Stochastic Modeling of a Fracture Networkin a Hydraulically Fractured Shale-Gas Reservoir

A Research Proposal

by

Adnene MHIRI

B.S., Ecole Polytechnique (2012)M.S., IFP-School (2014)

M.S., Texas A&M University (2014)

Chair of Advisory Committee: Dr. Thomas A. BlasingameCommittee Member : Dr. Walter B. AyersCommittee Member : Dr. Maria A. Barrufet

Submitted to the Office of Graduate Studies ofTexas A&M University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

May 2014

Major Subject: Petroleum Engineering

1. Abstract

Decline curve rates of shale gas reservoirs that are produced under a constant wellbore pressure are still to be

understood, which is partly due to the complexity of hydraulic fractures patterns. This work introduces a novel

approach to model the hydraulic fractures in shale reservoirs using a stochastic method called random-walk. The

approach aims to capture a part of the “complexity” of a fracture that has been generated after a hydraulic

fracturing treatment and that may be observed on the micro-seismic measurements.

Some sensitivity analyses will be performed on the generated fractures pattern characteristics including the

extent, the tendency to split and the number of branches. The effects of those aspects will be reviewed and

resolved with standard model of a plane hydraulic fracture. The pressure distribution will be analyzed in order to

understand how the complex-pattern effect is observed only in the early production times. The feasibility and the

advantages of a full-scale reservoir and well model will then be investigated.

2. Objectives

The overall objectives of this work are to:

Construct a computer model that simulates probabilistic random-walk, mono and multi-branched

hydraulic fracture networks that propagates in a shale-gas reservoir;

Analyze the effect of stochastic fractures network and to determine whether such random networks can

render the performance behavior observed in such reservoirs. A sensitivity analysis on the fracture shape,

tendency to bifurcate and number of bifurcations will be carried to study the impact of the modeled fracture

network on the production performance (rates and pressures);

Study the effect of those fractures on the pressure propagation to determine the time-frame at which the

pattern effect should be observed and then to focus on those observed times to better understand and

comprehend the effect of each pattern feature;

Use the results of the approach to better explain the flow path from the reservoir to the wellbore; and

Justify the use of the single plane fracture approach in numerical simulation versus the extended

complex hydraulic fracture pattern.

3. Present Status

Prior to the numerical simulator era, analytical and semi-analytical flow solutions for a single-fractured reservoir

were developed by (Gringarten et al. 1974) and then by (Blasingame and Poe 1993). For a multiple-fractured

horizontal well, a single equivalent-fracture was conventionally used until later analytical approaches (Medeiros

et al. 2006) were used to solve the flow equations. Other more sophisticated analytical and semi-analytical

models evolved to improve the existing forecasting methods (Anderson et al. 2010; Bello and Wattenbarger

2008; Mattar 2008). Unfortunately, those types of simplified approaches generalize the complexity of the

physical problem. In fact, the non-linearity that characterizes the gas diffusivity equation cannot be solved

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analytically without the use of the pseudopressure and pseudotime transformations. In contrast, numerical

simulation solutions provide approximate solutions to highly-complex, non-linear differential equations using

finite-difference and/or finite-element methods.

Induced hydraulic fractures in shale-gas and tight reservoirs are commonly seen as 2D structures that propagate

in the plane of maximum principal stress direction σ max (Geertsma and De Klerk 1969; Hossain et al. 2000;

Perkins and Kern 1961; Weng et al. 2011) explained the unpredictability of fractured reservoir as a complex

interaction between the natural fracture network and the hydraulically induced fractures. Micro-seismic data

reveal an apparent complexity of hydraulic fractures (Li et al. 2012). Therefore, the approach used to model

natural fracture patterns as well as their interactions with hydraulic fractures is very important. Two sets of

orthogonal fractures are conventionally proposed in numerical simulations (Meyer and Bazan 2011; Olorode et

al. 2013; Xu et al. 2010). Olson (2008) developed a more sophisticated fracture network using empirical crack

propagation laws.

Maxwell et al. (2006) showed that microseismic events could indicate fracture density. This activity may be seen

as a complex interaction between natural fractures and hydraulic fractures. One of the challenges of shale-gas

production is to be able to use the microseismic data to construct a fracture pattern that, once modeled, matches

the production curves. The ultimate goal is to be able to predict the future decline in these reservoirs. While

conceptual hydraulic fractures are still modeled in reservoir simulation as extended planes, one may wonder

whether the hypothesis of planar fractures accurately describes the “real” hydraulic fracture path and if this

description has an impact on the production and on reserve estimation.

4. Novel Approach

Two-dimensional fracture propagation models (Daguier et al. 1996; Mastorakos et al. 2003) use a probabilistic

approach for a tensile (mode I) fracture growth. The method assumes that a fracture grows from an initial point

(hydraulic fluid injection for example) into a media where heterogeneities are uniformly distributed. In simple

terms, we assume that the XY-plane contains a series of discrete integer points (x,y)∈ z × z and a fracture can

grow from a point to the next following the minimum critical fracturing stress σ c. Considering that the

mechanical properties are distributed in the two-dimensional porous media according to a certain probabilistic

law, the fracture growth direction could be randomly directed according to those distributed laws. As a first

approach, we suppose that those properties (critical stress) are uniformly distributed, and therefore, the

probability for the fracture to move in each direction is constant. The process is mathematically known under the

name of random walk. To summarize, the hypothesis of this work are:

Homogeneous distribution of geomechanical properties:

— In-situ stress

— Fracture initiation pressure

— Elastic moduli (Shear modulus and Poisson’s ratio)

No interaction with natural fractures:

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— Natural fractures are not modeled

— The pattern shape does not depend on the arrangement of natural fractures

Fractures are modeled in a 2-D plan (invariance of the structure vertically along the z-axis)

A possible application of this method would be to generate a sufficient number of random fracture patterns that

minimizes the differences with the micro-seismic mapping image, for example.

a. Mathematical description

The fracture trajectory may be seen as a 2-dimensional discrete density function, F ( x , y ), that is defined in a

certain volume V as follows:

¿..............................................................................................................................................................(1)

The function is assumed to start from the origin

F ( x0 , y0 )=F (0,0 )=1........................................................................................................................(2)

then, we iteratively define the function for a certain step n∈N

{ F ( xn+ 1 , yn+1 )=F ( xn+1 , yn ) with a probability P x+1

F ( xn+1 , yn+1 )=F ( xn , yn+1 ) with a probability P y+1

F ( xn+1 , yn+1 )=F ( xn , yn−1 ) witha probability P y−1

...............................................................(3)

Where the probabilities to progress in each of the special directions verify

P x+1+P y+1+P y−1=1..........................................................................................................................(4)

The following distributions are used to generate a random fracture path. Figure 1 gives the “possible”

progression paths starting from a certain point (2,-2).

P x+1=13

P y+ 1=13

................................................................................................................................................(5)

P y−1=13

4

1 1 2 3 4x

4

2

2

4y

Figure 1 — Random path for a given step.

The main advantage of a probabilistic method is the ability to generate a wide range of "possible" realizations.

The observed patterns of fractures usually include branches that intersects with the parallel networks of natural

fractures. Those are supposedly responsible of "linking" the sets of natural fractures (primary and secondary)

and therefore an important aspect of the flow in shale reservoirs.

b.Preferential growth directions

The defined probabilities define the direction of the fracture. For example, if the probabilities to progress along

the y-axis are equal

P y+ 1=Py−1...........................................................................................................................................(6)

then the fracture is expected to oscillate around the x-axis and the expected value of the yn coordinates of the

points where the density function is a fracture is zero.

E ( yn )=1× P y+1+(−1 ) × P y+1=0....................................................................................................(7)

However, when the fracture is chosen to have a preferential progression toward a positive value of y>0, then

P y+ 1 is slightly higher than P y−1

P y+1>P y−1............................................................................................................................................(8)

In this case, the expected value of E ( yn ) is positive, and the fracture will have the tendency to grow with a

positive slope. For the same number of iterations,N=1000, the upper plot of

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shows that the first fracture tends to remain parallel to the x-axis (between y=-33 and y=12) when the

probabilities to grow in the positive and negative y are equal P y+ 1=P y−1. In the lower Figure 2 case, the

probability for the fracture to go in the positive y-direction is slightly higher (P x+1=Py +1=25

, P y−1=15

), and

there is a steady increase (positive slope) of the fracture.

50 100 150 200 250 300x

30

20

10

10

y

100 200 300 400x

50

100

150

y

Figure 2 — Fracture preferential growth direction after 1000 iterations.

c. Possibility of splitting

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Figure 3 — Fractures branching in a shale formation (From Colorado School of Mines AAPG

website).

The fracture patterns in shale (or in other materials) (Figure 3) show that the fracture has the tendency to split

into sub-fractures and not only necessarily propagates in “lines”. It also. While this process could entirely be

described by a fully coupled geomechanical/flow simulations, we will investigate the possibility of having what

we shall call branched fractures or fractures that have one or more splitting stages.

This branching character may be captured by the random-walk process. At some randomly generated step, the

fracture has the tendency to split to "upper" and "lower" branches. Starting from that point, the propagation

continues as if we had to separate fractures whose propagation is governed by the same probability laws.

Figure 4 gives different "possible" realization using the same growth probabilities and the same number of

iterations (N=1000) and for a bifurcation that occurs after a randomly generated step (1<k<N ).

For 1<i<k

{P x+1=12

P y+1=14

P y−1=14

..............................................................................................................................................(9)

For k<i<N

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{ P x+1=25

P y +1=25

for theupper branch∧P y+1=15

for the lower branch

P y−1=15

for theupper branch∧P y−1=25

for thelower branch

..........................................(10)

Then, a multi-branching process (limited to 4 in this study for computational simplification purposes). Figure 5

illustrates different "possible" realizations using the same growth probabilities and the same number of

iterations(N=1000) with 4 bifurcation stages that occur randomly and for a randomly generated number of

steps for each branch. After each bifurcation, the upper and lower branches progress according the probability

law defined previously (for a single branch). Qualitatively, the computed 2D networks look like the Figure 3

observed patterns.

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100 200 300 400x

100

50

50

100

y

100 200 300 400x

40

20

20

40

60

80

y

100 200 300 400x

150

100

50

50

100

150

y

Figure 4 — Random walk with uniform probabilities for 1000 iterations with branching.

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50 100 150 200 250 300 350x

150

100

50

0

50

100

150

y

100 200 300 400x

150

100

50

0

50

100

150

y

100 200 300 400x

150

100

50

0

50

100

150

y

Figure 5 — Random walk with uniform probabilities for 1000 iterations with 4 branching stages.

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d.Exporting the generated structure to a numerical simulator

The generated fracture patterns, which are also a series of points in a 2-D space, can be transferred to a numerical

simulator by transforming it into a mesh. The mesh can be generated by considering that each point of the

fracture is the center of a grid-cell. On a rectangular uniform XYZ grid, a code MESHMODIFIER is created: it

takes a uniform matrix grid and transforms iteratively the fracture cells from “Shale” media to “Fract” media.

The original matrix box is chosen to have a very refined grid-block size in the XY plan (

dx=dy=0.01 m ,dz=20 m). A smaller grid size would have been more convenient for computational

purposes. However, a fracture that has an aperture greater than one centimeter would not have been realistic.

Smaller fracture aperture can still be performed using the same grid-size with a lower value of fracture porosity.

Out of the region where the fracture is assumed to grow (∆ x=∆ y=2.5 m , ∆ z=20 m), we use geometrically

increasing grid-sizes to achieve a large enough area to perform the simulations. The actual final extent of the

reservoir is:

∆ x=∆ y=∆ z=20 m......................................................................................................................(11)

Figure 6 shows a gridded fracture network on a 2-D grid (Nx=Ny=250 , Nz=1) for a 3-branched fracture

network. Figure 7 shows the entire reservoir 2-D grid (Nx=300 ,Ny=350 , Nz=1), the region where the

fractures are assumed to grow is highlighted.

To first understand the effect of those kinds of fractures on the production curves (rate), we assume a well that

produces from the base of the fracture (or its origin) to the reservoir. The study of the Stimulated Reservoir

Volume (SRV) and the behavior of the production curves can give us insights on how this kind of approach

coupled with a more comprehensive hydraulic-fracture simulation (including primary and secondary sets of

natural fractures) would impact decline curves. The purpose of this preliminary study is to investigate the effect

of those complex-shaped fractures and to eventually justify their use.

e. Reservoir simulation

The generated fracture patterns are 2-D structures invariant vertically (along the z direction). Therefore, the

wellbore is taken as the base cell of the fracture network (defined as the origin point). In a more comprehensive

reservoir model, this structure is repeated for negative values of x (non-necessarily identically to the one of

positive x). The extent is limited to 2.5 meters, because this method requires a very fine grid cells and, therefore,

a considerable amount of computational power. As a matter of fact, if we multiply the region extent in the x and

y direction by 2, the number of cells in the model must go from 105,000 to 250,000.

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To understand the effect of different fracture properties as:

The fracture tortuosity (which is controlled by the ratio of the probabilities of going in the x versus in

the y directionsPx+1

P y+1+Py−1)

The occurrence of the branching (early of late occurrence)

The orientation of the branches (controlled by the ratio of probabilities of the progression along the y

directionP y+1

P y−1)

The number of branches

All results are then compared with those of a plane fracture which is taken to be the base case simulation. Based

on the analysis of the pressure distribution evolution, we observed that the effect of the pattern would

“disappear” after a relatively short production period (less than 20 days in this case). Therefore, the simulations

have been conducted until 100 days, which is convenient both numerically and to understand the effect of the

fractures characteristics. However, one should be particularly cautious when analyzing the results of the

simulations. In fact, the effect is observed before 20 days because of the lateral extent of the constructed fractures

(contained in a 2.5m by 2.5m area). For much larger volumes, the effects are expected to as long as the pressure

has not become uniform in what we may call the pattern “envelope” (the minimal closed surface that contains the

entire pattern).

Figure 6 — Simulated fracture network on a 2D Grid.

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Figure 7 — Whole grid with a refined fracture area and progressively coarsening grid-size.

Eventually, the complex fractures that have been generated will extend in the outer reservoir region, starting

from a planar hydraulic fracture (macro-fracture). Gildin et al. (2013) used a non-uniform induced permeability

field from the fracture to the outer reservoir. A possible way to justify that permeability decrease is to use the

same kind of randomly generated fractures and consider that they drain the near vicinity of the y=0 plane. Away

from the main hydraulic fracture, the production comes only from the primary and secondary sets of natural

fractures. However, the flow is “improved” near the main hydraulic fracture due to the connections that have

been initiated between natural fractures and hydraulic fractures.

The whole idea behind this approach lies on the observation that the permeability in the near-vicinity of the

planar macro-fracture is higher than permeability far from the fracture. Thanks to the randomly generated multi-

branched fractures, we can build a comprehensive model that captures the permeability enhancement near to

main planar hydraulic macro-fracture.

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5. Results

a. Non-Branched patterns

Figure 8 — Mass rate evolution for a shale gas reservoir produced from 3 non-branched fracture

patterns and compared to the planar fracture case until 100 days of production.

Figure 9 — β derivative of the mass rate evolution for a shale gas reservoir produced from 3 non-

branched fracture patterns and compared to the planar fracture case until 100 days of

production.

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Figure 10 — Planar fracture pressure maps after 1x101s, 1x102s, 1x103s, 1x104s and 1x105s.

Figure 11 — Non-branched fracture pattern 1 pressure maps after 1x101s, 1x102s, 1x103s, 1x104s and 1x105s.

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b.Mono-Branched patterns

Figure 12 — Mass rate evolution for a shale gas reservoir produced from 5 mono-branched fracture

patterns and compared to the planar fracture case until 100 days of production.

Figure 33 — β derivative of the mass rate evolution for a shale gas reservoir produced from 5 mono-

branched fracture patterns and compared to the planar fracture case until 100 days of

production.

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Figure 14 — Mono-branched fracture pattern 3 pressure maps after 1x101s, 1x102s, 1x103s, 1x104s and 1x105s.

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c. Dual-Branched patterns

Figure 45 — Mass rate evolution for a shale gas reservoir produced from 5 dual-branched fracture

patterns and compared to the planar fracture case until 100 days of production.

Figure 56 — β derivative of the mass rate evolution for a shale gas reservoir produced from 5 dual-

branched fracture patterns and compared to the planar fracture case until 100 days of

production.

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Figure 17 — Dual-branched fracture pattern 3 pressure maps after 1x101s, 1x102s, 1x103s, 1x104s and 1x105s.

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d.Tri-Branched patterns

Figure 18 — Mass rate evolution for a shale gas reservoir produced from 5 tri-branched fracture

patterns and compared to the planar fracture case until 100 days of production.

Figure 19 — β derivative of the mass rate evolution for a shale gas reservoir produced from 5 tri-

branched fracture patterns and compared to the planar fracture case until 100 days of

production.

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Figure 20 — Tri-branched fracture pattern 1 pressure maps after 1x101s, 1x102s, 1x103s, 1x104s and 1x105s.

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e. Quad-Branched patterns

Figure 21 — Mass rate evolution for a shale gas reservoir produced from 5 quad-branched fracture

patterns and compared to the planar fracture case until 100 days of production.

Figure 22 — β derivative of the mass rate evolution for a shale gas reservoir produced from 5 quad-

branched fracture patterns and compared to the planar fracture case until 100 days of

production.

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Figure 23 — Quad-branched fracture pattern 1 pressure maps after 1x101s, 1x102s, 1x103s, 1x104s and 1x105s.

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6. Toward a characterization of the number of branches

1. The pressure maps in of Fig’s 14, 17, 20 and 23 are used to identify the period where the interferences

between the fractures seems to cause the observed effect on the decline curves.

2. The number of branches of the hydraulic fractures seem to influence the early time behavior of the

decline rate curve (between 100s and 100000s). In fact, we can observe that the mass decline rate curve

has a characteristic slope value (linear decline) during that early time (Fig’s 12, 15, 18 and 21). The β-

derivative confirms that observation as the derivative is progressively increasing with the number of

branches (Fig’s 13, 16, 19 and 22).

4. After the fracture interference effect (after 100000s of production), all the patterns exhibit a

convergence toward the planar fracture case. A significant part of the incremental recovery that is due to

the areal extent of the stochastic patterns compared to the planar fracture comes before the converged

half-slope linear flow.

3. The other geometric features such as tortuosity, the placement of the branches and spacing between the

branches, seem to have no significant impact on the decline rate curves. However, the production

performance (cumulative recovery facture) is naturally enhanced as the stochastic fractures have

different volumes.

7. Recommendations and Future Work

We suggest the following tasks for future work in the analysis/modeling of well performance behavior in shale-

gas reservoirs.

1. Use 3-D modeling to simulate fractures that extend from the wellbore and propagate in a plane orthogonal to

the wellbore from which propagate those branched- fractures. It would be a more "realistic" modeling of the

hydraulic fracture.

2. Generalize the proposed approach to the case of a multiply-fractured horizontal well, and analyze the effect

of fracture interference behavior(s).

3. Populate the reservoir model with natural fractures (possibly randomly generated), the interactions between

those and the natural fracture may significantly change the understanding of the flow and the consequences

of the approach.

4. Develop methods and workflows that help to estimate the extent and properties of random fracture patterns

(number of branches, extent of the fracture, and direction of growth …).

5. Generate an exhaustive number of production-profiles based on this stochastic fracture model to possibly

establish empirical "basis models" for production performance from these types of fracture sets.

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8. Organization of the Research

The outline of the proposed research is as follows:

Chapter I — Introduction

Statement of the problem Objectives of the research Assumptions and added value of the work Experimental setting parameters Flow simulator and pattern creation Confrontation of the results with the conventional planar fracture model Validation of the results with commercial software Organization of the research

Chapter II Literature Review

Fractured reservoirs analytical and numerical models Present status of complex hydraulic fracture pattern modeling Research hypothesis and type of investigated fractures Novel approach to model 2-D fractures: random walk Incompatibility of the created geometries with grid refining methods

Chapter III Development of New Approximate Stochastic Fracture Propagation

Model

Mathematical description Preferential growth directions Possibility of splitting Exporting the generated grid to the numerical simulator Fracture generation algorithm Reservoir model and simulation

Chapter IV Influence of the Stochastic Fracture characteristics

Methodology to understand the flow behavior and to assess the performance of the stochastic fractures Planar hydraulic fracture model: a conventional approach to benchmark the results Non-Branched fracture patterns Mono-branched fractures patterns Dual-branched fractures patterns Tri-branched fractures patterns Quad-branched fractures patterns Summary of the findings Limitations of the model and possible future declinations

Chapter V Summary and Conclusions

Summary Conclusions Recommendations for Future Research

Appendix

References

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Nomenclature

σ max = Maximum principal stress, Pa or MPa [Force/Surface]

σ c = Minimum critical fracturing stress, Pa or MPa [Force/Surface]

( x , y , z ) = First, second and third coordinate in an orthogonal Cartesian coordinate system, m [Length]

F ( x , y )= Discrete density function that describes the fracture, [1]

n = Integer that describes the iterative definition of the density function

N = Integer that describes the total number of iteration that defines the density function

k = Random integer that describes the number of iterations after which the fractures splits

P x+1 = Probability for the fracture to progress along the positive x direction by 1 increment, [1]

P y+ 1 = Probability for the fracture to progress along the positive y direction by 1 increment, [1]

P y−1 = Probability for the fracture to progress along the negative y direction by 1 increment, [1]

dx = Grid size along the x-direction, m [Length]

dy = Grid size along the y-direction, m [Length]

dz = Grid size along the z-direction, m [Length]

∆ x = Reservoir size along the x-direction, m [Length]

∆ y = Reservoir size along the y-direction, m [Length]

∆ z = Reservoir size along the z-direction, m [Length]

Nx = Number of grid-blocks in the x-direction

Ny = Number of grid-blocks in the y-direction

Nz = Number of grid-blocks in the z-direction

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References

Anderson, D.M., Nobakht, M., Moghadam, S. et al. 2010. Analysis of Production Data from Fractured Shale Gas Wells. Society of Petroleum Engineers. DOI: 10.2118/131787-MS.

Bello, R.O. and Wattenbarger, R.A. 2008. Rate Transient Analysis in Naturally Fractured Shale Gas Reservoirs. Society of Petroleum Engineers. DOI: 10.2118/114591-MS.

Blasingame, T.A. and Poe, B.D., Jr. 1993. Semianalytic Solutions for a Well with a Single Finite-Conductivity Vertical Fracture. Society of Petroleum Engineers. DOI: 10.2118/26424-MS.

Daguier, P., Bouchaud, E., and Lapasset, G. 1996. Model of Crack Propagation in Heterogeneous Materials Local Approach. Le Journal de Physique IV 6 (C6): C6-483-C486-490.

Geertsma, J. and De Klerk, F. 1969. A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures. DOI: 10.2118/2458-PA

Gildin, E., Valko, P.P., and Fuentes-Cruz, G. 2013. Analyzing Production Data from Hydraulically Fractured Wells: The Concept of Induced Permeability Field. Society of Petroleum Engineers. DOI: 10.2118/163843-MS.

Gringarten, A.C., Ramey, H.J., Jr., and Raghavan, R. 1974. Unsteady-State Pressure Distributions Created by a Well with a Single Infinite-Conductivity Vertical Fracture. DOI: 10.2118/4051-PA

Hossain, M., Rahman, M., and Rahman, S. 2000. Hydraulic Fracture Initiation and Propagation: Roles of Wellbore Trajectory, Perforation and Stress Regimes. Journal of petroleum science and engineering 27 (3): 129-149.

Li, Y., Wei, C., Qin, G. et al. 2012. Optimizing Hydraulic Fracturing Design for Shale Gas Production through Numerical Simulations. Society of Petroleum Engineers. DOI: 10.2118/157411-MS.

Mastorakos, I., Gallos, L.K., and Aifantis, E.C. 2003. Computer Simulation of Discrete Crack Propagation. Journal of the Mechanical Behavior of Materials 14 (1): 9-22.

Mattar, L. 2008. Production Analysis and Forecasting of Shale Gas Reservoirs: Case History-Based Approach. Society of Petroleum Engineers. DOI: 10.2118/119897-MS.

Maxwell, S.C., Waltman, C., Warpinski, N.R. et al. 2006. Imaging Seismic Deformation Induced by Hydraulic Fracture Complexity. Society of Petroleum Engineers. DOI: 10.2118/102801-MS.

Medeiros, F., Ozkan, E., and Kazemi, H. 2006. A Semianalytical, Pressure-Transient Model for Horizontal and Multilateral Wells in Composite, Layered, and Compartmentalized Reservoirs. Society of Petroleum Engineers. DOI: 10.2118/102834-MS.

Meyer, B.R. and Bazan, L.W. 2011. A Discrete Fracture Network Model for Hydraulically Induced Fractures - Theory, Parametric and Case Studies. Society of Petroleum Engineers. DOI: 10.2118/140514-MS.

Olorode, O., Freeman, C.M., Moridis, G. et al. 2013. High-Resolution Numerical Modeling of Complex and Irregular Fracture Patterns in Shale-Gas Reservoirs and Tight Gas Reservoirs. DOI: 10.2118/152482-PA

Olson, J. 2008. Multi-Fracture Propagation Modeling: Applications to Hydraulic Fracturing in Shales and Tight Gas Sands. In The 42nd US rock mechanics symposium (USRMS): American Rock Mechanics Association.

Perkins, T.K. and Kern, L.R. 1961. Widths of Hydraulic Fractures. DOI: 10.2118/89-PA

Weng, X., Kresse, O., Cohen, C.E. et al. 2011. Modeling of Hydraulic Fracture Network Propagation in a Naturally Fractured Formation. Society of Petroleum Engineers. DOI: 10.2118/140253-MS.

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Xu, W., Thiercelin, M.J., Ganguly, U. et al. 2010. Wiremesh: A Novel Shale Fracturing Simulator. Society of Petroleum Engineers. DOI: 10.2118/132218-MS.

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