OPTIMIZATION STRATEGIES

FOR SHALE GAS ASSET DEVELOPMENT

A THESIS

SUBMITTED TO THE DEPARTMENT OF

ENERGY RESOURCES ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

Jamal Cherry

June 2016

c© Copyright by Jamal Cherry 2016

All Rights Reserved

ii

I certify that I have read this thesis and that, in my opinion, it is fully

adequate in scope and quality as partial fulfillment of the degree of

Master of Science in Petroleum Engineering.

(Louis J. Durlofsky) Principal Adviser

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Abstract

With the recent boom in US shale gas production, optimal development of these as-

sets has become a topic of significant interest. In addition to the complex physics

typically associated with shale plays, field development optimization can be a chal-

lenging problem itself when binary, integer and continuous variables are concurrently

present. For complex problems of this nature, it is appropriate to use simulation-based

optimization to search the solution space.

Prior work has shown that shale gas reservoirs can be simplified, through applica-

tion of a history-matching-like tuning procedure, from a locally refined, dual-porosity,

dual-permeability model that considers desorption and non-Darcy effects, to a surro-

gate model without desorption, non-Darcy corrections or refined grids. This surrogate

model is simply a single-porosity, single-permeability model with tuned parameters

in the stimulated reservoir volume (SRV). The two tuning parameters, SRV porosity

and permeability multipliers, are determined through a history matching process that

minimizes the difference in production between the full and surrogate models. With

an appropriately tuned surrogate model, optimization function evaluations can be

performed at a low computational cost. Here we utilize PSO-MADS (particle swarm

optimization - mesh adaptive direct search), a hybrid global-local optimization algo-

rithm, to find the optimal tuning parameters for the surrogate model and to find an

optimal field development plan. During the asset development optimization we con-

sider five decision variables per well: well location, lateral length, number of fracture

stages, bottomhole pressure, and finally whether or not to drill that well at all.

In this work, we integrate PSO-MADS, coupled-geomechanics, and new decision

variables into the workflow. The results demonstrate the efficiency and utility of using

v

proxy-based optimization for shale gas asset development. The results also indicate

that geomechanics can have an effect on the optimal development plan and should

be considered during optimization. The overall optimization framework developed in

this study should be applicable for a wide range of shale development projects.

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Acknowledgments

First, I would like to thank God for His continued blessings. Everything that is good

in my life comes from Him.

I would like to express sincere gratitude to my adviser, Prof. Louis J. Durlofsky,

for his enthusiasm, patience and continued support during my two years at Stanford.

Prof. Durlofsky exemplifies a rare level of professionalism and attention to detail;

skills which I hope to embody in future endeavors.

I’m also extremely grateful for the support that I have received from Elnur Aliyev.

He was always willing to help and provided me with a tremendous amount of encour-

agement when I first arrived to Stanford. I would also like to thank Sergey Chaynikov

and Timur Garipov for their time, suggestions and many discussions pertaining to

my research.

Thanks are also due to my colleagues and friends in the department. Particularly,

I would like to thank Youssef Elkady, Jose Ramirez Lopez Miro, Sergey Klevtsov,

Scott McLaughlin, Patrick McCullough, Vinay Tripathi and many others for all their

laughter, friendship and support during my time at Stanford.

I’d also like to thank my parents, Drs. Glenn and Valerie Cherry, for providing

me with the inspiration and courage to pursue graduate studies. They have seen all

the ups and downs of the last two years and have been a rock of unwavering support.

Finally, I’d like to thank my girlfriend, Sierra Shumate, for always being there

and doing all of the little things that mean so much to me.

Additionally, I wish to acknowledge the financial support from the industrial af-

filiates of the Stanford Smart Fields Consortium.

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Contents

1 Introduction 1

2 Full-Physics and Surrogate Model 9

2.1 Full-Physics Simulation Model . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Gridding and Model Properties . . . . . . . . . . . . . . . . . 9

2.1.2 Coupled Flow and Geomechanics . . . . . . . . . . . . . . . . 15

2.2 Surrogate Model and Tuning Process . . . . . . . . . . . . . . . . . . 20

2.2.1 Surrogate Model Description . . . . . . . . . . . . . . . . . . . 21

2.2.2 Incorporating Geomechanics into the Surrogate Model . . . . 22

2.2.3 Tuning Process . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 PSO-MADS Optimization Algorithm . . . . . . . . . . . . . . . . . . 24

2.4 Field Development Optimization . . . . . . . . . . . . . . . . . . . . . 27

2.5 Integrated Workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Results and Discussion 35

3.1 Tuning Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Field Development Optimization Results . . . . . . . . . . . . . . . . 42

3.3 Marcellus Example Without Geomechanics . . . . . . . . . . . . . . . 46

viii

3.3.1 Integrated Workflow Results . . . . . . . . . . . . . . . . . . . 46

3.3.2 Sensitivity Results . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Marcellus Example With Geomechanics . . . . . . . . . . . . . . . . . 51

3.4.1 Tuning Results . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2 Integrated Workflow Results . . . . . . . . . . . . . . . . . . . 55

4 Concluding Remarks 59

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 60

ix

List of Tables

2.1 Reservoir Properties for the Full-Physics Marcellus Model . . . . . . . 15

2.2 Field Development Economic Parameters . . . . . . . . . . . . . . . . 30

2.3 Field Development Decision Variables and Constraints . . . . . . . . 31

3.1 Marcellus and Barnett Reservoir Parameters . . . . . . . . . . . . . . 36

3.2 Initial Guess and Resulting Optimal Tuning Parameters for Six-well

and Four-well Configurations . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Summary of Tuning Results . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Summary of Barnett Field Development Optimization . . . . . . . . . 44

3.5 Optimal Tuning Parameters for the Integrated Workflow Example . . 48

3.6 Marcellus Geomechanical Properties for Coupled Flow-Geomechanics

Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7 Optimal Tuning Parameters for Six-well Configuration with and with-

out Geomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.8 Summary of Tuning Results with Geomechanics . . . . . . . . . . . . 55

3.9 Optimal Tuning Parameters for the Integrated Workflow Example with

Geomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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List of Figures

1.1 Projected shale gas growth to 2040 (from [12]) . . . . . . . . . . . . . 2

2.1 Illustration of full-physics horizontal well model. The wellbore is shown

in red with the hydraulic fracture in each stage shown in blue . . . . 11

2.2 Natural fracture network in a Marcellus shale outcrop (from Engelder

et al. [13]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Effect of desorption on gas production in Marcellus shale (from Heller

and Zoback [20]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Fracture conductivity as a function of effective stress in Marcellus shale

(from McGinley et al. [28]) . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Normalized fracture permeability as a function of effective stress for

fractures in Marcellus shale (from McGinley et al. [28] and Cipolla

et al. [9]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Normalized permeability as a function of pore pressure (adapted from

McGinley et al. [28]) . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xi

2.7 PSO and MADS iterations for an optimization problem with two vari-

ables. Curves represent contours of the objective function value (note

local and global optima, with the latter indicated by a red star). (a)

PSO iteration k, (b) PSO iteration k+1, (c) switch to MADS using

best PSO particle (from Isebor and Durlofsky [22]) . . . . . . . . . . 26

2.8 Flowchart of PSO-MADS hybrid algorithm (from [22]) . . . . . . . . 28

2.9 Schematic of the integrated workflow . . . . . . . . . . . . . . . . . . 33

3.1 Tuning results for a six-well development plan . . . . . . . . . . . . . 38

3.2 Progress of the tuning optimization for the six-well configuration . . . 39

3.3 Results using initial guess for the six-well configuration . . . . . . . . 40

3.4 Tuning results for a four-well development plan . . . . . . . . . . . . 41

3.5 NPVs for optimal configurations at each well count . . . . . . . . . . 42

3.6 NPVs for optimal configurations at each well count (circles). The star

indicates the NPV of the best variable well count case . . . . . . . . . 43

3.7 Permeability and pressure map of the best optimum from the variable

well count case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Progress of the optimization during the variable well count case . . . 46

3.9 Progress of the optimization during the integrated workflow. Stars

indicate tuning/re-tuning . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.10 Permeability map of the base-case configuration for the integrated

workflow without geomechanics . . . . . . . . . . . . . . . . . . . . . 49

3.11 Permeability map for optimal configuration from integrated workflow

without geomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xii

3.12 Final pressure map for optimal configuration from integrated workflow

without geomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.13 Permeability maps for sensitivity cases . . . . . . . . . . . . . . . . . 52

3.14 Tuning results for a six-well development plan with geomechanics . . 54

3.15 Progress of the optimization during the integrated workflow with ge-

omechanics. Stars indicate tuning/re-tuning . . . . . . . . . . . . . . 57

3.16 Permeability map for optimal field development plan with geomechanics 57

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xiv

Chapter 1

Introduction

Over the last decade shale gas has been the fastest growing source of natural gas in

the US [29]. Due to advances in horizontal drilling, multi-stage hydraulic fracturing

and a favorable economic environment (from the early 2000s to mid-2014), producers

have been able to unlock an expansive resource class previously thought to be un-

recoverable. The EIA [12] estimates that the US has more than 1,864 trillion cubic

feet of technically recoverable natural gas across numerous shale plays. As shown

in Fig. 1.1, shale gas production is projected to continue its strong growth, and will

remain a leading source of natural gas for the US over the next 25 years.

Shale plays have revolutionized how we view hydrocarbon reservoirs. What was

once viewed as the source rock, tight and unproducible, is now the reservoir rock

itself. Shale gas transport is characterized by extremely low permeability values and

complex physics that pose a challenge to production. In order to efficiently produce

from shale formations, hydraulic fracturing is typically employed. The fracturing

process entails high pressure pumping to hydraulically crack the rock and propagate

fractures. These hydraulically-induced fractures are held open with a proppant, and

1

Fig. 1.1: Projected shale gas growth to 2040 (from [12])

can intersect with existing natural fractures to create complex networks. The fracture

network generated around the wellbore is generally known as the stimulated reservoir

volume, or SRV. As the SRV grows larger and more complex, productivity increases

along with the complexity of transport [26]. Cipolla et al. [9], Rubin [33] and Wu

et al. [43] discussed the various physical processes and conditions that are present in

shale gas transport. These include non-Darcy flow, nonlinear desorption, multiscale

heterogeneity, and stress-dependent fracture conductivity.

State-of-the-art numerical simulations of shale gas transport tend to incorporate

most, if not all, of the physics mentioned above. One of the important limiting

factors in most shale gas simulations is accurately representing fractures, which can

have apertures from around 3 mm to 0.3 mm. The discrete representation of a large

number of these fractures is extremely challenging computationally. Thus numerous

methods have been introduced that are aimed at reducing the computational effort

required while maintaining a reasonable degree of accuracy. A method introduced by

2

Rubin [33] employed a fracture pseudoization technique where a 2 ft wide grid block

was used to represent the fracture and the permeability was adjusted to maintain

actual fracture conductivity. With grid refinement around these fractures, production

and pressure profiles retained their accuracy with respect to more highly resolved

models, but required less runtime [33]. This general method is commonly applied in

most shale gas simulations.

The area where the literature diverges is on the best way to represent natural

fracture interaction in the simulation model. Some investigators used the dual-

permeability method with local grid refinement [9, 44]. Mayerhofer et al. [27] em-

ployed a stimulated reservoir volume technique with natural fractures explicitly mod-

eled. Wu et al. [43] used a hybrid approach with a combination of explicit hydraulic

fractures, dual-continuum and single-porosity modeling.

In the models used in this work, we explicitly represent the hydraulic fractures, as

is done in most recent studies, but we model the natural fracture network using a dual-

porosity, dual-permeability treatment. We also utilize a coupled flow-geomechanics

formulation to model effective stress changes in the reservoir. As effective stress

increases, the formation deforms, which can affect porosity and fracture conductivity.

Current practice in shale field development entails more of a manufacturing type

approach, where empirical and historical data are relied upon to define drainage

volumes and new wells are spaced and completed accordingly. Methods for shale

field development optimization have begun to gain traction in recent years, though

these approaches are still not widely used. Generally, these methods focus on the

optimization of one well and primarily consider completion design optimization [4, 34]

while holding other variables constant.

3

Optimization strategies for single horizontal wells in shale (completion design,

lateral length, etc) have also been presented. Gorucu and Ertekin [16] and Shelley

et al. [35] used artificial neural networks trained by historical performance of off-

set wells to predict the performance of new well configurations in the same field.

Extending this method to a new field might prove difficult due to the empirical

nature of the approach. Bhattacharya and Nikolaou [6] introduced a fairly compre-

hensive optimization workflow that considered well spacing and completion design

using an analytical model to predict well production. This method should be ap-

plicable for the development of new fields, but it does not consider geomechanics or

complex physics such as desorption and non-Darcy flow. Yu and Sepehrnoori [44]

developed a simulation-based well optimization workflow that accounted for complex

physics and geomechanics. They identified the most impactful parameters in shale

gas simulation through a sensitivity study and used those parameters in a response

surface methodology (RSM). RSM is a mathematical technique that generates a sim-

ple (response-surface) model from a set of simulation results. Further optimization

is then performed on the response-surface model to obtain the optimal well design.

This workflow is again limited to a single well configuration, however.

Neglecting the joint nature of the decision variables associated with multiwell

field development can lead to sub-optimal solutions [6]. A few investigators have

considered multiple wells, various well configurations, and the potential for well-to-

well interactions through pressure or geomechanical interference. Gupta et al. [17]

investigated the effect of geomechanics on well spacing and completion design in a

case study, but they did not address the potential for variability in well length or

number of fracture stages per well. The effect of these parameters on the net present

4

value (NPV) of the field was also not considered.

Wilson [41] and Wilson and Durlofsky [42] combined a number of the ideas pre-

sented above into a general workflow. They applied this simulation-based optimiza-

tion workflow to a field of five wells. Using a Generalized Pattern Search (GPS)

algorithm, they optimized well location, length and number of fracture stages for

each well. These studies did not, however, include geomechanical effects, nor did

they consider optimizing the number of wells in the field. It is clear that shale field

development optimization is advancing, but it still lags when compared to field de-

velopment optimization for conventional fields, as we now discuss.

Optimization techniques have been applied for conventional oil and gas problems

in various ways. Methods to optimize the placement of injection and production wells

[31, 18], and their respective well controls [7], have been presented. These two opti-

mizations were typically considered individually, but are more appropriately viewed

jointly. Specifically, Zandvliet et al. [46] and Forouzanfar and Reynolds [15] demon-

strated that optimal locations depend on how the well is controlled, and vice versa.

Bellout et al. [5] and Isebor et al. [23, 24] introduced joint optimization methods for

well placement and well controls. Isebor et al. [23, 24] provided a general methodology

for field development that optimizes the number and type of wells (injector or pro-

ducer), their locations, the drilling sequence, and time-varying well BHP profiles. This

work utilized a hybrid global-local optimization algorithm and demonstrated its effi-

ciency when applied to challenging Mixed Integer Nonlinear Programming (MINLP)

problems in conventional oil fields. The hybrid optimizer, PSO-MADS, combines

the beneficial elements of a local optimization method, MADS (Mesh Adaptive Di-

rect Search), and a global method, PSO (Particle Swarm Optimization), to search

5

through complex solution spaces that may contain many local optima.

This thesis aims to advance shale field optimization by incorporating recent meth-

ods used in conventional field development. Here, we will combine aspects of previ-

ously mentioned work into an automated, simulation-based framework that can find

an optimal development plan for a new field. The framework includes PSO-MADS

optimization, and considers well number, well locations, number of fracture stages per

well and bottomhole pressures (BHPs). The simulations include discrete fractures,

complex physics and, in some cases, the effect of geomechanics.

We build on the work of Wilson and Durlofsky [42], where it was shown that

shale gas reservoir simulation models can be simplified from a locally refined, dual-

porosity, dual-permeability model that considers desorption and non-Darcy effects,

to a surrogate (or proxy) model without desorption, non-Darcy corrections or refined

grids. The surrogate model is simply a single-porosity, single-permeability model

with tuned parameters in the stimulated reservoir volume. The two tuning param-

eters, SRV porosity and permeability multipliers, are determined through a history

matching process that minimizes the difference in gas production between the origi-

nal full-physics and surrogate models. With an appropriately tuned surrogate model,

optimization function evaluations can be performed at a low computational cost.

Here, instead of GPS as was used in [42], we utilize PSO-MADS both to find the

optimal tuning parameters for the surrogate model and to find an optimal field de-

velopment plan. During the field development optimization we consider five decision

variables per well: well location, lateral length, number of fracture stages, bottom-

hole pressure, and finally whether or not to drill that well at all. Previously, the

development plan was constrained to include a specified number of wells, but here

6

the drill/do not drill variable provides the ability to optimize the number of wells in

the field. This, along with the BHP optimization, augments the decision variables

considered in [42]. Another important extension relative to [42] is the incorporation

of geomechanical effects in the full-physics and surrogate models. Our ultimate goal

is to present a general shale gas field development framework that finds an optimal

development plan in hours rather than days or weeks.

The outline of this thesis is as follows. In Chapter 2 we describe the full-physics

and surrogate models for the non-geomechanical and geomechanical cases. Overviews

of both the tuning and field development optimization processes, and their integration

into a single workflow, are also provided. In Chapter 3 we present a number of ex-

ample cases. These examples are based on both the Barnett and the Marcellus plays,

and demonstrate the performance of our procedures without and with geomechanical

effects. Finally, we provide concluding remarks, and suggestions for future work, in

Chapter 4.

7

8

Chapter 2

Full-Physics and Surrogate Model

In this chapter we describe the full-physics and surrogate simulation models. We

will explain the tuning process, which links the two models and enables efficient

optimization. We next describe the field development optimization. Finally, the

integrated workflow will be presented.

All simulations in this work are performed using CMG’s GEM simulator [10].

2.1 Full-Physics Simulation Model

In this section we will describe the full-physics model. We first provide a general model

description, and then discuss the integration of the coupled-geomechanics feature.

2.1.1 Gridding and Model Properties

One of the plays modeled in this work is the Marcellus shale, and we describe our mod-

eling procedure with reference to this case. The reservoir parameters for the Marcellus

are adapted from Yu and Sepehrnoori [45]. The reservoir area of interest is 106,000 ft

9

× 5300 ft × 162 ft. This domain is represented on a Cartesian grid of dimensions

106 × 53 × 1. In the full-physics model we use a dual-porosity, dual-permeability

approach. Gas desorption and corrections for non-Darcy flow in hydraulic fractures

are also included. All wells in the model are horizontal (parallel to the y-axis) and

hydraulic fractures extend perpendicular to the wellbore (in the x-direction). One

of the primary difficulties in shale gas simulation is accurately representing flow and

pressure drop in the areas near hydraulic fractures, where typical fracture widths are

on the order of 3 mm. Modeling grid blocks at this scale would cause our simulation

runtime to increase significantly. Following the method introduced by Rubin [33], a

hydraulic fracture is represented as a 2 ft wide grid block with an effective permeabil-

ity value that maintains the actual fracture conductivity (F c = wf × kf , where F c is

fracture conductivity, wf is the fracture aperture and kf is the fracture permeability).

Here, we employ a simple bi-wing planar hydraulic fracture representation where the

fracture half length, xf , is 500 ft, and each hydraulic fracture is surrounded by a

logrithmically spaced, locally refined grid, as shown in Fig. 2.1.

10

Fig. 2.1: Illustration of full-physics horizontal well model. The wellbore is shown inred with the hydraulic fracture in each stage shown in blue

11

The grid blocks around each hydraulic fracture and along the wellbore are logarith-

mically refined in order to resolve the large pressure drop and transient flow behavior

that occurs in these areas [33, 9]. The refined area around each 2 ft hydraulic fracture

represents a fracture stage. Each fracture stage is representative of a single perfora-

tion cluster that contains seven perforations, which are distributed along the length

of the stage. While only the hydraulic fractures are modeled explicitly in this work,

there is also a network of perpendicular joints, or natural fractures, that increase the

effectiveness of hydraulic fracturing. Gas shales in the Marcellus are known for their

orthogonal regional joint sets, indicated as J1 and J2 in Fig. 2.2, which were created

by increasing fluid pressure as organic matter reached thermal maturity [13]. Due to

the relatively uniform nature of the J1 and J2 joints (Fig. 2.2), the natural fracture

network contribution is accounted for using the the dual-porosity, dual-permeability

feature, with a fracture spacing of 100 ft in the x and y-directions. Fig. 2.2 shows a

much smaller natural fracture spacing in an outcrop, but we approximate their effect

with 100 ft spacing (at depth) as in Cipolla et al. [9].

As previously mentioned, non-Darcy flow corrections are applied in the blocks

representing hydraulic fractures. The Forchheimer-modified Darcy’s law is used to

describe this effect:

−∇P =µ

kmv + βρv2, (2.1)

where P is pressure, µ is viscosity, v is Darcy velocity, km is matrix permeability,

ρ is phase density and β is the Forchheimer correction, which is determined using a

correlation introduced by Evans and Civan [14]:

β =1.485× 109

φmk1.021m

, (2.2)

12

Fig. 2.2: Natural fracture network in a Marcellus shale outcrop (from Engelderet al. [13])

where φm is matrix porosity. As previously mentioned, fracture widths are effectivized

in the simulation model from their actual width to 2 ft grid blocks. In order to

properly model non-Darcy flow in a 2 ft grid block, we also incorporate the correction

for non-Darcy flow suggested by Rubin [33].

Gas desorption is included in the simulation model through use of the extended

Langmuir isotherm model for multicomponent adsorption [2, 19]. The general equa-

tion is given as:

ωc =ωc,maxBc yc,gP

1 + P∑j

Bj yj,g, (2.3)

where ωc is the number of moles of adsorbed component c per unit mass of rock, B is

the parameter for the Langmuir isotherm relation, ωc,max is the maximum number of

moles of adsorbed component c per unit mass of rock and yj,g is the molar fraction of

adsorbed component c in the gas phase. The sum is over the adsorbed components.

13

Fig. 2.3: Effect of desorption on gas production in Marcellus shale (from Heller andZoback [20])

In this work, however, we only consider a single methane component, so the sum-

mation is not needed. Here Bc and ωc,max were taken from laboratory data specific

to the Marcellus [20] (detailed simulation parameters are given in Chapter 3). As

the reservoir pressure decreases, gas is released from the solid surface. This provides

additional gas for production while also maintaining pressure in the reservoir [20]. We

see in Fig. 2.3 that desorption plays little to no role in production until the reservoir

pressure drops below 2000 psi. This is consistent with the findings of Cipolla et al. [9],

who showed that desorption in the Marcellus provided a 10% increase in produced

gas over a 30-year period.

There are a variety of gridding techniques (unstructured, explicit natural frac-

tures), additional physics (Knudsen diffusion, non-Darcy flow in natural fractures)

and additional completion design effects (type of proppant, proppant loading levels)

14

that could be incorporated into the full-physics model. Although our model does not

include these treatments or effects, we believe the workflow described here would also

be applicable to systems of this level of complexity.

Table 2.1: Reservoir Properties for the Full-Physics Marcellus Model

Reservoir Property Value

Grid Dimension 106 × 53 × 1Grid Cell Dimension 100 × 100 × 162 ftReservoir Depth 8593 ftInitial Reservoir Pressure 4726 psiMatrix Porosity, φm 6%Matrix Permeability, km 0.0006 mdFracture Half Length, xf 500 ft

2.1.2 Coupled Flow and Geomechanics

Hydraulic fractures serve as highly conductive pathways through which gas flows from

the reservoir to the wellbore. The conductivity is initially created by high pressure

pumping which initiates and propagates fractures, but the conductivity is maintained

by proppant particles lodged in the fractures during the hydraulic fracturing process.

As the reservoir is depleted and the normal effective stress on the fractures increases,

the fractures begin to close and their conductivity decreases. This effect is particu-

larly prevalent in the Marcellus shale, which is considered a ductile shale due to its

relatively low Young’s modulus. This makes it a challenge to keep fractures propped

as pressure decreases [1]. Wu et al. [43], Yu and Sepehrnoori [45] and Rubin [33]

have all shown the detrimental effects on production from the stress sensitivity of

hydraulic fractures in shale gas simulation. More specifically, cumulative production

15

from a Marcellus well can be decreased by as much as 60% over a 30-year period [9].

Thus it is clearly important to include geomechanics in our full-physics model in order

to capture how changes in effective stress impact fracture conductivity in hydraulic

and natural fractures.

As previously mentioned, this work was accomplished using CMG’s GEM simu-

lator, which contains a coupled flow-geomechanics feature. GEM does not allow the

use of refined geomechanical grids, so we used an independent geomechanics grid that

directly overlies the flow grid.

GEM iteratively couples the reservoir flow equations with the geomechanical cal-

culations. In the isothermal case, it treats porosity as a function of pressure and total

mean stress (σm) using the following formula developed by Tran et al. [38]:

φn+1 = φn + (c1 + c2a1)(P − P n), (2.4)

where:

c1 =1

V 0b

(dVpdP

+ VbαcbdσmdP

), (2.5)

c2 = − VpV 0b

αcb, (2.6)

a1 =2

9

E

(1− ν)αcb, (2.7)

where φ is porosity, cb is bulk compressibility (1/psi), Vp is pore volume (m3), E is

Young’s modulus (psi), Vb is bulk volume (m3), V 0b is initial bulk volume, α is the

Biot number, and ν is Poisson’s ratio.

Additionally, GEM allows for the input of a table describing permeability as a

16

function of mean effective stress. Ideally, the permeability would be a function of the

normal effective stress on the fracture grid block, but GEM only accepts permeability

modifications as a function of mean effective stress. With this approach, we are still

able to model stress-based conductivity degradation in fracture grid blocks.

The boundary conditions for the geomechanical problem are based on seismic

data [37, 30, 21]. The overburden stress gradient was taken as 1.07 psi/ft, and the

minimum horizontal stress was specified as 6015 psi with a stress anisotropy of 5%.

Due to generally high overburden stress at production depths, hydraulic fractures

will grow perpendicular to the plane of minimum principal stress, and are vertical at

reservoir depth [13].

As shown in Fig. 2.4, this makes a substantial difference in how fractures respond

to changes in effective stress. McGinley et al. [28] conducted a number of experiments

to establish how conductivity changes with effective stress in propped hydraulic frac-

tures in the Marcellus shale. Fracture conductivity was determined by measuring

the pressure drop of nitrogen gas through a modified API cell. Their study focused

on testing a number of horizontally and vertically fractured outcrop samples from

the Ellmsport, Pennsylvania area of the Marcellus. In this work, we used the ver-

tical fracture conductivity data, from the data provided by McGinley et al. [28], as

the basis for modeling permeability as a function of mean effective stress. A similar

dataset for natural fracture conductivity was obtained from Cipolla et al. [9]. The

full conductivity data for hydraulic fractures from [28] is shown below in Figure 2.4.

To incorporate this fracture conductivity data into the model, we build a nor-

malized conductivity chart for the hydraulic and natural fractures. Our treatment is

17

Fig. 2.4: Fracture conductivity as a function of effective stress in Marcellus shale(from McGinley et al. [28])

similar to the procedure described by Wilson [40] and Yu and Sepehrnoori [45]. In or-

der to determine the initial conductivity for a hydraulic fracture, we subtract the pore

pressure from the initial total stress. Our initial conductivity is determined by taking

the initial mean total stress (7176 psi) minus the initial reservoir pressure (4726 psi),

which gives 2450 psi. The conductivity value at 2450 psi in Fig. 2.4 is taken to be

the initial hydraulic fracture conductivity. We now find the minimum conductivity

value by subtracting the minimum BHP value, 535 psi, from the initial mean total

stress, 7176 psi. The given data do not extend to this value, so we extrapolate to

find the value of fracture conductivity at 6641 psi. The data between 2450 psi and

6641 psi are then normalized, and deemed representative of how hydraulic fracture

conductivity, as a function of effective stress, decreases during the simulation.

The hydraulic fractures represented in this model have an initial permeability

18

value of 10,500 md and an aperture of 3.18 mm (0.125 in). These values are from

CMG [10]. As previously mentioned, we use 2 ft grid blocks to model hydraulic frac-

tures in the flow grid. This requires the use of an effective permeability value of 55 md

in order to maintain consistency with the actual hydraulic fracture conductivity. The

flow grid cells representing the hydraulic fracture do not change in width, therefore

fracture conductivity is only a function of fracture permeability (F c = wf×kf ). Thus

the normalized fracture conductivity plot is identical to the normalized fracture per-

meability plot. This logic is also applicable to the natural fracture data set where

the initial fracture permeability is 10 md with an aperture of 0.318 mm (note these

fractures have experienced mineralization [25]). Based on natural fracture spacing

in the dual-permeability, dual porosity model, the permeability of these fractures is

effectivized to 0.0001 md (as suggested by a CMG simulation engineer). Recall that

porosity changes in hydraulic fracture grid blocks are handled using Eqs. 2.4 to 2.7.

Thus, we capture the effects of increasing effective stress on wf .

This process was applied to the hydraulic fracture conductivity data from McGin-

ley et al. [28], as described above, and then to the natural fracture conductivity data

from Cipolla et al. [9]. The resulting normalized fracture permeability plots for hy-

draulic and natural fractures are shown in Fig. 2.5. This treatment is applied to the

hydraulic fracture and natural fracture cells in the simulation model.

By coupling the flow and geomechanical calculations (through Eqs. 2.4 to 2.7),

the simulator now incorporates the porosity changes that occur from stress and the

deformed matrix. It also accounts for the effect of closure stress on the conductivity

of hydraulic and natural fractures in the reservoir. A major disadvantage of running

19

Fig. 2.5: Normalized fracture permeability as a function of effective stress for fracturesin Marcellus shale (from McGinley et al. [28] and Cipolla et al. [9])

a coupled flow-geomechanics simulation is the increase in runtime. Our coupled-

geomechanics simulations displayed runtimes of 7-12 hours depending on the amount

of refinement in the flow grid, while simulations without coupled flow-geomechanics

were run in about 10 minutes. It is possible that the simulations with geomechanics

could be accelerated, but we expect them to remain slow relative to runs that do not

include geomechanical effects.

2.2 Surrogate Model and Tuning Process

In this section we will provide a description of the surrogate model. We will also

describe the process used to tune the surrogate model such that it provides results in

agreement with the full-physics model.

20

2.2.1 Surrogate Model Description

The surrogate model is a computationally inexpensive representation of the full-

physics model described in Section 2.1. Based on the work of Wilson and Durlofsky [42],

we use a single-porosity, single-permeability model, without grid refinement, desorp-

tion or non-Darcy flow effects, as the basis for our surrogate model. In order to achieve

satisfactory agreement between the two models, we will determine permeability and

porosity multipliers, Mk and Mφ respectively, for the original matrix parameters, km

and φm:

φs = Mφ × φm, (2.8)

ks = Mk × km. (2.9)

The stimulated reservoir parameters, ks and φs, are applied only in the stimulated

reservoir volume (SRV) around each well. The SRV is a rectangular area around the

well that spans the length of the wellbore and extends perpendicularly out from the

wellbore a distance xf . A hydraulic fracture stage is now represented by a single

perforation along the wellbore in the SRV. Recall that in the full-physics model a

fracture stage is represented with a cluster of seven perforations distributed in the

refined grid space. In the surrogate model we do not have grid refinement, thus we

represent a stage as a single perforation in the SRV. The reservoir outside of the SRV

is modeled using the original reservoir properties, km and φm, as in the full-physics

model. As noted in [42], depending on the characteristics of the full-physics model, it

may be necessary to include additional parameters in the surrogate model. We now

discuss our treatment of geomechanics in the surrogate model.

21

2.2.2 Incorporating Geomechanics into the Surrogate Model

In order to accurately represent the production profile of the full-physics simulation

with coupled flow-geomechanics, we added two features: desorption and a pressure-

dependent permeability multiplier, Mg (P ). This multiplier is only applied in the

SRV and is meant to replicate the effects of stress-dependent fracture permeability in

the full-physics model. The augmented expression for ks is

ks(P ) = Mg(P )×Mk × km. (2.10)

Fig. 2.6: Normalized permeability as a function of pore pressure (adapted fromMcGinley et al. [28])

To remain consistent with the full-physics model, we convert the data in Fig. 2.4

from a function of effective stress to a normalized function of pore pressure, as shown

in Fig. 2.6. This is accomplished by assuming a Biot coefficient value of 1 and using

22

the relationship:

σ′ = σT − P, (2.11)

where σT is total stress, P is pressure and σ′ is effective stress. The areas outside the

SRV maintain their original permeability throughout the simulation. Adding these

two features increases the runtime of the surrogate model by about 50% to about

8 seconds, but this is insignificant when compared to the runtime of the full-physics

model with coupled flow-geomechanics (7-12 hours).

2.2.3 Tuning Process

In order to use the surrogate model in place of the computationally expensive full-

physics model, the surrogate model production must match, to within a small tol-

erance, the production observed in the full-physics simulation. In order to achieve

this goal, we use a process similar to history matching, where the multipliers in the

surrogate model, Mk and Mφ, are adjusted to achieve satisfactory agreement with

the full-physics model. To systematically search through the Mk-Mφ solution space

we employ the PSO-MADS optimization algorithm, introduced by Isebor et al. [24].

This algorithm will be described in Section 2.3.

We apply PSO-MADS to provide a set of tuning parameters that minimizes the

mismatch R,

R =T∑i=1

Nw∑k=1

(Qfpik −Qsur

ik

)2e−rti , (2.12)

where Qfpik and Qsur

ik are respectively the full-physics and surrogate model gas produc-

tion over time step i for well k. Here, Nw refers to the number of wells, T is the total

number of time steps, and r is the discount rate. In order to more directly relate the

23

function R to the quantity of primary interest, NPV, we include a discount factor,

e−rti , which essentially places a higher weighting on matching early gas production.

Here, ti is in months and r is in months−1.

With an appropriately tuned surrogate model, we are able to evaluate numerous

field development configurations, as will be discussed in Section 2.4. As the optimizer

performs function evaluations and the optimal configuration changes, it may become

necessary to “re-tune” the surrogate model, which entails updating the multipliers Mk

and Mφ. This will be further discussed in Section 2.5 when the integrated workflow

is described.

2.3 PSO-MADS Optimization Algorithm

Particle swarm optimization (PSO) is a population-based global stochastic search

method designed to mimic the behavior of swarms or flocks of biological organisms.

The method was originally developed by Eberhart and Kennedy [11]. The global

search capability in PSO decreases the chance that the optimizer will get trapped

at a poor local minimum in complex solution spaces. PSO involves a swarm of

possible solutions (particles), initially distributed randomly throughout the solution

space. In this work we use 2n particles, where n refers to the number of decision

variables. This number is user-specified and should increase as the complexity of the

optimization increases. At each iteration, the particles move in search of an objective

function increase. Particle movement is defined by a velocity vector involving three

mechanisms: social, cognitive and inertial. The social mechanism is based on the

objective function values of other nearby particles, and moves a given particle towards

those favorable locations. The cognitive mechanism moves particles towards the best

24

location experienced thus far by the given particle. Finally, the inertial mechanism is

simply determined by the particle velocity from the previous iteration. These three

components, with associated weightings plus randomization, direct each particle to its

new position. This process continues until there is a minimal amount of improvement

in the best solution between subsequent PSO iterations (or a maximum number of

iterations is reached). This is a brief description of the PSO method implemented in

Isebor et al. [23, 24]. For a detailed explanation of the method and its application to

oil field problems, the reader should refer to those publications.

Mesh adaptive direct search (MADS) is a stencil-based local search method which,

in many cases, converges to a local minimum. MADS, originally developed by Audet

and Dennis Jr [3], searches for a local optimum by polling the solution space around

the current best solution. If a better solution is found at one of the polled solutions on

the stencil, then that point becomes the center of the new stencil. If a better solution

is not found on the stencil, then MADS decreases the size of the stencil and polls the

resulting points. This process continues until the user-defined termination criteria is

reached (which is usually a minimum stencil size or maximum number of iterations).

This algorithm as described so far is similar to the GPS algorithm used in Wilson

and Durlofsky [42]. The key difference between GPS and MADS, however, lies in the

underlying mesh in MADS where the poll points are placed. After an unsuccessful

iteration, the underlying mesh decreases in size faster than the stencil. This leads

to polling points extending in a range of directions, which allows MADS to access

more possible polling directions than GPS. Audet and Dennis Jr [3] showed results

verifying that MADS is more robust than GPS for constrained problems. The reader

should refer to that paper, and to [23, 24], for more details.

25

(a) (b)

(c)

Fig. 2.7: PSO and MADS iterations for an optimization problem with two variables.Curves represent contours of the objective function value (note local and global op-tima, with the latter indicated by a red star). (a) PSO iteration k, (b) PSO iterationk+1, (c) switch to MADS using best PSO particle (from Isebor and Durlofsky [22])

26

PSO-MADS combines the individual advantages of the MADS and PSO tech-

niques by conducting some amount of global exploration (PSO) and subsequently

searching locally (MADS). Fig. 2.7 shows an illustrative example of PSO and MADS.

In the existing implementation [23, 24], we run PSO as long as each iteration provides

an improved objective function value. When PSO fails to provide sufficient improve-

ment we shift to MADS. MADS begins with a user-specified stencil size and polls

with the best PSO particle as the stencil center. MADS continues polling as long as

improved solutions are found with the existing stencil. Once MADS stops improv-

ing, we shift back to PSO and reduce the stencil size for the next MADS iteration.

Termination occurs when a minimum stencil size or a maximum number of function

evaluations is reached. Nonlinear constraints are handled using the filter method as

described in Isebor et al. [24]. The filter method aims to minimize constraint vi-

olations while also maximizing the objective function value, essentially acting as a

bi-objective optimization.

A flowchart for the PSO-MADS procedure is shown in Fig. 2.8. Since PSO and

MADS are both derivative-free optimization algorithms, they evaluate all particles or

stencil points in parallel in a single iteration.

2.4 Field Development Optimization

After finding the optimal tuning parameters, Mk and Mφ, we can now use the sur-

rogate model within the optimization workflow. In addition to the decision variables

suggested by Wilson and Durlofsky [42], well locations, well lengths and number of

fracture stages, we also consider a drill/do not drill decision (binary variable) for each

well. This enables the determination of the optimal number of wells to be drilled in

27

Fig. 2.8: Flowchart of PSO-MADS hybrid algorithm (from [22])

the reservoir. The optimizer can thus also return a “do nothing” result, where the

field is not developed because the given economic parameters lead to a negative NPV.

We have also added a BHP control for each well, which is represented as a continuous

variable. All wells are constrained to be parallel to one another, and to extend in

the y-direction. Thus the well location variable specifies the x-location of the well.

The well length variable, Lk, determines the length of the horizontal well in the y-

direction. The variable Nf,k specifies the number of fracture stages for a given well.

Fracture stages, represented by a single perforation in the surrogate model, are evenly

spaced along the length of the horizontal, with xf fixed at 500 ft.

Clearly there are many potential configurations that need to be evaluated in order

to find an optimal development plan. Once again we utilize PSO-MADS to search for

an optimal solution. The PSO-MADS algorithm used here is identical to that used

for the tuning process, as explained in Section 2.2.3. See [23, 24] for discussion of the

treatments for integer and binary variables.

28

The objective function we now seek to maximize is the NPV of the field:

NPV =T∑i=1

DCFi − Ccap, (2.13)

where DCF indicates discounted cash flow and the capital expenditure is repre-

sented by Ccap. The equation for DCF is based on the work of Williams-Kovacs and

Clarkson [39]. Discounted cash flow is given as:

DCFi = (1−X) [Qi(G− T )(1−Rp)− CLOE] e−rti , (2.14)

where Qi is the total production over time step i, T is the cost of transportation and

gathering per mscf, G is the gas price per mscf, X is the tax rate, Rp is the royalty

percentage, and CLOE represents the lease operating cost. The economic parameters

used in this work, except when otherwise indicated, are given in Table 2.2.

The capital expenditure term in Eq. 2.13 encompasses drilling, completion and

leasing costs:

Ccap = Cdrill + Cfrac + Clease. (2.15)

The drilling cost, $500 per lateral foot, accounts for the cost of drilling the vertical and

horizontal sections of each well. The completion cost is set at $150,000 per fracture

stage, and the leasing cost is $1000/acre. Since each configuration considered is based

on the same field area, Clease is fixed at $1.28M. Drilling and completion costs are

based on investor presentations from a number of producers currently operating in

the Marcellus [8, 32].

The optimization problem is much more challenging here than the training process,

29

Table 2.2: Field Development Economic Parameters

Economic Parameter Value

Gas Price $4.00/mscfDrilling $500/ftCompletion $150,000/stageTransportation & Gathering $0.70/mscfOperating $30/well/dayRoyalty 15%Tax 30%Discount Rate 10%

although we use the same PSO-MADS algorithm. Here we have 50 variables, instead

of two, and multiple variable types (integer, continuous, binary) to describe each

field development scenario. The u vector below shows the decision variables for field

development optimization:

u =

[D1, x1, Nf,1, L1, B1, · · · DN , xN , Nf,N , LN , BN

]. (2.16)

Each well is described by a set of five variables. The variable Dk is binary and

determines whether or not well k is drilled. The variables xk, Nf,k, Lk are all integer-

based (recall that locations are defined on a simulation grid) and describe the x-

location, number of fracture stages and length of well k, respectively. Additionally,

the continuous variable Bk defines the bottomhole pressure for well k.

The dimension of u is controlled by the maximum number of wells, and in this

work we consider a maximum of ten wells. Large numbers of decision variables and

different variable types, as appear in this work, can lead to a complex solution space

with potentially many local minima. This type of problem is characterized as a Mixed

30

Integer Nonlinear Programming (MINLP) problem, which represents a complex class

of optimization problems, but PSO-MADS is applicable for such problems. A full list

of the decision variables and their respective constraints is shown in Table 2.3.

Table 2.3: Field Development Decision Variables and Constraints

Decision Variable Constraints

Drill/Do Not Drill, Dk

∑Nk=1Dk ≤ 10

Well Location, xk xk − xk−1 ≥ 2xfFracture Stage Count, Nf,k 2–50Lateral Length, Lk 1000–5000 ftMinimum BHP, Bk 535 psi

2.5 Integrated Workflow

In Sections 2.2.3 and 2.4 we described two separate optimization problems. The first

problem, a tuning process, is an optimization that seeks to minimize the difference

between the full-physics and surrogate models through two decision variables: Mk

and Mφ. The second optimization, a field development optimization, uses Mk and

Mφ to describe the SRV around each well. The second optimization focuses on finding

the optimal field development plan defined by u.

We now describe the integrated workflow, which combines these two optimizations.

Initially, we tune the surrogate model based on a specified (initial guess) base-case

scenario. We then begin the field development optimization, which proceeds until a

user-defined termination criterion is reached. In this work, the termination criteria

are set as a maximum number of function evaluations or a minimum MADS stencil

size. After a termination criterion is reached, the current-best field configuration

31

is run in the full-physics model. If there is greater than 5% NPV improvement in

the full-physics model since the last tuning, we re-enter the tuning process with the

current-best field configuration. This is similar to the initial tuning process, but

instead of matching the initial-guess base-case, we match the full-physics production

from the current-best field configuration. After this “re-tuning,” we re-enter the

field development optimization. Once there is minimal improvement (less than 5%)

in NPV between subsequent full-physics simulations, we exit the workflow with an

optimal field development plan u. The percent improvement can of course be varied.

A flow chart of the full field development optimization workflow is shown in Fig. 2.9.

32

Fig. 2.9: Schematic of the integrated workflow

33

34

Chapter 3

Results and Discussion

In this chapter we present results using the methods and workflow discussed in Chap-

ter 2. First, we will show results from the tuning process for a Marcellus shale

example. We will then present results from a field development optimization based

on a Barnett shale case. Next, we will show results from the integrated workflow

based on the Marcellus. None of these examples includes geomechanical effects. In

our final example, we will present tuning and workflow results for a Marcellus shale

case that includes geomechanics.

As previously mentioned, CMG GEM, which is a compositional simulator, was

used in this work. The Peng-Robinson fluid model with a single methane component

was used for all cases. Additionally, the relative permeability was described using

Corey curves with an exponent of 2.0 for both the gas and water phases. There is no

water production in the model as the initial and minimum water saturation are both

set to 30%.

35

Table 3.1: Marcellus and Barnett Reservoir Parameters

Reservoir Property Marcellus Barnett

Pressure 4726 psi 3800 psiTemperature 175◦F 180◦FMatrix Porosity, φm 6% 4%Matrix Permeability, km 0.0006 md 0.0001 mdLangmuir Volume 28.3 scf/ton 88 scf/tonLangmuir Pressure 556.2 psi 440 psiMinimum Well BHP 535 psi 1000 psiSimulation Time 10 years 10 years

3.1 Tuning Results

Tuning the surrogate model is a critical part of this work. As discussed in Sec-

tion 2.2.3, the tuning process utilizes PSO-MADS to find optimal tuning parameters,

Mk and Mφ, which minimize R in Eq. 2.12. This enables the use of a surrogate model

that runs in around 5 seconds as opposed to a full-physics model which requires at

least 10 minutes to run. The results in this section are based on a Marcellus shale

reservoir, with parameters detailed in Table 3.1.

We will refer to two separate well configurations in this section: a six-well config-

uration and a four-well configuration. The six-well configuration is based on a result

shown later in Section 3.3.1. Each of the six wells is drilled to full length, operated

at the minimum BHP and contains 15 fracture stages. The four-well configuration

contains four wells drilled to 3500 feet. Each well is operated at the minimum BHP

and has 12 fracture stages. Both the four-well and six-well configurations represent

the types of solutions encountered during optimization.

Tuning results for the six-well configuration are shown in Fig. 3.1. Comparisons

of monthly gas rates and cumulative production are shown in Fig. 3.1(a) and (b),

36

respectively, for the full-physics and surrogate models. The difference in cumulative

production between the models at 10 years is just over 1%, and the difference in NPV

is $0.73 MM (out of a NPV of $62.6 MM for the full-physics model).

Table 3.2: Initial Guess and Resulting Optimal Tuning Parameters for Six-well andFour-well Configurations

Tuning Parameter Mφ Mk

Initial Guess 1 17Nw = 6 0.89 70Nw = 4 0.79 78

The progress of the tuning optimization for the six-well configuration is shown in

Fig. 3.2. The tuning process required less than 120 runs of the surrogate model (only

one full-physics run is required) to generate the match shown in Fig. 3.1. Note that

most of the error is eliminated over the first 30 function evaluations (runs). Fig. 3.3

shows the results using the initial guess of the tuning process. The initial guess and

optimal solution for Mφ and Mk for this case are shown in Table 3.2. Note that Mφ is

less than unity, which means it decreases the porosity in the SRV. This reduction is

critical to matching the production decline that begins at around 10 months. A higher

or lower porosity value shifts the break point appearing at 10 months to the right

or left. The permeability multiplier, Mk, increases the permeability in the SRV to

about 70 times the original matrix permeability. This increase in permeability allows

the surrogate model to capture the effects of high-conductivity, hydraulically-induced

fractures as well as the natural fracture network. Both of these fracture types are

modeled in the full-physics model, but they do not appear (explicitly) in the surrogate

model.

The results presented so far demonstrate that we are able to efficiently replicate

37

(a) Gas rate comparison

(b) Cumulative production comparison

Fig. 3.1: Tuning results for a six-well development plan

38

Fig. 3.2: Progress of the tuning optimization for the six-well configuration

full-physics field production for a particular six-well configuration (shown later to be

near optimal) through surrogate model tuning. It is equally important to be able to

match other cases that may be encountered during the optimization. In Fig. 3.4, we

show a tuning result for the four-well configuration described previously. A summary

of the tuning results is shown in Table 3.3. We see that the tuning process provides

accurate parameters for both cases. As is evident in Table 3.2, the Mφ and Mk values

for the two configurations are similar.

39

Table 3.3: Summary of Tuning Results

Model, Configuration NPV ($MM) Cum. Production (bcf)

Full-Physics, Nw=6 62.6 62.3Surrogate, Nw=6 61.9 61.6Full-Physics, Nw=4 31.1 31.0Surrogate, Nw=4 30.1 30.0

Fig. 3.3: Results using initial guess for the six-well configuration

40

(a) Gas rate comparison

(b) Cumulative production comparison

Fig. 3.4: Tuning results for a four-well development plan

41

3.2 Field Development Optimization Results

We now present optimization results using a tuned surrogate model based on the

Barnett shale reservoir parameters given in Table 3.1. Our goal here is to assess the

performance of the optimization with a variable number of wells. First, we specify

Nw = 6, where Nw is the number of wells. The optimizer then provides xk, Nf,k, Lk

and Bk for each of the six wells. We run the optimization three times (since PSO-

MADS is a stochastic optimizer) and plot the resulting NPVs in Fig. 3.5. This process

was repeated for Nw = 7, 8, 9 and 10. Each of these four cases was again run three

times, and the resulting NPVs are plotted in Fig. 3.5. The spread in NPVs within

each case is caused primarily by slight variations in the number of fracture stages

per well. The resulting optimal configuration for each case involved wells drilled

to the maximum length (5000 feet) and operated at the minimum BHP (535 psi).

Optimization NPV results for each run are provided in Table 3.4.

Fig. 3.5: NPVs for optimal configurations at each well count

42

The maximum NPV in these results ($35.7 MM) occurs with Nw = 8. We see

from Fig. 3.5 and Table 3.4 that there is a clear optimum when eight wells are drilled.

Given this result, we should expect PSO-MADS to also select eight wells when we

run a variable well count case (i.e., we simply specify Nw ≤ 10).

Results for this case are shown in Table 3.4. In each of the three runs, PSO-MADS

found an optimum Nw of 8. Fig. 3.6 shows the best optimum from the variable well

count runs (marked by a yellow star), and we see that it is very close to the best

optimum found in the study. Considering the added complexity of treating binary

categorical variables (drill/do not drill), achieving an average NPV that is 3% lower

than the best optimum (for the case with Nw set to 8) is a satisfactory result.

Fig. 3.6: NPVs for optimal configurations at each well count (circles). The starindicates the NPV of the best variable well count case

The permeability map from the best optimum in the variable well count case is

shown in Fig. 3.7(a). Red areas are stimulated regions where we apply the tuning

parameters, Mk and Mφ, and blue areas are the unstimulated region of the reservoir.

43

Table 3.4: Summary of Barnett Field Development Optimization

Nw=7 Nw=8 Nw=9 Variable

NPV Run 1 ($MM) 31.5 35.7 33.1 34.5 (Nw=8)NPV Run 2 ($MM) 31.9 34.9 34.0 33.6 (Nw=8)NPV Run 3 ($MM) 31.4 34.2 33.8 33.9 (Nw=8)

Average ($MM) 31.6 35.0 33.6 34.0

The SRV created around each well in the field is where most of the production occurs,

as is evident from the final pressure plot in Fig. 3.7(b). We see the optimal solution

corresponds to an extensive SRV, or red area, in the field. With more than eight

wells, we would increase the SRV and the ultimate recovery of gas, but NPV would

decrease because the cost of the additional well exceeds the increased revenue from

gas production.

In Fig. 3.8, we show the progress of the optimization in the variable well count

case. Well count is also plotted. Major increases in NPV occur early in the optimiza-

tion, as the well count increases from the initial value of Nw = 2. After 500 function

evaluations the optimal configuration contains ten wells, and after 1800 function eval-

uations the well count remains constant at eight wells. Further NPV improvements

are the result of optimizing other decision variables, such as Nf,k and Lk.

This example demonstrates the performance of a new optimization feature that

controls the number of wells drilled in the field. The results for this case indicate that

this capability is working as expected. In all subsequent examples, we will optimize

the number of wells along with the other decision variables.

44

(a) Permeability map

(b) Final pressure map

Fig. 3.7: Permeability and pressure map of the best optimum from the variable wellcount case

45

Fig. 3.8: Progress of the optimization during the variable well count case

3.3 Marcellus Example Without Geomechanics

We now present results from the integrated workflow. The first example is based on

the Marcellus and does not consider geomechanics.

3.3.1 Integrated Workflow Results

The reservoir parameters for this case are shown in Table 3.1. As discussed in Sec-

tion 2.5, the integrated workflow combines the tuning process and field development

optimization into a framework that allows us to efficiently find an optimal field de-

velopment plan. The initial tuning process is completed on a base-case scenario. In

this example, we use the configuration shown in Fig. 3.10, which contains four evenly

spaced wells all operating at a BHP of 535 psi. Each well is drilled to 3500 feet

and contains 12 fracture stages. After completion of this initial tuning process, we

46

start the field development optimization, as shown in the schematic in Fig. 2.9. In

general, multiple workflow runs should be performed due to the stochastic nature of

PSO-MADS, but from here on we show the results for only a single run.

The workflow completes three field development optimization runs before reach-

ing the minimal improvement termination criteria. The progress of the optimization

during the integrated workflow is shown in Fig 3.9. Here we see that the base-case

configuration (shown in Fig. 3.10) gives an NPV of $30.1 MM. After one field develop-

ment optimization run, the NPV of the field increases by 65% to $49.7 MM. Over the

duration of the first field development optimization, the optimal configuration adds

two wells to the base-case and lengthens all wells to 5000 ft from 3500 ft. The second

field development optimization improves the NPV primarily through optimizing the

well location and fracture stage count of each well. All wells in the final optimal

configuration are drilled to full length with an average of 16 fracture stages per well

and are operated at the minimum BHP. The integrated workflow increases the NPV

of the field by 108% relative to the base-case. The final optimal configuration has an

NPV of $62.1 MM and is shown in Fig. 3.11. The final pressure map is displayed in

Fig. 3.12.

Function evaluations (simulations) during field development optimization are run

in parallel using 100 nodes. The integrated workflow required just under 23,000 func-

tion evaluations in total leading to a CPU time of less than 20 minutes. The expensive

part of the workflow lies within the tuning/re-tuning process, which on average re-

quired 12-15 CPU minutes per tune/re-tune. The optimal tuning parameters at each

stage are shown in Table 3.5. We see that the tuning parameters show limited, but

not insignificant, variability over the workflow. This shows that as field configurations

47

change, re-tuning is indeed required to keep the full-physics and surrogate models in

agreement.

Overall, the integrated workflow was able to increase the NPV of the base-case by

108% in just over an hour of computation time. This example shows the utility and

efficiency of combining the tuning process and field development optimization into a

single workflow.

Fig. 3.9: Progress of the optimization during the integrated workflow. Stars indicatetuning/re-tuning

Table 3.5: Optimal Tuning Parameters for the Integrated Workflow Example

Case Mφ Mk

Base-case 0.79 78.8Re-tuning #1 0.82 68.0Re-tuning #2 0.86 71.7Re-tuning #3 0.87 72.1

48

Fig. 3.10: Permeability map of the base-case configuration for the integrated workflowwithout geomechanics

3.3.2 Sensitivity Results

We now consider sensitivity to gas price and completion cost. We first increase the

gas price from the base-case value of $4.00/mscf to $7.00/mscf. All other economic

parameters remain unchanged. The permeability map for the resulting optimal field

configuration is shown in Fig. 3.13 alongside the original optimal configuration from

the example shown in Section 3.3.1. With the increase in gas price, the optimal

field development plan NPV increases to $181.2 MM. Relative to the original optimal

configuration, this case includes two additional wells and increases the average number

of fracture stages per well from 16 to 23. This leads to a 191% increase over the

original optimal configuration NPV of $62.1 MM with gas at $4.00/mscf.

The next sensitivity case investigates the effect of increasing the completion cost

from $150k per stage to $550k per stage. The resulting optimal development plan now

has an NPV of $42.1 MM and is shown in Fig. 3.13. Here, we see that the optimizer

49

Fig. 3.11: Permeability map for optimal configuration from integrated workflow with-out geomechanics

Fig. 3.12: Final pressure map for optimal configuration from integrated workflowwithout geomechanics

50

now reduces the average number of fracture stages per well from 16 to 10. This

reduction leads to a 33% decrease in NPV from the original optimal configuration

($62.1 MM).

3.4 Marcellus Example With Geomechanics

As discussed in Section 2.1.2, geomechanics can have a major impact on gas produc-

tion in shale plays with low Young’s moduli, such as the Marcellus. In this example,

we model this effect and assess its ramifications on field development optimization.

The full-physics reservoir parameters used here are identical to the Marcellus reser-

voir parameters used in the previous non-geomechanics example. These parameters

are given in Table 3.1.

In order to model geomechanical effects we incorporate a coupled flow-geomechanics

feature into the full-physics model as discussed in Section 2.1.2. This feature captures

the impact of effective stress changes on porosity and fracture conductivity. These

effects are neglected in the non-geomechanics model. In addition to the input data

given in Section 2.1.2, other pertinent geomechanical properties for the Marcellus are

listed in Table 3.6.

Table 3.6: Marcellus Geomechanical Properties for Coupled Flow-Geomechanics Sim-ulations

Geomechanical Property Value

Young’s Modulus 2.32× 106 psiBiot Coefficient 1.0Poisson’s Ratio 0.283

51

(a) Original optimal configuration ($4.00/mscf gas)

(b) Optimal configuration for $7.00/mscf gas

(c) Optimal configuration for higher completion cost ($550k per stage)

Fig. 3.13: Permeability maps for sensitivity cases

52

3.4.1 Tuning Results

The addition of geomechanical effects creates complications in the surrogate models.

After some experimentation, we determined that in order to best reproduce the full-

physics production profile, the surrogate model needed to include desorption and a

pressure-dependent permeability multiplier, Mg(P ), applied in the SRV. As discussed

in Section 2.2.2, these additions add about 3 seconds to the runtime of the surrogate

model, but ultimately provide a satisfactory match between the full-physics and sur-

rogate models. Based on results that will be presented in Section 3.4.2, we show the

tuning results for a six-well configuration in Fig. 3.14 (the tuned parameters are given

in Table 3.7). Each of the six wells is drilled to full length, operated at the minimum

BHP, and contains 25 fracture stages.

Although the six-well non-geomechanics and six-well geomechanics configurations

are slightly different, the effect of geomechanics on production is evident. The rate

at which field production declines in Fig. 3.14(a) compared to the non-geomechanics

case in Fig. 3.1(a) is a clear indication of the detrimental effects geomechanics can

have on production. The geomechanics field production declines much more rapidly,

and ultimately produces about 30% less gas over ten years. Even with this difference,

the modified surrogate model (which includes Mg(P )) is able to accurately capture

the production profile of the full-physics model with geomechanics. Fig. 3.14 shows

the match results for a six-well configuration. The agreement for this case is high,

and the NPVs for both models are within 1% of each other.

The permeability multipliers here (Table 3.7) are lower than the optimal values

for the non-geomechanics case in Section 3.1. As discussed earlier, the permeability

in the SRV decreases as a function of pressure based on the introduction of Mg(P )

53

(a) Gas rate comparison

(b) Cumulative production comparison

Fig. 3.14: Tuning results for a six-well development plan with geomechanics

54

as given in Eq. 2.10. A summary of the geomechanics matching results is provided in

Table 3.8.

Table 3.7: Optimal Tuning Parameters for Six-well Configuration with and withoutGeomechanics

Case Mφ Mk

Six-well (Geom.) 0.98 55Six-well (No Geom.) 0.89 70

Table 3.8: Summary of Tuning Results with Geomechanics

Model, Configuration NPV ($MM) Cum. Production (bcf)

Full-Physics, Nw=6 30.0 48.1Surrogate, Nw=6 29.8 48.2

3.4.2 Integrated Workflow Results

In this section we incorporate geomechanics into the integrated workflow. All other

parameters are the same as in the previous example. The initial guess for the field de-

velopment optimization is the optimal configuration found for the non-geomechanics

case shown in Fig. 3.11. Once again, multiple workflow runs should be performed,

but here we show the results for a single run.

The progress of the optimization is shown in Fig. 3.15. The workflow com-

pletes just over 12,500 function evaluations, which requires around 16 minutes of

CPU time. Similar to the non-geomechanics integrated workflow example, the tun-

ing process is the most computationally expensive portion of the workflow. Here we

spend about 33 hours tuning due to the expensive nature of the full-physics coupled

55

flow-geomechanics simulation. We see that the tuning parameters remain similar

throughout the integrated workflow, as shown in Table 3.9.

Table 3.9: Optimal Tuning Parameters for the Integrated Workflow Example withGeomechanics

Case Mφ Mk

Base-case 1.03 57.1Re-tuning #1 1.01 54.2Re-tuning #2 0.99 53.7

After two field development optimization runs, the NPV increases by about 8%

to $28.7 MM. The increase in NPV over the base-case is primarily driven by the

increase in the number of fracture stages per well. The optimal field configuration

with geomechanics contains an average of 8 more fracture stages per well (24) relative

to the optimal solution from the non-geomechanics case (16 per well). This result is

consistent with the expected effect of adding geomechanics to the simulation model.

Geomechanical effects reduce the efficiency of the hydraulic fractures, so the optimal

configuration now requires more fracture stages to stimulate more of the reservoir.

The geomechanics optimal field development plan is shown in Fig. 3.16. This example

indicates that geomechanics can have an impact on the optimal field development plan

and should be considered during optimization.

56

Fig. 3.15: Progress of the optimization during the integrated workflow with geome-chanics. Stars indicate tuning/re-tuning

Fig. 3.16: Permeability map for optimal field development plan with geomechanics

57

58

Chapter 4

Concluding Remarks

4.1 Conclusions

In this thesis, we built on the shale field development framework introduced by Wilson

and Durlofsky [42]. We switched the optimization algorithm from GPS to PSO-

MADS, added new decision variables (drill/do not drill and BHP variables) to the

problem statement, and incorporated coupled flow-geomechanics. Using PSO-MADS

enables us to utilize the global search benefits of PSO in conjunction with a local

search (MADS). The integrated workflow uses a surrogate model in the place of a

full-physics model for function evaluations. However, in order to use the surrogate

model it must be representative of the full-physics model. We therefore utilized a

tuning process to ensure that the models are in essential agreement. This process

is applied for cases with and without geomechanics. PSO-MADS is used to find the

optimal tuning parameters and to find an optimal field development plan with the

tuned surrogate model.

A number of example cases were presented using reservoir parameters based on the

59

Barnett and Marcellus plays. First, using a Marcellus example without geomechanics,

we showed that the tuned surrogate model was able to closely match the NPV of the

full-physics model. A similar result was later shown for the geomechanics (Marcellus)

case.

Next, a Barnett example was used to validate the new drill/do not drill feature.

Specifically, in all three runs performed, the optimization correctly provided the op-

timal number of wells for the field (as determined through a comprehensive search).

Based on the results noted above, we established that the two separate optimiza-

tion problems, tuning and field development, were being solved effectively. The third

example, based on the Marcellus, demonstrated the application of the integrated

workflow. Through a combination of tuning and field development optimization we

determined optimal development plans in a matter of hours of elapsed time. Finally,

we used a Marcellus example to investigate the impact of geomechanics on optimiza-

tion results. Using the surrogate model in place of the full-physics model is especially

useful in cases with geomechanics where full-physics runs can take 7-12 hours. We

observed a difference in the optimal configurations between the geomechanics and

non-geomechanics cases, leading us to conclude that geomechanics should be consid-

ered when creating an optimal field development plan.

4.2 Suggestions for Future Work

There are a number of directions in which future research could be performed. One

extension would be to add heterogeneity to the model. An additional step would be

the introduction of multiple geologic models, which would allow us to maximize the

expected value of the field while accounting for geological uncertainty. Both of these

60

extensions should be compatible with the overall framework.

An optimal re-tuning strategy should also be developed such that the framework

can intelligently determine when re-tuning is required. This may be done through

the use of an appropriate error estimator. Finally, the impact of geomechanics on

the optimal drilling sequence should be investigated. Depletion of the reservoir and

subsequent increases in effective stress from existing wells may impact the propagation

and conductivity of hydraulic fractures. Using the coupled flow-geomechanics feature

should allow us to optimize drilling sequence, along with the other parameters. In

combination, these additions might allow for shale field development optimization to

be completed in a closed-loop process similar to the method developed by Shirangi

and Durlofsky [36] for conventional reservoirs.

61

62

Nomenclature

Abbreviations

BHP Bottomhole pressure

GPS Generalized Pattern Search

MADS Mesh Adaptive Direct Search

MINLP Mixed Integer Nonlinear Programming

mscf 1000 standard cubic feet

NPV Net present value

PSO Particle Swarm Optimization

SRV Stimulated reservoir volume

Symbols

β Forchheimer correction

u Set of optimization variables

v Darcy velocity

63

µ Viscosity

φm Matrix porosity

φs SRV porosity

ρ Phase density

σ′ Effective stress

σT Total stress

Bk BHP of well k

CLOE Lease operating cost

Dk Drill/do not drill variable

e−rti Discount factor

F c Fracture conductivity

G Gas price

kf Fracture permeability

ks SRV permeability

Lk Lateral length of well k

Mφ Porosity multiplier

Mk Permeability multiplier

Nw Number of wells

64

Nf,k Number of fracture stages in well k

P Pressure

Qfpik Full-physics gas production over time step i for well k

Qsurik Surrogate model gas production over time step i for well k

Qi Cumulative production over time step i

r Interest rate

Rp Royalty percentage

T Total number of time steps

Tc Transportation and gathering cost

wf Fracture aperture

X Tax rate

xk Well location

xf Hydraulic fracture half length

Subscripts

c Component c

i Time step

s Stimulated property

k Well

65

66

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