Embed Size (px)
Citation preview
University of Calgary
PRISM: University of Calgary's Digital Repository
Graduate Studies The Vault: Electronic Theses and Dissertations
2019-05-07
Improved Design and Analysis of Diagnostic Fracture
Injection Tests
Zanganeh, Behnam
Zanganeh, B. (2019). Improved Design and Analysis of Diagnostic Fracture Injection Tests
(Unpublished doctoral thesis). University of Calgary, Calgary, AB.
http://hdl.handle.net/1880/110330
doctoral thesis
University of Calgary graduate students retain copyright ownership and moral rights for their
thesis. You may use this material in any way that is permitted by the Copyright Act or through
licensing that has been assigned to the document. For uses that are not allowable under
copyright legislation or licensing, you are required to seek permission.
Downloaded from PRISM: https://prism.ucalgary.ca
UNIVERSITY OF CALGARY
Improved Design and Analysis of Diagnostic Fracture Injection Tests
by
Behnam Zanganeh
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING
CALGARY, ALBERTA
MAY, 2019
© Behnam Zanganeh 2019
ii
Abstract
Diagnostic Fracture Injection Tests (DFITs) have become commonplace in low-permeability
(unconventional) reservoirs to obtain parameters used in hydraulic fracture stimulation design
and reservoir characterization including minimum in-situ stress, initial reservoir pressure and
reservoir permeability.
The current understanding of the parameters that impact successful DFIT design and
analysis is limited. A DFIT exhibits very complex physical behavior, with various mechanisms
active at the same time, including those related to wellbore, fracture, leakoff and reservoir flow.
Therefore, the observed trends in field data are not often predicted using existing analytical
methods, and some common signatures cannot be interpreted. This underscores the need for a
systematic simulation study of DFIT responses where all the active mechanisms are captured
simultaneously. Furthermore, the required shut-in time to acquire reliable DFIT data for
estimation of minimum in-situ stress and reservoir pressure may be excessive, ranging from days
to weeks or months.
In this study, a fit-for-purpose coupled reservoir-geomechanics model is used to simulate
DFITs and generate synthetic pressure responses under various conditions. The validity of the
simulation model is confirmed by comparison to field data. Progressive fracture closure is
presented as an alternative closure mechanism, and the primary pressure derivative (PPD) is
identified as a powerful tool to estimate fracture closure. The effect of wellbore storage, leakoff
rate and dynamic fracture geometry on pressure response is investigated, and their signatures are
identified. These findings are used to explain and analyze field data in major unconventional
plays in western Canada.
iii
In order to accelerate the test and reduce shut-in time, a new DFIT procedure which
combines the injection period with an ultra-low rate flowback is presented. Two successful field
trials of this modified procedure are reported in this work.
Finally, a conceptual method is presented for estimation of reservoir pressure in pump-
in/flowback tests. This method utilizes rate transient analysis techniques to account for variations
in pressure and flowback rate. This method is validated with numerical simulation and a field
trial.
iv
Acknowledgements
I wish to express my gratitude and appreciation to the following people who made a significant
contribution to this dissertation and my academic and professional development:
Dr. Christopher Clarkson for his support and mentorship throughout the completion of this
work. I have enormous appreciation for his unfailing academic and practical support. Without his
commitment in conducting the field trials, completion of this work would not have been possible.
Dr. Jack Jones for his ongoing support, invaluable perspectives and comments. It was truly
an honor to learn from Dr. Jones.
Michael Sullivan for his inspiration, mentorship, encouragement and sharing his broad
knowledge and experience.
Bob Bachman, Dr. Hassan Hassanzadeh, Dr. John Foster and Dr. Rachel Lauer for serving
as members of my advisory and defense committee, and for their comments and constructive
criticism.
Robert Hawkes for his contribution in two of the field trials. Also, Don Bresee, Dr. Mark
McClure , Brett Miles, Ali Esmail, Kirby Nicholson, and Grace Guo for their technical feedback.
Dr. Mason MacKay for his direct contribution to chapter 6 of this thesis.
NSERC, BP, Dassault Systemes Simulia, Seven Generations Energy, Chevron and
ConocoPhillips for their support throughout my studies.
My friends and colleagues in the Tight Oil Consortium and the Department of Chemical
and Petroleum Engineering at the University of Calgary.
Last by not least, my wife and my best friend Atena Vahedian, for her ongoing support of
my life, education and work.
ii
Dedication
To
Atena, Maryam & My Grandparents
iii
Table of Contents
Abstract .................................................................................................................... ii Table of Contents .................................................................................................... iii
List of Tables ........................................................................................................... vi List of Figures ........................................................................................................ vii
Chapter 1: Introduction ....................................................................................................1 1.1 Problem Statement ..................................................................................................1 1.2 DFIT Procedure .......................................................................................................2
1.3 Literature Review ....................................................................................................4 1.3.1 Holistic fracture diagnostics. ..........................................................................7
1.3.1.1 Leakoff mechanisms. ..............................................................................9
1.4 Objectives ...............................................................................................................15
1.5 Organization of Dissertation .................................................................................16 1.6 Nomenclature .........................................................................................................17
Chapter 2: Theory and Methods ....................................................................................18
2.1 Simulation Model ...................................................................................................18 2.1.1 Porous media deformation. ...........................................................................18
2.1.2 Fluid flow in porous media. ..........................................................................19
2.1.3 Cohesive zone model (CZM). ........................................................................19
2.1.3.1 Fracture initiation and propagation. ...................................................20 2.1.4 Fluid flow within the fracture. .....................................................................24
2.1.4.1 Tangential flow. ....................................................................................24 2.1.4.2 Leakoff model. .......................................................................................25
2.2 Analysis Plots ..........................................................................................................26
2.2.1 PTA diagnostic plot. ......................................................................................26 2.3 After Closure Analysis...........................................................................................28
2.3.1 Horner analysis. .............................................................................................29
2.3.2 Nolte’s method for after-closure analysis. ...................................................30 2.4 Nomenclature .........................................................................................................31
Chapter 3: Reinterpretation of Fracture Closure Dynamics During Diagnostic Fracture
Injection Tests .........................................................................................................34 3.1 Introduction ............................................................................................................34 3.2 Model setup and description. ................................................................................34 3.3 Results and Discussion...........................................................................................38
3.3.1 Base Case. .......................................................................................................38 3.3.1.1 Progressive fracture closure. ................................................................40
3.3.2 Model 2. ..........................................................................................................43 3.3.3 Model 3. ..........................................................................................................45 3.3.4 Model 4. ..........................................................................................................45
3.3.5 Field Example 1. ............................................................................................47 3.3.6 Field Example 2. ............................................................................................49
iv
3.3.7 Field Example 3. ............................................................................................50 3.4 Conclusions .............................................................................................................52 3.5 Nomenclature .........................................................................................................54
Chapter 4: Reinterpretation of Flow Patterns During DFITs Based on Dynamic
Fracture Geometry, Leakoff and Afterflow .........................................................55 4.1 Introduction ............................................................................................................55 4.2 Model Setup and Description................................................................................56 4.3 Results and Discussion...........................................................................................59
4.3.1 Model 1. ..........................................................................................................59
4.3.1.1 Zone 1: wellbore storage dominance and fracture expansion. ...........59 4.3.1.2 Zone 2: transition to leakoff dominance with moving hinge-closure (tip
extension)................................................................................................60
4.3.1.3 Zone 3: leakoff dominance. ..................................................................62
4.3.1.4 Zone 4: progressive fracture closure. ...................................................62 4.3.1.5 Zone 5: residual leakoff with residual afterflow. ................................63
4.3.2 Model 2. ..........................................................................................................63
4.3.2.1 Zone 5: residual leakoff without afterflow. .........................................64 4.3.2.2 Zone 6: reservoir flow dominated. ........................................................64
4.3.2.3 Zone 7: reservoir boundary and derivative effects. .............................66 4.3.3 Model 3. ..........................................................................................................67
4.3.4 Field example 1. .............................................................................................69 4.3.5 Field example 2. .............................................................................................70
4.3.6 Field example 3. .............................................................................................71 4.4 Conclusions .............................................................................................................72 4.5 Nomenclature .........................................................................................................74
Chapter 5: A New DFIT Procedure and Analysis Method: An Integrated Field and
Simulation Study .....................................................................................................75
5.1 Introduction ............................................................................................................75
5.2 Procedures and Analysis Methods .......................................................................78 5.2.1 DFIT with ultra-low rate flowback. .............................................................78
5.2.2 Conceptual model for pump-in\flowback tests. ..........................................79 5.3 Results .....................................................................................................................83
5.3.1 Field examples of DFITs with ultra-low rate flowback. ............................83 5.3.1.1 Field example 1 (FE1). .........................................................................84 5.3.1.2 Field example 2 (FE2). .........................................................................87
5.3.2 Simulation results for the conceptual model. ..............................................90 5.3.2.1 Simulation model 1 (SM1). ...................................................................90 5.3.2.2 Blind Test. .............................................................................................92
5.3.3 Field example 3 (FE3): application of the conceptual model to pump-
in/flowback tests. ............................................................................................95
5.4 Discussion ...............................................................................................................98
5.5 Conclusions ...........................................................................................................101
v
Chapter 6: DFIT Analysis in Low Leakoff Formations: A Duvernay Case Study ..102 6.1 Introduction ..........................................................................................................102 6.2 Geological Overview ............................................................................................103
6.2.1 Duvernay Formation. ..................................................................................103
6.2.2 Evidence for episodic fracture growth in the Duvernay. .........................105 6.3 Problems with Application of Conventional DFIT Analysis Methods to the
Duvernay. ............................................................................................................110 6.4 Model Description and Setup..............................................................................113 6.5 Results and Discussion.........................................................................................115
6.5.1 Model 1: tip extension. ................................................................................115 6.5.2 Model 2: pre-existing fractures and tip extension. ...................................117 6.5.3 Field Example 1. ..........................................................................................118
6.5.4 Field Example 2. ..........................................................................................119 6.6 Conclusions ...........................................................................................................121 6.7 Nomenclature .......................................................................................................122
Chapter 7: Conclusions .................................................................................................123
7.1 Contributions and Conclusions ..........................................................................123 7.2 Recommendations for Future Work ..................................................................126
References .......................................................................................................................128
Copyright Permissions...................................................................................................140
vi
List of Tables
Table 1.1 - Identification of flow regimes using derivative slope on log-log plot of pressure
difference versus equivalent time functions (Bachman et al. 2012). .................................... 14
Table 3.1 - Base Case model simulation properties ...................................................................... 37
Table 3.2 - Differences in model setup and simulation settings ................................................... 37
Table 4.1 - Base Case model simulation properties ...................................................................... 58
Table 4.2 - Differences in model setup and simulation settings ................................................... 59
Table 4.3 - Comparison of input reservoir permeability and initial pore pressure in Model 2
with estimated values using radial flow (Horner) analysis ................................................... 66
Table 5.1 - Comparison between the simulation model inputs and analysis results for SM1 ...... 92
Table 5.2 - Comparison between the Blind Test simulation inputs and analysis results .............. 95
Table 5.3 - Comparison between the field examples of this study (DFIT with ultra-low rate
flowback) and previously published conventional (pump-in/shut-in) DFIT data in the
Montney Formation. ........................................................................................................... 100
vii
List of Figures
Figure 1.1 - Typical pressure response during DFIT (Padmakar 2013). ........................................ 4
Figure 1.2 - Flow regimes for a hydraulically-fractured well (Cinco-Ley and Samaniego-V
1981). ...................................................................................................................................... 5
Figure 1.3 - Representation of Carter leakoff model for a dynamic propagating fracture
(Bachman et al. 2012). ............................................................................................................ 6
Figure 1.4 - Combination of diagnostic plots to identify fracture closure during normal
leakoff behavior (Barree et al. 2009). ..................................................................................... 8
Figure 1.5 - Leakoff mechanism signatures on G-function combination plot. ............................. 10
Figure 1.6 – Comparison of fracture closure pick based on Compliance Method (point A) and
Holistic Method. ................................................................................................................... 12
Figure 1.7 - Log-log diagnostic plot for an idealized DFIT (Marongiu-Porcu et al. 2011).......... 13
Figure 2.1 - (a) A traction-separation behavior for normal opening mode (Mode-I) in a
cohesive element; (b) Schematic of one wing of a planar fracture (consisting of cohesive
elements) that demonstrates damage evolution in cohesive elements (Abaqus Analysis
User’s Guide 2016). .............................................................................................................. 23
Figure 2.2 - Plan view schematic of one wing of the fracture showing different components
of fluid flow in the fracture ................................................................................................... 24
Figure 2.3 - A typical PTA diagnostic log-log plot used for identification of fracture closure
and flow regimes in this thesis. ............................................................................................. 26
Figure 2.4 - The calculation of Bourdet-derivative ...................................................................... 28
Figure 2.5 - Radial flow analysis using Horner plot ..................................................................... 29
Figure 2.6 - Radial flow analysis using radial flow time function based on Nolte’s method ....... 31
Figure 3.1 - Plan view schematic of the simulation model setup. ................................................ 36
Figure 3.2 - Base Case model: (a) pressure profile during injection and fracture propagation;
(b) aperture profile during injection and illustration of tip extension during the early
shut-in period. ....................................................................................................................... 39
Figure 3.3 - Base Case model: fracture aperture behavior at x=1 during falloff with (a) time
and (b) pressure. .................................................................................................................... 40
viii
Figure 3.4 - Accumulated leakoff volume and effective stress profile at fracture face along
the fracture. ........................................................................................................................... 41
Figure 3.5 - Progressive fracture closure from the tip of the fracture to the perforation. ............. 41
Figure 3.6 - Base Case model: before-closure analysis using G-function and derivative plots.
Left side figure is zoomed in to illustrate tip closure. ........................................................... 42
Figure 3.7 - Base Case model: before-closure analysis using PTA plot. ...................................... 43
Figure 3.8 - Model 2: fracture aperture behavior during falloff with (a) time and (b) pressure. .. 44
Figure 3.9 - Model 2: before-closure analysis using (a) G-function (b) PTA plot. ...................... 44
Figure 3.10 - Model 3: before-closure analysis using (a) G-function (b) PTA plot. .................... 45
Figure 3.11 - Model 4: before-closure analysis using PTA plots. ................................................ 46
Figure 3.12 - Field Example 1: before-closure analysis using PTA plot. ..................................... 48
Figure 3.13 - Field Example 1: before-closure analysis using the G-function plot. ..................... 49
Figure 3.14 - Field Example 2: before-closure analysis using PTA plot (modified from
Hawkes et al., 2013; Well 058L). ......................................................................................... 50
Figure 3.15 - Field Example 3: (a) effect of ambient temperature variation on derivative plot
(b) removing the effect of ambient temperature using Eq. 3.1 and BCA based on height
recession signature. ............................................................................................................... 51
Figure 3.16 - Field Example 3: before-closure analysis using PTA plots. ................................... 52
Figure 4.1 - Plan view schematic of the simulation model setup. ................................................ 57
Figure 4.2 – (a) PTA plots for Model 1; (b) Total leakoff rate and afterflow during falloff ........ 61
Figure 4.3 - Illustration of (a) fracture expansion during wellbore storage dominance (Zone
1); (b) fracture tip extension during moving hinge-closure (Zone 2) ................................... 62
Figure 4.4 - (a) PTA plots for model 2; (b) Total leakoff rate and afterflow during falloff. ........ 65
Figure 4.5 - PTA plots for Model 3 demonstrating progressive closure when the closed part
of fracture has finite conductivity; b) Total leakoff rate and afterflow during falloff .......... 68
Figure 4.6 - Interpretation of Field Example 1 based on afterflow, leakoff and fracture
dynamics. .............................................................................................................................. 70
ix
Figure 4.7 - Interpretation of Field Example 2 based on afterflow, leakoff and fracture
dynamics. .............................................................................................................................. 71
Figure 4.8 - Interpretation of Field Example 3 based on conductivity of closed fracture
(modified from Houzé et al. 2017)........................................................................................ 72
Figure 5.1 The diagnostic plot of flowing pressure vs. flowback time to identify fracture
closure ................................................................................................................................... 76
Figure 5.2 - Possible flow-regimes during flowback of fracturing fluids from MFHWs in
cross section and plan view of a single fracture.................................................................... 80
Figure 5.3 - Conceptual model for flowback analysis after fracture closure, and the expected
sequence of flow patterns and their characteristic slopes. .................................................... 82
Figure 5.4 - Pressure and rate profile during injection, flowback and early shut-in for FE1. ...... 84
Figure 5.5 - Fracture closure picks for FE1 using G-function combination plots. ....................... 85
Figure 5.6 - PTA diagnostic plots for FE1 including before-closure and after-closure data. ....... 86
Figure 5.7 - Pressure and rate profile during injection, flowback and early shut-in for FE2. ...... 87
Figure 5.8 - G-function combination plots for FE2. ..................................................................... 88
Figure 5.9 - PTA diagnostic plots for FE2 including before-closure and after-closure data. ....... 89
Figure 5.10 - After-closure analysis plots for FE2 using a) Horner time; b) Radial flow time
function ................................................................................................................................. 89
Figure 5.11 - Pressure and rate profile for SM1. .......................................................................... 91
Figure 5.12 - Flowback diagnostic plots for SM1. ....................................................................... 92
Figure 5.13 - Early time pressure and rate profile of the blind experiment. ................................. 94
Figure 5.14 - Flowback diagnostic plots for the Blind Test. ........................................................ 94
Figure 5.15 - Pressure and flowback rate profiles for FE3. The flowback process was
conducted using a choke at wellhead resulting in a variable flowback rate. ........................ 96
Figure 5.16 – Fracture closure identification for FE3. Closure pressure was picked as the
intersection of two straight lines on the pressure curve. ....................................................... 97
Figure 5.17 - Flowback diagnostic plots for FE3. The unit slope trend indicates pseudo-
steady state fluid bank depletion. The start of unit slope trend is selected to estimate the
initial reservoir pressure at 7.88 MPa (wellhead). ................................................................ 98
x
Figure 6.1 - Microseismic clusters showing episodic fracture propagation. .............................. 106
Figure 6.2 - (a) Stereonet representation of natural fracture orientations observed from image
logs in the horizontal leg within the Duvernay Formation. Fracture planes are shown as
great circles while the poles to the fractures are plotted as points. (b) Mohr’s circle
representation of normal and shear stresses resolved onto fractures under the estimated
in-situ stress conditions. A Mohr-Coulomb envelope with no cohesion and 20 degree
friction angle is plotted to show how close to failure the fracture system is. (c) Natural
fracture system within the Duvernay Formation as exposed in outcrop. Fluid alteration
(steel blue) follows the fracture network with some leakoff occurring into the rock
matrix. ................................................................................................................................. 109
Figure 6.3 - Pressure profile during injection for (a) Field Example 1; (b) Field Example 2. ... 111
Figure 6.4 - G-function and PTA diagnostic plots for Field Example 1. ................................... 112
Figure 6.5 - Simplified schematics of (a) Model 1; (b) Model 2. ............................................... 114
Figure 6.6 - (a) PTA diagnostic plots; (b) pressure profile and G dP/dG curves. Tip extension
phases are shown with the dotted squares. .......................................................................... 116
Figure 6.7 - Plan view of a single wing of the fracture showing pressure gradient inside the
fracture and tip extension phases during falloff. ................................................................. 117
Figure 6.8 - (a) Pressure profile during injection and early shut-in time for Model 2; (b)
Propagation as the primary fracture hits and activates a pre-existing fracture ................... 118
Figure 6.9 - PTA plots for Field Example 1 showing the interpretation based on tip extension
cycles. .................................................................................................................................. 119
Figure 6.10 - PTA plots for Field Example 2. ............................................................................ 120
1
Chapter 1: Introduction
1.1 Problem Statement
Horizontal drilling, coupled with multi-stage hydraulic fracturing, has proven to be an effective
solution for producing oil and natural gas at economic rates from ultra-low permeability
shale/tight oil and gas formations. Hydraulic fracturing is a method of well stimulation in which
large volumes of fracturing fluid are injected into rock formations at very high pressures to
fracture the rock and create flow paths for stored hydrocarbons (King, 2012). According to the
U.S. Energy Information Administration (EIA), 6.44 million barrels per day of crude oil, or
about 59% of total U.S. crude oil production in 2018, were produced directly from tight oil
resources. In Canada, tight oil production doubled between 2011 and 2014, from 0.2 to 0.4
million barrels per day, with most production coming from Alberta and Saskatchewan (EIA,
2015).
The overall production performance of multi-fractured horizontal wells (MFHWs) depends
on the fracturing treatment and in-situ properties such as reservoir pressure and permeability.
Conventional well testing methods (such as drawdown-buildup) for estimation of initial reservoir
pressure and permeability are not often successful in ultra-low permeability shale/tight
formations due to the excessive times required to reach a radial flow period. The optimal design
of a hydraulic fracturing treatment itself requires in-depth understanding of formation
geomechanical properties and in-situ stresses, particularly minimum horizontal stress.
The Diagnostic Fracture Injection Test (DFIT), also known as a Minifrac and Fracture
Calibration Test, has become the standard pre-stimulation method for determination of minimum
2
horizontal stress, initial reservoir pressure and effective reservoir permeability in unconventional
reservoirs.
The current understanding of the parameters that impact successful DFIT design and
analysis is limited. As will be discussed in the following sections, several methods are presented
in the literature for analysis of DFIT data. However, due to uncertainties associated with the
dynamics of fracture growth, fracture geometry, leakoff mechanism, fracture closure and after-
closure flow regimes, there is no consensus in the petroleum engineering community on how to
analyze these tests. A DFIT exhibits very complex physical behavior, with various mechanisms
active at the same time, including those related to wellbore, fracture, leakoff and reservoir flow.
Therefore, it is not surprising that the observed trends in field data are not predicted using
existing analytical methods, and some common signatures cannot be interpreted. This
underscores the need for a systematic simulation study of DFIT responses where all the active
mechanisms are captured simultaneously.
Furthermore, the required time to acquire reliable estimates of minimum horizontal stress and
reservoir pressure may be excessive; for some DFITs, this information may require days, weeks
or months to acquire. Therefore, reducing the overall DFIT duration is of significant importance
in development of unconventional reservoirs.
1.2 DFIT Procedure
A DFIT is an injection-falloff test conducted before a hydraulic fracturing treatment. The goal of
a DFIT is to fracture the rock and create a short hydraulic fracture during injection of high-
pressure fluid (usually water), and then record the pressure response during the shut-in period.
The pressure response is analyzed to obtain fracturing treatment parameters such as breakdown
3
pressure, minimum in-situ stress (fracture closure pressure) and leakoff characteristics. If the
falloff period is sufficient, good estimates of formation permeability and reservoir pressure can
also be obtained. These parameters can assist completion and reservoir engineers to optimize the
fracturing treatment and predict future well performance.
A typical DFIT test sequence is shown in Figure 1.1 and is summarized below (Cramer and
Nguyen, 2013):
1. Initially the wellbore is filled with water. Water is injected at surface with a preferably
constant rate, and the wellbore pressure increases until it reaches the breakdown
pressure.
2. At the breakdown point, a hydraulic fracture is created and propagated into the
formation. Water injection continues until the pressure response is stabilized indicating
that fracture propagation has stopped.
3. At the shut-in time, wellhead pressure immediately drops to instantaneous shut-in
pressure (ISIP) due to wellbore and near wellbore friction.
4. During shut-in, water inside the hydraulic fracture leaks off to the surrounding
formation resulting in a pressure drop inside the fracture. Eventually, hydraulic fracture
pressure is not high enough to keep the fracture open, and fracture closure occurs. This
pressure is called fracture closure pressure which is considered to be equivalent to the
minimum principal stress.
5. After fracture closure, pressure falloff continues as the pressure transient penetrates into
the formation which may result in linear and/or radial flow regimes.
Based on the aforementioned sequence, the pressure response during a DFIT is analyzed in
two periods: before-closure and after-closure. The main objective of before-closure analysis
4
(BCA) is identifying fracture closure pressure, which is considered to be equivalent to minimum
in-situ stress. After-closure analysis (ACA) is used to estimate reservoir permeability and initial
reservoir pressure.
Figure 1.1 - Typical pressure response during DFIT (Padmakar 2013).
1.3 Literature Review
Cinco-Ley and Samaniego-V (1981) identified the following four types of flow patterns, in the
presence of a static hydraulic fracture propagated from a wellbore, during production and
injection (Figure 1.2):
• Fracture linear flow: Most of the fluid entering the wellbore is the result of the
expansion of the system within the fracture and the flow pattern is linear (Figure 1.2(a)).
This occurs at very early time and may be masked by wellbore storage effects.
• Bi-linear flow: Two linear flows occur simultaneously in the presence of finite
conductivity fractures including linear flow within the fracture and linear flow in the
formation (Figure 1.2(b)).
5
• Formation linear flow: Occurs only in the case of infinite conductivity fractures where
there is no pressure gradient within the fracture (Figure 1.2(c)).
• Pseudo-radial flow: After a sufficiently long flow period, the fracture acts as an
expanded wellbore, and the drainage pattern can be considered circular (Figure 1.2(d)).
Figure 1.2 - Flow regimes for a hydraulically-fractured well (Cinco-Ley and Samaniego-V
1981).
Howard and Fast (1970) demonstrated experimentally that for dynamic propagation of a
hydraulic fracture during a fracturing treatment one dimensional flow into the formation occurs.
This is known as the Carter leak-off model (Figure 1.3), which dictates that the leakoff rate at a
point along the fracture is a function of the time (τ) at which the fracture reaches that point:
6
( , )( )
Carterl
Cu x t
t x=
−, (1.1)
where, ul is leakoff velocity at point x and time t, CCarter is fluid loss coefficient, t is time elapsed
since start of pumping and τ(x) is the time when fracture is created at point x. The coefficient
CCarter depends on fracturing fluid, filter cake and formation parameters; and it is usually
determined experimentally.
Figure 1.3 - Representation of Carter leakoff model for a dynamic propagating fracture
(Bachman et al. 2012).
The fracturing pressure analysis was pioneered by Nolte (1979) who introduced the G-
function based on material balance and the Carter leakoff model. The G-function is related to
injection and shut-in times as below:
1.5 1.516
( ) [(1 ) 1]3
G
= + − − , (1.2)
where, δ is the dimensionless shut-in time defined as the ratio of shut-in time (∆t) over the
injection time (tp).
7
1.3.1 Holistic fracture diagnostics. Barree and Mukherjee (1996), Barree (1998) and
Barree et al. (2009) recommended a combination of diagnostic plots including G-function,
square-root of shut-in time (√𝑡) and log-log plot of pressure change from ISIP (∆P) versus shut-
in time for identification of fracture closure in DFITs. This combination of plots for the case of
normal leakoff behavior is presented in Figure 1.4. Normal leakoff is observed when the
reservoir system permeability is constant, fracture propagation stops after shut-in, and fracture
surface area contributing to leakoff remains constant during closure (Barree et al. 2009).
According to Barree’s method, on the plot of bottomhole pressure versus the G-function,
fracture closure can be identified as the deviation from the horizontal trend on the derivative
curve (dP/dG) or deviation of the semi-log derivative (GdP/dG) from a straight line passing
through the origin (Figure 1.4(a)). Also, if bottomhole pressure is plotted versus the square-root
of shut-in time (Figure 1.4(b)), the inflection point indicates fracture closure. The inflection point
is found as the point of maximum amplitude of the first derivative (dP/d√𝑡). According to Barree
et al. (2009), a fracture closure point must satisfy both the G-function and √𝑡 requirements.
The log-log plot of pressure change versus shut-in time is also shown in Figure 1.4(c). The
pressure difference curve and its semi-log derivative are parallel immediately before fracture
closure. The separation of these parallel lines indicates fracture closure.
8
Figure 1.4 - Combination of diagnostic plots to identify fracture closure during normal leakoff
behavior (Barree et al. 2009).
9
1.3.1.1 Leakoff mechanisms. Barree et al. (2009) also recommended the use of the G-
function plot to characterize leakoff mechanisms (Figure 1.5). As stated earlier, normal leakoff
(Figure 1.5(a)) during fracture closure is characterized by a constant pressure derivative (dP/dG)
and a straight line trend of semi-log derivative (GdP/dG).
Pressure-dependent leakoff (PDL) indicates the presence and activation of secondary
fractures around the main fracture. PDL causes additional leakoff by providing a larger surface
area exposed to matrix, and it is identified by a hump in the semi-log derivative that lies above
the straight line trend of the normal leakoff (Figure 1.5(b)).
A concave up or belly shape trend on the semi-log derivative (Figure 1.5(c)) indicates
transverse storage or fracture height recession. Transverse storage also indicates the presence of
secondary fractures except that they provide pressure support to the main fracture, rather than
additional leakoff in the case of PDL. Fracture height recession occurs if the fracture propagates
into impermeable layers above or below the target formation. In this case, only the area of the
fracture which is in communication with the permeable target zone contributes to leakoff.
Therefore, the leakoff rate is slower compared to the normal case.
If fracture propagation continues during the shut-in period, a concave down curvature on
the semi-log derivative (Figure 1.5(d)) can be observed. This signature is very similar to PDL,
and it is difficult to distinguish between fracture tip extension and PDL.
10
Figure 1.5 - Leakoff mechanism signatures on G-function combination plot. (a) normal leakoff:
identified by constant pressure derivative and a straight line trend of semi-log derivative; (b)
pressure-dependent leakoff: identified by a hump in the semi-log derivative that lies above the
straight line trend; (c) transverse storage or fracture height recession: identified by a concave up
trend on the semi-log derivative; (d) fracture tip extension: identified by a concave down
curvature on the semi-log derivative (Barree et al. 2009).
Gu et al. (1993) and Nolte et al. (1997) demonstrated the possibility of observing after-
closure reservoir linear and pseudo-radial flow, and their application to the estimation of fracture
geometry and reservoir properties.
11
Van Dam et al. (2002) studied closure mechanisms in elastic and plastic rocks by
conducting experiments on different rock types. They observed that fracture closure happens at
the tip first, and then towards the wellbore. They reported a break on a plot of pressure versus G-
function due to the decrease of fracture compliance at closure, and argued that, instead of a
deviation from a straight line on the pressure derivative (with respect to G-function), the local
minimum should be selected as closure pressure. Further, they stated that plastic deformation can
cause a fracture to remain open in the vicinity of the wellbore.
McClure et al. (2014) and McClure et al. (2016) investigated the effect of fracture
compliance on pressure behavior using a numerical simulator. Their modeling of after-closure
compliance behavior was based on the Barton-Bandis (1985) model. They demonstrated that the
previous “fracture height recession” signature presented by Barree et al. (2009) is the natural
behavior of closure caused by a change in fracture compliance. Based on this method, closure is
recognized as the start of an upward deviation on the 𝐺𝑑𝑃
𝑑𝐺 curve (Figure 1.6). McClure et al.
(2016) argued that previously-presented Holistic Method (Barree et al. 2009) tends to
underestimate the closure pressure.
12
Figure 1.6 – Comparison of fracture closure pick based on Compliance Method (point A) and
Holistic Method (point B; McClure et al. 2014).
Mohamed et al. (2011) and Marongiu-Porcu et al. (2011) presented a model to predict the
falloff pressure trend of an idealized DFIT and identify fracture closure using standard pressure
transient diagnostic and interpretation plots. A log-log diagnostic plot of the basic falloff
response shape predicted by their model is shown in Figure 1.7. The semi-log derivative of the
falloff pressure change with respect to the superposition time shows the following straight line
trends:
• 3/2 slope indicating closure-dominated flow. The fracture closure is identified by a
deviation of the semi-log derivative from the 3/2 slope trend.
• 1/2 slope representing the after-closure formation linear flow
• A horizontal trend at the late time representing radial flow
13
Figure 1.7 - Log-log diagnostic plot for an idealized DFIT (Marongiu-Porcu et al. 2011).
Bachman et al. (2012) presented a workflow based on the pressure transient approach and a
combination of diagnostic plots to identify various flow regimes before and after fracture closure
(Table 1.1). They also recommended the following procedure to pick fracture closure:
• In the presence of Carter leakoff, the end of Carter leakoff flow regime indicates fracture
closure.
• If Carter leakoff is not seen, the end of any linear flow regime indicates fracture closure.
• The test cannot be interpreted if Carter leakoff or linear flow is not observed.
14
Table 1.1 - Identification of flow regimes using derivative slope on log-log plot of pressure
difference versus equivalent time functions (Bachman et al. 2012).
Van Den Hoek (2016) and McClure et al. (2016) demonstrated limitations of superposition
time and Agarwal’s time (1980) for analyzing DFITs where the pumping time is small. They
stated that the semi-log derivative with respect to superposition time and Agarwal’s time starts to
exceed one at shut-in times equal to roughly one-tenth of the pump time and that the 3
2 slope is
not related to fracture closure.
Liu and Ehlig-Economides (2015) presented analytical models to represent before-closure
non-ideal behaviors. Van Den Hoek (2016) presented a PTA approach for modeling pressure
behavior in DFITs and waterflood-induced fractures based on simplified numerical and
analytical solutions. In the latter work, fracture growth rate was a predefined input into the
simulator, the Carter model was used as the leakoff model for the DFIT, and closure was
15
modeled as a gradual decline of fracture compliance representing a combination of “hinge”
(constant length) and “zipper” (length recession) closure.
1.4 Objectives
While the current analytical methods in the literature provide insight into certain parameters,
their validity and accuracy are questionable due to fundamental assumptions being violated. A
DFIT exhibits very complex physical behavior, with various mechanisms active at the same
time, including those related to wellbore, fracture, leakoff and reservoir flow. Therefore, it is not
surprising that the observed trends in field data are not predicted using existing analytical
methods, and some common signatures cannot be interpreted. This underscores the need for a
systematic simulation study of DFIT responses where all the active mechanisms are captured
simultaneously.
The primary objective of this thesis is to improve DFIT design and analysis through
fundamental simulation study of DFIT responses. A coupled reservoir flow-geomechanics model
is used to simulate DFITs and generate synthetic pressure responses under various operational,
reservoir and stress conditions. Once the validity of the simulation model is confirmed, the
following topics will be addressed:
• Explain field observations based on synthetic responses.
• Investigate the applicability and limitations of conventional methods for DFIT analysis.
• Identify consistent signatures for fracture closure.
• Explain non-ideal behaviors and identify their signatures such as tip extension and false
radial flow.
16
• Optimize test design in order to accelerate the test and estimate closure and reservoir
pressure in a short period of time without delaying the development plan.
1.5 Organization of Dissertation
This dissertation consists of seven chapters and follows the paper format. The chapters of this
dissertation have been presented as either journal papers or at conferences. Copyright permission
for re-publication has been acquired from respective publishers. A brief description of each
chapter is provided below.
Chapter 1, the current chapter, introduces the problem, presents a short summary of the
DFIT procedure and literature review, and defines the objectives of this study.
Chapter 2 describes the simulation approach and presents a review of the analysis methods
and diagnostic plots used in this study.
Chapter 3 focuses on modeling DFITs using a coupled reservoir-fracture simulator and on
generating synthetic DFIT responses to explain field observations. Progressive fracture closure is
presented as an alternative closure mechanism. Also, the primary pressure derivative (PPD) is
presented as a powerful tool to identify fracture closure.
Chapter 4 builds on the previous chapter by interpreting the full spectrum of flow patterns
observed during a DFIT. The effect of wellbore storage, leakoff rate and dynamic fracture
geometry on pressure response is investigated, and their PTA signatures are identified.
Chapter 5 presents a new DFIT procedure that accelerates fracture closure and the required
time to observe radial flow regime. Two successful field trials of this modified procedure are
reported in this chapter. Also, a conceptual method is presented for estimation of reservoir
pressure in pump-in/flowback tests that is validated with numerical simulations and a field trial.
17
Chapter 6 explains DFIT responses in the Duvernay Formation. The Duvernay Formation
possesses several properties that may complicate DFIT analysis. Two scenarios are presented to
explain the overall DFIT behavior in the Duvernay. The validity of each scenario is examined
using coupled reservoir-geomechanics simulation.
Chapter 7 is a summary of the overall work; lists the conclusions of this dissertation and
provides a discussion of future work.
1.6 Nomenclature
∆P = Pressure difference between shut-in pressure and pressure at time t, Pa
∆t = Shut-in time, sec
Ccarter = Leakoff coefficient, m.sec-0.5
G = G-function, dimensionless
ISIP = Instantaneous shut-in pressure, Pa
P = Pressure, Pa
t = Elapsed time, sec
tD Dimensionless time, dimensionless
tc = Fracture closure time, sec
teb = Bilinear equivalent time, sec
tec = Carter equivalent time, sec
tel = Linear equivalent time, sec
ter = Radial equivalent time, sec
tp = Pumping time, sec
ul = Leakoff velocity, m/sec
Xf = Hydraulic fracture half-length, m
Greek Variables
∆P = Pressure difference between shut-in pressure and pressure at time t, Pa
∆t = Shut-in time, sec
δ = Dimensionless shut-in time, dimensionless
τ Exposure time, sec
18
Chapter 2: Theory and Methods
In this chapter, key components of the simulation model used in this study are discussed. The
analysis plots and their corresponding calculations are described. Furthermore, the most common
methods for after-closure analysis are presented.
2.1 Simulation Model
In this thesis, a customized fully-coupled reservoir flow and geomechanics simulator (Abaqus
Analysis User’s Guide 2016) is used to generate synthetic DFIT responses. The customized
model is capable of simulating all the physical processes involved in a typical DFIT including:
porous media deformation; fluid flow inside the reservoir; hydraulic fracture initiation,
propagation and closure (based on the Cohesive zone method); compliance change before and
after closure; residual fracture aperture and conductivity; dynamic wellbore storage and
afterflow; fluid flow inside the fracture and fluid interaction between the fracture and the
reservoir (leakoff). The governing equations behind these physical processes are summarized in
what follows.
2.1.1 Porous media deformation. The porous media deformation and pore fluid flow is
governed by the poroelasticity theory (Biot 1955). The constitutive poroelastic equation for an
isotropic rock mass under isothermal conditions is given by (Zielonka et al. 2014):
0
0
22 ( ) ( )
3ij ij sm ij bm kk ijG K G P P − = + − − − , (2.1)
where σij is the total stress tensor (i,j=x,y,z), εij is the total strain tensor, εkk is the summation of
strains, α is Biot’s coefficient, Gsm and Kbm are the dry elastic shear and bulk moduli, δij is the
Kronecker delta function, P is the pore pressure, and the superscript 0 represents the reference
19
configuration. The dry elastic shear and bulk moduli are related to Young’s modulus (E) and
Poisson’s ratio (ν) based on the following equations:
21
sm
EG
v=
+, (2.2)
21 2
bm
EK
v=
−. (2.3)
In Abaqus, total stresses and strains are transformed into Terzaghi effective stresses (σ′)
and effective strains (ε′) based on the following definitions for a fully saturated rock:
ij ij ijP = + , (2.4)
0
1( )
3ij ij ij
bm
P PK
− = − − . (2.5)
2.1.2 Fluid flow in porous media. The constitutive behavior for fluid flow in porous
media combining the continuity equation and Darcy’s law is given as:
21 matrixP kP
M t t
= −
, (2.6)
where kmatrix is the rock permeability, μ is the fluid viscosity, M is the Biot’s modulus and α is the
Biot’s coefficient. The poroelastic constants, M and α, are defined as:
1 1
s bmK K
−= , (2.7)
0 01
f sM K K
−= + , (2.8)
where Ks is the porous medium solid grain bulk modulus, Kf is the pore fluid bulk modulus and
Φ0 is the initial porosity.
2.1.3 Cohesive zone model (CZM). The cohesive zone model (CZM) for modeling
crack propagation was originally proposed by Dugdale (1960) and Barenblatt (1962). Recently,
20
this method has been successfully used for modeling hydraulic fracture initiation and
propagation during fracturing treatment (Yao 2012; Shen and Cullick 2012; Shin and Sharma
2012; Chen 2012; Zielonka et al. 2014; Haddad and Sepehrnoori 2015). Some of the advantages
of the CZM compared to conventional linear elastic fracture modeling are: it avoids a singularity
at the crack tip; the location of the crack tip is not an input and is a direct output of the solution;
and it is capable of modeling fracture tip material softening in quasi-brittle rocks such as shale
(Chen 2012; Haddad and Sepehrnoori 2015). The CZM has been implemented in the Abaqus®
finite element program and validated analytically and experimentally by Zielonka et al. (2014)
and Searless et al. (2016).
2.1.3.1 Fracture initiation and propagation. With the CZM, the fracture is modeled as a
gradual separation between two material (rock) surfaces. This separation is modeled as a
progressive degradation of cohesive strength along the cohesive layer, which is a pre-defined
surface embedded in the rock and follows a traction-separation law (Abaqus Analysis User’s
Guide 2016; Chen 2012). With traction-separation behavior, prior to damage initiation, cohesive
elements exhibit an initial reversible linear elastic response in terms of an elastic constitutive
matrix that relates the nominal stresses to the nominal strains and separations across the cohesive
interface as below:
1n nn ns nt n
s ns ss st s coh coh
coh
t nt st tt t
t K K K
t t K K K K KT
t K K K
= = = =
, (2.9)
where t is the nominal traction stress vector on the cohesive zone interface that consists of three
components in three-dimensional models. The subscripts n, s, and t represent normal, shear 1 and
shear 2 directions, respectively. Kcoh is elasticity matrix of the cohesive layer, is the stain
21
vector, is the separation vector and Tcoh is the original thickness of the cohesive element and
usually taken as unity (Haddad and Sepehrnoori 2015). The off-diagonal terms in the elasticity
matrix are zero for the uncoupled behavior between the normal and shear components. The
nominal strains and corresponding separations are correlated as follows:
nn
cohT
= , (2.10)
ss
cohT
= , (2.11)
tt
cohT
= . (2.12)
Several fracture initiation criteria are present in the literature (Abaqus Analysis User’s
Guide 2016) including maximum nominal stress criterion for fracture initiation that is used in
this study, and can be represented as:
0 0 0
max , , 1n s t
n s t
t t t
t t t
=
, (2.13)
where 𝑡𝑛0, 𝑡𝑠
0, 𝑡𝑡0 represent the peak values of the nominal stress when the deformation is either
purely normal to the cohesive layer interface (Mode-I) or purely in the first or the second shear
direction (Mode-II and Mode-III), respectively. The symbol < > returns the same value when its
argument is positive; and returns zero for negative values of its argument. This symbol is used to
signify that a pure compressive stress state does not initiate a fracture.
After fracture initiation, the deviation of cohesive element from the elastic behavior is
described as below:
22
(1 ) 0
0
n n
n
n n
D t tt
t t
− =
, (2.14)
(1 )s st D t= − , (2.15)
(1 )t tt D t= − , (2.16)
where 𝑡, 𝑡,𝑡 are the stress components predicted by the elastic traction-separation behavior for
the current strains without damage, and D is the scalar damage variable that represents the
overall damage in a cohesive element. The parameter D has an initial value of 0 and increases to
1 during damage evolution.
There are two options to define damage evolution, evolution based on the critical energy
release rate (Gc; also known as the fracture energy or cohesive energy) or evolution based on the
maximum effective displacement at complete failure (δf). The behavior of damage variable (D)
between fracture initiation and complete failure is governed by a softening law. In simulations
of this study, the fracture energy is specified as a cohesive layer property; and an exponential
softening law is used as described below:
0
f
mc
tdD
G G
=
− , (2.17)
where D is the scalar damage variable, t is the traction, δ is the effective displacement, Gc is the
critical fracture energy, G0 is the elastic energy at damage initiation, δ0 is the displacement at
damage initiation and δf is the displacement at complete failure. The fracture toughness or stress
intensity factor is related to critical fracture energy based on the Irwin’s formula (Irwin 1957):
21IC c
EK G
v=
−, (2.18)
23
where KIC is the fracture toughness or stress intensity factor in Pa.m0.5, Gc is the critical fracture
energy in Pa.m, v is Poisson’s ratio and E is Young’s modulus in Pa.
Figure 2.1(a) shows a traction separation behavior for normal opening mode (Mode-I).
Prior to fracture initiation, cohesive elements exhibit a reversible linear elastic response
(characterized by the normal cohesive layer stiffness, Knn) until the normal traction reaches the
maximum tensile strength (tn0) that satisfies the maximum nominal stress criterion for fracture
initiation in Mode-I. After fracture initiation, the cohesive traction evolves from the maximum
strength (tn0) down to zero where the element is fully damaged with the separation of δf. The
damage evolution follows an exponential softening trend, and it is governed by the critical
energy release rate (Gc) that is equal to the area under traction-separation curve. Figure 2.1(b) is
a schematic of fracture propagation in a cohesive layer, demonstrating the fully damaged
cohesive zone (filled with fracturing fluid) and the fracture process zone (damage initiation and
evolution).
Figure 2.1 - (a) A traction-separation behavior for normal opening mode (Mode-I) in a cohesive
element; (b) Schematic of one wing of a planar fracture (consisting of cohesive elements) that
demonstrates damage evolution in cohesive elements (Abaqus Analysis User’s Guide 2016).
24
Different guidelines have been presented for selecting the cohesive element mesh sizes and
stiffness. Turon et al. (2007) suggested the following equation for the cohesive layer stiffness:
cohnn
adjacent
EK
t
= , (2.19)
where Knn is the normal stiffness of the cohesive layer, E is the Young’s modulus of the material
and tadjacent is thickness of the adjacent sub-laminate. They recommended αcoh values much larger
than 1 (αcoh >>1). Zielonka et al. (2014) and Searless (2016) used αcoh =100, and presented close
matches with analytical solutions and experimental results. Haddad and Sepehrnoori (2015)
derived an optimum αcoh value of 60 (assuming tadjacent = 1) by conducting a sensitivity analysis.
2.1.4 Fluid flow within the fracture. Figure 2.2 is a plan view schematic of a wing of a
fracture showing the components of fluid flow in the fracture. There are two components of fluid
flow inside the fracture: tangential flow within the fracture gap (qfrac) and normal flow (leakoff)
from the fracture to the surrounding rock (qleak).
Figure 2.2 - Plan view schematic of one wing of the fracture showing different components of
fluid flow in the fracture
2.1.4.1 Tangential flow. The tangential flow is formulated based on Poiseuille's law:
3
( )12
f
frac
Pwq x
x
= −
, (2.20)
25
where w is the fracture opening, μ is the fluid viscosity and Pf is the fluid pressure along the
fracture length.
2.1.4.2 Leakoff model. The Carter leak-off model (Eq. 1.1; Howard and Fast 1957) has
been used extensively for modeling of hydraulic fracture propagation and in the analytical
solutions for DFIT analysis. It is a one dimensional pressure-independent leak-off model,
applicable to high-viscosity fracturing fluids causing the formation of a low permeability filter
cake on the fracture walls. The Carter leakoff coefficient depends on filter cake created on
fracture walls, and it is not related to formation properties such as reservoir permeability and
pressure.
In this study, a leakoff model based on the solution of the 1-D diffusion equation in a half-
space with an imposed pressure history at the boundary is used (Sarvaramin and Garagash 2015):
0 ( )
4 4( , ) ( , )( , )
( )
t
l l
leak
t x
S SP x t dt P x tq x t
t t t t x
=
− − , (2.21)
where qleak(x,t) is leakoff rate at point x, S is the storage coefficient and αl is the hydraulic
diffusivity. Unlike the Carter model, this model (Sarvaramin and Garagash 2015) is pressure-
dependent, and it is directly related to formation properties such as permeability, porosity and
total compressibility. The final approximation of this model is similar to Howard and Fast
(1957)’s solution for leakoff of low viscosity fracturing fluids, except that the pressure difference
term is not constant, and it is a function of time and location. This leakoff model is coupled with
the Abaqus solver through UFLUIDLEAKOFF user subroutine and a Fortran code (Abaqus User
Subroutines Reference Guide 2016).
26
2.2 Analysis Plots
In this thesis, several pressure transient plots are used for before- and after-closure
analysis. In the following, the time functions and derivative calculations used in the analysis
plots are presented.
2.2.1 PTA diagnostic plot. In pressure transient analysis derivative curves are used to
identify flow regimes and estimate some parameters (e.g. permeability or skin) corresponding to
specific flow regimes. Figure 2.3 shows a typical PTA diagnostic plot used in this thesis for
analysis of falloff data during a DFIT. It is a log-log plot of three derivative curves, including the
Primary Pressure Derivative, PPD, the Bourdet-derivative with respect to Agarwal's time,
𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 and the Bourdet-derivative with respect to shut-in time, ∆𝑡
𝑑𝑃
∆𝑡 plotted against
the shut-in time.
Figure 2.3 - A typical PTA diagnostic log-log plot used for identification of fracture closure and
flow regimes in this thesis.
27
Primary pressure derivative (PPD) was developed by Mattar and Zaoral (1992) to
differentiate between reservoir and wellbore effects in welltest analysis. PPD is the slope of the
pressure-time curve on Cartesian coordinates and it is defined as:
( )
( )
d PPPD
d t
=
, (2.22)
where ΔP is the pressure difference between pressure at the end of pumping and the falloff
pressure; and Δt is the shut-in time. In welltest analysis, the PPD for any reservoir response
should be a constant or decreasing (Mattar and Zaoral 1992). As will be discussed in Chapter 3,
the PPD is identified as a powerful tool to estimate fracture closure.
The Bourdet-derivative (Bourdet et al. 1989) is a method to calculate and smooth the semi-
log derivative of pressure difference (ΔP) with respect to a time function. As demonstrated in
Figure 2.4 to calculate the derivative at point i, one point before and one point after point i are
used with distances of ΔX1 and ΔX2, respectively. Then, the derivative is estimated as follows:
1 22 1
1 2
2 1
i
P PX X
X XDer
X X
+
= +
. (2.23)
In this thesis, ΔX1 and ΔX2 are considered to be equal and are referred to as the “derivative
window”. The derivative window controls the amount of smoothing that is used to reduce noise
in field data.
28
Figure 2.4 - The calculation of Bourdet-derivative (retrieved from Fekete.com)
Agarwal (1980) recommended a time transformation to analyze buildup data using
drawdown methods. The Agarwal time, also known as radial equivalent time, is defined as
follows:
𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 =𝑡𝑝∆𝑡
𝑡𝑝+∆𝑡 , (2.24)
where tp is the injection or pumping time and Δt is the shut-in time during a DFIT.
2.3 After Closure Analysis
The goal of after-closure analysis is to obtain reasonable estimates of reservoir permeability and
initial pressure. This can be achieved if the pressure falloff, after closure, is long enough to
establish reservoir radial flow regime. The signature of radial flow on PTA diagnostic plot is a
horizontal straight line (slope=0) on the Bourdet-derivative with respect to Agarwal's time
(𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙) and a straight line with negative unit slope (slope=-1) on the Bourdet-
derivative with respect to shut-in time, ∆𝑡𝑑𝑃
∆𝑡.
29
2.3.1 Horner analysis. Horner time is a superposition time function based on radial flow
equation that is used to analyze buildup data following a constant rate drawdown period. The
same methodology can be used to analyze radial flow period in a DFIT. Horner time for a DFIT
is defined as:
p
Horner
t tt
t
+ =
, (2.25)
where tp is the pumping time and Δt is the shut-in time.
Once a radial flow signature is observed on PTA diagnostic plot, a plot of falloff pressure
versus Horner time can be used to estimate permeability and reservoir pressure. Figure 2.5
demonstrates an example for radial flow analysis using Horner plot. The radial flow period
appears as a straight line with a slope of mHorner and an intercept of initial reservoir pressure (Pi).
The reservoir permeability can be estimated using mHorner as follows:
162.6Horner
qBk
m h
= , (2.26)
where k is reservoir permeability in md, q is injection rate in bbl/day, B is formation volume
factor in STB/bbl, μ is fluid viscosity in cP, h is fracture height in feet, and mHorner is the slope of
straight line in psi.
Figure 2.5 - Radial flow analysis using Horner plot
30
2.3.2 Nolte’s method for after-closure analysis. The radial flow regime can be
identified on a log-log plot of falloff pressure minus reservoir pressure (P(t)-Pi) versus square of
the linear flow time function (FL2) defined by Nolte et al. (1997) and expanded on by Talley et
al. (1999) and Barree et al.(2009) as follows:
12( , ) sin
cL c
tF t t
t
− =
, (2.27)
where, Δt is shut-in time and tc is the fracture closure time.
A guess for initial reservoir pressure (Pi) is required to calculate pressure difference.
However, the shape of the semi-log derivative (FL2 d∆P/dFL
2) used for flow regime identification
is independent of the pressure guess.
After-closure linear and radial flow regimes exhibit 1/2 and 1 slopes, respectively, on the
semi-log derivative curve. Once radial flow regime is identified, data points in this period can be
used to estimate permeability and initial pressure by defining the radial flow time function (FR)
as:
2)
1 16( , ) ln(1 )
4 (
cR c
c
tF t t
t t = +
−. (2.28)
As shown in Figure 2.6 a Cartesian plot of falloff pressure versus radial flow time function (FR)
yields a straight line with slope equal to mR and intercept of Pi based on the equation below:
( ) ( , )i R R cP t P m F t t− = . (2.29)
The reservoir transmissibility (kh/μ) can be calculated by:
251,000R c
kh Qt
m t= , (2.30)
31
where, Qt is fluid volume injected during the test in bbls, k is permeability in md, h is pay net
thickness in feet, μ is fluid viscosity in cP, tc is in minutes and mR is in psi.
Figure 2.6 - Radial flow analysis using radial flow time function based on Nolte’s method
A few other after-closure analysis methods are published in the literature to estimate
reservoir permeability and initial pressure (Soliman et al., 2005; Craig and Blasingame, 2006).
However, it is still a requirement that the pressure falloff data be recorded long enough to reach
the radial flow period in order to obtain an estimate of reservoir permeability and pressure.
2.4 Nomenclature
D Cohesive layer damage variable, dimensionless
E = Young’s modulus, Pa
FL = Linear flow time function, dimensionless
FR = Radial flow time function, dimensionless
Gc Critical energy release rate, Pa.m
Gsm Dry shear modulus, Pa
ISIP = Instantaneous shut-in pressure, psi
k = Absolute permeability, md
Kbm = Bulk modulus, Pa
32
Kcoh Cohesive layer stiffness matrix, Pa
Kf = Pore fluid bulk modulus, Pa
KIC Fracture toughness, Pa.m0.5
Knn Mode-I cohesive layer stiffness, Pa
Ks = Solid grain bulk modulus, Pa
Kss Mode-II cohesive layer stiffness, Pa
Ktt Mode-III cohesive layer stiffness, Pa
M Biot’s modulus, Pa
P = Pressure, Pa
Pc = Closure pressure, Pa
Pf = Fluid pressure along the fracture length, Pa
Pi = Initial reservoir pressure, Pa
PISIP = Instantaneous shut-in pressure, Pa
qfrac Gap flow rate within the cohesive element, m3/sec
qinj Injection rate into the cohesive element, m3/sec
qleak Leakoff rate from cohesive element, m3/sec
Q = Fluid volume injected during the test, bbls
S Storage coefficient, Pa-1
t Traction vector, Pa
tadjacent Thickness of the adjacent sub-laminate, m
tAgarwal Agarwal time function, sec
tc = Fracture closure time, sec
Tcoh Thickness of the cohesive element, m
teb = Bilinear equivalent time, days
tec = Carter equivalent time, days
tel = Linear equivalent time, days
tHorner = Horner time, dimensionless
tn0 Maximum tensile strength, Pa
tp = Pumping time, sec
w Fracture opening (aperture), m
33
μ = Fluid viscosity, cP
Greek Variables
α Biot’s coefficient, dimensionless
αcoh Coefficient for calculation of cohesive layer stiffness, dimensionless
αl Hydraulic diffusivity, m2/sec
Cohesive layer separation vector, m
δ0 Cohesive layer separation at complete failure, m
δf Cohesive layer separation at complete failure, m
δij Kronecker delta function, dimensionless
∆P = Pressure difference, Pa
Δt Shut-in time, sec
Cohesive layer strain vector, dimensionless
εij Total strain tensor, dimensionless
ε'ij Effective strain tensor, dimensionless
σij = Total stress tensor, Pa
σ'ij = Effective stress tensor, Pa
ν Poisson’s ratio, dimensionless
Φ Porosity, dimensionless
34
Chapter 3: Reinterpretation of Fracture Closure Dynamics During Diagnostic Fracture
Injection Tests1
3.1 Introduction
In this chapter, a fit-for-purpose, fully coupled stress-pore pressure simulation model in
Abaqus® is used to simulate diagnostic fracture injection tests (DFITs) and generate before
closure pressure responses. The cohesive zone method described in Chapter 2 is used to model
fracture initiation, propagation and closure. The pressure-dependent leakoff model presented in
Chapter 2 is coupled with the Abaqus solver through a user subroutine. The model is used to
generate synthetic pressure responses. The simulated responses are used to explain field
observations, and to propose a new concept: progressive fracture closure.
A key finding is that for planar fractures, closure is a transient process, starting from the tip
of the fracture to the vicinity of the wellbore. This process is referred to as “progressive fracture
closure”. Different estimates of closure pressure will be obtained early and late in this process.
Several field cases are presented which exhibit progressive fracture closure. A consistent closure
signature can be identified for these cases using the primary pressure derivative. The common
signature referred to as “fracture height recession/transverse storage” is reinterpreted to be
caused by this phenomenon.
3.2 Model setup and description.
A 2D plane-strain model is used to model hydraulic fracture initiation, propagation and closure
in a DFIT. Figure 3.1 provides a schematic of the simulation model. Minimum and maximum
1 This chapter is a modified version of a paper presented at SPE Western Regional Meeting held in Bakersfield,
California, 23 April 2017 as: Zanganeh, B., Clarkson, C.R., and Hawkes. R.V., 2017. Reinterpretation of Fracture
Closure Dynamics During Diagnostic Fracture Injection Tests. In SPE Western Regional Meeting. Society of
Petroleum Engineers. Copyright approval has been obtained from SPE.
35
horizontal stresses are acting in the Y and X directions, respectively. In the Base Case model,
wellbore storage is neglected, and the fluid is injected directly at the injection point (perforation).
In a more complex scenario, a block representing the wellbore storage is considered. The fluid is
injected into the wellbore block, which is connected to the model using pipe elements. The pipe
elements also model possible pressure drops during the test.
In order to consider the appropriate far-field boundary conditions and model propagation
of the fracture in an infinite medium, the model is surrounded by infinite elements. Infinite
elements are used in cases where the modeled region is small in size compared to the
surrounding medium (formation), and they provide large stiffness values at the boundaries of the
simulation model (Abaqus Analysis User’s Guide 2016). Chen (2012) and Haddad and
Sepehrnoori (2015) have discussed the advantages of using infinite elements in improving the
accuracy and runtime of the simulation. For well-testing applications, we have observed
improvements in the representation of pressure transient behaviors, particularly in after-closure
flow regimes and removal of unrealistic boundary effects, using infinite elements.
36
Figure 3.1 - Plan view schematic of the simulation model setup.
The cohesive elements are embedded in the formation rock, and act as the potential
pathway for hydraulic fracture growth. During the injection of high-pressure fluid, once the
required criteria (as discussed in Chapter 2) are reached, the cohesive elements can break and act
as a hydraulic fracture. During shut-in, the fracture aperture reduces to closure aperture (wclosure),
which is a predefined input into the simulator and it is defined so that the hydraulic fracture
remains conductive after closure. After closure, the aperture can further decrease below the
wclosure, but it will always be a positive value, meaning the fracture retains a positive aperture and
hydraulic conductivity. In low permeability formations, even a few microns of residual aperture
results in an infinite conductive fracture relative to matrix permeability.
Table 3.1 lists properties of the Base Case simulation model. In order to compare
simulation results with field observations, four other models with the same reservoir and rock
37
mechanical properties, but different settings, are considered (Table 3.2). In Model 2, the cohesive
layer stiffness is reduced to observe its effect on fracture compliance and pressure behavior. In
Model 3, the first cohesive element connected to the injection point remains open during shut-in
and closure. In Model 4, wellbore storage is included that is comparable with real field data.
Table 3.1 - Base Case model simulation properties
Input parameter Value
Permeability, md 0.025
Initial porosity, % 10
Young's modulus, GPa 20.7
Poisson's ratio 0.25
Total compressibility, kPa 4.68×10-6
Shmin, MPa 38.6
Shmax, MPa 44.1
Initial pore pressure, MPa 24.8
Fracture fluid viscosity, cP 1
Maximum tensile strength, MPa 1.25
Pumping time, sec 120
Pumping rate (per unit thickness), m3.sec-1 1.5×10-4
Cohesive layer stiffness, GPa.m-1 2070
Table 3.2 - Differences in model setup and simulation settings
Base Case Model 2 Model 3 Model 4
Cohesive layer stiffness, GPa.m-1 2070 20.7 2070 2070
Displacement boundary condition applied
to the first cohesive element No No Yes No
Wellbore storage and pipe elements No No No Yes
38
3.3 Results and Discussion
In the following, synthetic before-closure signatures are generated for the Base Case and Models
2-4 and interpreted for various physical phenomena. Field cases are then analyzed in the context
of the simulation model results to understand the signatures that commonly occur in the field.
3.3.1 Base Case. Figure 3.2(a) illustrates the pressure behavior during injection for the
Base Case simulation model. Because no wellbore is included in the Base Case model, pressure
increases quickly during injection, until it reaches the breakdown point pressure. At this point,
damage initiates (fracture initiation), and then propagates along the cohesive element (fracture
propagation). The smooth pressure behavior during propagation indicates proper mesh size of the
model and cohesive elements. Figure 3.2(b) illustrates the fracture aperture profile during
propagation and early shut-in time. Fracture tip extension occurs for approximately 1 meter after
shut-in.
39
Figure 3.2 - Base Case model: (a) pressure profile during injection and fracture propagation; (b)
aperture profile during injection and illustration of tip extension during the early shut-in period.
Figure 3.3(a) illustrates fracture opening at an element along the length of the fracture
(x=1) during pressure falloff. As the fluid inside the fracture leaks off to the surrounding
formation and pressure inside the fracture decreases, the fracture opening gradually reduces to
the closure aperture. The fracture aperture further reduces after the closure, but will always be a
positive value, meaning the fracture retains a finite aperture and conductivity even after closure.
As demonstrated in Figure 3.3(a), the rate of aperture change with time before closure,
(𝑑𝑤
𝑑𝑡)
𝑏𝑒𝑓𝑜𝑟𝑒−𝑐𝑙𝑜𝑠𝑢𝑟𝑒, is significantly different from after closure behavior, (
𝑑𝑤
𝑑𝑡)
𝑎𝑓𝑡𝑒𝑟−𝑐𝑙𝑜𝑠𝑢𝑟𝑒. The
simulated response is similar to the experimental results of Van Dam et al. (2002). Based on the
continuity equation of fluid flow in the fracture, the change of 𝑑𝑤
𝑑𝑡 at the time of closure causes an
abrupt change in pressure. The same behavior is observed for change of the aperture with
40
pressure (Figure 3.3(b)) in our simulated case. This is in agreement with the concept of
compliance contrast at closure presented previously by Van Dam et al. (2002) and McClure et al.
(2014).
Figure 3.3 - Base Case model: fracture aperture behavior at x=1 during falloff with (a) time and
(b) pressure.
3.3.1.1 Progressive fracture closure. Figure 3.4 illustrates the accumulated leakoff
volume from the fracture face along the fracture length for a typical time during falloff (e.g. 15
minutes). The accumulated leakoff volume is higher at the perforation compared to at the
fracture tip. Due to the low permeability of the formation, this results in a non-uniform pressure
profile and effective stress (on fracture walls) along the fracture length (Figure 3.4). The
effective stress is maximum at the fracture tip (x=11) and gradually decreases along the fracture
(up to x=5) where it remains nearly constant up to the perforation (x=0). Therefore, fracture
closure first occurs at the tip and then moves toward the wellbore, which is referred to herein as
“progressive fracture closure”. Figure 3.5 illustrates the fracture aperture over time during
progressive closure. As expected, closure first occurs at the tip (x=11) moving toward x=5,
where fracture suddenly closes up to perforation (x=0).
41
Figure 3.4 - Accumulated leakoff volume and effective stress profile at fracture face along the
fracture.
Figure 3.5 - Progressive fracture closure from the tip of the fracture to the perforation.
The progressive closure causes an increase in the pressure derivative. Figure 3.6 illustrates
the pressure falloff trend, and its semi-log derivative (𝐺𝑑𝑃
𝑑𝐺), on a G-function plot. Using the
Carter leakoff model, we would expect a linear trend on the 𝐺𝑑𝑃
𝑑𝐺 curve; however, due to the
42
pressure-dependency included in the leakoff model used herein (Eq. 2.21), the 𝐺𝑑𝑃
𝑑𝐺 curve slightly
deviates from the straight line with a concave downward trend. The start of the upward deviation
(derivative spike) on the 𝐺𝑑𝑃
𝑑𝐺 plot is correlated with tip closure, and the end of derivate spike
indicates a fully closed fracture (closure at the wellbore).
Figure 3.6 - Base Case model: before-closure analysis using G-function and derivative plots. Left
side figure is zoomed in to illustrate tip closure.
The PTA plot in Figure 3.7 includes the Primary Pressure Derivative (PPD), Bourdet-
derivative with respect to Agarwal’s time (𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙) and Bourdet-derivative with
respect to shut-in time (∆𝑡𝑑𝑃
∆𝑡). The early time fluctuations on the derivative plots are caused by
fracture tip extension. According to Mattar and Zaoral (1992), the PPD for any type of reservoir
effect or flow regime will always either be a constant or decreasing, and any non-reservoir effect
will appear as an increase in the PPD. The non-reservoir effects in the Base Case model are
fracture tip extension and fracture closure. After the early time tip extension signature
(fluctuations), the start of PPD violation (upward trend) is caused by fracture tip closure. The end
of PPD violation corresponds to a fully closed fracture. Closure pressure may be estimated at the
start and end of the PPD violation as 38.7 MPa and 38.3 MPa, respectively. In real field tests, it
43
is likely that stress conditions near the wellbore are significantly affected by drilling and
completion operations. Therefore, the difference between closure pressure at the perforation and
fracture tip can be larger.
Consistent with previous findings (Van Den Hoek 2016; McClure et al. 2016) it is
observed that 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 starts to deviate from ∆𝑡
𝑑𝑃
∆𝑡 at ∆𝑡 = 0.1 𝑡𝑝, and the closure time is
not long enough for the 3
2 slope to establish.
Figure 3.7 - Base Case model: before-closure analysis using PTA plot.
3.3.2 Model 2. The main difference between this model and the Base Case is that the
𝛼𝑐𝑜ℎ
𝑡𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ratio used in the cohesive layer stiffness calculation (Eq. 2.19) is reduced from 100 to 1.
As shown in Figure 3.8, even after closure, th fracture aperture continues to decrease from the
44
closure aperture to a residual aperture. The difference between (𝑑𝑤
𝑑𝑡)
𝑏𝑒𝑓𝑜𝑟𝑒−𝑐𝑙𝑜𝑠𝑢𝑟𝑒 and
(𝑑𝑤
𝑑𝑡)
𝑎𝑓𝑡𝑒𝑟−𝑐𝑙𝑜𝑠𝑢𝑟𝑒is also less than for the Base Case.
Similar to the Base Case, progressive closure from the tip of the fracture to near the
wellbore causes a clear break in pressure, and a spike in the pressure derivative on the G-
function and PTA plot; however, its magnitude is smaller than for the Base Case model (see
Figure 3.9). Fracture tip closure starts at 39.4 MPa, wellbore closure occurs at 38.4 MPa, and the
progressive closure period lasts for 5 minutes.
Figure 3.8 - Model 2: fracture aperture behavior during falloff with (a) time and (b) pressure.
Figure 3.9 - Model 2: before-closure analysis using (a) G-function (b) PTA plot.
45
Small values of cohesive layer stiffness are not realistic. But even for low cohesive layer
stiffness, a small change in aperture behavior along the length of fracture causes an abrupt
change in pressure behavior. While the progressive fracture closure signature (or previously
known as height recession/transverse storage) is usually observed in field data, a very sharp
increase in derivative for a fully closed fracture is not common. With the next two models, a few
scenarios are presented that can reduce this sudden pressure change.
3.3.3 Model 3. In this model, the first cohesive element connected to the injection point
remains open during closure. This was achieved by applying appropriate boundary conditions to
the first element. Figure 3.10 presents BCA analysis plots for Model 3. The overall pressure
behavior (Figure 3.10(a)) is smooth and similar to field observations. Also, the progressive
closure signature (upward deviation in derivative plots) is smooth (Figure 3.10(b)).
Figure 3.10 - Model 3: before-closure analysis using (a) G-function (b) PTA plot.
3.3.4 Model 4. In this case, wellbore storage is included in the model by addition of a
well block and pipe elements. Figure 3.11 provides pressure transient analysis for this model,
without any smoothing applied to the derivative. As demonstrated by the schematics included in
the figure, at early time, the afterflow caused by wellbore storage increases both the fracture
46
aperture and length, identified by a unit slope on the Bourdet-derivative. The transition from the
unit slope occurs when the fracture walls start to close (the so-called hinge closure suggested by
Van Den Hoek 2016). Due to low leakoff, this can cause additional tip extension during the early
closure (abrupt changes and fluctuations) as fluid is pushed toward the tip of fracture. After this
transition, leakoff becomes the dominant factor affecting the pressure transient and similar flow
regimes to previous models appear.
Figure 3.11 - Model 4: before-closure analysis using PTA plots.
In the presence of wellbore storage, the first fracture element connected to the wellbore can
stay open without applying any boundary conditions (Model 3). This can be explained with
reference to the continuity equation. Initially it was assumed that the qinj term is zero during shut-
in. In the presence of afterflow caused by wellbore storage, the qinj term is no longer zero and can
act to reduce the abrupt change in the 𝑑𝑤
𝑑𝑡 term during closure. Therefore, progressive fracture
47
closure stops at the wellbore without causing an abrupt change in pressure and derivatives (as
with the Base Case model and Model 2). Also, as mentioned earlier, Van Dam et al. (2002)
demonstrated that plastic deformation can cause the fracture to remain open in vicinity of the
wellbore. While plastic deformation is not modeled in this study, it can be another reason why a
very sharp spike in derivative plots is not observed in field data. Nonetheless, the start of
progressive closure (tip closure) can be used to estimate closure pressure and minimum
horizontal stress.
As suggested by Cramer and Nguyen (2013), using downhole shut-in accelerates fracture
closure and time to reach radial flow after closure. These advantages can be explained with
Model 4. In the presence of wellbore storage, fracture length and volume increase significantly
during the early falloff period. With larger fracture length, a much longer time is required for
radial flow to be established in the formation. Further, re-supply of fluid into the fracture due to
afterflow, coupled with a larger volume, delays fracture closure significantly. In addition to the
benefits suggested by Cramer and Nguyen (2013), reducing afterflow by using downhole shut-in
results in a clearer signature of fracture closure (abrupt change in pressure and derivatives).
In many field cases, a unit slope on the Bourdet-derivative is observed. Hawkes et al.
(2013) introduced a composite permeability concept to explain after-closure behavior. Bachman
et al. (2015) expanded this idea and stated this behavior can happen before or after closure. A
simpler explanation for unit slope after closure is the effect of wellbore storage. Depending on
the re-supply rate of fluid from the wellbore, and its magnitude relative to leakoff, a unit slope
after closure can exist.
3.3.5 Field Example 1. The analysis for this dataset is provided in Figure 3.12 and
Figure 3.13. The pressure values are recorded using surface gauges. No smoothing is used for the
48
purpose of comparison with Model 4. The overall trend and signatures are very similar to Model
4 except that the progressive closure period is longer. The early unit slope (Figure 3.12) and
fluctuations indicate fracture volume increase and tip extension due to afterflow. The transition
from a unit slope represents the start of hinge closure. Similar to Model 4, tip extension occurs
during this period as hinge closure pushes the fluid to the tip of the fracture. Progressive fracture
closure starts with PPD violation (or upward deviation on derivative plots) at 22.7 MPa and ends
at 19.5 MPa. The progressive closure period lasts around 5 hours and the difference between
closure pressure estimates is significant. Importantly, the most effective diagnostic for
identifying progressive closure is the PPD.
Figure 3.12 - Field Example 1: before-closure analysis using PTA plot.
49
Figure 3.13 - Field Example 1: before-closure analysis using the G-function plot.
3.3.6 Field Example 2. This well has been analyzed previously by Hawkes et al. (2013;
Well 058L). Again, the overall trend is very similar to Model 4 (Figure 3.14). The upward
deviation at the end of the 3
2 slope was of particular interest in the previous analysis, and no
explanation was provided at the time for this feature.
The early time unit slope indicates fracture expansion and tip extension due to wellbore
storage. At the end of the unit slope, hinge closure starts and is accompanied by tip extension.
After this period (considering that falloff time in this case is large enough), the 3
2 slope appears
on the Bourdet-derivative. Consistent with our modeling results, the upward deviation at the end
of the 3
2 slope, that is consistent with the start of the PPD violation, corresponds to the
progressive fracture closure period. The start of the PPD violation indicates tip closure (52.8
MPa) and the end of PPD violation represents a fully closed fracture (50.9 MPa). The
50
progressive closure period lasts 7.5 hours with the pressure at tip closure interpreted as closure
pressure.
Figure 3.14 - Field Example 2: before-closure analysis using PTA plot (modified from Hawkes
et al., 2013; Well 058L).
3.3.7 Field Example 3. In this test, pumping time was relatively long (19 minutes).
Combined with the low permeability of the formation, this probably resulted in a long fracture
requiring more time to close. The pressure values are recorded using surface gauges and
significant ambient temperature variations between day and night occurred during the falloff
period. As shown in Figure 3.15(a), there are severe fluctuations in the G-function derivative
plot. These functions can result in misinterpretation of the falloff period (e.g. near wellbore
complexities, tip extension, pressure dependent leakoff, etc.); however, when ambient
51
temperature is plotted, it is observed that pressure fluctuations are a direct effect of ambient
temperature variation. To remove the temperature effect from the surface pressure data, the
following equation is used to account for fluid volume change with pressure and temperature:
0( )f
corrected reading
f
P P T TC
= − − , (3.1)
where Preading is the recorded pressure at surface, T is the current ambient temperature, T0 is the
ambient temperature at time of shut-in, βf is the thermal expansion coefficient of wellbore fluid
and Cf is the compressibility of wellbore fluid. In this equation, temperature dependency of βf
and Cf is neglected; inclusion of their temperature dependence can further increase its accuracy.
As demonstrated in Figure 3.15, applying Eq. 3.1 to recorded pressure values reduces the
ambient temperature effect significantly without using a larger derivative window. Closure
pressure is estimated to be 21.5 MPa, based on conventional interpretation of fracture height
recession/transverse storage.
Figure 3.15 - Field Example 3: (a) effect of ambient temperature variation on derivative plot (b)
removing the effect of ambient temperature using Eq. 3.1 and BCA based on height recession
signature.
Figure 3.16 provides before-closure analysis using the PTA plot to illustrate the
progressive closure concept. Although closure time is much longer compared to pumping time, a
52
3
2 slope is not clear in this test. After the tip extension period, the start of the PPD violation at
25.2 MPa represents tip closure. Due to the very low permeability of the formation, and large
fracture created during pumping, the progressive closure period is very long (around 6 days). The
end of the PPD violation at 23.3 MPa indicates a fully closed fracture. The closure pressure
estimate based on the conventional interpretation of height recession (Figure 3.15(b)) is 21.5
MPa.
Figure 3.16 - Field Example 3: before-closure analysis using PTA plots.
3.4 Conclusions
The concept of compliance change at the time of closure is validated in this study. We
demonstrated that even for a soft material with low cohesive layer stiffness (not realistic) this
feature exists.
53
For planar fractures, closure is a transient process, starting from the tip of the fracture to
the vicinity of the wellbore. We refer to this process as “progressive fracture closure”. The
common signature referred to as “fracture height recession/transverse storage” is reinterpreted to
be caused by this phenomenon. Progressive fracture closure explains the increase of Bourdet-
derivative after the 3/2 slope. It can also provide information about the leakoff characteristics
and stress distribution around the fracture from tip of the fracture to vicinity of the perforation.
The primary pressure derivative (PPD) is a powerful tool for identifying progressive
fracture closure. The start of a PPD violation corresponds to tip closure, and the end of the
violation indicates full closure near the wellbore and perforation. Other PTA derivative plots are
useful in the identification of flow regimes before the start of progressive closure and explaining
other PPD violations. For instance, PPD violations happening before closure during the unit
slope on the Bourdet-derivative, and subsequent transition to zero slope during hinge closure, are
caused by fracture expansion and tip extension due to wellbore storage.
In previous studies, the magnitude and duration of the tip extension period has been
underestimated. The use of the cohesive zone model in this study enabled us to capture this
process accurately and the simulations are consistent with field observations. The model results
illustrate that tip extension can occur during the early fracture expansion period due to wellbore
storage and during hinge closure (and can last for several hours).
The afterflow caused by wellbore storage can mask the 𝑑𝑤
𝑑𝑡 change at closure, and its
transition from the tip of the fracture to near the wellbore. Further, it may result in excessive tip
extension and a larger fracture which delays after closure radial flow. Therefore, removing
54
wellbore storage using downhole shut-in not only accelerates closure and radial flow, but also
provides a clear signature of compliance change and progressive closure.
A simple method is presented for removing the effect of ambient temperature change on
pressure values without applying large derivative windows for smoothing. The temperature
effect may be interpreted incorrectly as tip extension or pressure dependent leakoff. Further,
using large derivative windows results in loss of accuracy and removal of signatures such as tip
extension and PPD violation.
3.5 Nomenclature
Field Variables
Cf Wellbore fluid compressibility, Pa-1
G G-function time, dimensionless
P Pressure, Pa
qinj Injection rate into the cohesive element, m3/sec
Shmax Maximum horizontal stress, Pa
Shmin Minimum horizontal stress, Pa
tAgarwal Agarwal’s time, dimensionless
tadjacent Thickness of the adjacent sub-laminate, m
t Elapsed time, sec
tp Pumping time, sec
w Fracture opening (aperture), m
wclosure Closure aperture, m
Greek Variables
αcoh Coefficient for calculation of cohesive layer stiffness, dimensionless
βf Thermal expansion coefficient, 0C-1
Δt Shut-in time, sec
55
Chapter 4: Reinterpretation of Flow Patterns During DFITs Based on Dynamic Fracture
Geometry, Leakoff and Afterflow1
4.1 Introduction
The goal of this chapter is to explain the full spectrum of flow patterns observed before and after
closure during DFITs by considering the dynamic nature of fracture geometry, variable leakoff
rate and afterflow volume caused by wellbore storage.
The cohesive zone model in Abaqus is used to simulate hydraulic fracture propagation and
closure. The customized leakoff model (described in Chapter 2) accounts for variable leakoff rate
as a function of reservoir properties, fracture pressure, fracture surface area and exposure time.
The afterflow is modeled by including a wellbore volume and accounting for wellbore storage.
Results are compared to field data to explain the full spectrum of flow patterns and fracture
dynamics observed in pressure transient analysis of DFITs.
The overall falloff period is interpreted, using PTA diagnostic plots, for relative
magnitudes of afterflow, leakoff rate and fluid flow in the formation. Initially, afterflow is high,
resulting in fracture expansion, which is characterized by a unit slope on the Bourdet-derivative
plot. The afterflow does not necessarily end after the unit slope terminates; the end of fracture
expansion is signaled by a characteristic hump on the derivative plot. During fracture expansion,
the afterflow decreases and the leakoff rate increases due to the larger fracture area. When the
leakoff rate dominates over afterflow, fracture closure mechanics can be conceptualized as a
moving hinge-closure, where the fracture volume reduces, and fracture tip extension occurs as
1 This chapter is a modified version of a paper presented at SPE Hydraulic Fracturing Technology Conference &
Exhibition held in The Woodlands, Texas, 23-25 January 2018 as: Zanganeh, B., Clarkson, C.R., and Jones, J.R.,
2018. Reinterpretation of Flow Patterns During DFITs Based on Dynamic Fracture Geometry, Leakoff and
Afterflow. In SPE Hydraulic Fracturing Technology Conference & Exhibition. Society of Petroleum Engineers.
Copyright approval has been obtained from SPE.
56
fluid is pushed to the tip of fracture (indicated by fluctuations in the derivative). The transition
from afterflow to leakoff dominance and the moving hinge-closure manifest as a semi-horizontal
trend on the Bourdet-derivative. Subsequently, a progressive fracture closure occurs gradually
along the fracture, identified by an increasing trend or a sharp decline on the primary pressure
derivative, depending on the conductivity. Different estimates of closure pressure will be
obtained early and late in this process. The pressure behavior immediately after full closure is
observed to be affected by the residual leakoff and the continuing afterflow. Once all of these
fracture, wellbore and leakoff processes are abated, the reservoir response is observed.
This chapter provides a clear understanding of the different mechanisms affecting pressure
behavior during DFITs for tight reservoirs in order to arrive at more reliable estimates of
fracturing parameters and reservoir properties. As an example, mechanisms leading to false
before- and after-closure radial flow identification are explained.
4.2 Model Setup and Description
Figure 4.1 provides a schematic of the simulation model that is built based on the modeling
approach described in Chapter 2. Minimum and maximum horizontal stresses are acting in the Y
and X directions, respectively. The cohesive elements are embedded in the formation rock - they
are assumed non-existent in the model until fracture initiation and propagation criteria are
reached, at which time these elements act as the potential pathway for hydraulic fracture growth.
57
Figure 4.1 - Plan view schematic of the simulation model setup.
Three different simulation models are presented in this chapter, each focusing on a certain
scenario with different input settings. Table 4.1 lists the main parameters that are constant among
all three models. The main differences of the three models are listed in Table 4.2. Simulation
settings are chosen to represent different possible scenarios observed in field examples.
Model 1 has a very high wellbore storage coefficient, requiring longer pumping time to
pressurize the wellbore. Also, cohesive zone properties (i.e. stiffness, initiation criteria and
evolution law) are selected to favor tip extension. In Model 2 and 3, the wellbore storage
coefficient is reduced, and an exponential softening law for the cohesive zone is chosen to reduce
the possibility of tip extension and compliance contrast during closure. In Model 3, advanced
58
formulated cohesive elements, COD2D4P, are used (Abaqus Analysis User’s Guide 2016). This
type of element support transitions from Poiseuille flow to Darcy flow and the other way around,
as fracture propagates during pumping or closes during falloff. This enables modeling of the
closure process where the closed portion of the fracture has finite conductivity. Each model is
initialized using the Geostatic Stress State step in Abaqus (Abaqus Analysis User’s Guide 2016).
This step is used to verify that the input stress field is in equilibrium with the input properties and
applied boundary conditions, and to iterate, if necessary, to obtain equilibrium. This is the reason
why stress values in Table 4.2 are different for the three models.
Table 4.1 - Base Case model simulation properties
Input parameter Value
Permeability, md 0.025
Young's modulus, GPa 20.7
Poisson's ratio 0.25
Total compressibility, kPa 4.68×10-6
Initial pore pressure, MPa 24.8
Initial porosity, % 10
Fracture fluid viscosity, cP 1
Pumping rate (per unit thickness), m3.sec-1 2×10-4
Maximum tensile strength, MPa 1.25
59
Table 4.2 - Differences in model setup and simulation settings
Model 1 Model 2 Model 3
Wellbore storage Coefficient, m3/MPa 3×10-3 4×10-6 4×10-5
Shmin, MPa 38.6 30.2 30.3
Shmax, MPa 44.1 38.3 38.3
Tip extension Significant present Negligible
Conductivity of closed fracture infinite infinite finite
4.3 Results and Discussion
In the following, synthetic pressure responses are presented for Models 1 to 3 and interpreted for
various physical phenomena. Field cases are then analyzed in the context of the simulated results
to understand the signatures that commonly occur in the field.
4.3.1 Model 1. The main purpose of this model is to demonstrate the effect of variable
afterflow and leakoff rates before and during fracture closure on fracture dynamics and PTA
plots. Figure 4.2(b) presents the plot of afterflow and total leakoff rate from the fracture surface
during the shut-in period. Based on the trend of total leakoff rate, and its relative magnitude with
respect to afterflow rate, the overall falloff period on PTA plots (Figure 4.2(a)) is divided into the
following zones:
4.3.1.1 Zone 1: wellbore storage dominance and fracture expansion. While afterflow is
decreasing during this period, its magnitude is much larger than the total leakoff rate. As a result,
the fracture continues to grow in all directions (aperture, length and height in case of a 3D
model). A conceptual schematic of this phenomenon is illustrated in Figure 4.3(a). Because the
total leakoff is a function of fracture surface area (Eq. 2.21), enhanced surface area due to
60
fracture expansion increases the total leakoff rate during this period. This zone follows a unit
slope trend on PTA plots (Figure 4.2(a)). It must be noted that afterflow does not end at the end
of this unit slope. The characteristic hump at the end of unit slope indicates that fractured
expansion has stopped.
4.3.1.2 Zone 2: transition to leakoff dominance with moving hinge-closure (tip
extension). This period starts when the value of the total leakoff rate exceeds the afterflow rate
(Figure 4.2(b)). At this point, fracture hinge closure (reduction in fracture aperture) starts while
afterflow is still present; the fracture is conductive and there is a process zone at the tip of
fracture. Therefore, as fluid is pushed toward the tip of fracture, it can cause additional tip
extension/fracture growth (Figure 4.3(b)). The abrupt changes and fluctuations on PTA plots
represent this phenomenon (Figure 4.2(a)). Again, total leakoff rate increases as fracture surface
area is extended. This zone can last for 1 to 2 log-cycles, and, depending on its duration, it can
show up as a semi-horizontal trend on Bourdet-derivatives accompanied with fluctuations.
61
Figure 4.2 – (a) PTA plots for Model 1; (b) Total leakoff rate and afterflow during falloff
62
4.3.1.3 Zone 3: leakoff dominance. During this period, the total leakoff rate is larger
than the afterflow rate. Also, it follows a decreasing trend due to reduced fracture pressure and
higher shut-in time (pressure and time terms in Eq. 2.21). The overall trend on Bourdet-
derivatives is smooth, but does not necessarily follow a 3/2 slope due to pressure dependency.
Figure 4.3 - Illustration of (a) fracture expansion during wellbore storage dominance (Zone 1);
(b) fracture tip extension during moving hinge-closure (Zone 2)
4.3.1.4 Zone 4: progressive fracture closure. This concept is illustrated in detail in
Zanganeh et al. (2017). That study noted that, for planar fractures, closure is a transient process,
starting from the tip of the fracture to the vicinity of the wellbore (perforations).
Hydraulic fracture retains a residual aperture after mechanical closure. Depending on the
value of residual aperture and matrix permeability, the closed fracture can either have finite or
infinite conductivity. In Model 1, the closed portion of the fracture has infinite conductivity. As a
result, progressive fracture closure causes an upward deviation on the Bourdet-derivative plots
and PPD curve (PPD violation). The start of the PPD violation corresponds to tip closure, and the
end of the violation indicates full mechanical closure near the injection point. In contrast, for the
finite conductivity closed fracture case, progressive closure causes a downward deviation on
PTA plots. This will be discussed in detail for Model 3.
63
4.3.1.5 Zone 5: residual leakoff with residual afterflow. As mentioned earlier, the closed
fracture retains a residual aperture even after mechanical closure. The fracture pressure is also
slightly elevated above initial formation pressure. Because the pressure dependency of leakoff is
accounted for in the simulation model (Eq. 2.21), the leakoff mechanism is active as long as
fracture pressure is elevated above formation pressure, even after mechanical closure. Further, as
illustrated in Figure 4.2(b), the afterflow continues after closure with its value being comparable
to the residual leakoff (same order of magnitude). During this period, a unit slope trend again
appears on the 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 plot. The derivative plot deviates from the unit slope as residual
afterflow becomes negligible.
4.3.2 Model 2. This model is used to investigate long time pressure behavior (after-
closure) and the corresponding flow patterns and time zones developed during falloff. PTA plots
and the afterflow/leakoff combination plot for Model 2 are shown in Figure 4.4(a) and Figure
4.4(b), respectively.
As observed in Figure 4.4(b), afterflow is much smaller than total leakoff rate from the
beginning of the falloff data, and Zone 1 (as described in Model 1) does not occur with this
model. Moving hinge-closure occurs briefly during Zone 2, causing an abrupt fluctuation on all
derivative plots (Figure 4.4(a)). Except for the duration of tip extension, Bourdet-derivative plots
follow a unit slope trend until 𝛥𝑡 = 6 𝑠𝑒𝑐 = 0.1 𝑡𝑝 , where they deviate from this trend. The 3/2
slope trend appears at the end of leakoff dominance (Zone 3; Figure 4.4(a)), where 𝛥𝑡 > 10 𝑡𝑝.
Progressive closure starts at the tip of fracture (Δtc1=0.36 hr) and causes an upward deviation on
all derivative plots (including PPD violation) as the closed part of the fracture remains infinitely
64
conductive. Progressive closure ends at Δtc2= 0.58 hr. The rest of the falloff period can be
divided into 3 other zones as follows:
4.3.2.1 Zone 5: residual leakoff without afterflow. As illustrated in Figure 4.4(b),
residual leakoff continues until Δt = 104 seconds. Compared to Zone 5 with Model 1, there is no
residual afterflow in this case. Residual leakoff without considerable afterflow appears as a semi-
horizontal trend (m=0) on 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 curve, and negative unit slope trend (m=-1) on the
∆𝑡𝑑𝑃
∆𝑡 plot (Figure 4.4(a)). This period may be interpreted incorrectly as formation radial flow,
and therefore incorrectly used to estimate reservoir permeability and pressure.
4.3.2.2 Zone 6: reservoir flow dominated. During this period, the fracture is finally
closed (static) and leakoff/afterflow are negligible. The falloff period can now be analyzed
similarly to buildup/falloff tests, where the expected sequence is formation linear flow followed
by formation radial flow. With Model 2, formation linear flow is not observed, and the pressure
transient approaches radial flow after 106 seconds (Figure 4.4(a)).
65
Figure 4.4 - (a) PTA plots for model 2; (b) Total leakoff rate and afterflow during falloff.
66
Table 4.3 compares the model input reservoir permeability and initial pressure with values
estimated using conventional Horner analysis for both Zone 5 and Zone 6. Falloff analysis of
Zone 6 (true formation radial flow) provides an accurate estimate of reservoir permeability and
initial pressure as compared to the input data. Contrarily, analysis of Zone 5 (residual leakoff;
false radial flow) results in a significant overestimate of permeability and initial pressure.
Table 4.3 - Comparison of input reservoir permeability and initial pore pressure in
Model 2 with estimated values using radial flow (Horner) analysis
Permeability, md Initial pore pressure, MPa
Model input 0.0250 24.800
Zone 5 0.1400 25.840
Zone 6 0.0255 24.812
4.3.2.3 Zone 7: reservoir boundary and derivative effects. The pressure transient reaches
the boundaries of the simulation model after 2×106 seconds, causing a downward deviation on all
derivative plots (Figure 4.4(a)). At the end of the falloff data, pressure changes become very
small with values being beyond the accuracy of the simulator (or pressure gauges in field data).
This causes fluctuations in all derivative calculations.
In some of the field data (e.g. Field Example 2), where pressure is recorded using surface
gauges, ambient temperature changes can significantly affect pressure recordings and derivative
calculations once pressure changes become very small.
Depending on the size of formation/model and reservoir permeability, Zone 7 can occur
much earlier.
67
4.3.3 Model 3. This model demonstrates the effect of closed fracture conductivity on
pressure behavior during progressive closure. As demonstrated earlier with Model 1 and Model
2, when the closed part of fracture remains infinitely conductive, progressive closure causes a
PPD violation (upward deviation on Bourdet-derivative plots). Model 3 is designed so that, after
closure, fracture permeability is of the same order as matrix permeability (finite conductivity). In
this case, progressive closure causes a sharp decline on the PPD curve (or downward trend on
Bourdet-derivative plots).
As illustrated in Figure 4.5(a) using PTA plots of Model 3 output, progressive closure
starts at the tip of the fracture at Δt = 120 seconds and causes a sharp decline on PPD curve and
downward deviation on the Bourdet-derivatives. The fracture fully closes near the injection point
at Δt = 300 seconds.
In Model 3, after full mechanical closure (Zone 5; Figure 4.5(b)), closed fracture has finite
conductivity and both residual leakoff and residual afterflow are present for almost one log-
cycle. This scenario appears as an unusual trend on 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 curve (almost 7/2 slope),
and unit slope trend (m=-1) on the ∆𝑡𝑑𝑃
∆𝑡 plot (Figure 4.5(a)). This period may be interpreted
incorrectly as Carter leakoff (Nolte flow).
The rest of falloff data is interpreted similarly to Models 1 and 2.
68
Figure 4.5 - PTA plots for Model 3 demonstrating progressive closure when the closed part of
fracture has finite conductivity; b) Total leakoff rate and afterflow during falloff
69
4.3.4 Field example 1. The analysis for this dataset (conducted in the Montney
Formation) is presented in Figure 4.6. A small derivative window of 0.001 is used in derivative
calculations. The overall trend and signatures are very similar to Model 1, except that the
progressive closure period is longer in field data because the simulation model is 2D. All of the
predicted falloff zones are present in this dataset. Fracture expansion and afterflow dominance
ends after 3.5 minutes (Zone 1). Additional tip extension occurs during Zone 2, with severe
fluctuations occurring in the derivatives (moving hinge-closure). Similar to Model 1, this
transition period follows a semi-horizontal trend. Progressive fracture closure (Zone 4) starts
with a PPD violation (or upward deviation on Bourdet-derivative plots) at 22.7 MPa and ends at
19.5 MPa. The progressive closure period lasts around 5 hours and the difference between
closure pressure estimates is significant. Both residual afterflow and residual leakoff are present
after closure (Zone 5). The reservoir response finally appears when the residual afterflow and
leakoff are abated, and formation linear flow appears (m=0.5) at the end of the falloff period.
70
Figure 4.6 - Interpretation of Field Example 1 based on afterflow, leakoff and fracture dynamics.
4.3.5 Field example 2. This DFIT is conducted in the Montney Formation with pressure
values recorded using surface gauges. The interpretation for this dataset is provided in Figure
4.7. The derivative window used for smoothing is 0.001. The overall trend and signatures are
very similar to Model 2. Zones 1 to 6 are clearly present during the falloff period. The early unit
slope in Zone 1 indicates fracture expansion and afterflow dominance. All derivative plots
exhibit significant fluctuations in Zone 2, indicating moving hinge closure/tip extension. The 3/2
slope is not clear during the leakoff dominance period. Progressive closure starts at the PPD
violation (Pc1=17.1MPa and Δt=0.95 hr) and ends at 15.3 MPa (Δt=2.1 hr).
After closure residual leakoff (without afterflow) is clearly observed in this data set,
demonstrated by a semi-horizontal trend on 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 or the negative unit slope on ∆𝑡
𝑑𝑃
∆𝑡
curve. As mentioned earlier, this can be interpreted incorrectly as formation radial flow, which in
71
turn would result in an overestimate of permeability. The reservoir flow dominated period (Zone
6) starts after 100 hours, with formation linear flow (m=0.5) developing at the end of the falloff
data. The late time pressure data are significantly affected by ambient temperature changes
during the day and night resulting in severe fluctuations on derivative plots.
Figure 4.7 - Interpretation of Field Example 2 based on afterflow, leakoff and fracture dynamics.
4.3.6 Field example 3. The interpretation for this dataset is provided in Figure 4.8. The
PTA diagnostic plots are modified from Houzé et al. (2017; p.563). The overall trend and
signatures are very similar to Model 3 demonstrating the effect of closed fracture conductivity on
pressure response.
The PPD curve is not shown for this data set. However, it is expected to follow a similar
trend as Model 3 (Figure 4.5). The fracture expansion and afterflow dominance ends shortly after
shut-in (Zone 1). The transition to leakoff dominance is smooth and no tip extension occurs in
72
Zone 2. The 3/2 slope trend is not observed during the leakoff dominance period (Zone3).
Progressive closure starts at at Δt = 0.6 hr. Since the closed portion of fracture has finite
conductivity, it causes a sharp decline and downward deviation on the derivative plots (Zone 4).
The behavior after fracture closure (Zone 5) is affected by residual leakoff and finite
conductivity of the closed fracture.
Figure 4.8 - Interpretation of Field Example 3 based on conductivity of closed fracture (modified
from Houzé et al. 2017).
4.4 Conclusions
In this Chapter, for the first time, the full spectrum of flow patterns and signatures observed
before and after closure during DFITs are explained by considering the dynamic nature of
73
fracture geometry, variable leakoff rate and afterflow. The overall falloff period is divided into 7
zones that can be summarized as follows:
1. Zone 1: Fracture expansion occurs due to wellbore storage dominance. Total leakoff rate
follows an increasing trend.
2. Zone 2: Early transition from wellbore storage to leakoff dominance can be
conceptualized as a moving hinge-closure causing tip extension. Again, total leakoff rate
follows an increasing trend. This zone may result in a semi-horizontal trend on Bourdet-
derivatives (false before-closure signature of radial flow)
3. Zone 3: Total leakoff rate starts to decrease as fracture pressure is reduced and shut-in
time increases.
4. Zone 4: Mechanical fracture closure starts at the tip of fracture, moving towards the
vicinity of the injection point (progressive closure). Depending on the conductivity of
the closed part of fracture, this causes an upward deviation (PPD violation) or downward
deviation (sharp decline in PPD) on Bourdet-derivatives.
5. Zone 5: It is possible for leakoff and afterflow to continue after closure (residual
afterflow and leakoff). Depending on the conductivity of the closed fracture, this period
can be misinterpreted as Carter leakoff or reservoir radial flow causing an overestimate
of permeability and initial pressure (false after-closure signature of radial flow).
6. Zone 6: Reservoir dominated behavior is observed when the fracture is static, and
leakoff and afterflow are negligible compared to flow inside the reservoir.
7. Zone 7: Boundary dominated effects may be observed if the boundaries of the reservoir
are reached. Also, as pressure changes become very small, derivative calculations are
affected.
74
4.5 Nomenclature
G G-function time, dimensionless
m Slope of straight line, dimensionless
P Pressure, Pa
Pc1 Closure pressure at the start of progressive closure, Pa
Pc2 Closure pressure at the end of progressive closure, Pa
qleak Leakoff rate, m3/sec
S Storage coefficient, Pa-1
Shmax Maximum horizontal stress, Pa
Shmin Minimum horizontal stress, Pa
tAgarwal Agarwal’s time, dimensionless
tp Pumping time, sec
Greek Variables
Δt Shut-in time, sec
75
Chapter 5: A New DFIT Procedure and Analysis Method: An Integrated Field and
Simulation Study1
5.1 Introduction
In ideal scenarios, a significant amount of information can be obtained from DFITs to aid
in reservoir characterization and hydraulic fracture stimulation design efforts. However, the
required time to obtain reliable estimates of key reservoir properties, in particular reservoir
pressure and permeability, is significant and can range from a few days to weeks or even months.
Modified DFITs in the form of a pump-in/flowback test have been demonstrated in the
literature to accelerate the fracture closure process. Varela and Maniere (2016) provided a
detailed history of pump-in/flowback test and its evolution during the past few decades. The
procedure and analysis method was pioneered by Nolte (1979), Nolte and Smith (1981) and
Smith (1985). The procedure consists of injecting fracturing fluid at sufficient pressure to create
a fracture in the formation, followed by flowing back the well at a constant rate through a surface
choke. The proposed flowback rate was 25% of the injection rate. Those authors suggested a
Cartesian plot of flowing pressure vs. flowback time (Figure 5.1) to analyze the data. They
identified fracture closure by a characteristic reversal of curvature, and picked the closure
pressure at the end of the first straight line trend on pressure curve (point A in Figure 5.1).
Shlyapobersky et al. (1988) proposed picking closure at the start of second slope (point C
in Figure 5.1). Soliman and Daneshy (1991) determined fracture closure pressure and fracture
volume based on mass balance and compressibility equations. Those authors suggested that,
1 This chapter is a modified version of a paper published in the Journal of Natural Gas Science and Engineering as:
Zanganeh, B., Clarkson, C.R., Hawkes R.R., and Jones J.R., 2019. A New DFIT Procedure and Analysis Method:
An integrated field and simulation study. Journal of Natural Gas Science and Engineering, 63, 10-17. Copyright
approval has been obtained from the journal.
76
because fracture volume changes during the flowback period as the fracture closes, well testing
techniques (pressure transient analysis) are not applicable. They identified a gradual closure
process from point A to C in Figure 5.1 with point C being the lower bound of closure pressure.
Plahn et al. (1997) conducted numerical simulation and recommended using the pressure value at
the intersection of two slopes (point B) as the closure pressure.
Figure 5.1 The diagnostic plot of flowing pressure vs. flowback time to identify fracture closure
(Savitski and Dudley 2011). The points A, B and C represent closure pressure picks based on
different methods.
Savitski and Dudley (2011) discussed the possibility of choked flow (due to restrictions
near the wellbore) with continuous flowback in shallow formations with smaller wellbore
storage. To address this issue, they proposed reducing the inflow rate from the fracture either by
increasing the wellbore compressibility or by reducing the flowback rate. However, they did not
77
observe significant improvements using these test modifications. Therefore, they introduced a
new procedure where, instead of continuous flowback, flowback was conducted in small
increments of volume.
While the aforementioned pump-in/flowback techniques provide reliable results for
closure pressure and insight into fracturing design, the after-closure data and analysis is ignored.
To the authors’ knowledge, many of the current DFITs in North America are conducted with the
ultimate goal of obtaining reservoir pressure. Estimation of reservoir pressure relies on after-
closure data and the use of pressure transient analysis (PTA).
In the analysis of conventional DFITs (pump-in/shut-in), Zanganeh et al. (2018)
demonstrated the significant effect of wellbore storage and residual leakoff in DFITs both
before- and after-closure. The afterflow caused by wellbore storage, especially in deeper
completions with larger wellbore volumes, delays fracture closure by providing additional
pressure support. This in turn may cause fracture expansion or tip extension, and affect pressure
behavior and PTA signatures. Those authors also showed that residual leakoff from the
mechanically closed fracture results in a false radial flow signature, causing overestimation of
reservoir pressure and permeability. These undesirable effects can be avoided using a downhole
shut-in system. However, down-hole shut-in is expensive and operationally challenging.
In this chapter, DFITs combined with an ultra-low rate flowback are proposed as an
alternative to conventional pump-in/flowback or pump-in/shut-in tests. The main advantage of
the proposed procedure is accelerating fracture closure process without sacrificing the after-
closure flow regimes and derived parameters. The ultra-low rate flowback procedure can be
considered to be analogous to downhole shut-in. The application of this new DFIT procedure for
estimating closure pressure and reservoir properties is demonstrated with two field examples in
78
the Montney Formation. Using this procedure, fracture closure happens between 1 to 2 hours.
Also, in one of the two field trials, radial flow regime is established in 4 days; and reservoir data
is obtained in a very short period of time. As presented in the discussion section, compared to
conventional DFITs conducted in the same area, both fracture closure and radial flow times are
accelerated significantly using this procedure.
Furthermore, a conceptual method based on flowback analysis is presented for pump-
in/flowback tests (with medium-to-high flowback rates) to estimate reservoir pressure. This
concept is validated with synthetic results of two fit-for-purpose numerical simulators. This
method is applied to a field trial, and reservoir pressure is estimated with continuous flowback
data in only 1 hour. The results of the field trial are in very good agreement with an independent
conventional DFIT in an offset well.
5.2 Procedures and Analysis Methods
5.2.1 DFIT with ultra-low rate flowback. The proposed DFIT procedure consists of an
injection period followed by a highly controlled flowback at surface until fracture closure occurs.
Then, the flowback is stopped, and pressure falloff is monitored. The recommended flowback
rate is less than 0.1% of the injection rate. The primary objective of the ultra-low rate flowback
is to minimize wellbore storage and afterflow from wellbore to the fracture. Compared to
conventional DFITs (pump-in/shut-in), there is a flowback period of few hours between the end
of pumping and shut-in. Therefore, this procedure can be referred to as pump-in/flowback/shut-
in. As will be shown in the Results section, this procedure accelerates the closure process, as
well as the appearance of formation linear or pseudo-radial flow, the latter of which can be
analyzed for reservoir information. Furthermore, with such a small flowback rate at surface, the
79
flowback rate at the sandface is approximately zero, and therefore conventional PTA diagnostics
plots can be used to analyze before- and after-closure data.
5.2.2 Conceptual model for pump-in\flowback tests. Clarkson and Williams-Kovacs
(2013) presented diagnostic tools (Figure 5.2) and analytical procedure for analysis of flowback
fluids immediately after fracturing operations in multi-fractured horizontal wells (MFHW)
completed in tight oil reservoirs. They suggested that the initial production on flowback after the
hydraulic fracture stimulation occurs only in the fracture and consist only of the fracturing fluid.
A short fracture transient flow (Flow Regime 1; FR1) period is followed by fracture boundary
dominated flow (Flow Regime 2; FR2), the latter of which is the dominant flow regime. After a
short period of single-phase (fracture fluid) flow, formation fluid breakthrough occurs, followed
by formation linear flow (Flow Regime 3; FR3).
80
Figure 5.2 - Possible flow-regimes during flowback of fracturing fluids from MFHWs in cross
section and plan view of a single fracture (after Clarkson and Williams-Kovacs 2013). Figures
a) and b) show transient radial (FR1) and boundary-dominated flow (FR2) of fracturing fluid
identified in figure c) with RNP' slopes of 0 and 1, respectively. After the breakthrough of
formation fluids, formation transient linear flow to the fracture happens (figures d and e)
identified with a RNP' slope of ½ (f).
The conceptual model presented herein for analysis of pump-in/flowback tests with
medium rates (1% to 10% of the injection rate) is similar to the model suggested by Clarkson
and Williams-Kovacs. The main difference is that, in flowback operations of MFHWs, there
may be a shut-in period of a few days between the hydraulic fracture treatment and the start of
flowback. This shut-in period provides time for fluid and pressure dissipation in the formation.
However, in a pump-in/flowback test, the flowback process starts shortly after the injection.
Therefore, it is expected that a high-pressure region exists around the fracture, referred to herein
81
as a “fluid bank”. The schematics in Figure 5.3 illustrate the expected sequence of flow patterns
after fracture closure, and during flowback, based on this concept. The flowback diagnostic plot
used to identify flow regimes is a log-log plot of water rate normalized pressure (RNP) and its
derivative (RNP') with respect to material balance time, defined below:
( )
( )
reference wf
w
P P tRNP
q t
−= , (5.1)
( )
( )
p
c
w
W tt
q t= , (5.2)
'ln c
dRNPRNP
d t= , (5.3)
where Pwf flowing bottomhole pressure in MPa, tc is material balance time in days, qw is water
rate in m3/days, Wp is cumulative water production in m3 and Preference is the reference pressure in
MPa that is considered to be the flowing pressure at the start of flowback. If only the after-
closure data are analyzed, as it is the case for simulation models in the next section, fracture
closure pressure is used as the Preference.
The conditions required to observe the sequence of flow regimes in Figure 5.3 are:
• The fracture should remain conductive after closure.
• No significant pressure drop occurs near the wellbore.
82
Figure 5.3 - Conceptual model for flowback analysis after fracture closure, and the expected
sequence of flow patterns and their characteristic slopes. a) Wellbore and closed fracture
depletion; b) Fluid bank depletion; c) Breakthrough of the formation fluid and formation linear
flow
Wellbore and fracture depletion occur as the first flow regime after fracture closure. This
results in a unit slope observed with RNP'. Next, there is a transition period caused by inflow of
the fracturing fluid from the fluid bank. Once the pressure transient reaches the boundary of
fluid bank, the behavior is similar to a pseudo-steady state, appearing as another unit slope on the
RNP' plot. Based on the analysis of several synthetic simulation responses, it is suggested that
83
the flowing pressure at the start of this period is a good estimate of initial reservoir pressure. If
the flowback process continues beyond fluid bank depletion, formation fluid breakthrough
occurs and may appear as a half slope on the RNP' plot if formation linear flow is occurring.
5.3 Results
In the following, two field examples are presented to demonstrate the application and analysis of
the modified DFIT procedure using an ultra-low rate flowback period. Then, simulation results
are generated to validate the conceptual model for analyzing pump-in/flowback tests.
Furthermore, the application of this analysis method in estimating reservoir pressure is
demonstrated with a field example.
5.3.1 Field examples of DFITs with ultra-low rate flowback. The field examples are
conducted in the toe section of two horizontal wells drilled from the same pad in the Montney
Formation. The main goals of the tests were to determine if closure pressure estimates could be
accelerated relative to conventional pump-in/shut-in tests, and with comparable closure pressure
results, and to evaluate the quality of the flowback data when applying a very small (constant)
flow rate. It was also hoped that reservoir information (e.g. reservoir pressure) could be obtained
from the tests. For the second test, an estimate of initial reservoir pressure and transmissibility
was obtained using a short fall-off period after flowback was terminated. In order to reduce near
wellbore complexities and provide connectivity between wellbore and formation, the toe port
was acidized before the test. Water was injected as the fracturing fluid at a rate of 1 m3/min. A
special turbine was used for the flowback process to ensure a constant rate was maintained.
There was a short delay between the shut-in time and the start of flowback, and the flowback rate
was set at the lowest limit of the turbine at 0.3 Liters/min.
84
5.3.1.1 Field example 1 (FE1). Figure 5.4 provides the pressure profile during injection,
flowback and shut-in periods for FE1. The injection period was about 10 minutes, with a total of
10 m3 water injected into the formation after breakdown. Pressure values were recorded with
surface gauges, and converted to bottomhole pressure (BHP) using the hydrostatic pressure
corresponding to a vertical depth at 2930 meters. The flowback process initiated 45 minutes after
the end of pumping, and lasted for 340 minutes.
Figure 5.4 - Pressure and rate profile during injection, flowback and early shut-in for FE1.
Closure pressure picks based on straight line trend (Barree et al. 2009) and compliance
change (McClure et al. 2016) on G-function combination plots are shown in Figure 5.5. The
compliance method results in a closure pressure estimate of 51.3 MPa. In contrast, the deviation
from straight line trend on the GdP/dG curve results in a closure pressure of 47.6 MPa. The
overall data quality is good, and trend and signatures are consistent with conventional DFITs
(pump-in/shut-in).
85
Figure 5.5 - Fracture closure picks for FE1 using G-function combination plots.
PTA diagnostic plots, including the Primary Pressure Derivative, PPD (Mattar and Zaoral,
1992), the Bourdet-derivative with respect to Agarwal's time, 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 (Bourdet et al.
1989; Agarwal 1980) and the Bourdet-derivative with respect to shut-in time, ∆𝑡𝑑𝑃
∆𝑡, are shown in
Figure 5.6. The overall trend is what is expected in this reservoir, and is similar to conventional
DFITs (pump-in/shut-in) conducted in this area. Fracture closure is picked based on the concept
of progressive fracture closure (Chapter 3) using the PPD curve. The start of the PPD violation
(positive slope) indicates tip closure at 50.2 MPa, and the end of the PPD violation represents a
fully closed fracture (mechanical closure) at 47.2 MPa. The 3/2 slope trend, the end of which has
been used by several authors (Mohamed et al. 2011; Marongiu-Porcu et al. 2011) as an
indication of fracture closure, is not observed on derivative plots, and cannot be used to identify
86
fracture closure in this example. The flowback period and shut-in durations after fracture closure
are not long enough to observe formation linear or pseudo-radial flow regimes.
Figure 5.6 - PTA diagnostic plots for FE1 including before-closure and after-closure data.
Importantly, for this example, a full (mechanical) closure pressure estimate was obtained in
just 1.5 hours. As will be discussed later, this is a significant time reduction compared to
conventional (pump-in/shut-in) tests performed in this area.
87
5.3.1.2 Field example 2 (FE2). Figure 5.7 provides the pressure profile during injection,
early shut-in, and flowback periods for FE2. Compared to FE1, injection time was intentionally
increased to 16 minutes in order to create a larger fracture. The pressure falloff was monitored
for about 90 hours after the flowback process. Pressure values were recorded with surface
gauges, and converted to BHP using the hydrostatic pressure corresponding to a vertical depth of
2915 meters.
The G-function combination plots are shown in Figure 5.8. Closure pressure is estimated to
be 50.3 MPa using compliance method, and 47.7 MPa using the deviation from the straight line
trend on the GdP/dG curve.
Figure 5.7 - Pressure and rate profile during injection, flowback and early shut-in for FE2.
88
Figure 5.8 - G-function combination plots for FE2.
The PTA diagnostic plots are provided in Figure 5.9. A closure pick based on deviation
from 3/2 slope results in a closure pressure estimate of 47.9 MPa. In this field example, a PPD
violation is not observed due to the large size of fracture, and possibly a pressure gradient inside
the fracture. In this case, the end of fracture closure is picked using the start of the downward
deviation on the PPD curve at 47.0 MPa. The overall fracture closure time is 130 minutes. Two
log cycles after fracture closure, during the post-flowback shut-in period, a horizontal trend
appears on the derivative curve indicating formation pseudo-radial flow. This flow regime is
analyzed to estimate initial reservoir pressure and reservoir transmissbility, using Horner analysis
and Nolte’s method (Nolte et al. 1997). These results and the corresponding plots are given in
Figure 5.10. Both methods provide consistent values of initial pressure (~40.7 MPa) and
transmissiblity (~ 27 mDm/cP).
89
Figure 5.9 - PTA diagnostic plots for FE2 including before-closure and after-closure data.
Figure 5.10 - After-closure analysis plots for FE2 using a) Horner time; b) Radial flow time
function
90
5.3.2 Simulation results for the conceptual model. A customized 2D fully-coupled
stress-pore pressure simulator is used herein to generate synthetic DFIT/flowback responses,
referred to as ‘Simulation Model 1’ (SM1). The modeling approach is described in detail in
Chapter 2.
In order to further validate the proposed design and the analysis method, synthetic pump-
in/flowback data was provided by a third party using a combined reservoir and hydraulic
fracturing simulator (ResFrac®). The simulation model and approach is described in McClure
and Kang (2018). Compared to the simulation model used in this study, ResFrac® provides 3D
simulation and more rigorous wellbore modeling and operational constraints. The provided data
included pressure and rate history at surface and bottomhole conditions. The formation
properties including stress values and initial pore pressure were unknown until the diagnostic
analysis was completed. Therefore, this model is referred to as the ‘Blind Test’.
5.3.2.1 Simulation model 1 (SM1). Figure 5.11 provides the pressure and rate profile
during injection and flowback for SM1. The injection rate is downscaled for a 2D model at 80
cc/min. Flowback starts immediately after pump-in at 10% of the injection rate. Fracture closure
is selected as the intersection point of two straight lines on the pressure curve (Figure 5.11),
which results in a closure pressure estimate of 33.2 MPa. The value of closure pressure is used as
the reference pressure in calculation of RNP and RNP' (Eq. 5.1). In this model, it is not possible
to switch to a constant pressure constraint during the flowback process. Therefore, the simulation
terminates once BHP reaches a predefined value of 16 MPa.
91
Figure 5.11 - Pressure and rate profile for SM1. Negative and positive rates indicate injection
and flowback, respectively. The flowback rate was set at 10% of the injection rate. Closure
pressure was picked as the intersection of two lines on the pressure curve.
The flowback diagnostic plot for after-closure data of SM1 is shown in Figure 5.12.
Initially, the RNP' curve follows a unit slope indicating wellbore and fracture depletion. Then,
there is a transition to the second unit slope which indicates pseudo-steady state and depletion in
the fluid bank. The start of second unit slope, occurring 37 minutes after flowback, is selected to
provide the initial reservoir pressure (Pi = 25.4 MPa). The RNP' curve deviates from a unit slope
at late time as formation fluid breakthrough occurs. Table 5.1 compares the simulation model
inputs and analysis results for SM1; the derived parameters are in very good agreement with
model input, suggesting the analysis methods to derive closure and reservoir pressure are robust.
The conceptual model provides a very good estimate of initial reservoir pressure using flowback
data in less than 1 hour.
92
Figure 5.12 - Flowback diagnostic plots for SM1. The first unit slope represents wellbore and
fracture depletion. Then, there is a transition to the second unit slope which indicates pseudo-
steady state and depletion in the fluid bank. The start of second unit slope is selected to provide
the initial reservoir pressure at 25.4 MPa.
Table 5.1 - Comparison between the simulation model inputs and analysis results for SM1
Parameter Input Analysis Error %
Minimum in-situ stress, MPa 33.1 33.2 0.3
Initial reservoir pressure, MPa 24.8 25.4 2.4
5.3.2.2 Blind Test. The provided data were converted to SI units for consistency with the
other examples and is given in Figure 5.13. The injection rate was set at 795 Liters/min (5 bpm)
with a desired flowback at constant rate of 79.5 Liters/min (0.5 bmp). The overall flowback time
is 200 hours. Formation fluid breakthrough occurs after 2 hours. It takes a minute for the
constant flowback rate to be established. Once the BHP reaches a certain level after about 15
93
minutes, production constraint changes from constant rate to constant pressure. Fracture closure
is selected as the intersection of two straight lines on pressure curve (Figure 5.13). This results in
closure pressure and time of 56.19 MPa and 10 minutes, respectively. These values are used as
the initial values in the calculation of RNP.
The flowback diagnostic plot for after-closure data is shown in Figure 5.14. Initially, the
RNP' curve follows a unit slope, indicating wellbore and fracture depletion. This is followed by a
sharp transition to the second unit slope corresponding to pseudo-steady state depletion in the
fluid bank. The quality of derivative curves is not very good during the second unit slope
because the data output frequency is not constant over the simulation period, with data output
resolution decreasing after fracture closure. The start of second unit slope is used to select initial
reservoir pressure (Pi = 47.93 MPa), obtained after 27.5 minutes of flowback time. The RNP'
curve deviates from unit slope as formation fluid breakthrough occurs. After breakthrough, the
RNP' curve follows a half slope indicating formation linear flow.
94
Figure 5.13 - Early time pressure and rate profile of the blind experiment. Injection rate was set
at 795 Liters/min. Closure pressure was picked as the intersection of two lines on the pressure
curve.
Figure 5.14 - Flowback diagnostic plots for the Blind Test. The first unit slope represents
wellbore and fracture depletion. Then, there is a transition to the second unit slope which
95
indicates pseudo-steady state and depletion in the fluid bank. The start of second unit slope is
selected to provide the initial reservoir pressure at 47.93 MPa. After breakthrough, formation
linear flow is observed with a half slope.
After conducting the flowback analysis, minimum in-situ stress and initial pressure values
were provided and used to compare against the analysis results provided above (Table 5.2). The
estimated initial reservoir pressure (47.93 MPa) is in very good agreement with the model input
value of 48.26 MPa.
Table 5.2 - Comparison between the Blind Test simulation inputs and analysis results
Parameter Input Analysis Error %
Minimum in-situ stress, MPa 57.23 56.19 1.8
Initial reservoir pressure, MPa 48.26 47.93 0.7
5.3.3 Field example 3 (FE3): application of the conceptual model to pump-
in/flowback tests. This field example is conducted in the toe section of a horizontal well in an
unconventional (low permeability) reservoir in western Canada with approximate true vertical
depth of 2007 meters. The main goal of the field trial was to determine if the reservoir pressure
can be estimated using the conceptual model presented earlier. To validate the analysis results,
an independent conventional DFIT (injection/falloff) was conducted by the operating company
on an offset well at a similar depth. This conventional DFIT resulted in a closure pressure
estimate of 36 MPa, and an initial reservoir pressure value of 27 MPa. The overall test duration
for the conventional DFIT was about 1 month.
Figure 5.15 presents pressure (wellhead) and flowback rate profiles for FE3. Pressure
values were recorded at wellhead with a surface gauge. Water was injected at about 500
96
Liters/min to initiate and propagate a hydraulic fracture. Immediately after the injection, the
flowback process started at wellhead using a choke management system. This resulted in a
variable flowback rate during the flowback process. The flowback process continued for 80
minutes with the average flowback rate of about 25 Liters/min. The frequency of rate
measurement was 5 seconds.
Figure 5.15 - Pressure and flowback rate profiles for FE3. The flowback process was conducted
using a choke at wellhead resulting in a variable flowback rate.
Figure 5.16 demonstrates fracture closure identification using the curvature on pressure
trend. The closure is selected as the intersection of two straight lines on pressure curve. This
results in a closure pressure of 15.6 MPa (wellhead) that occurs 38 minutes after the start of
flowback process.
97
Figure 5.16 – Fracture closure identification for FE3. Closure pressure was picked as the
intersection of two straight lines on the pressure curve.
The flowback diagnostic plot for FE3 is shown in Figure 5.17. The quality of derivative
curves is not very good due to significant variations in the flowback rate and low sampling
frequency of 5 seconds. Nonetheless, similar to the simulated cases, both RNP and RNP'
converge towards a unit slope indicating fluid bank depletion. The start of unit slope trend is
used to select initial reservoir pressure at 7.8 MPa (wellhead) after 61 minutes of the flowback
process.
Assuming a hydrostatic gradient of 10 kPa/m, the estimated bottomhole values of closure
pressure and reservoir pressure are 35.7 MPa and 27.9 MPa, respectively. These values are in
very good agreement with the results of the conventional DFIT on the offset well. The main
advantage is that these estimates are obtained within only 1 hour of the flowback process with
overall test duration (pump-in/flowback) being less than 2 hours. On the other hand, about 1
98
month of pressure falloff data was required to come up with the similar estimates in the
conventional DFIT.
Figure 5.17 - Flowback diagnostic plots for FE3. The unit slope trend indicates pseudo-steady
state fluid bank depletion. The start of unit slope trend is selected to estimate the initial reservoir
pressure at 7.88 MPa (wellhead).
5.4 Discussion
In order to demonstrate the value of the new DFIT procedures used in this study (DFIT with
ultra-low rate flowback), with respect to accelerating closure time and obtaining after-closure
results, a comparison with previously published conventional DFIT test interpretations in the
Montney Formation is provided in Table 5.3. The published data include 2 examples from
Bachman et al. (2012), 4 examples from Hawkes et al. (2013) and 2 examples from Zanganeh et
99
al. (2017). These authors used different methods to select fracture closure (G-function, Bourdet-
derivative, PPD variation) or identify different closure mechanisms (secondary fractures, tip
closure/full closure, etc.). Therefore, regardless of the analysis method, only the final closure
time of primary fracture is quantitatively compared. If the closure time is not clearly mentioned
by the authors, an approximate value based on the provided plots is used.
Overall, it is observed that fracture closure is significantly accelerated with the proposed
method of ultra-low rate flowback. Further, none of the previous tests observed after-closure
formation radial flow during the falloff period, and hence unique pore pressure/reservoir
permeability values could not be obtained. However, FE2 in the current study resulted in a radial
flow signature after only 4 days of falloff time. Hence, in one of the two modified tests
performed herein, reservoir data could be obtained in a very short period of time.
It must be noted that, even though these tests were conducted in the same formation, there
are uncertainties associated with the target area, depth, wellbore configuration, pumping
schedule and the operator. A more meaningful comparison should include performing
conventional pump-in/shut-in tests, and the proposed modified DFIT procedure, in the same
well. For this purpose, the following test sequence is proposed for future work to provide a more
robust comparison:
1) Perform a conventional DFIT (pump-in/shut-in) to estimate fracture closure pressure
and reservoir flow properties. The falloff period should be long enough to observe
formation radial flow.
2) Conduct a DFIT with ultra-low rate flowback at constant rate (less than 1% of the
injection rate). Compare the results (test duration, closure pressure, reservoir
transmissibility and reservoir pressure) with the previous test.
100
3) Perform a pump-in/flowback test with medium to high rates (10% of the injection rate).
Estimate closure pressure, and possibly reservoir pressure, based on the conceptual
model presented in this study, and compare them with previous steps. It is not
necessary to maintain a constant rate in this test because the analysis method accounts
for variation in flowback rate.
Table 5.3 - Comparison between the field examples of this study (DFIT with ultra-low rate
flowback) and previously published conventional (pump-in/shut-in) DFIT data in the Montney
Formation.
Source Name Total falloff time
(days)
Full closure time
(hours)
Radial flow
signature Comments
Bachman et
al. (2012)
Example 3 20 ≈24 No
Example 4 4 8.4 No
Hawkes et al.
(2013)
D58L 20 ≈12 No
O58L 20 ≈19 No
F58L 20 ≈30 No
E58L 20 ≈28 No
Closure
signature was
not clear
Zanganeh et
al. (2017)
Field Example 1 4 ≈18 No
Field Example 3 33 ≈200 No
Pumping time
was long (19
minutes) at
about 10
m3/min
This study
FE1 0.6 1.5 No
FE2 4 2.1 Yes
101
5.5 Conclusions
Two successful field trials of modified DFITs, for which an ultra-low rate flowback period was
implemented after pump-in, are reported in this work. The following advantages of this
procedure were demonstrated with these tests:
• Reduction of wellbore storage effects
• Acceleration of the closure process (and hence reduced times to obtain closure pressure
estimates), and start of formation pseudo-radial flow from which reservoir information
can be obtained
• Applicability of well-testing methods (PTA) for before- and after-closure analysis
Furthermore, a conceptual method is presented for estimation of reservoir pressure in
pump-in/flowback tests with medium to high flowback rate. The concept is validated with
synthetic results of two independent numerical simulators. The analysis method is applied to a
field trial, and reservoir pressure is estimated with continuous flowback data in less than 2 hours.
The analysis results of the field trial (based on the conceptual method) are in very good
agreement with an independent conventional DFIT conducted on an offset well.
102
Chapter 6: DFIT Analysis in Low Leakoff Formations: A Duvernay Case Study1
6.1 Introduction
Diagnostic Fracture Injection Test (DFIT) responses in some shale reservoirs, such as the
Duvernay shale in western Canada, are not consistent with those interpreted through traditional
analysis methods. Indeed, interpretation with traditional techniques may result in significantly
incorrect estimates of closure pressure, pore pressure and formation permeability. The goal of
this chapter is to explain the observed DFIT behaviours for selected Duvernay shale wells in
terms of low leakoff of fracturing fluid to the formation, activation of pre-existing fractures, and
tip extension during the test.
DFIT data in the Duvernay shale are analyzed using pressure transient analysis methods.
Two scenarios are presented to explain the overall falloff behavior; moving-hinge closure with
tip extension, and activation of secondary natural fractures. The validity of each scenario is
examined using rigorous coupled flow-geomechanical simulation, geological information and
geomechanical settings in the Duvernay Formation.
Due to extremely low leakoff, the main mechanism affecting pressure falloff during the
DFIT is pressure dissipation through the primary fracture created during injection. This results in
significant tip extension or activation of secondary fractures. The fluctuations and spikes
observed on G-function or pressure derivative plots are explained in the context of these
scenarios. The leakoff rate varies with the pressure change, and the enhanced fracture surface
1 This chapter is a slightly modified version of a paper presented at SPE Canada Unconventional Resources
Conference held in Calgary, Alberta, 13-14 March 2018 as: Zanganeh, B., MacKay, M.K., Clarkson, C.R., and
Jones , J.R., 2018. DFIT Analysis in Low Leakoff Formations: A Duvernay Case Study. In SPE Canada
Unconventional Resources Conference. Society of Petroleum Engineers. Copyright approval has been obtained from
SPE.
103
area, during tip extension. Therefore, the assumption of Carter leakoff, and the traditional closure
picks based on a straight-line tangent to the semi-log derivative on a G-function plot or 3/2 slope
on Bourdet-derivative plot are not valid. Due to very low matrix permeability and the additional
fracture length created through tip extension, it is unlikely that formation radial flow is
established during the test, compromising the ability to obtain a valid pore pressure or formation
permeability.
In the following, a brief overview of the Duvernay Formation geological and
geomechanical characteristics is provided, with a focus on those characteristics favoring episodic
fracture growth. A hydraulic fracturing treatment example with microseismic observations is
presented to illustrate this concept. The appropriateness of interpreting Duvernay DFITs with the
conventional interpretation methods is then examined. Two simulation models are built to
generate synthetic DFIT responses. Finally, in the Results and Discussion section, field data are
interpreted in the context of simulation results.
6.2 Geological Overview
6.2.1 Duvernay Formation. The Duvernay Formation is a Devonian aged shale
exploited for hydrocarbons within the Western Canadian Sedimentary Basin (WCSB).
Depositional conditions within shales results in marked vertical heterogeneity, which leads to
mechanical complexity (Harris et al. 2011). Within the Duvernay Formation, at least ten
microfacies are identified, each with their own natural fracture fabrics and elastic properties
(Knapp et al. 2017). This vertical mechanical heterogeneity favors the creation of bed-contained,
sub-vertical fracture systems (Cooke and Underwood 2010). Natural fractures within the
Duvernay are observed and inferred over a variety of scales from microfractures (Ghanizadeh et
al. 2015) to larger fractures (Fox 2015) and even fault systems (Chopra et al. 2017).
104
The in-situ stress conditions are interpreted in terms of the orientation and magnitude of
the effective stress. Through basin scale modelling work, the stress regime is thought to be in a
strike slip regime in much of the basin with thrust fault regimes encountered proximal to the
Rocky Mountain deformation zone (Reiter and Heidbach 2014). The regional stress in much of
the sedimentary basin trends with an azimuth of 047 as indicated through borehole breakout
directions and other geologic indicators (Reiter et al. 2014).
The Duvernay Formation has a characteristically high organic content with TOC values
ranging from 0.1 up to 11 percent (Rivard et al. 2014) which suggests significant potential for
hydrocarbon generation. In fact, it is the source rock for most of the hydrocarbons trapped within
the upper Devonian section of the WCSB (Creaney and Allan 1990). Hydrocarbon generation
can raise fluid pressures and may be the underlying cause of significant overpressure conditions
observed within the Duvernay (Fox and Soltanzadeh 2015, Davis and Karlen 2014). The highly
overpressured conditions create a geomechanical sensitivity that brings discontinuities to a state
of incipient failure. This phenomenon is directly observed through reported cases of induced
seismicity arising from treatment operations within the Duvernay Formation (Bao and Eaton
2016; Schultz et al. 2017).
The Duvernay Formation is primarily composed of mudstones and thus is associated with
low matrix permeability in the range of 3.7 × 10−7 to 1.2 mD (Ghanizadeh et al. 2015). This low
permeability suggests that significant leak-off into the matrix is unlikely over the timescale of a
DFIT. Instead, fracture fluid volume may be accommodated through crack tip extension of the
hydraulic fractures as well as accommodation through secondary natural fractures. This process
of rupture of the rock and subsequent flow of fluid into the fracture is theorised to occur in
temporally intermittent periods. Evidence of sporadic spatial-temporal evolution of fluid flow is
105
supported through various case studies of microseismicity in shales. For example, Goertz-
Allman et al. (2017) provide a case study where microseismicity occurs due to punctual
activation of faults which controlled fluid movement in the sub-surface.
6.2.2 Evidence for episodic fracture growth in the Duvernay. Evidence of
temporally-intermittent fracture behaviour is gleaned from microseismic observations during the
main hydraulic treatment of a nearby well within the Duvernay Formation. The microseismic
temporal evolution shows that the hydraulic fracture does not grow in a simple, steady and
continuous way, but rather the fracture expands to new areas of the reservoir over time. This
occurs episodically through growth phases followed by relative calm. In Figure 6.1(b), four
major microseismic clusters are identified during a hydraulic fracturing treatment stage, each
following a growth front envelope. While there is a trend to follow these pressure diffusion
fronts (Shapiro and Dinske 2009), microseismicity may jump ahead of the theoretical curve or
fall behind, indicating that fracture growth is not continuous and steady. Furthermore, multiple
microseismic clusters form within a single treatment stage, suggesting that rock deformation
processes follow an episodic behavior as well. Figure 6.1(b) also illustrates how the hydraulic
fracture accesses different parts of the reservoir throughout the treatment. The first hydraulic
growth cluster produces a large half-length towards the northwest. Subsequently, active
deformation occurs closer to the wellbore before finally moving towards the southeast. Even
though continual pumping occurred, the fracture did not continually grow in length, but rather
accessed different areas of the reservoir at different times.
106
Figure 6.1 - Microseismic clusters showing episodic fracture propagation. (a) Treatment
parameters showing consistent injection (top), yet non-consistent growth phases (bottom); (b)
microseismic events plotted in plan view and colored by time illustrating episodic spatial
temporal growth throughout the reservoir.
This intermittent behavior may be a response to the mechanical layering found within the
Duvernay Formation. Brenner and Gudmundsson (2004) described how hydrofractures become
arrested at mechanical contrasts between layers especially when propagating from a stiff layer
into a soft layer. This arresting behavior would mean that as a fracture grows, it may encounter a
difficult layer to propagate into; thus, a new area of the reservoir will be accessed instead.
Additionally, the aperture of the hydraulic fracture varies depending on the Young’s modulus of
the surrounding rock matrix and the fluid pressure within the fracture. Therefore, the hydraulic
fracture likely has a non-uniform aperture distribution resulting from the complex mechanical
heterogeneity. The heterogeneous mechanical contrast suggests that a hydraulic fracture will
undergo a period of dilation and extension as it extends in length. This transition between
dilation and rupture likely contributes to episodic fracture growth. Furthermore, soft elastic rock
107
will be able to store elastic potential from dilation of the fracture during injection. This
mechanical potential energy can be a driving mechanism for growth after the injection has
ceased if the pressure was not able to leak off due to low matrix permeability. This effect has
been observed through the generation of Krauklis waves, where fracture wall deformation
coupled to fluid flow creates resonances within the hydraulically connected fluid network
(Krauklis 1962). In this case, the dilation and subsequent collapse of fracture aperture is coupled
to fluid movement within the fracture (Liang 2017). The time scale that this process occurs on is
partially dependent on the elasticity of the surrounding rock mass, where low shear modulus
values of the rock mass produces larger time period fluid movement events (Tary et al. 2014).
This is because the rock can accommodate more elastic deformation and thus larger apertures
may be reached before the fracture constricts.
Abundant vertical natural fractures are observed in image logs taken from a horizontal leg
within the Duvernay Formation (Figure 6.2(a)). These natural fractures occur in specific
orientations called sets. The first set is an extension fracture and follows the orientation of
maximum horizontal stress. The second set is a shear fracture and occurs at an angle to the main
set. The third set is a cross joint perpendicular to the maximum horizontal stress. To calculate the
resolved normal and shear stress on the natural fractures, an understanding of the in-situ effective
stress field is required. In this case, we use a vertical stress based on 24 kPa/m and a pressure
gradient of 20 kPa/m located at 3 km depth. We choose the minimum horizontal stress to be 0.8
times the vertical while the maximum horizontal stress is 1.5 times the vertical stress. Using
tensor rotations, the normal and shear stresses are resolved for each fracture set and shown on a
Mohr’s circle (Figure 6.2(b)).
108
As fluid pressure is increased during injection, the normal effective stress in the fracture
system decreases within the fluid front. The extension fracture will accept fluid most easily as it
has the least resolved normal stress; however, when the linking cross joint or shear joint are
encountered, they will transmit fluid from one fracture to another. Before they can transmit the
fluid, there must be a buildup of fluid pressure because they require a higher normal stress to
open. Once the fluid builds up to sufficient level to overcome the normal stress, fluid may be
transmitted to another extension fracture which in turn will release the pressure to a lower state.
Thus, flow through the fracture system itself leads to episodic buildup and release of pressure as
different fracture systems are accessed. Figure 6.2(c) illustrates this effect in a Duvernay
Formation equivalent outcrop exposure at Roche Miette in Jasper National Park, Alberta. The
natural fracture system is comprised of multiple connecting fracture sets in which fluid flow
pathways are established. The low permeability of the matrix concentrates fluid into the fracture
system as observed from the mineralogical alteration front following the natural fracture system.
109
Figure 6.2 - (a) Stereonet representation of natural fracture orientations observed from image
logs in the horizontal leg within the Duvernay Formation. Fracture planes are shown as great
circles while the poles to the fractures are plotted as points. (b) Mohr’s circle representation of
normal and shear stresses resolved onto fractures under the estimated in-situ stress conditions. A
Mohr-Coulomb envelope with no cohesion and 20 degree friction angle is plotted to show how
close to failure the fracture system is. (c) Natural fracture system within the Duvernay Formation
as exposed in outcrop. Fluid alteration (steel blue) follows the fracture network with some
leakoff occurring into the rock matrix.
110
From the above summary, it is concluded that the Duvernay possesses several properties
that may complicate DFIT analysis: low matrix permeability, limiting leakoff from fractures; an
extensive natural fracture system; and fine-scale mechanical layering, the latter two properties
increasing the propensity for intermittent fracture growth that could affect the DFIT signature.
Direct observations of this behavior are illustrated through episodic spatial-temporal clustering
behavior of microseismicity during a hydraulic fracture treatment.
6.3 Problems with Application of Conventional DFIT Analysis Methods to the Duvernay.
Some DFIT responses in shale reservoirs, such as the Duvernay shale, are not consistent with
those interpreted through traditional analysis methods. Figure 6.3 provides pressure profiles
during injection for two DFITs conducted in the Duvernay Formation. There is no clear
breakdown, or a sharp pressure drop after the breakdown, as expected in more conventional
reservoirs. In fact, in some cases there is an increasing pressure trend after the breakdown even
though the injection rate is constant or decreasing (see Field Example 2, Figure 6.3(b)). Another
common signature observed in Duvernay DFITs is a large pressure drop at the time of shut-in,
e.g. 20 MPa and 25 MPa for Field Example 1 and 2, respectively. This significant pressure drop
is likely caused by another mechanism other than friction in the wellbore or tortuosity.
111
Figure 6.3 - Pressure profile during injection for (a) Field Example 1; (b) Field Example 2.
PTA and G-function diagnostic plots for Field Example 1 are provided in Figure 6.4. The
trend of 𝐺𝑑𝑃
𝑑𝐺 curve is not similar to any of the signatures proposed by Barree et al. (2009).
Except for the early time unit slope, no other straight line trend with a specific slope is observed
on any of the derivative plots. Based on the results of Chapter 4, the long transition after unit
slope, with no sharp characteristic hump, indicates that leakoff from fracture to surrounding
formation is not dominant. This is probably due to very low matrix permeability of the formation
coupled with small fracture surface area of the created fracture. Several fluctuations are observed
on all of the derivative plots, each lasting for a considerable period of time. In this case, there is
no signature of fracture closure based on any of the identification methods.
112
Figure 6.4 - G-function and PTA diagnostic plots for Field Example 1.
Based on the geological information, geomechanical settings and evidence for episodic
hydraulic fracture growth in the Duvernay Formation provided in the previous section, combined
with the anomalous DFIT observations just illustrated, two hypotheses are presented to explain
the DFIT behavior in the Duvernay formation; moving-hinge closure with tip extension, and
propagation through activation of secondary fractures. As will be discussed below, model-
generated synthetic results are compared with field data, and necessary modifications are made
113
to calibrate the model. The observations from the calibrated model are then used to explain field
behaviors.
6.4 Model Description and Setup.
A customized fully-coupled stress-pore pressure simulator (Abaqus Analysis User’s Guide
2016) is used herein to generate synthetic DFIT responses. The modeling procedure is described
in detail in Chapter 2 of this thesis and Zanganeh et al. (2017) and Zanganeh et al. (2018). The
cohesive zone method (CZM) is used to model hydraulic fracture initiation, propagation and
closure. With the CZM, the fracture is modeled as a gradual separation between two material
(rock) surfaces. This separation is modeled as a progressive degradation of cohesive strength
along the cohesive layer, which is a pre-defined surface embedded in the rock and follows a
traction-separation law. Model 1 assumes propagation of a single planar primary fracture (HF;
Figure 6.5(a)). However, in Model 2, there is a pre-existing network of natural fractures (NF;
Figure 6.5(b)), similar to the system observed in the Duvernay outcrop (Figure 6.2(c)). The
cohesive elements are embedded in the formation rock - they are assumed to be non-existent in
the model until fracture initiation and propagation criteria are reached, at which time these
elements act as the potential pathway for fracture growth. In both simulation models, a matrix
permeability of 25 nd was used.
114
Figure 6.5 - Simplified schematics of (a) Model 1; (b) Model 2.
In the previous simulation models of Chapter 3 and Chapter 4, tangential fluid flow inside
the fracture is calculated using Poiseuille's law:
3
12
f
frac
Pwq
x
= −
,
(6.1)
115
where w is the fracture opening, μ is the fluid viscosity and Pf is the fluid pressure along the
fracture length (x direction). For an injection fluid with low viscosity (i.e. water), even a small
fracture aperture of 0.1 mm, results in a fracture with large tangential permeability (conductivity)
without any significant pressure gradient inside the fracture. To achieve synthetic pressure
responses similar to the presented field examples in the Duvernay Formation, especially during
pumping and early shut-in time, the β coefficient is introduced into Poiseuille's formula (Eq.
6.1). After several sensitivity simulation runs, a β value of 1000 was selected as the optimum
value.
6.5 Results and Discussion
6.5.1 Model 1: tip extension. Figure 6.6(a) provides the simulated pressure profile, G-
function semi-log derivative and PTA plots during pressure falloff for Model 1. The overall
trends of the curves are similar to the Field Example 1 (Figure 6.4). At the time of breakdown,
the created fracture has finite conductivity with a considerable pressure gradient inside the
fracture. This results in the sudden pressure drop at the time of shut-in.
As demonstrated in Chapter 4, moving hinge closure occurs during the transition from
afterflow dominance (caused by wellbore storage) to leakoff dominance. Due to low matrix
permeability and the pressure gradient inside the fracture in the Model 1, coupled with the
presence of afterflow, the pressure front moves to the tip of fracture and causes tip extension of
the primary fracture. Pressurization and depressurization during tip extension can be repeated
several times (Figure 6.7) during the transition period before leakoff dominates the falloff
process. Tip extension phases are shown as fluctuations on pressure derivative plots. It must be
noted that in the simulation models, the magnitude and duration of these fluctuations are
116
controlled by the size of fracture elements which is uniform throughout the length of fracture.
However, in reality, the magnitude and duration of each tip extension phase can be different.
In Model 1, leakoff dominance starts after 4.5 log-cycles and fracture closure does not
occur.
Figure 6.6 - (a) PTA diagnostic plots; (b) pressure profile and G dP/dG curves. Tip extension
phases are shown with the dotted squares.
117
Figure 6.7 - Plan view of a single wing of the fracture showing pressure gradient inside the
fracture and tip extension phases during falloff. The fracture aperture is magnified 1000 times.
6.5.2 Model 2: pre-existing fractures and tip extension. As mentioned in the model
description and setup section, Model 2 uses a pre-existing network of natural fractures (NF),
similar to the system observed in the Duvernay outcrop. Pre-existing natural fractures can be
activated during injection and falloff and affect propagation direction and pressure response.
Figure 6.8(a) illustrates the pressure profile during injection and early shut-in period. There is no
significant pressure drop at the time of breakdown due to the low conductivity of the primary
fracture. As the primary fracture hits the pre-existing fracture, propagation temporarily stops
(upward trend on pressure profile) until the pre-existing fracture is activated. Then, the
118
propagation continues in the original direction perpendicular to the minimum horizontal stress
(Figure 6.8(b)).
The falloff behavior is similar to Model 1. Given the low matrix permeability and leakoff,
reactivation of secondary fractures can happen during the falloff period, too. The response on
pressure derivative plots will be similar to tip extension phases as illustrated in Model 1.
Figure 6.8 - (a) Pressure profile during injection and early shut-in time for Model 2; (b)
Propagation as the primary fracture hits and activates a pre-existing fracture (fracture apertures
are magnified 1000 times).
6.5.3 Field Example 1. The observed trends of Field Example 1 in Figure 6.4 can be
explained based on the simulation Model 1. The absence of a clear breakdown and the significant
pressure drop at the time of shut-in (Figure 6.3(a)) is caused by low conductivity and the large
pressure gradient inside the created fracture.
The PTA plots for this dataset are revisited in Figure 6.9. The long transition after the
early time unit slope, with no sharp characteristic hump, indicates that leakoff from the fracture
119
to surrounding formation is not dominant. This is due to low matrix permeability of the
formation coupled with low conductivity and pressure gradient inside the fracture. Three major
extension cycles are observed during the falloff period demonstrated by the fluctuations and
inflection points. The pressure changes at the end of fall period are very small as the leakoff is
low and no additional extension occurs. As a result, the end of falloff period is highly affected by
ambient temperature changes. No signature of fracture closure is observed after 200 hours.
Therefore, no reliable estimate of closure pressure and formation permeability can be made in
this example.
Figure 6.9 - PTA plots for Field Example 1 showing the interpretation based on tip extension
cycles.
6.5.4 Field Example 2. In this DFIT, a total volume of 6.8 m3 of fresh water was
injected during 3 minutes of pumping. The pressure values were recorded using surface gauges.
The increasing trend after breakdown while the injection rate is kept constant (Figure 6.3(b))
120
indicates the activation of secondary fractures in addition to the primary fracture during the
injection. Again, there is a long transition of about 2 log-cycles after the early unit slope on PTA
plots (Figure 6.10). There are two major opening cycles during this transition, caused by
activation of secondary fractures or extension of primary fracture during falloff, which creates
additional fracture surface area for fluid leakoff from fracture system to the formation. As a
result, leakoff dominance starts after about 2.2 hours. The slope of 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙 and ∆𝑡
𝑑𝑃
∆𝑡
plots during leakoff dominance is smaller than 3/2 and 1/2, respectively; and there is no evidence
of Carter leakoff. The closure is picked at the inflection point on ∆𝑡𝑑𝑃
∆𝑡 curve that corresponds to
change of slope on PPD and 𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙𝑑𝑃
𝑑𝑡𝐴𝑔𝑎𝑟𝑤𝑎𝑙curves, with closure pressure of 32.1 MPa and
closure time of 51.4 hours. Formation linear or radial flow are not observed after 300 hours of
shut-in time, and no reliable estimate of formation permeability or initial pressure can be made.
Figure 6.10 - PTA plots for Field Example 2.
121
6.6 Conclusions
DFIT responses in the Duvernay Formation for the cases studied are controlled by the following
mechanisms:
• Episodic fracture growth during injection and falloff. The hydraulic fracture does not
grow in a simple, steady and continuous way. Growth occurs episodically through growth
phases followed by relative calm.
• Low permeability of the matrix. This property delays the transition to leakoff dominance
in the falloff period, and results in long fracture closure times. Also, low leakoff favors
the episodic fracture growth (moving hinge-closure and tip extension) as it provides a
mechanism to release pressure and create additional fracture surface area.
• Long lasting afterflow (caused by wellbore storage) and transition period due to limited
leakoff.
• Low conductivity fractures. The fractures, both primary and pre-existing, have a low
conductivity resulting in pressure gradient within the fractures. This explains the unclear
breakdown point and pressure responses after the breakdown.
The above mechanisms result in complex leakoff and pressure behavior. Therefore, the
assumption of Carter leakoff, and the traditional fracture closure picks based on a straight-line
tangent to the semi-log derivative on a G-function semi-log derivative plot or 3/2 slope on
Bourdet-derivative plot are not reliable. Finally, it is very unlikely for the formation radial flow
to occur during short falloff periods of days or weeks.
122
6.7 Nomenclature
G G-function time, dimensionless
m Slope of straight line, dimensionless
P Pressure, Pa
Pf Fracture pressure (fluid pressure in the cohesive element), Pa
qfrac Gap flow rate within the cohesive element, m3/sec
Shmax Maximum horizontal stress, Pa
Shmin Minimum horizontal stress, Pa
ta Agarwal’s time, dimensionless
tp Pumping time, sec
w Fracture opening (aperture), m
Greek Variables
β Coefficient in the Poiseuille's formula , dimensionless
Δt Shut-in time, sec
μ Fracturing fluid viscosity, cP
123
Chapter 7: Conclusions
7.1 Contributions and Conclusions
The major contributions and conclusions of this dissertation are as follows:
1) For planar fractures, closure is a transient process, starting from the tip of the fracture to
the vicinity of the wellbore. We refer to this process as “progressive fracture closure”. The
common signature referred to as “fracture height recession/transverse storage” is
reinterpreted to be caused by this phenomenon.
2) The primary pressure derivative (PPD) is presented as a powerful tool for identifying
progressive fracture closure. The start of a PPD violation corresponds to tip closure, and the
end of the violation indicates full closure. Other PTA derivative plots are useful in the
identification of flow regimes before the start of progressive closure and explaining other
PPD violations. For instance, PPD violations happening before closure during the unit slope
on the Bourdet-derivative, and subsequent transition to zero slope during hinge closure, are
caused by fracture expansion and tip extension due to wellbore storage.
3) In previous studies, the magnitude and duration of the tip extension period has been
underestimated. The use of the cohesive zone model in this thesis enabled us to capture this
process accurately and the simulations are consistent with field observations. The model
results illustrate that tip extension can occur during the early fracture expansion period, due
to wellbore storage, and during the hinge closure (and can last for several hours).
4) A simple method is presented for removing the effect of ambient temperature change on
pressure values without applying large derivative windows for smoothing. The temperature
effect may be interpreted incorrectly as tip extension or pressure dependent leakoff. Further,
124
using large derivative windows results in loss of accuracy and removal of signatures such as
tip extension and PPD violation.
5) The full spectrum of flow patterns and signatures observed before and after closure
during DFITs are explained by considering the dynamic nature of fracture geometry, variable
leakoff rate and afterflow. The overall falloff period is divided into 7 zones that can be
summarized as follows:
• Zone 1: Fracture expansion occurs due to wellbore storage dominance. Total leakoff
rate follows an increasing trend.
• Zone 2: Early transition from wellbore storage to leakoff dominance can be
conceptualized as a moving hinge-closure causing tip extension. Total leakoff rate
follows an increasing trend. This zone may result in a semi-horizontal trend on
Bourdet-derivatives (false before-closure signature of radial flow).
• Zone 3: Total leakoff rate starts to decrease as fracture pressure is reduced and shut-in
time increases.
• Zone 4: Mechanical fracture closure starts at the tip of fracture, moving towards the
vicinity of the injection point (progressive closure). Depending on the conductivity of
the closed part of fracture, this causes an upward deviation (PPD violation) or
downward deviation (sharp decline in PPD) on Bourdet-derivatives.
• Zone 5: It is possible for leakoff and afterflow to continue after closure (residual
afterflow and leakoff). Depending on the conductivity of the closed fracture, this
period can be misinterpreted as Carter leakoff or reservoir radial flow, the latter
causing an overestimate of permeability and initial pressure (false after-closure
signature of radial flow).
125
• Zone 6: Reservoir dominated behavior is observed when the fracture is static, and
leakoff and afterflow are negligible compared to flow inside the reservoir.
• Zone 7: Boundary dominated effects may be observed if the boundaries of the
reservoir are reached. Also, as pressure changes become very small, derivative
calculations are affected.
6) A new DFIT procedure is presented in this thesis. The application of this new DFIT
procedure for estimating closure pressure and reservoir properties is demonstrated with two
field examples in the Montney Formation. Using this procedure, fracture closure happens
between 1 to 2 hours. Also, in one of the two field trials, the radial flow regime is established
in 4 days and reservoir data is obtained in a very short period of time. Compared to
conventional DFITs conducted in the same area, both fracture closure and radial flow times
are accelerated significantly using this procedure.
7) A conceptual method is presented for estimation of reservoir pressure in pump-
in/flowback tests with medium to high flowback rate. Using this method, reservoir pressure
is obtained within a few hours of the flowback process. The concept is validated with
synthetic results of two independent numerical simulators. A field trial is conducted and
analyzed based on this concept providing reservoir pressure estimate in only 2 hours of
operation.
8) Non-ideal DFIT responses in the Duvernay Formation are shown to be controlled by the
following mechanisms:
• Episodic fracture growth during injection and falloff. The hydraulic fracture does not
grow in a simple, steady and continuous way. Growth occurs episodically through
growth phases followed by relative calm.
126
• Low permeability of the matrix. This property delays the transition to leakoff
dominance in the falloff period, and results in long fracture closure times. Also, low
leakoff favors episodic fracture growth (moving hinge-closure and tip extension) as it
provides a mechanism to release pressure and create additional fracture surface area.
• Long lasting afterflow (caused by wellbore storage) and transition period due to
limited leakoff.
• Low conductivity fractures. The fractures, both primary and pre-existing, have a low
conductivity resulting in pressure gradient within the fractures. This explains the
unclear breakdown point and pressure responses after the breakdown.
7.2 Recommendations for Future Work
Based on the findings of this dissertation, the following recommendations are made for future
research on the topic:
1) The progressive fracture closure concept and its duration may have implications for
estimating fracture dimensions and leakoff coefficient. A rigorous analytical or semi-
analytical method can potentially use progressive closure duration as an in input to back-
calculate fracture length or leakoff coefficient.
2) More in-depth parametric 3D simulation is highly recommended to investigate the effect
of various mechanisms on pressure response and fracture closure including: stress regime
(i.e. normal or strike-slip), stress contrast between target formation and vertical layers,
vertical fracture growth and fracture height.
3) The flow patterns and their associated fracture dynamics presented in Chapter 4 are
applicable to the main hydraulic fracture stimulation treatment. If post fracturing pressure
127
data are monitored for a few minutes, they can be analyzed using the signatures presented
in Chapter 4. This has implications for stage-by-stage analysis in a multi-stage hydraulic
fracturing treatment.
4) The field trials of the new DFIT procedure presented in Chapter 5 were conducted in the
Montney Formation. It is recommended to apply this procedure in other formations to
validate the procedure or investigate possible issues.
5) Induced seismicity during hydraulic fracturing is a common issue in the Duvernay
Formation. A DFIT may be used as a diagnostic tool to evaluate the potential for induced
seismicity using the presented scenarios and signatures in Chapter 6 (tip extension and re-
activation of natural fractures).
6) The preliminary simulation results indicate that hoop stress and near wellbore
complexities can affect fracture orientation and pressure profile inside the fracture. It is
recommended to investigate closure behavior in the presence of near wellbore
complexities and to compare the relative magnitude of closure pressure with minimum
in-situ stress.
128
References
Abaqus Analysis User’s Guide 2016. Dassault Systemes Simulia Corp., Providence, RI.
Abaqus User Subroutines Reference Guide 2016. Dassault Systemes Simulia Corp., Providence,
RI.
Agarwal, R.G. 1980. A New Method to Account for Producing Time Effects when Drawdown
Type Curves are Used to Analyze Pressure Buildup and Other Test Data. Presented at the SPE
Annual Technical Conference and Exhibition, Dallas, Texas, 21-24 September. SPE-9289-MS.
http://dx.doi.org/10.2118/9289-MS
Bachman, R., Afsahi, B., and Walters, D. 2015. Mini-Frac Analysis in Oilsands and their
Associated Cap Rocks Using PTA Based Techniques. Presented at the SPE Canada Heavy Oil
Technical Conference, Calgary, Alberta, Canada, 9-11 June. SPE-174454-MS.
http://dx.doi.org/10.2118/174454-MS
Bachman, R.C., Walters, D.A., Hawkes, R.A. et al. 2012. Reappraisal of the G Time Concept in
Mini-Frac Analysis. Presented at the SPE Annual Technical Conference and Exhibition, San
Antonio, Texas, 8-10 October. SPE-160169- MS. http://dx.doi.org/10.2118/160169-MS
Bachman, R.C., Walters, D.A., Hawkes, R.A. et al. 2012. Reappraisal of the G Time Concept in
Mini-Frac Analysis. Presented at the SPE Annual Technical Conference and Exhibition, San
Antonio, Texas, 8-10 October. SPE-160169-MS. http://dx.doi.org/10.2118/160169-MS
Bao, X. and Eaton, D.W. 2016. Fault Activation by Hydraulic Fracturing in Western Canada.
Science 354:1406-1409. https://doi.org/10.1126/science.aag2583
129
Barenblatt, G. 1962. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Adv.
Appl. Mech. 7: 55–129. http://dx.doi.org/10.1016/S0065-2156(08)70121-2
Barree, R. D. 1998. Applications of Pre-Frac Injection/Falloff Tests in Fissured Reservoirs-Field
Examples. Presented at the SPE Rocky Mountain Regional/Low Permeability Reservoirs
Symposium, Denver, Colorado, 5-8 April. SPE-39932-MS. http://dx.doi.org/10.2118/39932-MS.
Barree, R. D. and Mukherjee, H. 1996. Determination of Pressure Dependent Leakoff and Its
Effect on Fracture Geometry. Presented at the SPE Annual Technical Conference and Exhibition,
Denver, Colorado, 6-9 October. SPE-36424-MS. http://dx.doi.org/10.2118/36424-MS.
Barree, R.D., Baree, V.L., and Craig, D.P. 2009. Holistic Fracture Diagnostics: Consistent
Interpretation of Prefrac Injection Tests Using Multiple Analysis Methods. SPE Production &
Operations 24(3): 396-406. SPE-107877-PA. http://dx.doi.org/10.2118/107877-PA
Barton, N., Bandis, S., and Bakhtar, K. 1985. Strength, Deformation and Conductivity Coupling
of Rock Joints. International Journal of Rock Mechanics and Mining Sciences & Geomechanics
Abstracts 22(3): 121–140. http://dx.doi.org/10.1016/0148-9062(85)93227-9.
Biot, M.A. 1955. Theory of Elasticity and Consolidation for a Porous Anisotropic Solid. J. Appl.
Phys. 26: 182–185.
Bourdet, D., Ayoub, J. A., and Pirard, Y. M. 1989. Use of Pressure Derivative in Well Test
Interpretation, SPE Formation Evaluation 4(2). https://doi.org/10.2118/12777-PA
130
Brenner, S. L. and Gudmundsson, A. 2004. Arrest and Aperture Variation of Hydrofractures in
Layered Reservoirs. Geological Society, London, Special Publications 231(1): 117-128.
https://doi.org/10.1144/GSL.SP.2004.231.01.08
Chen, Z.R. 2012. Finite Element Modeling of Viscosity-Dominated Hydraulic Fractures. J.
Petrol. Sci. Eng. 88–89: 136–144. http://dx.doi.org/10.1016/j.petrol.2011.12.021
Chopra, S., Sharma, R. K., Ray, A. K. et al. 2017. Seismic Reservoir Characterization of
Duvernay Shale With Quantitative Interpretation and Induced Seismicity Considerations—A
case study. Interpretation 5(2): T185-T197. https://doi.org/10.1190/INT-2016-0130.1
Cinco-Ley, H., and Samaniego-V, F., 1981. Transient Pressure Analysis for Fractured Wells.
Journal of Petroleum Technology. 33(09):1749-1766. https://doi.org/10.2118/7490-PA
Clarkson, C.R. and Williams-Kovacs, J.D. 2013. A New Method for Modelling Multi-Phase
Flowback of Multi-Fractures Horizontal Tight Oil Wells to Determine Hydraulic Fracture
Properties. Presented at the SPE Annual Technical Conference and Exhibition,New Orleans,
Louisiana, 30 September-2 October. SPE-166214-MS. https://doi.org/10.2118/166214-MS
Cooke, M.L. and Underwood, C.A. 2001. Fracture Termination and Step-Over at Bedding
Interfaces Due to Frictional Slip and Interface Opening. Journal of Structural Geology 23: 223–
238. https://doi.org/10.1016/S0191-8141(00)00092-4
Cramer, D. D. and Nguyen, D. H. 2013. Diagnostic Fracture Injection Testing Tactics in
Unconventional Reservoirs. Presented at the SPE Hydraulic Fracturing Technology Conference,
The Woodlands, Texas, 4–6 February. SPE-163863-MS. http://dx.doi.org/10.2118/163863-MS
131
Creaney, S and Allan, J. 1990. Hydrocarbon Generation and Migration in the Western Canada
Sedimentary Basin. Geological Society, London, Special Publications 50: 189–203.
Davis, M. and Karlen, G. 2014. A Regional Assessment of the Duvernay Formation: A World-
Class Liquids-Rich Shale Play: Geoconvention, Calgary, Alberta, Canada.
Dugdale, D. 1960. Yielding of Steel Sheets Containing Slits. J. Mech. Phys. Solids 8(2): 100–
104. http://dx.doi.org/10.1016/0022-5096(60)90013-2
EIA, 2015. U.S. Energy Information Administration, Annual Energy Outlook, Washington, DC.
www.eia.gov/forecasts/aeo/
EIA, 2019. U.S. Energy Information Administration, Annual Energy Outlook, Washington, DC.
https://www.eia.gov/outlooks/aeo/
Fox, A., Soltanzadeh, M., Watson, N. et al. 2015. The Link Between Lithology and Rock
Fractures in the Duvernay. 2015 Gussow Conference: Fine-Grained Rocks: Resources to
Reserves, Banff, Alberta, Canada, October 13-15.
Fox, A.D. and Soltanzadeh, M. 2015, A Regional Geomechanical Study of the Duvernay
Formation.Geoconvention, Calgary, Alberta, Canada.
Ghanizadeh, A., Bhowmik, S., Haeri-Ardakani, O. et al. 2015. A Comparison of Shale
Permeability Coefficients Derived Using Multiple Non-Steady-State Measurement Techniques:
Examples from the Duvernay Formation, Alberta (Canada). Fuel 140: 371–387.
https://doi.org/10.1016/j.fuel.2014.09.073
132
Goertz‐Allmann, B. P., Gibbons, S. J., Oye, V. et al. 2017. Characterization of Induced
Seismicity Patterns Derived from Internal Structure in Event Clusters. J. Geophys. Res. Solid
Earth 122: 3875–3894. http://dx.doi.org/10.1002/2016JB013731
Gu, H., Elbel, J.L., Nolte, K.G. et al. 1993. Formation Permeability Determination Using
Impulse-Fracture Injection. Presented at the SPE Production Operations Symposium, Oklahoma
City, Oklahoma, 21-23 March. SPE-25425-MS. https://doi.org/10.2118/25425-MS
Haddad, M., and Sepehrnoori, K. 2015. Simulation of Hydraulic Fracturing in Quasi-Brittle
Shale Formations Using Characterized Cohesive Layer: Stimulation Controlling Factors. Journal
of Unconventional Oil and Gas Resources 9: 65-83.
http://dx.doi.org/10.1016/j.juogr.2014.10.001
Harris, N. B., Miskimins, J. L. and Mnich, C. A. 2011. Mechanical Anisotropy in the Woodford
Shale, Permian Basin: Origin, Magnitude, and Scale. The Leading Edge 30(3): 284-291.
https://doi.org/10.1190/1.3567259
Hawkes, R., V., Anderson, I., Bachman, R., C. et al. 2013. Interpretation of Closure Stress in the
Montney Shale Using PTA Based Techniques. Presented at the SPE Hydraulic Fracturing
Technology Conference, The Woodlands, Texas, 4-6 February. SPE-163825-MS.
https://doi.org/10.2118/163825-MS
Houzé, O., Viturat, D., Fjaere, O.S. et al. 2017. Dynamic Data Analysis, V5.12.01, p. 563.
https://www.kappaeng.com/papers
133
Howard, C.C., and Fast, C.R. 1957. Optimum Fluid Characteristics for Fracture Extension. API
Drilling and Production Practice 24(1): 261-270. API-57-261.
Irwin, G. 1957. Analysis of Stresses and Strains Near the End of A Crack Traversing A Plate. J.
Appl. Mech., 24(1): 361-364.
Jung, H., Sharma, M.M., Cramer, D.D et al. 2016. Re-examining Interpretations of Non-ideal
Behavior During Diagnostic Fracture Injection Tests. J. Petroleum Science and Engineering
145(1): 114-136. https://doi.org/10.1016/j.petrol.2016.03.016
King, G.E. 2012. Hydraulic Fracturing 101: What Every Representative, Environmentalist,
Regulator, Reporter, Investor, University Researcher, Neighbor and Engineer Should Know
About Estimating Frac Risk and Improving Frac Performance in Unconventional Gas and Oil
Wells. Presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, TX,
6-8 February. SPE-152596-MS. https://doi.org/10.2118/152596-MS
Knapp, L.J., McMillan, J.M and Harris, N.B. 2017. A Depositional Model for Organic-Rich
Duvernay Formation Mudstones. Sedimentary Geology 347: 160–82.
https://doi.org/10.1016/j.sedgeo.2016.11.012
Krauklis, P. V. 1962. On Some Low-Frequency Vibrations of A Liquid Layer in An Elastic
Medium. J. App. Mathematics and Mechanics 26(6): 1685-1692. https://doi.org/10.1016/0021-
8928(62)90203-4
134
Liang, C., O’Reilly, O., Dunham, E. M. et al. 2017. Hydraulic Fracture Diagnostics from
Krauklis-Wave Resonance and Tube-Wave Reflections. Geophysics 82(3):D171-D186.
https://doi.org/10.1190/geo2016-0480.1
Liu, G., and Ehlig-Economides, C.A. 2015. Comprehensive Global Model for Before-Closure
Analysis of an Injection Falloff Fracture Calibration Test. Presented at the SPE Annual
Technical Conference and Exhibition, Houston, Texas, 28-30 September. SPE-174906-MS.
https://doi.org/10.2118/174906-MS
Marongiu-Porcu, M., Ehlig-Economides, C.A. and Economides, M.J. 2011. Global Model for
Fracture Falloff Analysis. Presented at the SPE North American Unconventional Gas Conference
and Exhibition, The Woodlands, Texas, 14-16 June. SPE-144028-MS.
http://dx.doi.org/10.2118/144028-MS
Mattar, L. and Zaoral, K. 1992. The Primary Pressure Derivative (PPD) A new Diagnostic Tool
in Well Test Interpretation, J. Can. Pet. Technol. 31(4). http://dx.doi.org/10.2118/92-04-06
McClure, M. and Kang, C. 2018. ResFrac Technical Writeup. https://arxiv.org/abs/1804.02092
McClure, M.W., Blyton, C.A.J., Jung, H. et al. 2014. The Effect of Changing Fracture
Compliance on Pressure Transient Behavior During Diagnostic Fracture Injection Tests.
Presented at the SPE Annual Technical Conference and Exhibition, Amsterdam, The
Netherlands, 27–29 October. SPE-170956-MS. http://dx.doi.org/10.2118/170956-MS
135
McClure, M.W., Jung, H., Cramer, D.D. et al. 2016. The Fracture-Compliance Method for
Picking Closure Pressure From Diagnostic Fracture-Injection Tests. Soc. Pet. Eng. 21(4): 1321-
1339. SPE-179725-PA. http://dx.doi.org/10.2118/179725-PA
Mohamed, I.M., Nasralla, R.A., Sayad, M.A. et al. 2011. Evaluation of After-closure Analysis
Techniques for Tight and Shale Gas Formations. Presented at the SPE Hydraulic Fracturing
Technology Conference and Exhibition, The Woodlands, Texas, 24-26 January. SPE-140136-
MS. http://dx.doi.org/10.2118/140136-MS
Nolte, K.G. 1979. Determination of Fracture Parameters From Fracturing Pressure Decline.
Presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, 23-26
September. SPE-8341-MS. http://dx.doi.org/10.2118/8341-MS
Nolte, K.G. and Smith, M.B. 1981. Interpretation of Fracturing Pressures, Journal of Petroleum
Technology 33(9). https://doi.org/10.2118/8297-PA
Nolte, K.G., Maniere, J.L., and Owens, K.A. 1997. After-Closure Analysis of Fracture
Calibration Tests. Presented at the SPE Annual Technical Conference and Exhibition, San
Antonio, Texas, 5-8 October. SPE-38676-MS. https://doi.org/10.2118/38676-MS
Nordgren, R.P. 1972. Propagation of a Vertical Hydraulic Fracture. SPE Journal. 12(4): 306–
314. https://doi.org/10.2118/3009-PA
Perkins, T.K. and Kern, L.R., 1961. Widths of Hydraulic Fractures. Journal of Petroleum
Technology. 13(9): 937–949. https://doi.org/10.2118/89-PA
136
Plahn, S.V. , Nolte, K.G., Thompson, L.G. et al. 1997. A Quantitative Investigation of the
Fracture Pump-In/FIowback Test, SPE Production and Facilities 12(1).
https://doi.org/10.2118/30504-PA
Poulsen, D.K. 1997. A General Theory of Minifracturing. Presented at the SPE Production
Operations Symposium, Oklahoma City, Oklahoma, 9-11 March. SPE-37407-MS.
https://doi.org/10.2118/37407-MS
Reiter, K., O. Heidbach, D. Schmitt, K. et al. 2014. A Revised Crustal Stress Orientation
Database for Canada. Tectonophysics 636: 111–124. https://doi.org/10.1016/j.tecto.2014.08.006
Rivard, C., Lavoie, D., Lefebvre, R. et al. 2014. An Overview of Canadian Shale Gas Production
and Environmental Concerns. International Journal of Coal Geology 126: 64-76.
https://doi.org/10.1016/j.coal.2013.12.004
Sarvaramini, E., and Garagash, D.I. 2015. Breakdown of a Pressurized Fingerlike Crack in a
Permeable Solid. J. Appl. Mech 82(6). http://dx.doi.org/10.1115/1.4030172
Savitski, A. and Dudley J.W. 2011. Revisiting Microfrac In-situ Stress Measurement via Flow
Back - A New Protocol. Presented at the SPE Annual Technical Conference and Exhibition,
Denver, Colorado, 30 October-2 November. SPE-147248-MS. https://doi.org/10.2118/147248-
MS
Schultz, R., R. Wang, Y.J. Gu, et al. 2017. A Seismological Overview of the Induced
Earthquakes in the Duvernay Play near Fox Creek, Alberta. J. Geophys. Res. Solid Earth 122:
492-505. http://dx.doi.org/10.1002/2016JB013570
137
Searles, K.H., Zielonka, M.G., Ning, J. et al. 2016. Fully-Coupled 3D Hydraulic Fracture
Models: Development, Validation, and Application to O&G Problems. Presented at the SPE
Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 9-11 February. SPE-
179121-MS. http://dx.doi.org/10.2118/179121-MS
Shapiro, S. A. and Dinske, C. 2009. Fluid Induced Seismicity: Pressure Diffusion and Hydraulic
Fracturing. Geophysical Prospecting 57(2): 301-310. http://dx.doi.org/10.1111/j.1365-
2478.2008.00770.x
Shen, X., Cullick, S. 2012. Numerical Modeling of Fracture Complexity with Application to
Production Stimulation. Presented at the SPE Hydraulic Fracturing Technology Conference and
Exhibition, The Woodlands, Texas, 6–8 February. SPE-151965-MS.
http://dx.doi.org/10.2118/151965-MS
Shin, D.H., Sharma, M.M. 2014. Factors Controlling the Simultaneous Propagation of Multiple
Competing Fractures in a Horizontal Well. Presented at the SPE Hydraulic Fracturing
Technology Conference and Exhibition, The Woodlands, Texas, 4–6 February. SPE-168599-MS.
http://dx.doi.org/10.2118/168599-MS.
Shlyapobersky, J., Walhaug, W., Sheffield, R. E. et al. 1988. Field Determination of Fracturing
Parameters, for Overpressure Calibrated Design of Hydraulic Fracturing. Presented at the SPE
Annual Technical Conference and Exhibition, Houston, Texas, 2-5 October. SPE-18195-MS.
https://doi.org/10.2118/18195-MS
Smith, M. B. 1985. Stimulation Design for Short, Precise Hydraulic Fractures, Society of
Petroleum Engineers 25(3). https://doi.org/10.2118/10313-PA
138
Soliman, M.Y. and Daneshy, A.A. 1991. Determination of Fracture Volume and Closure
Pressure From Pump-In/Flowback Tests. Presented at the SPE Middle East Oil Show, Bahrain,
16-19 November. SPE-21400-MS. https://doi.org/10.2118/21400-MS
Tary, J., Baan, M. and Eaton, D. 2014. Interpretation of Resonance Frequencies Recorded
During Hydraulic Fracturing Treatments. J. Geophys. Res. Solid Earth 119: 1295-1315.
http://dx.doi.org/10.1002/2013JB010904
Turon, A., Davila, C.G., Camanho, P.P. et al. 2007. An Engineering Solution for Mesh Size
Effects in The Simulation of Delamination Using Cohesive Zone Models. Eng. Fract. Mech. 74:
1665–1682. http://dx.doi.org/10.1016/j.engfracmech.2006.08.025
Van Dam, D.B., Papanastasiou, P., and de Pater, C.J. 2002. Impact of Rock Plasticity on
Hydraulic Fracture Propagation and Closure. SPE Production & Facilities, 17(3): 149 – 159.
SPE-78812-PA. http://dx.doi.org/10.2118/78812-PA
Van Den Hoek, P. 2016. A Simple Unified Pressure Transient Analysis Method for Fractured
Waterflood Injectors and Minifracs in Hydraulic Fracture Stimulation. Presented at the SPE
Annual Technical Conference and Exhibition, Dubai, UAE, 26-28 September. SPE-181593-MS.
http://dx.doi.org/10.2118/181593-MS
Varela, R. A. and Maniere, J. L. 2016. Successful Dynamic Closure Test Using Controlled Flow
Back in the Vaca Muerta Formation. Presented at the SPE Argentina Exploration and Production
of Unconventional Resources Symposium, Buenos Aires, Argentina, 1–3 June. SPE-180997-MS.
https://doi.org/10.2118/180997-MS.
139
Yao, Y. 2012. Linear Elastic and Cohesive Fracture Analysis to Model Hydraulic Fracture in
Brittle and Ductile Rocks. Rock Mech. Rock Eng. http://dx.doi.org/10.1007/s00603-011-0211-0
Zanganeh, B., Clarkson C.R., and Hawkes, R.V. 2017. Reinterpretation of Fracture Closure
Dynamics During Diagnostic Fracture Injection Tests. Presented at the SPE Western Regional
Meeting, Bakersfield, California, 23-27 April. SPE-185649-MS. https://doi.org/10.2118/185649-
MS
Zanganeh, B., Clarkson C.R., and Jones, J.R. 2018. Reinterpretation of Flow Patterns During
DFITs Based on Dynamic Fracture Geometry, Leakoff and Afterflow. Presented at the SPE
Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 23-25 January. SPE-
189840-MS. https://doi.org/10.2118/189840-MS
Zielonka, M. G., Searles, K. H., Ning, J., and Buechler, S. R. Development and Validation of
Fully-Coupled Hydraulic Fracturing Simulation Capabilities. 2014 Simulia Community
Conference, Providence RI 17.
140
Copyright Permissions
Chapter 3
141
Chapter 4
142
Chapter 5
143
Chapter 6
![Distinctive Flow Regions in Crossform Fracture …...pressure derivative diagnostic plot and is used to determine the fracture conductivity [7]. Thirdly, pseudo-radial flow with fractures](https://img.pdfslide.net/doc/110x75/5e81acb32224f159c448c147/distinctive-flow-regions-in-crossform-fracture-pressure-derivative-diagnostic.jpg)
