relationship between Nusselt number and entropy production

In section 7.4.1, the relationship between Nusselt number and entropy production was

described. In the experiment shown in figure 7.5, it was observed that the finite values

of entropy production in convective systems increase for convection in higher permeable

layers (fig. 7.5). We will now evaluate the relationship between entropy production and

Nusselt number in this system.

The non-dimensional Nusselt number is defined as the ratio of total heat flow to conduc-

tive heat flow (eq. 7.11). For a homogeneous system of equal thickness with impermeable

boundaries as considered here, the Nusselt number can be estimated from the ratio of the

mean temperature gradient over the boundary to the temperature gradient over the whole


0 5000 10000 15000 20000 25000

0.0

0.2

0.4

0.6

0.8

1e�101E132E133E135E131E122E123E125E121E11

Time [a]

<s

>

Figure 7.5.: Average specific entropy production during the onset of convection for the same boundaryconditions but with different permeabilities; in all cases, entropy production first increases, then goesthrough a maximum and decreases again and converges to a finite value > 0. This finite valueincreases for higher permeabilities when more vigorous convection and higher flow velocities can beexpected. The high fluctuations for high permeability values are caused by numerical instabilities.


system (e.g. Holzbecher, 1998), as already done in section 5.3.3:

Nu ≈1L

∫

L∂T∂z

dxTmax−Tmin

H

, (7.14)

where Tmax and Tmin are fixed temperatures at bottom and top of the system L the area

of the base of the system and H the thickness. Applying this method to the last time step

of the convection simulations presented above (fig. 7.5), we obtain an estimation of Nusselt

numbers in the system and can compare these to the average specific entropy production,

presented in figure 7.6.

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1e�10

Nu

<s

>

1E-13

2E-133E-13

5E-13

1E-12

2E-12

3E-12

5E-12

1E-11

Figure 7.6.: Entropy production for increasing convection and increased heat transfer, described bythe Nusselt number. The results suggest a linear relationship between Nusselt number and entropyproduction, as expected from theory. Results for higher permeabilities are unreliable due to numericalinstabilities.

The results show a correlation between Nusselt number and entropy production, with

a clear linear relationship for the lower permeability scenarios, up to k = 2 · 10−13 m2.

Since inertial terms are neglected in our calculations, we attribute the deviation for higher

permeabilities to numerical instabilities. Nevertheless, the proportionality between Nusselt

number and entropy production as expected from theory (Regenauer-Lieb et al., 2010) is

obtained in the results of the numerical study, providing an indication for the suitability

of the measure to classify the thermodynamic state of the system.

7.5. Discussion

The results show that average thermal entropy production can be applied as a measure

to compare the thermodynamic state of a coupled thermo-hydraulic system. Thermal


entropy production describes the state of a system with respect to applied boundary

conditions, through a comparison of heat transport into and out of the system. If a

system is in steady state, entropy production converges to a stable value. Specifically, if

a system is in a conductive steady state, then the entropy production is 0. The example

simulations show that entropy production increases during the onset of convection, but

then decreases again as the system stabilizes. The experiments of the onset of convection

showed that the average entropy production in a system converges to a finite value during

equilibration. A motivation for this study was to determine a single measure to classify an

entire hydrothermal system. This measure is here obtained with average specific entropy

production < s >.

In the context of the thesis, the evaluation of thermal entropy production as a measure

to visualize and analyze flow fields is in symmetry to the application of information entropy

to analyze the geological model. The similarity of the term“entropy” in both cases is more

than pure coincidence. As stated in the chapter on information entropy (chapter 4), a close

relation between information and thermodynamic entropy exists (e.g. Ben-Naim, 2008).

For the purpose of the work presented here, the main feature of both measures is that

they are in a similar way useful to classify the uncertainty state of a system with a single

measure. In the case of information entropy, this measure quantifies the uncertainties in

a geological model. Thermal entropy production describes uncertainties related to the

dynamic state with a single measure. This single measure can then be applied for a

comparison of different flow realizations.

The logical next question is then how uncertainties in the geological model, classified

with the information entropy, influence the predictions of the flow system, classified with

the thermal entropy production. In other words: does a higher geological model uncer-

tainty lead to higher flow uncertainties? This compelling question is evaluated in the next,

and final, chapter of the thesis.

8. Effect of Geological Data Quality

on Geothermal Flow Fields


simulation





simulations




Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7



Influence of uncertainty ingeological data on long-term

flow predictions

Chapter 8

124 8. Effect of Geological Data Quality on Geothermal Flow Fields

Overview The following chapter brings together the single aspects of previous chapters

in a case study to evaluate the effect of geological uncertainties on simulated flow fields.

The case study is an extension of the geothermal resource model of the North Perth Basin,

presented in (chapter 6). In that chapter, this model was used to evaluate how different

possible geological scenarios influence the simulated flow field. The single models for the

geological scenarios were created manually and results were analyzed qualitatively. In

this chapter, a whole suite of different geological models is automatically created with

the stochastic uncertainty simulation presented in chapter 3. The uncertainties in these

geological scenarios are analyzed with the information entropy method, introduced in

chapter 4. All geological models are automatically processed to geothermal simulations

with the integrated workflow described in chapters 5 and 6. The results of the whole suite

of hydro-thermal simulations are then analyzed with standard statistical methods.

For the case study considered here, it is finally evaluated how geological uncertainites,

evaluated with the information entropy described in chapter 4, correlated with uncertain-

ties in the flow fields, considered here from distributions of average entropy productions,

introduced in chapter 7.


8.1. Introduction

As discussed before in this thesis, no method exists yet to evaluate the influence of uncer-

tainties in the geological data on predicted fluid and heat flow fields. The hypothesis of the

work presented here is that it is possible to combine stochastic simulations of geological

models with coupled hydrothermal flow simulations to evaluate the influence of uncer-

tainties in geological data on a geothermal system simulation. The methods of stochastic

geological modeling are combined with the integrated framework for geologic modeling and

flow simulations to automatically generate a suite of simulated fluid and heat flow fields,

all probable in the range of geological uncertainties. The relevant aspects of the methods

developed before are briefly reviewed and it is then described how they are combined into

one framework. As a test of feasibility, the methods will be tested with a coupled hydro-

thermal simulation in the North Perth Basin, Western Australia. Uncertainties for several

geological scenarios will firstly be evaluated with probabilistic measures in 1-D and 2-D

sections. Finally, the global measures of information entropy and entropy production will

be applied to compare a whole set of simulated geological models and flow fields.


8.2.1. Combination of methods to generate multiple flow realizations

According to the abstract notation used in the previous chapters, the aim is to generate

a set of n flow fields {F} = {F1,F2, . . .Fn}, directly from the input data set K of geo-

logical observations and probability distributions p(k) for each observation k ∈ K. This

is achieved with a combination of the methods detailed in previous chapters:

➤ The stochastic simulation of structural geological models, described in chapter 3, is

used to generate a set {M} of realizations of the geological model, based on multiple

datasets {K} determined from the initial geological observations k ∈ K and assigned

probability distributions p(K):

{M} = {M1,M2, . . .Mn} = f1 ({K} , αk, βℓ) (8.1)

This step can be completely automated assuming that parameters of the interpola-

tion function, αk, and the space partitioning βℓ are constant.

➤ The the integrated workflow for geological modeling and flow simulations, described

in chapter 5 and 6, enables a flow simulation on the basis of a geological model

M, rock properties assigned to the geological units P, boundary conditions Bq and

simulation settings γr, equations (6.2) to (6.7). Keeping all settings constant, a flow

field can directly be simulated for a given geological input data set K:

F = fW (K) (8.2)


It is straight-forward to combine these two methods into one framework, enabling an

automatic generation of multiple possible flow fields as a function of geological input data

set and assigned probability distributions:

{F} = fW ({K}) (8.3)

Instead of one result of the flow simulation, a set n simulation results is obtained, one for

each geological data set. Furthermore, during the process of the workflow, a geological

model is constructed for each geological data set. We therefore obtain two types of results

that can be subjected to an uncertainty estimation:

➤ The set {M} of geological model realizations; these models can be used to analyze

and visualize uncertainties in the structural geological model.

➤ The set {F} of possible flow fields which can be used to identify the influence of

geological uncertainties on specific model predictions and the whole flow system.

Both types of uncertainties will be evaluated to obtain a detailed insight into the uncer-

tainties associated with this study.

8.2.2. Visualization and analysis of uncertainties

Uncertainties in both the geological model and simulated fluid and heat flow fields are

analyzed with standard statistical methods and the information entropy and entropy pro-

duction measures described above (chapters 4 and 7). For the geological model, estimated

are:

➤ The indicator probability density function for every geological formation in the model

PF (G), equation (4.6);

➤ The cell information entropy H(G) for every grid cell as a measure of the uncertainty

state of this cell (eq. 4.2);

➤ Total model entropy HT and the fuzziness for the single geological units Hm to

describe the uncertainty state of the model and the single geological units (equations

4.3 and 4.4).

These measures are described in detail in chapter 4. The analysis of the simulated flow

fields is more complex as several variables have to be considered. For the purpose of this

case study, the following analyses are performed:

➤ Mean, standard deviation and coefficient of variability are estimated for the simu-

lated fields for every grid cell;

➤ The average specific entropy production 〈s〉 for the whole model is determined as

measure of the thermodynamic state of the whole system (eq 7.7).

In both cases, the last type of analysis is a measure to describe uncertainties in the system

as a whole.


8.2.3. Integration of uncertainty analysis into one workflow

All methods that are required to perform the uncertainty study were implemented in mod-

ules in the scripting language Python and a combination of the single elements into one

workflow is straightforward. The complete workflow is outlined in fig. 8.1. The analysis

starts with the initial data set K of geological observations (surface contact points and ori-

entation measurements) that were used to construct the model, and associated probability

density functions p(K). Depending on the problem, the same geological data set can be

applied to test several geological uncertainty scenarios. These different scenarios can be

defined by different probability distributions assigned to the geological data, but they can

also include different types of stratigraphic relationships between the geological units or

the consideration of additional faults or other structures. Based on the geological data and

assigned probability distributions for each scenario, a range of probable input data sets

{K} is generated with a stochastic method. For each of these data sets, a geological model

{M} is created. These models are then processed to two types of uncertainty evaluations:

(i) the geological uncertainty and (ii) the uncertainty of the simulated flow fields.

The geological uncertainty can directly be evaluated from the different realizations of the

geological models. The first step is to determine the indicator probability density function

for each geological unit, equation (4.6). From these functions, measures of probabilities for

each geological unit are derived and a range of standard statistical evaluations, for example

modes of P (10), P (50), P (90), or minimum and maximum possible extent can directly be

determined and visualized. However, these measures relate to one specific geological unit

and not to the overall uncertainty in the geological model.

The first step for the uncertainty evaluation of the flow fields is the simulation of a set

{F} of flow field realizations for every geological model. Results of these flow simulations

can then be subjected to standard statistical evaluations, as for the geological uncertainty.

For example, an expected value for simulated temperatures or flow velocities in each cell

can be calculated.

In the last chapter above (chapter 7), it was demonstrated that the specific thermal

entropy production s can be applied to characterize the thermodynamic state of the system,

as a means of visualizing the state of each cell, and also as a measure of the whole system,

〈s〉. These measures will be evaluated for each of the simulated flow fields.

Finally, results from uncertainty evaluations of the geological models and the simulated

flow fields can be compared. The first obvious step is to directly compare uncertainties, for

example along a vertical profile through the models. This analysis provides an insight into

uncertainties at specific locations. However, this type of uncertainty is not directly useful

to evaluate uncertainties related to the dynamics in the whole system. These insights

might be obtained with a comparison of total information entropy Hm for a geological

model scenario with the set of average entropy productions {〈s〉} for each of the simulated

flow fields in this scenario. Important to note is that the information entropy represents

the uncertainties for a set of geological models, whereas average entropy production is

associated with a single realization of flow fields for each of the geological models.


Statistical analysisStatistical analysis

Geological data set

Scenario 1 Scenario 2 Scenario m

Uncertainty Simulation

Information Entropy

Flow simulations

Entropy Production

Geological Modeling

K, p(K)

P1(50), ... P2(50), ... Pm(50), ...

K1, p(K1) K2, p(K2) Km, p(Km)

{K1} = {K11, . . . K1n} {K1} = {K21, . . . K2n} {Km} = {Km1, . . . Kmn}

{F1} {F2} {Fm}

{〈s1〉} {〈s2〉} {〈sm〉}

{M1} {M2} {Mm}

H1

E1, ...

H2

E2, ...

Hm

Em, ...

Figure 8.1.: Schematic workflow for uncertainty simulation and analysis in a combined geologicalmodeling and flow simulation study


8.3. Case study: North Perth Basin, Western Australia

8.3.1. Regional context, model scenarios and data quality

The influence of uncertainties in geological data on simulated flow fields will be analyzed in

a case study of the Dandaragan Trough in the North Perth Basin, Western Australia. The

base model considered here is the same as in chapter 6. The geological model represents

a part of the North Perth Basin (fig. 6.4a). The regional context and the data basis for

the model were described in the chapter before, as well as representative rock properties

for the geothermal simulation and the applied boundary conditions. A difference to the

models before is that for the following examples, the thermal boundary conditions were

here chosen to reflect a higher basal heat flow of 0.085 W/m2, more adapted to estimations

for this region (Chopra and Holgate, 2005). However, it has to be noted that the purpose

of the study presented is not to provide a detailed geothermal resource study of this area

but to present, analyze, and discuss uncertainties related to geological data that typically

occur in this and other similar settings.

In the previous study, presented in chapter 6, it was evaluated how different realizations

of the geological model, all possible within the range of uncertainties, influence the accuracy

of simulated geothermal flow fields. As in the above example, different geological scenarios

of uncertainties are considered:

Scenario 1: The structural setting within the basin is assumed to be precise but the depth

to the geological contacts is uncertain (fig. 8.2a);

Scenario 2: An additional sub-basin exists within the Dandaragan Trough, enclosed by

normal faults with similar strike to the basin bounding faults. This addi-

tional sub-basin is hypothetical and, therefore, the position of it is uncertain

(fig. 8.2b);

Scenario 3: This scenario is similar to Scenario 2, but in addition to the uncertainty of

position of the sub-basin, here also the width is considered uncertain.

These uncertainties reflect the main structural uncertainties that were identified by Mory

and Iasky (1996) for this part of the basin.

The previous study of chapter 6 is here extended with an additional uncertainty sim-

ulation of the geological models. In the previous study, the different scenarios were con-

structed manually. Here, a full range of possible realizations of the geological model is

generated for each scenario, applying the combination of geological modeling and flow

simulation update presented above (sec. 8.2.3).

The different geological models are generated with the stochastic uncertainty simula-

tion presented in chapter 3. With this approach, possible geological input data sets are

generated from an original base set and the probability distributions assigned to each

measurement. As described before, Mory and Iasky (1996) state that the position of ge-

ological contacts at depth is increasingly uncertain due to a lack of seismic imaging for


(a) Original model

Permian Yarragadee Cretaceous

12 k

m

63 km

(b) Additional faults in the basin

Hypothetical faults E and W

12 k

m

63 km

Figure 8.2.: Geological model of the Dandaragan Trough, North Perth Basin: (a) original model withcontinuous sedimentary layers within the Dandaragan Trough, as presented by Mory and Iasky (1996),and (b) with an additional small hypothesized sub-basin within the Dandaragan Trough.

the deeper parts of the basin. This geological uncertainty is here represented with a nor-

mal distribution assigned to data points with increasing standard deviation towards the

deeper parts of the basin. A normal distribution is considered as representative for this

type of uncertainty. However, other types of probability distributions are applicable for

other geological data uncertainties (sec. 3.2.3).

The values of standard deviation for the data points in the basin are presented in

table 8.1. The probability distributions for data points affect the vertical position for

geological units and lateral position for faults. For the case study performed here, the

same probability distribution is assigned to data points of the same geological unit as they

are considered here as belonging to the same type of geological uncertainty. This is not

a general case and not a requirement of the method itself. In fact, individual probability

distributions can be assigned to every data point separately.

For each of the three geological scenarios considered here, three different schemes of

probabilities are applied (see tab. 8.1): (A) a scheme with low standard deviations assigned

to the geological data points, (B) with intermediate, and (C) with high standard deviations.

This is done here in order to compare the sensitivity of simulated flow fields for different

ranges of geological data uncertainty.

For each of the three geological scenarios with the three discretization schemes, 20

geological models were generated, resulting in a total of 180 structural geological models.

In a first step, the uncertainties of the geological models are analyzed for each scenario


Scenario Cretaceous Yarragadee Leseur Permian Faults Sub-basin width

1 A 50 150 225 300 n.a. n.a.1 B 100 300 450 600 n.a. n.a.1 C 150 450 675 900 n.a. n.a.

2 A 50 150 225 300 1000 fixed2 B 100 300 450 600 1500 fixed2 C 150 450 675 900 2000 fixed

3 A 50 150 225 300 1000 variable3 B 100 300 450 600 1500 variable3 C 150 450 675 900 2000 variable

Table 8.1.: Quality of the geological data, assigned as standard deviations [m] to the position values;Three different geological scenarios are considered: (1) basic structures are identical to the originalmodel, (2) an additional sub-basin exists, bounded by two faults, and with an uncertain position;(3) similar to the second scenario, with an additional uncertainty of the sub-basin width. For eachof these scenarios, three different probability schemes were applied: (A) low standard deviation, (B)intermediate standard deviation, and (C) high standard deviation. Normal distributions are assignedto the data points with the mean taken from the original model (n.a. = not applicable for this case)

before coupled fluid and heat flow simulations are performed for each realization of the

geological model.

8.3.2. Analysis of uncertainties in the geological model

Probabilistic analysis

The result of the stochastic geological modeling is a set of probable realizations of the

geological model. A straightforward way to estimate uncertainties in this set of realizations

is to determine the probability indicator function for each geological unit (eq. 4.6). For the

purpose of numerical processing, all geological realizations are mapped on a regular grid

with 250 x 25 x 200 cells, the same spatial discretization that was used for the simulations

in chapter 6. The probability indicator functions are evaluated for the grid cells.

The probability density function for the Yarragadee unit in the uncertainty scenario

3C (tab. 8.1) is presented in figure 8.3a as a function of depth in a vertical 1-D profile.

The result provides a direct measure of the probability to obtain this specific unit at any

depth. The same type of analysis and visualization can be performed in a 3-D view of the

probability function, presented in figure 8.3a. In the upper figure, the whole probability

field for this geologic unit is shown. Obviously, a the unit does not occur in the main part

of the model range. For a better insight into the distribution of probabilities, a filter can

be applied to remove low probability values. This is done for the representation in the

lower figure of 8.3b where only values greater than P(50) are shown. In this figure, more

details about the distribution of the unit in space are visible, including the decrease of

probability at the upper and lower boundaries of the unit.

These representations are based on simple probabilistic estimates and yet provide a

good insight into the probability distribution of one geological unit in space. The analysis

can, for example, directly be applied to evaluate probabilities of the geology at a potential


(a) p(G8), 1-D profile

0.0 0.5 1.0 1.5 2.0 2.5

H

0.0 0.5 1.0

pG

0.0 0.5 1.0-11000

-9000

-7000

-5000

-3000

-1000

1000p8

(b) p(G8), 3-D view

Figure 8.3.: Probability density for the indicator function of the Yarragadee formation, scenario 3C:(a) probability of obtaining the Yarragadee formation in a vertical 1-D profile through the center ofthe model; (b) 3-D view for the same indicator function.

drilling site and it provides a probabilistic estimation of the volume for a geological unit

in a region.

Information entropy of the geological model

The probabilistic estimation for a single geological unit is useful to estimate uncertainties

associated with one unit, but not to capture uncertainties in the geological model in a

more general way. For this purpose, the information entropy is applied to the model. At

first, the information entropy for each cell in the model is calculated based on the indicator

probability fields of the single geological units with equation (4.2). A single measure for

each cell in the model is obtained that classifies the uncertainty state of this cell.

The information entropy for a vertical 1-D profile at the center of the model is presented

in figure 8.4. The left diagram shows the same 1-D vertical profile as in figure 8.3a

of the probability for the Yarragadee formation. The diagram in the center shows the

probability density functions for all geological units. Obviously, this representation is

not useful for cases where several probability density functions are overlapping, that is

where various geological units could occur (for example at label C in figure 8.4). However,

the information entropy, shown in the right diagram, provides an insight into the model

uncertainties, combining the information of all probability density functions. For example,

at around negative 2000 meters (label A), the probability for one geological unit is 1

and no uncertainty exists at this specific point, reflected by an information entropy of


A

C

B

³1 ³4³2 ³3

P (G8) P (G1) − P (G8) H

z

Figure 8.4.: Information entropy along a vertical 1-D profile in the center of the model for scenario 3C;The left figures shows the probability of one specific geological unit (here: Yarragadee formation),P (G8). This representation is useful for one unit but can be quickly confusing when all units areconsidered (middle figure). Calculating the information entropy from all probabilities provides a clearpicture of uncertainties in the model: where one unit has got a probability of 1, H is zero, thereis no uncertainty (label A). For every value H > 0, at least two units can occur. If two values areexactly equally probable, then H = 1 (label B), and accordingly for higher values: at label C, 4 unitsare equally probable.

zero. Around label B, two units are probable and the information entropy increases.

The measure is exactly 1.0 at the point where both units are equally probable, reflecting

the highest possible uncertainty for two occurrences (see fig. 4.1). The diagram can be

interpreted accordingly for more overlapping unit probabilities: at the depth of label C,

four geological units are almost equally probable and the information entropy reaches a

values of 2. The simple relationship between the information entropy and the minimum

possible units can be generalized as the condition:

N ≥ 2H , (8.4)

stating that, given a value of H, at least N outcomes are possible where N is the next

lowest natural number below or equal to 2H . The vertical dashed lines in the diagram

reflect these different levels of uncertainty. The measure of information entropy therefore

provides a clear measure for uncertainties with a statement about the minimum of possible

occurrences.

Similar diagrams at the same position are presented for all geological scenarios in fig-

ure 8.5. The different rows denote the different geological scenarios 1-3. The left column


contains results for low variability in the geological data (i.e. high data quality) and the

right column show results for a high data variability.

The information entropy for the minimum data case of the first scenario (1A) is mostly

below 1, indicating that a minimum of two units is probable, whereas for the maximum

uncertainty scenario (1C) at several locations more than two units are probable. The re-

sults for the geological scenario with an additional sub-basin of fixed width are presented

in the middle row and the results where, additionally, the width of the sub-basin is uncer-

tain are presented in the lowest row. Results for minimum cases in the left column (2A

and 3A) show a similar pattern to the minimum scenario without faults (1A). However, at

most points below a depth of 4000 meters at least two geological units are probable. As

expected, information entropy increases notably for the maximum uncertainty scenarios

with additional faults in the basin (2C and 3C). For those two cases, more than three units

are probable for a substantial depth range.

The representation of information entropy along a vertical profile shows that, in general,

uncertainties are increasing with depth. This can of course be expected as the assigned

standard deviations increase with depth (tab. 8.1). This visual inspection of the results

suggests that the uncertainties in the model scenarios 2 and 3 are higher than those

for scenario 1. However, only a 1-D profile is considered here. In order to compare

uncertainties of scenarios in the whole system, 3-D visualizations for scenario 1C and 3C

are presented in figure 8.6. In the upper figures, (a) and (b), all cells with an information

entropy of zero are transparent. This representation provides a spatial overview of the

uncertainty that at least two geological units are probable. The lower figures, (c) and

(d), show only those parts in the models with an information entropy of H > 1. Here, at

least two geological units can occur. The influence of the additional faults on the model

uncertainty is clearly visible in figure 8.6d as the information entropy locally increases to

values of H > 2, in these areas, at least four geological units can occur.

The examples in the 1-D profiles in figure 8.5 show that the additional faults lead locally

to very high uncertainties. However, it is not directly visible if the total uncertainty of

the model is higher for the case of an additional fault. One possibility to derive a detailed

insight into the differences between two scenarios would be to create a difference plot in

3-D, as presented in chapter 4. Another possibility, also discussed before, is to calculate

the total entropy of each scenario, as a single measure classifying the state of uncertainty

(eq. 4.4). To compare uncertainties of geological units in different scenarios, the unit

fuzziness can be calculated (eq. 4.3). In figure 8.7, the total model information entropy

and unit fuzziness for three geological units is presented. As expected, the model entropy

increases within each scenario for increasing standard deviations assigned to the data

points (e.g. from 1A to 1C). The figure also shows that the model entropy is higher for the

fault scenarios, but no clear difference is visible between the two fault scenarios, 2A–C and

3A–C. The same behavior is here reflected with the fuzziness values for different geological

units. As Leseuer is the deepest and Yarragadee the shallowest unit of those shown in

the diagram, the unit fuzziness clearly reflects the fact that uncertainty is increasing with


(a) Scenario 1A

0.0 0.5 1.0 1.5 2.0 2.50.0 0.5 1.00.0 0.5 1.0-11000

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1000HP (G1)-P (G9)P (G8)

(b) Scenario 1C

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(c) Scenario 2A

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(d) Scenario 2C

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(e) Scenario 3A

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(f) Scenario 3C

0.0 0.5 1.0 1.5 2.0 2.50.0 0.5 1.00.0 0.5 1.0-11000

-9000

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1000HP (G1)-P (G9)P (G8)

Figure 8.5.: Information entropy in 1-D profiles for different scenarios: (a) and (b) for scenarios 1Aand 1C, the original geological model, (c) and (d) for scenarios 2A and 2C, the scenario with anadditional sub-basin of fixed width, and (e) and (f) for an additional sub-basin with variable width.The scale of the vertical axis is meter.


(a) Scenario 1C, H > 0

(b) Scenario 3C, H > 0

(c) Scenario 1C, H > 1

(d) Scenario 3C, H > 1

Figure 8.6.: Comparison of information entropy in two scenarios, with and without the hypotheticalsub-basin (1C and 3C). Figures (a) and (b) show only those parts of the model with an informationentropy H > 0, those areas where at least two geological units are probable. Figures (c) and (d)show areas for H > 1: in these areas at least three units are probable. The additional uncertaintydue to the hypothetical faults is clearly visible in figure (d), with areas where even five units areprobable in some areas (H > 2). For comparison, the black line in the center of the model indicatesthe position of the vertical profile in figure 8.5.


depth.

The evaluation of information entropy in the geological models showed how the quality of

the geological data influences the model uncertainty. In the next step, it will be evaluated

how these variations in geological data quality influence the simulated flow fields.

Scenario 1 Scenario 2 Scenario 3

Figure 8.7.: Comparison of total information entropy (upper line) and unit fuzziness for differentgeological scenarios (lower lines); The total information entropy and the unit fuzziness increasewithin each scenario for increasing data uncertainty (tab. 8.1). Also, results for scenario 1 and 2 aredifferent, whereas results for scenario 2 and 3 are very similar. Comparison of unit fuzziness showsthat uncertainty is increasing with depth (Leseuer is the deepest of the three units, Yarragadee theshallowest).

8.3.3. Analysis of uncertainties in the flow fields

After the sets of geological models are created, all geological models are processed to

coupled fluid and heat flow simulations with the integrated workflow, described in chapter

6. The same discrete versions of the geological models that have been created for the

geological uncertainty study before can be utilized. According to the workflow represented

in figure 8.1, a set of flow simulation results {Fi} is obtained for each geological scenario i

from the set of geological models. The geothermal simulations were performed in two steps:

An initial simulation of the conductive steady state, followed by the transient simulation

of coupled fluid and heat flow in the system for a time period of 100,000 years. This

scheme was chosen based on preliminary studies that showed that the temperature and

flow fields reached a state where no significant changes occurred over time frames of several

thousand years after 100,000 years. However, no strict steady state termination condition

was applied. A full output of all simulation variables is obtained from 25 time steps in the

model with logarithmically increased time spacing.

The obtained flow results can be analyzed in several ways. In a first step, results from


one scenario can be compared to evaluate sensitivities of the flow field with respect to

the geological uncertainties of one scenario. Then, the results of different scenarios can

be compared to determine the influence of the scenarios of geological uncertainty on the

flow simulations. However, a visual comparison for all simulation results quickly becomes

inconvenient as, including all transient simulation results, 4500 flow results are obtained.

A straight-forward possibility to compare all flow realizations for one scenario is to

visualize all results of one variable in a vertical profile through the model. The relevant

outcomes in the hydrothermal context considered here are temperature and flow velocity.

These represent important parameters for geothermal resource evaluation (e.g. Muffler and

Cataldi, 1978; Gringarten, 1978; Tester et al., 2006). However, the evaluated variables

depend on the studied problem.

(a) Temperature realizations

0 100 200 300 400-11000

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1000

0 150 300 0.0 0.2 0.4 0.6

Depth

[m]

T [◦C]T [◦C] Variability

~� ��A

~� ��B

~� ��C

(b) Flow velocities (absolute)

0.0 0.8 1.6 2.41e�8-11000

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-7000

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1000

0 1 2 31e�8 0.8 1.6 2.4

Depth

[m]

Variability|~v| [m/s]|~v| [m/s]

Figure 8.8.: Simulated flow fields in the center of the model for scenario 2C: (a) Temperatures, and(b) Fluid velocity (absolute); In both profiles, the left diagram shows all results for this scenario,the middle diagram estimated mean and standard deviation, and the right diagram the coefficient ofvariability. A high variability in temperature, most visible in the coefficient of variability, correlatesto areas of high flow velocity (labels A and B). However, at label in the upper part of the model,considerable variation exists for simulated temperatures whereas flow velocity and standard deviationare relatively low (label C).

Observations of temperatures along a vertical profile are presented in figure 8.8a, and

of the absolute value of velocity in figure 8.8b. In both figures, the left profile shows all

simulation results for one specific geological scenario, the middle profile the estimation of

mean and standard deviation and the right profile the coefficient of variability. Fluctua-

tions in the temperature field, obvious in the left plot of all flow results, coincide mainly

with depth ranges where flow velocities are highly fluctuating, for example at the depth

ranges indicated with the black dotted lines at label (A) and (B). These fluctuations are

also visible in the coefficient of variability for temperature. However, in the upper part of

the model (label C), temperature variability is high whereas flow velocities are lower.

The variations in the temperature profiles are well represented in the coefficient of

variability plot and this analysis type will be applied for further data analyses. However,


the coefficient of variance for velocities seems to be heavily influenced by small variations

of very low velocities at depth. Therefore, for the visualization of the fluid flow field, the

plot of mean plus two standard deviations will be used below.

The correlation of a high variability in temperatures with areas where fluid flow is

changing can be related to advective heat transport processes. As the fluid flow field is

affected by variations in the structural model, the advective transport field is changed,

leading to high temperature variations for different geological models. However, high

fluctuations close to the surface are also potentially augmented by the boundary conditions

applied here.

Interesting to note is the high temperature variability in the upper part of the model

around label (C). The different temperature profiles (left plot in figure 8.8) exhibit almost

a bipolar distribution with a very high temperature gradient. This affect could be caused

by convection, and the differences in the temperature profiles caused by changes in the

convection system. In order to analyze the flow field with respect to these effects, 2-D

sections will be examined. Figure 8.9 shows a cross-section plot in E-W direction through

the center of the model for mean temperature and coefficient of variability of temperature

for scenario 2C. The thin black overlying lines show geological contacts for one realization

of the geological models, and the vertical black line indicates the position of the vertical

profiles inspected above (fig. 8.8). Mean values for all simulated temperatures are pre-

sented in figure 8.9a. Specifically in the upper layers, lateral variations are visible. These

differences in the simulated temperature fields can also clearly be seen in the representa-

tion of the coefficient of variability, figure 8.9b. In addition to the same pattern of high

variability in the layers close to the surface, the lateral extent of the variations is here

clearly visible. The temperature field changes most within the upper layers of the basin

and around the faults.

Figure 8.10 shows cross-sections for mean and standard deviation of absolute velocities.

They show a similar behavior, with highest mean values and most differences in the upper

layers of the model (fig. 8.10). Mean values of absolute flow velocities (fig. 8.10a) reveal

that very high flow velocities exist in the uppermost layer (Cretaceous) and at the bottom

of the thick layer (the Yarragadee formation). The standard deviation of velocities in

figure 8.10b indicates that differences in the flow velocity mainly occur near geological

surfaces.

The observed variations in temperature and velocity are solely due to uncertainties in

the geological model and therefore associated with the geological data quality. Therefore,

the variations observed here already show a clear sensitivity of the flow fields with respect

to geological data quality. Interesting to note is that velocities and temperatures are

not only varying near the uncertain geological boundaries, but that the changes in the

structural setting affect the whole fluid and heat flow fields.


(a) Mean temperature

(b) Coefficient of variability

Figure 8.9.: Temperature (a) and coefficient of variability (b) for scenario 2C in an E-W cross-sectionthrough the center of the model. The black lines show one simulated geological model of this scenario.Mean temperatures (a) show some indication of convection with lateral temperature differences inthe upper part of the model, better visible in the representation of the coefficient of variability (b).The black line indicates the position of the vertical profiles in figure 8.8. Dimensions of the sectionare as in the examples before (fig. 8.2)


(a) Mean value of absolute flow velocities

(b) Standard deviation of absolute flow velocities

Figure 8.10.: Mean (a) and standard deviation (b) of absolute velocities in an EW cross-section throughthe center of the model. Mean velocities show that, for all simulated flow fields of this scenario, theflow is mainly constrained to the upper layers in the system. The standard deviation of velocities(b) shows that changes in the flow fields mainly correlate with geological surfaces. As in the figurebefore, the black line indicates the position of the vertical profiles and section dimensions are asbefore (fig. 8.2)


8.3.4. Flow uncertainties for different geological scenarios

The above examples show how the flow fields are affected by the quality of the input data

for one geological scenario. Next, it will be evaluated if different geological scenarios, with

different ranges of data quality, affect the flow fields in a different way.

In a first step, the plots along the vertical profile in the center of the model can be

compared for information entropy of the geological model and uncertainties in the flow

fields, evaluated with the standard deviation of absolute velocity and the coefficient of

variability of temperatures. A comparison of these three measures for the min/max data

quality cases of all geological scenarios from table 8.1 is presented in figure 8.11.

The results for the first scenario (1A) show a correlation between positions of uncertainty

in the geological data and high mean and standard deviation values for velocity in the

central part of the model. The vertical scale is identical to the plots before (e.g. fig.

8.8). The temperature variability is highest near the surface, correlating with the highest

value for flow standard deviations. However, this peak is not directly correlated to a

high geological uncertainty but related to boundary conditions, but this is not further

evaluated in this case study. Important are the higher geological uncertainty at depth. It

is interesting to note that uncertainties at great depth do not have a significant influence

on temperature and flow velocity as it mainly affects impermeable geological layers (see

tab. 6.1). The second scenario, 1C, has been described above (sec. 8.3.3). Comparison

to the geological uncertainty here highlights the correlation between variations in flow

velocity and geological uncertainty.

In the second scenario, shown in the second row of figure 8.11, additional faults exist

in the basin, leading to a small sub-basin. The width of this basin is here considered

constant (see sec. 8.3). Comparison of flow fields and geological uncertainties for scenario

2 (fig. 8.11c and 8.11d) show a similar behavior for the temperature variability as scenario

1C. However, the peak in the central part of the model for the standard deviation of

the flow simulation is not as pronounced as before. An interesting point is that for this

scenario, differences for the high quality data case (2A) and the low quality data case (2C)

show a similar pattern. This is not the case in scenario 1.

The third scenario is structurally similar to the second, but the width of the sub-basin

varies, as well. Results for this scenario are presented in the third row of figure 8.11.

Results are similar to the scenario with fixed width. A difference is that the velocity peak

in the upper part of the model is less pronounced. However, as described before, this is

due to a boundary effect. For the case of a high data quality, Scenario 3A in figure 8.11e

is very similar to the simulated values of scenario 2.

The results before already indicated that the geological structure influences the flow

field. This was, for example, obvious from the temperature variability plot in figure 8.8

and is in accordance to theoretical considerations (e.g. Nield and Bejan, 2006). In order

to evaluate if, and how, the state of the system is affected by geological uncertainties, a

measure is required that characterizes the dynamics of the whole system. The measure

of entropy production will next be applied to compare simulation results for the different


(a) Scenario 1A

0 1 2 3T 1e 8

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pG

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(b) Scenario 1C

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(c) Scenario 2A

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(d) Scenario 2C

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(e) Scenario 3A

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(f) Scenario 3C

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pG

0.0 0.5 1.0-11000

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1000p8

H

Figure 8.11.: Comparison of simulated flow fields for different geological scenarios and the informationentropy of geological uncertainties, in 1-D vertical profiles through the center of the model. Inputdata uncertainties of he scenarios are described in table 8.1. For each subfigure, left plots are velocitymagnitude plus two standard deviations, middle plots are coefficients of variance of temperature andright plots show the geologic information entropy.


scenarios.

8.3.5. Entropy production as a measure of the system state variability

In chapter 7 it was shown that thermal entropy production can be applied to characterize

hydrothermal flows. Entropy production is defined as the ratio between the change of

heat and the temperature. For a hydrothermal system as considered here, the conductive

heat transport contributes to the entropy production as an irreversible process (Ozawa

et al., 2003). An important aspect of entropy production is that it is directly related to

uncertainties in a hydrothermal flow field. The measure has been applied to study transient

effects during the equilibration phase of a conductive system (sec. 7.3.2). During the

equilibration phase when the temperature field is disturbed, an average non-zero entropy

production is observed. However, once the system reaches a conductive steady state, the

entropy production is zero (fig. 7.1).

In a further example, entropy production was evaluated for the equilibration phase

of a convective system (sec. 7.4.3). It was observed that entropy production initially

increases towards a maximum value, then decreases again and converges to a finite, non-

zero value when the convective system is in steady state. Furthermore, the finite value

is proportional to the Nusselt number, describing the heat transfer through the system.

This proportionality could be evaluated for simple convective systems (sec. 7.4.4). The

average entropy production therefore provides a measure of the thermodynamic state and

can be applied to evaluate if convection exists in the system.

For the case study considered here, the average specific entropy production was evaluated

for each of the three data quality ranges for scenarios 1 and 2 (tab. 8.1). These two

scenarios were chosen because they showed significant differences in the mean information

entropy of the geological models (fig. 8.7). As the information entropy is a measure of the

geological uncertainties, these scenarios will be applied to test how uncertainties in the

geological model and in the simulated thermodynamic state of the system correlate.

In analogy to the experiments for the onset of convection (sec. 7.4.3), the average specific

entropy production during the equilibration phase for all fluid simulations of the geological

scenarios 2A and 2C is presented in figure 8.12. All curves in both scenarios show the same

behavior that was observed during the equilibration phase of the simple convection models

(fig. 7.5): initially, entropy production increases towards a maximum value, then decreases

again and converges towards a finite, non-zero value. Also, the specific entropy production

for both scenarios is comparable. However, the ranges of the maximum value of entropy

production show a higher variability for scenario 2C, the case of lower data quality and

higher geological model uncertainty (fig. 8.7). In a next step, it will now be evaluated if

the entropy production values of the last time step differ between the considered scenarios.

The entropy production values for all models from scenarios 1 and 2 are analyzed with

box and whisker plots in figure 8.13. The median value (red lines in figure 8.13) for the

finite value of entropy production does not differ significantly for different quality ranges in

the input data. This result could be expected as all simulated geological models are based


(a) Entropy production for scenario 2A

(b) Entropy production for scenario 2C

Figure 8.12.: Entropy production during the equilibration phase for all flow models of scenarios 2Aand 2C: in both cases the entropy initially increases to a maximum value, then decreases again andconverges towards a finite, non-zero value. In the initial phase of high entropy production, the resultsfor the scenario with a higher geological uncertainty (2C) show a higher variability than for scenario(2A). The entropy production values for the last time step are compared in figure 8.13.

on the same geological base model. However, the initial geological model of the second

scenario is structurally different to the first scenario (fig. 8.2). Therefore, the median value

is not necessarily identical. However, a comparison of median values in figure 8.13 provides

no indication for a significant difference in the thermodynamic states of both geological

scenarios. This suggests that the overall dynamic state of the system is not affected by

the difference of the structural models. However, the spread of entropy production values

increases within each scenario. This observation suggests that the distribution of obtained

entropy production values is related to the geological uncertainty.

As the thermal entropy production provides one scalar value to describe the state of the

flow field, the distribution of average entropy production can be evaluated as a measure

of uncertainty of the flow prediction. In order to investigate the relationship between the

uncertainty of the flow field with the uncertainty of the geological model, the spread of

entropy production is evaluated as the difference between the 25th and the 75th percentile

for each scenario in fig. 8.13. The spread of entropy productions is presented with the

information entropy of the geological model in figure 8.14.


1A 1B 1C 2A 2B 2C0.0

0.5

1.0

1.5

2.0

2.51e�10

〈s〉


Figure 8.13.: Comparison of the finite values of entropy production for all simulations of scenario 1and 2 in a box and whisker plot representation. The median value is represented with a red line, therange of a box determines 25th to 75th percentile. Outliers are shown as (+). The median value doesnot change significantly, but the spread of the data increases in each scenario for higher geologicalmodel uncertainty.

A comparison of the behavior of both measures for the different geological scenarios in

figure 8.14 suggests that changes of geological uncertainty coincide with changes of flow

field uncertainty. Furthermore, the uncertainty of the geological model and the flow field

increases with a decreasing quality of geological data (tab. 8.1) from scenarios A-C.

8.4. Discussion

The results of this study show that both, the geological model and the simulated subsurface

flow fields are affected by a varying quality of geological input data and that, in fact,

uncertainties in the geological model and uncertainties in simulated flow fields respond in

a similar way to perturbations in the geological model, resulting from uncertainties in the

geological data.

The effect of changing geological data qualities, here described as different probability

distributions assigned to the geological input data, was also visible in a direct inspection

of the probability of geological units and in plots of standard deviation and variability

of simulated temperature and flow fields. However, even if these measures provide an

indication about the uncertainty at a specific point in the model, they can not be applied

to evaluate the uncertainty in the whole system. Therefore, the system-based measures

that were developed in previous chapters and were used here to evaluate the uncertainties

related to the whole system. The mean model information entropy, described in chapter 4,

was applied to evaluate the state of uncertainty in the geological model with a single value.

In order to characterize the flow system, the average entropy production (chapter 7) was

calculated for the hydrothermal flow fields. As the average specific entropy production



Figure 8.14.: Geological uncertainty (information entropy) and flow field variation (spread of averageentropy production) for different geological scenarios; Changes of information entropy for the differentgeological scenarios coincide with changes in the spread of entropy productions.

describes the state of the whole system with a single measure, the distribution of average

entropy results for different flow realizations can be taken as a measure of uncertainty for

the flow prediction.

The comparison of uncertainties for geological models and simulated flow fields is based

on an integrated workflow that combines geological uncertainty simulation for structural

geological models directly with hydrothermal flow simulations. Based on a geological

input data set and probability distributions that are assigned to each data value, a range

of possible geological models is generated. All of these models are then processed to

a hydrothermal flow simulation. An important point of this workflow is that both the

geological uncertainty and the flow field uncertainty are directly determined for probability

distributions of the geological input data, and not from a perturbed structural geological

model. Therefore, uncertainties can here be related to observations of geological surface

contacts and orientation measurements. This is a novelty and a specific strength of the

method.

The uncertainties of the geological model and the flow fields were evaluated for a 3-D

geological model of the North Perth Basin, illustrating that the methods allow an analysis

of uncertainties in a realistic setting and that typical geological uncertainties of data

resolution at depth, and of different possible structural interpretations, can be considered.


9. Key Findings and Further Outlook

9.1. Key findings

The main contribution of this thesis is the application of system-based measures to classify

uncertainties in geological models and in subsurface flow fields. Information entropy has

been used to evaluate uncertainties in geological models, and thermal entropy production

has been used to analyze uncertainties related to hydrothermal flow. As these measures

have a fundamental theoretical basis and are related to the internal state of the system,

they can be interpreted quantitatively and, consequently, give uncertainties a meaning.

Information entropy values are directly related to the state of uncertainty of a geological

model. For a point within the model, information entropy is a measure of the minimum

number of geological units that could occur at its location. If the information entropy is

zero, only one unit is possible and no uncertainty exists. If the value is greater than zero,

at least two units are probable. If it increases above 1, three units can occur. In general the

measure provides a weight of probability for different states. A strong point of the method

is that it gives a entropy measure for the state of the entire model and therefore lends

itself as a robust measure to quantitatively compare uncertainties in difference models.

In a similar sense, the thermal entropy production provides a quantitative measure

of the thermodynamic state of a hydrothermal system. When the entropy production

is zero, the system must be in a conductive steady state for a closed system. If the

entropy production is larger than zero, the system can be in a convective or transient

conductive state. For higher values of entropy production, the convective units show

higher complexities and, hence, uncertainty of the hydrothermal field increases. Moreover,

the average model entropy production gives a measure of the convective vigor that can be

expected in the system. This is directly related to the efficiency of heat transfer over the

system. The measure is therefore not only useful for a classification of different models,

but also has a quantitative meaning for the productivity of heat that can be harvested

from a particular setting. This valuable information, derived from a three-dimensional

distributed domain, is encapsulated within a scalar value allowing a simple comparison of

a vast range of different models.

In the case of a realistic complex system, both measures have been applied to study

the influence of geological data quality on the uncertainty of geological model and flow

predictions. This analysis has only been possible due to a newly developed workflow that

150 9. Key Findings and Further Outlook

integrates geological modeling and hydrothermal flow simulations. A key finding of this

study is that the geological uncertainty and the flow uncertainties can be subsumed in two

interrelated measures: (a) information entropy, and (b) the spread of the thermal entropy

production. Since the measures provide single values that characterize uncertainties, they

provide a promising path for physically based data compression in data intensive modeling.

9.2. Further outlook

The results of the thesis and the developed methods open up the way to a wide range of

further studies and applications in the fields of thermodynamic systems classification, geo-

logical uncertainty quantification and integrated geological and hydrothermal simulation.

Some specific areas of further research are briefly presented here.

9.2.1. System classification with thermodynamic measures

Analysis of uncertainties in geological units

In the case study, the average entropy production of the whole model was compared

for different geological scenarios (fig. 8.13) and the correlation of the spread of entropy

productions with the geological uncertainty was evaluated (fig. 8.14). In both cases, the

entire model was considered. A logical follow-up would be to perform those studies for

single geological units within the system. A visual inspection of the flow fields (fig. 8.9)

showed that convection was mainly constrained to some geological units. A comparison

of the geological uncertainties in single units, evaluated with the unit fuzziness (eq. 4.3),

and the mean entropy production for one unit could provide interesting insights into the

effect of uncertainties on the dynamics of an open sub-system.

Specific entropy production, Carnot efficiency and geothermal resources

In chapter 7 it was shown that the entropy production is related to heat transfer, evaluated

with the Nusselt number, and to linear stability, described with the Rayleigh number

(Nield and Bejan, 2006). In fact, theoretical considerations show that, for a normalization

of entropy production by conductive heat, the constant of proportionality is the Carnot

efficiency (Regenauer-Lieb et al., 2010). Entropy production can therefore be related to the

amount of work that can be extracted from a hydrothermal system. This relationship has

potential implications for geothermal resource evaluations. To date, geothermal resources

are evaluated based on the thermal heat stored in the system (Muffler and Cataldi, 1978).

This measure has several disadvantages, mainly because it can not provide an insight into

the heat that can be extracted from a reservoir (see e.g. publications in appendix). As

entropy production is related to the Carnot efficiency, it provides a compelling new path

for geothermal resource estimations based on available work.


Evaluation of relative importance of boundary conditions on flow field

uncertainty

In the simulations presented in the case study, temperature boundary conditions were

applied (sec. 8.3). It is well known that boundary conditions can greatly influence simu-

lated flow fields (e.g. Franke et al., 1987). An important path of further research could

be to evaluate the sensitivity of the flow field uncertainty with respect to the boundary

conditions.

Regenauer-Lieb et al. (2010) showed that upper and lower bounds for the total entropy

production of a system can be derived from applied boundary conditions. The theory has

been formulated for coupled thermo-hydro-mechanical-chemical systems and is based on

the consideration of thermodynamic forces and thermodynamic fluxes. The flux multiplied

by the forces reveals the irreversible entropy production. This quantity is related to the

applied boundary conditions: if a thermodynamic flux boundary is applied, an upper

bound of entropy production is obtained. Force boundary conditions reveal a lower bound.

In the case of hydrothermal simulations considered here, the according thermodynamic

forces are differences in pressure and temperature, and the fluxes are heat and fluid flux.

Applying the concept described above, upper and lower bounds of entropy production can

be obtained for each simulated flow field from a comparison of simulations with the different

boundary conditions. If this estimation is combined with the analysis of the influence of

geological uncertainties on flow simulations presented in this thesis, it will yield an insight

into the relative importance of applied boundary conditions on the uncertainty evaluation

of flow fields.

9.2.2. Integrated geological and hydrothermal simulation

Geologically based flow calibration

An important element of the integrated workflow for geological modeling and hydrothermal

simulations that was developed in this thesis is that it relates simulated flow fields directly

with geological input data. An interesting extension of the workflow is therefore to apply

it for a calibration of the flow simulation based on geological data quality.

As a very first test of feasibility, results of the flow simulations in the case study of

chapter 8 were compared with a temperature measurement in a petroleum well for three

different geological scenarios (see tab. 8.1): the original model with high data uncertainty

(1C), the subbasin model with fixed width and low data uncertainty (2A) and the same

model with high data uncertainty (2C). The position of the well in the model is indicated

in figure 6.4c. Temperature profiles are presented in figure 9.1 and the measurement is

shown with a black disc.

Comparison of the results for different scenarios shows that the mean simulated tem-

peratures for scenario 1C (middle diagram in fig. 9.1a) exactly predict the temperatures

at this well position. In contrast to this result, all simulated temperature fields of scenario

(2A) do not match the temperature measurement. However, if the same geological scenario


(a) Warro 1C

0 100 200 300 400-11000

-9000

-7000

-5000

-3000

-1000

1000

0 100 200 300 4000.00 0.15 0.30

Warro

TTT

z

(b) Warro 2A

0 100 200 300 400-11000

-9000

-7000

-5000

-3000

-1000

1000

0 100 200 300 400 0.0 0.1 0.2 0.3

Warro

TTT

z

(c) Warro 2C

0 100 200 300 400-11000

-9000

-7000

-5000

-3000

-1000

1000

0 150 300 0.0 0.2 0.4 0.6

Warro

TTT

z

Figure 9.1.: Simulated temperature profiles at the position of a petroleum well (Warro 1) for threedifferent geological scenarios. The black disc signifies the temperature measurement in the well.

with higher geological data uncertainties is considered (2C), the temperature measurement

is within a reasonable range of results. These preliminary results illustrate that the inte-

grated workflow can be applied to a flow field calibration with a consideration of geological

data.

Monte-Carlo simulation of geothermal resource estimations

As all methods developed in this work are integrated in a scripting framework, they can

easily be extended with further post-processing methods. In the appendix, methods are

presented to evaluate geothermal resource measures directly from simulated flow fields.

These methods have been combined with the workflow from geological data to flow simu-

lations and integrated into the inverse modeling framework iTOUGH2 (Finsterle, 1999).

This software provides a wide range of possibilities for parameter estimation and inverse

simulation. As a first test, iTOUGH2 was applied to control a Monte-Carlo simulation of

heat in place estimations from uncertain geological data.

Results of this first study are presented in figure 9.2. A full 3-D geological model was

constructed with uncertain depth of geological input data, shown in figure 9.2a. From

these uncertain input data, 220 geological model realizations were created, processed to

a geothermal simulation and the total heat in place was calculated for all simulated tem-

perature fields. The results of this study are shown in figure 9.2b.

Probabilistic evaluations of heat in place based on uncertainties in rock properties (heat

capacity and density) have already been performed by Meixner et al. (2010). Combining

their methods with the consideration of geological data uncertainties described here has a

great potential for comprehensive probabilistic resource evaluations.


(a) Geological data and uncertainties

Depth

2 k

m

E-W 4 km

(a) Geological data

(b) Results of one realization

(d) Results of Monte-Carlo simulation

Simulated temperatures3-D Geological model

Aquifer

Top Aquifer

Base Aquifer

Data points with ranges forMonte-Carlo Simulation

Total Heat in Place [J]

6*1017

7*1017

5*1017

8*1017

9*1017

Counts

10

20

30

0

20 50 80

Temperature [ C]o

Total heat in place for aquiferfor 220 simulated temperature fields

(b) Probabilistic estimation of total heat in place

Depth

2 k

m

E-W 4 km

(a) Geological data

(b) Results of one realization

(d) Results of Monte-Carlo simulation

Simulated temperatures3-D Geological model

Aquifer

Top Aquifer

Base Aquifer

Data points with ranges forMonte-Carlo Simulation

Total Heat in Place [J]

6*1017

7*1017

5*1017

8*1017

9*1017

Counts

10

20

30

0

20 50 80

Temperature [ C]o

Total heat in place for aquiferfor 220 simulated temperature fields

Figure 9.2.: Combination of methods developed for this thesis with the inverse modeling and parameterestimation package iTOUGH2: (a) Input data for the geological model with range of uncertainties(in vertical direction only); The lines represent one realization of the geological model. (b) Histogramof calculated heat in place for 220 realizations of the geological model; Only the geological inputparameters were changed, all other settings (rock properties, boundary conditions, etc.) were keptconstant.


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A. Improved Geothermal Resource

Estimation


simulation





simulations




Chapter 3

Chapter 6

Chapter 5

Chapter 4

Appendix

Chapter 8

Chapter 7



AppendixAppendix


168 A. Improved Geothermal Resource Estimation

Overview The methods that were developed to study uncertainties in geological data

and hydrothermal flow fields can directly be applied to practical problems. In fact, in

the course of the thesis, novel techniques were developed to optimize classical geother-

mal resource estimations. With one method, heat in place density maps, the classical

heat in place analysis is transferred into a density measure, providing a detailed spatial

evaluation of the stored heat below ground, instead of a simple bulk measure for a whole

resource area. Maps of sustainable pumping rates were developed on the concept of hy-

drothermal breakthrough and provide an insight into aspects of sustainability, directly as

an exploration measure.

As these methods are directly based on the results of simulated hydrothermal flow fields

and can be seamlessly integrated into the workflow from geological data to flow simulations

(chapter 6) and the geological uncertainty simulation (chapter 3). A combination of the

methods with the uncertainty simulation methods presented in the main part of the thesis

will lead to a probabilistic analysis of geothermal resources in future work.

The two appended papers have been published in conference proceedings.


Concept of an Integrated Workflow for Geothermal Exploration in Hot Sedimentary Aquifers

J. Florian Wellmann, Franklin G. Horowitz, Klaus Regenauer-Lieb Western Australian Geothermal Centre of Excellence, UWA-CSIRO-CURTIN

ARRC 29, Dick Perry Avenue 6165 Kensington [email protected]

Geothermal exploration is currently performed in different steps and on different scales, from the initial, large-scale resource estimation going down to local reservoir sustainability analysis for a specific application. With this approach, it is not possible to explore directly for requirements dictated by a geothermal application.

If we, for example, consider the exploration for a direct heat-use application we could require a pumping rate of 100 l/s at a minimum temperature of 70°C. Economic constraints could be a maximum drilling depth and the minimum years lifetime of the system. The direct map-based exploration for the best locations considering these constraints is not possible with the standard workflow.

We present here an approach to overcome this limitation. We combine geological modelling, geothermal simulation and reservoir estimation into one consistent location-based method. Outcomes of this integrated workflow are map-based reservoir and resource analyses that can directly be used as guidance in the exploration for the best possible location of a geothermal application. Our workflow is specifically developed for applications in hot sedimentary aquifers but can be extended to other geothermal settings.

Keywords: Geological Modelling, Geothermal Simulation, Direct Heat Use, Integrated Workflow, Hot Sedimentary Aquifers

Geothermal Exploration

Geothermal exploration for hot sedimentary aquifers usually consists of the following steps (not necessarily in this order):

Geological Modelling for a resource area

Resource Base Estimation in a large-scale target area (accessible and useful resources)

Market analysis and other local considerations (e.g. power lines, infrastructure)

Above-ground installation and technical application (direct heat use, power generation)

Detailed resource analysis in a smaller scale (economic resources for a specific application)

Local reservoir exploration and sustainability analysis

Financial modelling

Depending on the reservoir type, further analyses are necessary (e.g. stress-field, permeability optimisation, etc.). The single parts of this workflow are usually performed separately and in a sequential order. Our method combines the steps from geological modelling to sustainability analysis which are briefly described below.

Geological Modelling

A structural geological 3-D model is an important basis for geothermal exploration. It allows the visualisation of geological structures in the subsurface and can directly be used to identify relevant areas (e.g. from fault structures, etc.). Also, a 3-D geological model is the basis for other types of analyses, like the geothermal simulation.

A large variety of tools exist to construct geological models, ranging from map-based interpolation of structures (2.5-D methods, e.g. depth to basement maps interpolated from drillhole data) to full 3-D geological modelling that can consider complicated structures like reverse faulting or doming structures (Turner, 2006).

Geothermal Simulation

Numerical geothermal simulation is the next important step in the exploration. Based on physical constraints and subsurface data, a model of the temperature distribution below ground is simulated. This is the basis for the geothermal resource estimation and allows first estimates of drilling depth to a desired temperature.

Figure 1: Example of a simulated fluid and heat flow field. The section shows a contour map of temperatures, the plane is a temperature isosurface, and streamlines (gray) indicate fluid flow paths.

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Similar to geological modelling, a variety of different methods and codes are available for geothermal simulation. Main differences are the complexity of the simulation, i.e. from simple heat conduction simulation to coupled simulation of fluid and heat to complex multi-phase flow and reactive transport. (Kohl et al., 2007). The application of a code strongly depends on the geothermal reservoir type. In the case of hot sedimentary aquifers, fluid flow has to be considered as a heat transport mechanism and a suitable code should be used.

Geothermal Resource Base Estimation

Standard methods for the quality estimation of a geothermal resource are based on Muffler and Cataldi (1978). They describe several different approaches, most widely known is the volume method, often referred to as “heat-in-place”. The

total thermal energy contained in a volume V of

rock is estimated based on specific heat of rock

rc and fluid wc , porosity , density and a

temperature difference T :

! TVccH wwrrip !"#$ %&%&)1(

The calculation of heat-in-place is usually performed for an estimated total volume, mean temperature and porosity of a resource rock.

Other estimations are possible and depend on the geological situation and geothermal resource type.

The evaluated resource base has to be further subdivided (Fig. 2) into accessible heat, usually defined by the maximum depth of drilling (this is what is usually considered in a standard “heat-in-place” analysis). But not all heat from the accessible heat is actually useful, based on physical limitations, reservoir lifetime and legal and environmental considerations. Finally, only a fraction of the useful heat can be considered as economic, which Muffler and Cataldi (1978) define as the geothermal energy that can be extracted in the lifetime of a reservoir at costs comparable to other energy sources.

Figure 2: From the broad geothermal resource base to estimation of the economically useable resource (redrawn from Muffler and Cataldi, 1978).

Estimation of Extractable Energy

The amount of extractable heat depends on many geological, physical and technical factors. These are usually combined into a general “recovery factor” as a broad estimation.

For a hot sedimentary aquifer, Gringarten (1978)

defines a heat recovery factor, gR , as a the ratio

of extracted heat, TctQ ww !max , to the total

theoretically recoverable heat-in-place as given

above. Here t is the producing time, the

quantity maxQ is the maximum production flow

rate that can be maintained either indefinitely (for a truly sustainable system) or over the assumed economic lifetime of the geothermal system and

wwc is the volumetric heat capacity of water.

Writing

! ! V

tQ

ccR

wwrr

g

!"# max

/1

1

!!

we find the heat recovery factor is dominated by

tQ max for a porosity of : the recovery factor is

a function of time. The maximum sustainable

pumping rate maxQ for a doublet well (pumping

and re-injection) over a production time t can

be analytically estimated from heat and flow equations. Gringarten (1978) presents an analytical approximation and derives the following relationships for the pumping rate Q:

2

3Dt

h

c

cQ

ww

aa

!

"

#"

and

TsrD

Qw )/ln(

12 !

The first equation describes the pumping rate as a function of production time, thickness h of the aquifer and distance D between pumping and re-injection well. The second equation includes the maximum drawdown s, the well diameter rw and transmissivity T. Temperature is implicit in these equations as density of water and transmissivity are a function of temperature.

The combined solution of these equations provides an estimate of the maximum pumping

rate maxQ and the minimal distance D required

between the pumping and re-injection well in the aquifer to avoid a thermal breakthrough during the production lifetime of the doublet.

The result can be considered a very conservative estimate as an application may still be possible after thermal breakthrough for some time. Also, as soon as a natural hydraulic gradient is present, a

112


layout of the re-injection well downstream from the pumping well will increase the lifetime even more (Banks, 2009).

Limitations of the standard approaches

The presented standard methods to evaluate a geothermal resource and its sustainable application are performed on two different scales. Whereas the heat-in-place estimation is performed for a whole resource, the estimates for a sustainable pumping rate are performed on the local scale. It is not possible to derive a location-based analysis of heat in the subsurface (i.e. how is the total heat-in-place distributed in space) or to analyse a whole area for a required pumping rate (i.e. where can a certain pumping rate be obtained for a minimum time). Thus, the combined analysis of both factors is not possible for a whole resource region.

To overcome this limitation, we present an approach to down-scale the heat-in-place estimation for a regional analysis and to extend the Gringarten estimations to a whole area, all within the context of geological modelling and geothermal simulation.

Integrated Geothermal Exploration

Concept of workflow

In our workflow (Fig. 3), we combine the steps from geological modelling to resource and sustainability estimations.

Figure 3: workflow of our approach from geological model to efficiency estimation of geothermal application

The starting point for our workflow is a full 3-D geological model. We use GeoModeller (www.geomodeller.com) for the modelling as it is capable of dealing with complicated 3-D geological settings and provides a very fast and

efficient way to create realistic geological models directly based on input data (e.g. Calcagno, 2008). It is thus possible to quickly test several geological scenarios as the starting point for the geothermal simulation.

We link the geological model directly to a geothermal simulation code. The simulation is performed with a fully coupled fluid, heat and reactive transport simulation code (SHEMAT). All relevant physical properties are calculated as a function of temperature in each time step. It is also possible to include anisotropies in thermal conductivity and permeability (see Clauser, 2003 for a detailed description). The simulation code is thus capable of dealing with complex settings (from hot dry rock to hydrothermal) and has been applied to many geothermal simulations (e.g. Soultz-sous-Foret (France), Waiwera (New Zealand)).

Now, we process the results of the geothermal simulation further for two analyses: (1) the distribution of heat in the subsurface and (2) estimation of the sustainable pumping rates. The main difference to the standard approaches is that we create a map view of the distribution of both properties in the whole resource area.

The simulated temperature and fluid flow field and the distribution of physical properties in 3-D are then processed further with a set of programs to derive several characteristic parameters (e.g. transmissivity, mean water density, mean temperature of one formation at depth). Essentially, we analyse the physical properties in the subsurface at every location in space. This is then used as an input for the extended volumetric heat-in-place calculation (following Muffler and Cataldi, 1978) and the well doublet spacing and maximum pumping rate analysis from Gringarten (1978) and Banks (2009), as described above.

The distribution of temperatures, local heat-in-place and the evaluation of sustainable pumping rates in the resource area now directly allows the exploration for a suitable area given the characteristics of a geothermal application. For example, we can now identify areas in the map where we can achieve the required pumping rate for a given minimum temperature and available heat which, in the end, determines the economics of a geothermal application.

Example Model

We apply our workflow for geothermal resource estimation to a full 3-D geological model to local heat-in-place and sustainable pumping rate evaluation. The model is situated in a half-graben setting (Fig. 4). A large normal fault in the east off-sets the basement creating a basin. This basin is filled with several sedimentary formations that are furthermore displaced by normal faults, leading to an internal graben structure. The scale of the model is 8 km x 8 km x 5 km.

113


Figure 4: Simple geological model used for the application of the workflow. The structural setting is a half-graben structure; the basin is filled with sedimentary formations that are further cut by faults.

Figure 5: Selected results of our workflow. Analyses are performed for the second lowest sedimentary formation (light green in Fig. 4).

(a) local heat-in-place, normalised to m2. (b) Sustainable pumping rates [m3/s] for a production period of 30 years. We can clearly identify the most promising areas.

(a) Local heat-in-place (b) Sustainable pumping rates

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This structural set-up happens to be similar to areas in the Perth Basin and representative of geological settings in other sedimentary basins. Values for thermal conductivity and hydraulic properties are also similar to formations in the Perth Basin.

Results

The maps in Figure 5 show the most important results of our integrated workflow, i.e. the local heat-in-place and the maximum pumping rate for one formation. These maps are created in a GIS framework and can directly be used for a location-based analysis. We obtain the local heat-in-place in addition to the total heat-in-place which is usually estimated (it would be approximately 5.2E18 J in this example).

The map dimensions are the same as for the model (8 x 8 km). Displayed is the analysis for the second lowest sedimentary formation (light green in Fig. 4). We can see that most of the heat in place (Fig. 5a) is located within the Northern part of the graben. In the same area, we can obtain the highest sustainable pumping rates (here determined for a total lifetime of 30 years). The patterns coincide in this case as we are considering a simple structure with homogeneous permeabilities and thus pumping rates are strongly related to temperature (which is, in this simple case, also reflected by the local heat-in-place pattern). In other cases (e.g. in lower permeability settings like Enhanced Geothermal Systems), we might obtain a completely different picture for local heat-in-place and sustainable pumping rates.

Discussion

We presented an integrated geothermal resource evaluation workflow that combines and extends classical methods. Starting from a full 3-D geological model and relevant physical properties, we simulate the temperature and fluid flow fields and use these as a basis for a variety of estimations. Firstly, we calculate the overall heat-in-place, as defined in Muffler and Cataldi (1978). We extend this classical method to a location-based analysis to identify directly the position of a valuable resource. We also use the results of the simulation for an estimation of a well doublet scheme and sustainable pumping rates, after Gringarten (1978) and Banks (2009) and extend it to a resource-wide estimation. The main benefit of our workflow is that it directly combines these standard methods for a location-based geothermal resource and sustainability analysis.

As the results from our integrated workflow are location-/map-based, it is possible to combine them with other relevant location factors. We can, for example, combine our analyses with a map of the depth of a formation and a maximum drilling depth. Other map-based economic constraints

can directly be implemented, e.g. the distance to the market or available infrastructure. Our workflow thus opens up the way to an integration of geological, geothermal, technical and financial considerations within one combined framework.

Furthermore, our workflow can be extended to scenario testing. All the single steps in the workflow are linked. It is thus possible to directly test the effect of a change in the geological model or the physical properties on the estimation of the sustainable pumping rate. This is not possible with common standard approaches.

The results of our simple example model (Fig. 5) are based on several assumptions and simplifications (see Gringarten, 1978, and Muffler and Cataldi, 1978, for a detailed description of their assumptions). The calculated estimations have to be considered in the light of these assumptions. Still, in a recent review of these methods, Banks (2009) points out that they are applicable in many cases and provide a rather conservative estimate. We interpret the numbers as a guideline but the distribution in space as very valuable information as this directly points out the location of a probable geothermal resource.

Our approach is flexible and can be applied in simple and complex settings. The geological modelling is capable of dealing with complicated geological settings, like reverse faulting or overturned folding and doming structures (e.g. Calcagno, 2008). The geothermal simulation can be extended to include reactive transport and species transport (Clauser, 2003). Furthermore, the applied simulation code can also model pumping and re-injection. This will be implemented in the future into our workflow. It will thus be possible to directly validate the effect of long-term pumping in the fluid and heat flow field in complex geological settings in an identified target area.

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Estimates of sustainable pumping in Hot Sedimentary Aquifers: Theoretical considerations, numerical simulations and their

application to resource mapping

J. F. Wellmann*, F. G. Horowitz, L. P. Ricard, K. Regenauer-Lieb Western Australian Geothermal Centre of Excellence, The University of Western Australia, 35

Stirling Hwy, WA-6009 Crawley, Australia *Corresponding author: [email protected]

A method for a spatial analysis of potential sustainability for the early stage of exploration in Hot Sedimentary Aquifers (HSA) is presented here. Our analyses are based on well established estimations for the thermal breakthrough in a doublet well setting. We consider two significantly different scenarios: the placement of a well doublet in an aquifer without significant natural flow, and the case where a natural groundwater flow exists. We integrate these two analytical estimations into one workflow with geological modelling and geothermal simulation. As a result, we obtain spatial analyses of theoretical sustainable pumping rates for a whole resource area. These maps are specifically suitable for the early stage of exploration where a potential target area has to be determined based on limited information. We present the application of our method to a geothermal resource area in the North Perth Basin, from geological modelling, to the simulation of fluid and heat flow, and finally to map the analysis of sustainable pumping rates for one aquifer. The results contain a high degree of uncertainty, but indicate the distribution of future prospective areas. These maps can be combined with other spatial datasets, e.g. infrastructure. Also, as they are integrated into one workflow, an update of the analyses is directly possible when new data become available.

Keywords: Hot Sedimentary Aquifer (HSA); Resource Analysis; Sustainable Pumping Rates; Geothermal Simulation; Geological Modelling

Introduction

This paper presents a novel exploration method to identify geothermal prospects based on thermal and hydraulic properties of the subsurface. We combine estimates of sustainable pumping rates with simulations of fluid and heat flow, and derive maps of estimations for sustainable pumping rates.

Our work regards estimates of sustainability for well doublet systems. After a certain time tB, the reinjected cold water front may reach the extraction well and cool down the extracted temperature (Fig. 1, red curve). This will affect the geothermal application and, at some stage, rule out further effective usage of the site. An

estimation of this breakthrough time tB is required to evaluate the sustainability of a project.

Temperature development at production well

90

95

100

0 10000 20000 30000 40000 50000

production time [days]

tem

pe

ratu

re [

C]

flow parallel

no flow

flow perpendicular

Figure 1: Comparison of temperature development at the extraction well for three different scenarios: i) If no advective flow is present, the cold reinjected water may reach the production/ extraction well (red curve) and the temperature of the pumped water will decrease; ii) For advective groundwater flow perpendicular to the wells, the temperature decrease is significantly slower (green curve); and iii) For the case that the reinjection well is directly downstream of the production well, no thermal breakthrough occurs (blue curve).

Analytical estimates of a sustainable long-term use for geothermal installations have been applied for many years (e.g. Gringarten, 1978, Lippmann and Tsang, 1980). Most of the approaches are based on many simplifications and assumptions. They nonetheless deliver an important insight into the distribution of promising areas for sustainable flow in the subsurface, especially in the early exploration phase, as not only available temperature and heat in place are considered, but also hydraulic parameters like permeability and porosity.

Another standard tool in geothermal exploration is numerical simulation of subsurface fluid and heat flow. (See e.g. O’Sullivan et. al. 2001 for a detailed revision of applications.) A thoroughly performed study can deliver detailed insight into fluid and heat movement in the subsurface, within the usual limitations of data availability and model accuracy.

One problem with both estimations, analytical and numerical, is that they are usually only performed at one location, i.e. at a previously identified target, to evaluate its long term behaviour. We


propose here that it is useful to perform a raster analysis of sustainable pumping rates. This can be applied from the very first stages of geothermal exploration and subsequently refined during ongoing exploration, when more data become available.

We present here a method to perform these spatial analyses. Our approach is implemented in a complete framework covering geological modelling and fluid and heat flow simulations. The results we obtain have to be analysed critically as the many assumptions that go into the analysis prohibit an absolute interpretation of the results. For example, an estimated average pumping rate of 80 m

3/s for a lifetime of 30 years may contain a

high degree of uncertainty. But even if the total values might vary, we consider the general distribution of the analysis to be a valuable representation of potential target areas in a resource area.

Theoretical considerations and simulated examples

Here, we briefly review some of the commonly applied theoretical estimations of sustainability studies. All the presented estimations below are suitable for application in Hot Sedimentary Aquifer/ porous media systems. The situation is much more complex in fractured systems (EGS) and special considerations are necessary. For more detailed information, see the recent review of Banks (2009).

Analytical estimations

The longevity of a doublet well can be defined as the time it takes for the reinjected cool water to reach the extraction well indicated at the point when the extracted temperature starts to decrease (e.g. Fig. 1). We are applying this definition here, but it should be noted that this time defines the lower end of the usability. After the thermal breakthrough, the extracted temperature will decrease but possibly the application will still be usable (e.g. Lippmann and Tsang, 1980, Banks, 2009).

In this paper, we consider two cases for the estimation of a sustainable pumping rate, with and without advective background flow.

Well doublet without advective background flow

We firstly consider the case of hydraulic breakthrough. This is the time thyd the reinjected water takes to reach the extraction well. For simple cases (flow along the shortest path, homogeneous aquifer, no dispersion) the hydraulic breakthrough time (e.g. Hoopes and Harleman, 1967) can be evaluated for a pumping

rate Q, an aquifer with porosity ! , a thickness h

and a spacing D between extraction and reinjection wells as:

Q

Dhthyd

3

2

!"=

The hydraulic breakthrough (i.e. the time when the reinjected water reaches the extraction well) is not equal to the thermal breakthrough (the time when the cold temperature front reaches the extraction well). The temperature front is delayed by a retardation factor Rth (Bodvarsson, 1972) that depends on the thermal properties (specific heat c and density ") of the aquifer rock (a) and the

water (w):

ww

aa

th

c

cR

!

!

"

1=

Therefore, the time for the arrival of the thermal front at the extraction well can be calculated as:

ww

aa

thec

c

Q

hDt

!

!" 2

3=

Interestingly, the thermal breakthrough time in this case does not depend on the hydraulic conductivity / permeability of the aquifer, but only on geometric and thermal properties.

This estimation is based on many assumptions; the most important are:

1) Fluid properties are constant and do not depend on temperature.

2) The flow itself is steady-state, injection rate and temperature are constant and there is no mixing between the reinjected fluid and the native water.

3) The geometry of the aquifer is very simple: constant thickness, constant porosity and it is assumed to be horizontal.

4) Cap rock and bedrock of the aquifer are impermeable.

(For a further detailed discussion see e.g., Gringarten and Sauty, 1975.)

Apart from these conditions, another common assumption is that there is no heat transport from the aquifer into the surrounding rocks by conduction. This assumption is reasonable in many cases (see Gringarten and Sauty, 1975, for accurate criteria) and we will adopt it here as well.

Well doublet with advective background flow

If a native hydraulic gradient is present in the aquifer, the situation is more complex. It is now important to consider the well placement with respect to the natural advective groundwater flow v0 (Fig. 2). An analytical estimation for the thermal breakthrough can be derived for the case that the


A variety of software codes exists to perform these simulations (see O’Sullivan et al., 2001). We used SHEMAT (Simulator of HEat and MAss Transfer, Clauser and Bartels, 2003) for the resource scale simulations presented in this paper. (e.g. Fig. 3). To test the validity of the analytical estimation of breakthrough times, we simulated a well doublet (extraction and reinjection well) with SHEMAT and, additionally, with the petroleum reservoir engineering software TEMPEST.

The plots in Fig. 2 show the temperature distribution (colour map) and fluid flow vectors (grey arrows) in the subsurface. The three examples given relate to temperature decrease at the extraction well for three different scenarios, as given in Fig. 1.

In summary:

1) Well doublet in an aquifer without groundwater flow, after thermal breakthrough occurred. We can see that the flow field affects a wide area perpendicular to the direct connection between the wells.

2) Natural groundwater flow, the reinjection well is in the downstream direction of the extraction well. The temperature field is now disturbed by the natural groundwater flow. For the same pumping rate, well spacing, and simulation time, the cold temperature field does not reach the extraction well. No thermal breakthrough occurs.

3) Natural groundwater flow perpendicular to the connection line of the well doublet. The temperature field is again clearly affected by the groundwater flow field and the temperature decrease at the extraction well is slowed down.

Example: North Perth Basin

Geological model

We applied the analytical estimations presented above to exploration-scale simulations of fluid and heat flow. For the first stage of exploration, we consider these analytical assumptions as a valuable indication of potential geothermal target areas.

Figure 3: 3-D Geological model for a part of the North Perth Basin (In inset picture: model location=red square; black circle=Perth). Sedimentary formations are overlying basement which is offset by normal faults.

As an example here, we show the application of the method to an area in the North Perth Basin. The geology is characterised by thick sedimentary formations cut by normal faults in a graben setting (Fig. 3). The geological model was created with an implicit potential-field approach (Lajaunie, et. al. 1997), implemented in the GeoModeller software (Calcagno et al., 2008). The model is a simplified version of a more complex regional model.

Geothermal simulation

The geological model is directly processed to an input file for fluid and heat flow simulation with SHEMAT (see Clauser and Bartels, 2003). Rock properties (permeability, porosity, thermal conductivity and heat capacity) were assigned according to samples in this region where available. A strong anisotropy (horizontal / vertical = 10) was applied to all permeability values to achieve a more realistic model.

Figure 4 is a representation of the simulated fluid and heat flow field for the North Perth Basin model. The effect of fluid flow on the temperature distribution is clearly visible. The resulting temperature gradients appear reasonable and qualitatively in accordance with measured values in the area.

For a quantitative analysis of the results, the model has to be refined and adjusted further, especially at the borders (boundary conditions, see discussion). Respecting these current limitations to model verification, we next apply our resource analysis methods to this model. As all steps are integrated into one workflow, it is easily possible to update the model and all analyses later, when more data become available.


reinjection well is placed downstream of the extraction well (Lippmann and Tsang, 1980):

!"

#$%

&

''''+=

A

A

A

A

c

cvDt

ww

aa

the

41

4arctan

41

41

1)/( 0

(

(

)

where

02 hDv

QA

!=

This equation does not have a real solution when the natural groundwater velocity is above a critical value

hD

Qv

!"

2

0>

In this case, no thermal breakthrough will occur and the system is, in principle, completely sustainable and can be operated without time limitations.

Validity of the analytical solution

The analytical estimations of hydraulic and thermal breakthrough depend on many assumptions (see above). In a realistic setting, some effects might reduce the breakthrough time (hydraulic dispersion, heat conduction in the fluid phase) while others might lead to longer breakthrough times (heat resupply from surrounding beds, stratification). A careful examination of these effects is possible with numerical simulations of pumping and reinjection.

Consideration of pressure drawdown

In both cases presented above, with and without natural groundwater flow, we can consider a maximum pressure drawdown s at the extraction well as another criterion for the determination of a sustainable pumping rate. Gringarten (1978) presented a relationship obtained from potential theory:

wr

D

T

Qs ln

2!=

We can see from this equation that the pressure drawdown s depends on pumping rate Q, aquifer transmissivity T and well diameter rw, as can be expected, but also on the well spacing D, as the two wells interact and a smaller spacing leads to less drawdown.

Combined analysis

For a complete sustainability analysis for the well doublet, we might consider the thermal breakthrough time and a maximum pressure drawdown in the reservoir. Concerning the temperature breakthrough time, we want to have

a large well spacing D, but if we consider the pressure drawdown, a smaller spacing is more beneficial. The optimal value of D cannot be determined analytically, but numerical solutions can be applied (e.g. Kohl et al., 2003, Wellmann et al., 2009).

Numerical simulations

The theoretical estimations described above deliver a very useful estimation about potential geothermal targets. We therefore consider them as ideal for the early exploration stage. But as they depend on many assumptions and simplifications, numerical simulations of subsurface fluid and heat flow have to be applied to derive a more realistic insight into the sustainability of the system.

Figure 2: Fluid and heat flow representations for the three example scenarios described below, and presented in Fig.1. In the first example, without advective flow, we can see that the reinjected cold water (reinjection well: blue triangle) reaches the extraction well (red triangle) after a certain time, and the produced temperature decreases. Example 2: for natural advective flow in the direction of the injection well, the cold temperature fan does not reach the extraction well, and the system is completely sustainable. Example 3: for flow perpendicular to the wells, less cold water reaches the extraction well, and the temperature decrease is reduced.


Figure 4: Visualisation of the temperature and fluid flow field simulated for the 3-D geological model of the North Perth Basin. The orange isosurface shows the depth to 120oC. The black tubes indicate fluid flow pathways. General flow direction is N-S.

Novel resource analysis methods

Next we use the simulated fluid and heat flow field to estimate different aspects of geothermal resource sustainability. All the examples here are performed for the oldest sedimentary formation (see Fig. 3, light green unit). The minimum lifetime to consider a geothermal project site as “sustainable” is assumed to be 30 years for the following analyses.

Key novel aspects of all our analyses are:

1) We perform the analyses directly on the basis of the simulated fluid and heat flow field for the resource area, linked to the 3-D geological model.

2) All aspects (maximum sustainable pumping rate, heat in place, pressure drawdown) are evaluated on a spatial basis, i.e. we derive 2-D maps of these properties showing their distribution.

3) All relevant steps are integrated into one workflow, it is therefore readily possible to update the geological model and the geothermal simulation when more data become available.

Maximum sustainable pumping rate, without consideration of advective flow

Following the definition of the theoretical breakthrough time for the thermal front given above, we estimate a maximum pumping rate that could be expected for a doublet system. Spatial analysis is performed step-by-step at every point in space (Fig. 5).

Figure 5: Maximum pumping rates for a doublet system with 800 m spacing and a lifetime of 30 years. All other properties required for the estimation of the pumping rate (see equations above) are directly taken from the simulation (e.g. density) and the model (e.g. formation thickness).

Consideration of natural groundwater flow

In Fig. 4 we can see that groundwater flow is present in this regional scale model. If we hypothetically placed a well doublet in one of the flow areas, it is possible to determine a pumping rate where a thermal breakthrough will, theoretically, never occur (see Example 2 in Fig. 2). If we then apply this analysis again at every point in our model, we can derive a spatial analysis of these pumping values (Fig. 6).

Figure 6: Estimation of a maximal pumping rate for the theoretical case of no thermal breakthrough (Example 2 in Fig. 2), in the presence of advection. This is of practical significance as areas with a high value can also be expected to allow a higher sustainable pumping rate (for a project durance less than infinity).

Combination with other important factors

These spatial analyses can now be used in combination with other relevant factors for geothermal exploration, e.g. mean temperature at depth for a target formation, or maps of heat in place density, to display the spatial distribution of subsurface heat in place (Wellmann et al., 2009). As all results are in map view, they can easily be included in a GeoInformationSystem (GIS) and


combined with, for example, infrastructure considerations.

Discussion

We have shown that it is possible to combine analytical considerations of resource sustainability, with geothermal fluid and heat flow simulations. Our approach enables a direct spatial analysis of relevant factors for geothermal exploration in Hot Sedimentary Aquifers. The major advantage is that geothermal prospects can be identified based on physical reasoning (in the context of geological modelling), geothermal simulation, and ideally, all available data. We propose that this method is a step forward for the identification of geothermal target areas from a regional analysis.

The example presented above for the North Perth Basin is performed for a resource-scale model, representative for an early stage of geothermal exploration. It contains a high degree of uncertainty and the determined numbers for sustainable pumping rate are probably not quantitatively correct. But as they are based on a full 3-D integration with geological knowledge, physical simulation and all available data, we can interpret the results spatially, i.e. identify areas which should be analysed more carefully. This is a major advantage to standard resource estimation methods, e.g. heat in place, where only one value for the whole resource area is determined.

In a realistic project scenario, the next steps would be to refine the model and adjust boundary conditions carefully to the local setting. But as we have integrated all relevant modelling, simulation and resource analysis steps into one workflow, an update at every stage is easily possible.

We recognise that the results are subject to a large degree of uncertainty. Two ways we will address this in future work, will be to combine this workflow with an uncertainty simulation of geological modelling (Wellmann et al., 2010), and with sensitivity studies of the geothermal simulation to derive a quantitative evaluation of the sustainability map quality.

Acknowledgements

This work was enabled by an Australian International Postgraduate Research Scholarship (IPRS) and a Top-up Scholarship from Green Rock Energy Ltd. The Western Australian Geothermal Centre of Excellence (WAGCoE) is a joint centre of Curtin University, The University of Western Australia and the CSIRO and funded by the State Government of Western Australia.

We are very thankful for the comments of an anonymous reviewer.

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