1

Direct Construction and History

Matching of Ensembles of Coarse-Scale Reservoir

Models

Céline Scheidt1, Jef Caers

1 and Yuguang Chen

2

1

Department of Energy Resources Engineering

Stanford University 2

Chevron Energy Technology Company, San Ramon, CA

1. Introduction

Despite many efforts, there is still substantial disconnect between reservoir

modeling at the geology and geophysics (G&G) level on one hand, and reservoir

engineering (history matching, optimization) on the other hand. In an ideal situation,

multiple geological models (possibly 10s or 100s) would be constructed integrating

various sources of uncertainty, such as depositional systems (geological scenarios),

data uncertainty (in well-logs and 3D seismic), structural uncertainty (faulting and

layering) and uncertainty about facies and petrophysical properties.

These high-resolution models need to be upscaled and then history matched. In

reality, CPU limitations prevent the construction of multiple history-matched models.

Often, one only history matches a single model and most times, this is done without

regard of the intended geological spatial continuity. In other words, while a history

match may be achieved, this often comes at the cost of geological consistency, hence

prediction power. Recall that the ill-posed nature of the history matching inverse

problem makes it possible to create such geologically inconsistent reservoir models.

The availability of such multiple models is critical for uncertainty quantification,

risk assessment and optimization task, since optimizations or decisions based on one

model often leads to overly optimistic scenarios. The fundamental problem therefore

lies in how to deal with a large set of reservoir models and how to carry such a set

forward during history matching, optimization and reservoir model updating. Direct

flow simulation of the multiple models on a high-resolution grid is often

computationally prohibitive. Various combinations of upscaling and downscaling

steps (see Caers, 2005 for an overview) have been proposed to address the issue of

the needed change in grid resolution. However neither is fully satisfactory in

efficiency nor effectiveness as they often require several levels of automation (such as

2

in the upscaling step) that may not be practical. The much heralded ensemble Kalman

Filtering is suitable for the modeling of an ensemble of reservoir models, but is too

limited in scope and applicability due to its underlying theoretical assumptions.

The objective of this work is to develop a workflow which addresses the issue of

multiple history matched models through ensemble-level reservoir modeling. The

history matching is performed on coarse-scale models where flow simulation is

feasible, while being under geological control (consistent with fine-scale data). This

short paper briefly describes the methodology and provides some preliminary results

on this newly initiated research in collaboration with Chevron.

2. Proposed Workflow

Caers et al. (2009) describes distance-based techniques that provide a

parameterization based on an application-tailored distance between any two reservoir

models. The size of the resultant distance matrix (i.e., number of realizations) is much

less than that of a covariance matrix (i.e., number of high-resolution grid cells), as

encountered in traditional geostatistical modeling and ensemble Kalman filters.

Classical KL expansion and kernel-based techniques are then applied to the distance

matrix to allow new realizations to be generated and/or adjusted to new data (Scheidt

et al. 2009). The distance-based approach is purpose-driven and is general to account

for different sources of uncertainty, i.e., it can handle any type of continuous and

discontinuous variables (e.g., facies, petrophysics, structure, etc.).

To construct new realizations that match history, we use the approach described

in Caers et al. (2009), which consists of solving a so-called post-image problem. This

approach requires performing a flow simulation for each model, which may be too

expensive computationally for high-resolution grids. To address this, we propose

using the ensemble-level upscaling techniques proposed in the work of Chen and

Durlofsky (2008). In these techniques, flow-based upscaling is applied to only a few

high-resolution models, and statistical procedures are employed to generate the

multiple coarse-scale models. The flow simulations would then be performed on a

coarse grid, rendering the problem more tractable.

The workflow we propose is as follow (Figure 1):

1. Construction of an ensemble of fine-scale reservoir models.

2. Upscale the ensemble of fine-scale reservoir models (either the ensemble-

level uspcaling (Chen and Durlofsky, 2008) or any other traditional

uspcaling approaches (e.g., Durlofsky, 2005) can be applied).

3. Perform flow simulations on the ensemble of coarse-scale models.

4. Calculate the distance between any two models – the distance is defined as

the difference of the response of interest obtained from the simulations on

the coarse-scale models.

3

5. Map the ensemble of realizations into a metric space using Multi-

dimensional scaling, as well as the historical data.

6. Map the ensemble of realizations into a feature space where we define a

parameterization of the ensemble of models.

7. Solve the post-image problem (see Caers et al., 2009)

8. Reconstruct coarse and fine-scale history matched realizations

Note that once the post-image problem is solved, new realizations of coarse-scale

permeabilities, transmissibilities, porosities, and upscaled well indices can be

constructed without any additional flow simulations. In the above workflow, step 3 is

evidently the most CPU demanding, but these are the only flow simulations required

in the workflow, and they are performed on the coarse-scale models. One very

attractive aspect of this workflow lies in its ability to construct simultaneously the

coarse and fine-scale models, and at the cost of running flow simulations only on the

coarse grids. The combination of the ensemble upscaling and ensemble

parameterization allows a direct modeling at the coarse scale, while still being

consistent with the fine-scale data and geological heterogeneity (if the upscaling and

parameterization are sufficient).

3. Preliminary Results

The method has been applied successfully in cases where the log permeability is

Gaussian. We considered two different permeability distributions as shown in Figure

2. Case 1 involves permeabilities correlated in the x direction (Figure 2a), while in

case 2 the permeability is correlated at 45 degrees from the x-axis (Figure 2b). For

each case, 100 realizations of fine-scale permeabilities (with dimensions of 100 ×

100) are generated. There are 2 wells (1 production and 1 observation well) in the

reservoir (as shown in Figure 2), and oil is produced under primary depletion. The

objective is to history match pressure at the observation well.

The fine-scale models are uniformly coarsened to 20 × 20 (for case 1) and 10 ×

10 (for case 2). We apply extended local permeability (or transmissibility) upscaling

(Durlofsky, 2005) to each of the 100 realizations. Coarse-scale permeabilities are

generated for case 1; and transmissibility upscaling is applied to coarsen the models

in case 2. In addition, a flow-based near-well upscaling technique is applied in both

cases to generate an upscaled well index for the production well for each realization.

In the workflow described in Section 2, flow simulations are performed only on

the coarse-models. On the coarse scale, either permeability or transmissibility, as

well as the upscaled well index are history matched; while the history matched fine-

scale permeability is also constructed at the same time. Here, 10 new history-matched

models are generated in each case, without running any additional flow simulations.

Figure 3 shows an example, i.e., the results obtained for case 2. For this case, when

generating new history matched models, we generate coarse-scale transmissibilities

4

and well indices, but permeabilities on the fine scale. Note that new models at both

the coarse and fine scales are generated, and both coarse and fine-scale simulations

match history (as shown in Figure 3). The results for case 1 are equally well, though

they are not shown here.

4. Future Work

We propose a workflow to construct multiple history matched coarse-scale

models. Distance-based techniques are applied, and the distance is constructed from

the simulations of the response of interest, i.e. from the simulations on the coarse-

scale models. The methodology does not require further flow simulations to find

multiple history-matched models. In addition, the proposed workflow can reconstruct

simultaneously the models on coarse grid and its associated fine grid from the post-

image problem, ensuring consistency between the coarse and fine-scale models.

We are currently working on more complex cases, particularly channelized

realizations where the uspcaling is more challenging as well as the post-image

optimization and reconstruction of the fine-scale models. Also, running flow

simulation on the entire ensemble (typically 100s of models) may not be desirable. In

that case we are working with proxy distances such as Scheidt and Caers (2008)

where models can be selected for flow simulation based on a distance that is fast to

calculate.

References

Caers, J. (2005), Petroleum Geostatistics, Society of Petroleum Engineers.

Caers, J., Scheidt, C. and Park, K. (2009), Modeling Uncertainty in Metric Space,

SCRF report 22.

Chen, Y. and Durlofsky, L. (2008), Ensemble-Level Upscaling for Efficient

Estimation of Fine-Scale Production Statistics, SPEJ 106086, v13, 4, 400-411.

Durlofsky, L. (2005), Upscaling and Gridding of Fine Scale Geological Models

for Flow Simulation, Paper presented at the 8th International Forum on Reservoir

Simulation, Iles Borromees, Stresa, Italy, June 20-24

Scheidt, C., and Caers, J. (2008), Representing Spatial Uncertainty Using

Distances and Kernels. Mathematical Geosciences, DOI:10.1007/s11004-008-9186-

Scheidt C., Park, K. and Caers J. (2009), Defining a Random Function from a

given set of realizations, SCRF report 22.

5

Distance: Difference in coarse - simulations

ϕϕϕϕ

MDS

Feature Space

F

Feature Space

F

Euclidean Space (MDS)

R

Euclidean Space (MDS)

R

Tx* (or Kx*) Ty* (or Ky*) WI*

K-L Exp.

Φαx =)( newϕ

Pre-imageϕϕϕϕ-1

. . .NR

. . .NR

NR

P1

O1

New HM models

Initial models

. .

Fine-scale K. . Tx* (or Kx*) Tx* (or Kx*) WI*

0 200 400 600 800500

1000

1500

2000

2500

3000

3500

Time (days)

Pre

ssure

at 01

Coarse scale Simulations

Figure 1: Proposed workflow to construct an ensemble of history-matched coarse-

scale models.

6

P1

O1

P1

O1

(a) (b)

Figure 2: Example of realization of fine-scale log permeabilities for: (a) case 1,

(b) case 2. P1 represents a production well, and O1 designates an observation well.

Flow simulation on new

coarse-scale trans. and WI

Flow simulation on new fine-scale perms.

Figure 3: Preliminary results (case 2): 10 history-matched models for both

coarse-scale and fine-scale results. The green dots represent the reference production

data, the blue curves designate the pressure at well O1 as a function of the time for all

100 initial models, and red curves designate the pressure at well O1 as a function of

the time for the 10 new history-matched models.