OPTIMIZATION STRATEGIES FOR SHALE GAS ASSET DEVELOPMENT A THESIS SUBMITTED TO THE DEPARTMENT OF ENERGY RESOURCES ENGINEERING OF STANFORD UNIVERSITY

Transcription

1 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

2 c Copyright by Jamal Cherry 2016 All Rights Reserved ii

3 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 iii

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5 Abstract With the recent boom in US shale gas production, optimal development of these assets has become a topic of significant interest. In addition to the complex physics typically associated with shale plays, field development optimization can be a challenging 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 application 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 surrogate 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 algorithm, to find the optimal tuning parameters for the surrogate model and to find an optimal field development plan. During the asset development optimization we consider 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

6 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. vi

7 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 encouragement 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 affiliates of the Stanford Smart Fields Consortium. vii

8 Contents 1 Introduction 1 2 Full-Physics and Surrogate Model Full-Physics Simulation Model Gridding and Model Properties Coupled Flow and Geomechanics Surrogate Model and Tuning Process Surrogate Model Description Incorporating Geomechanics into the Surrogate Model Tuning Process PSO-MADS Optimization Algorithm Field Development Optimization Integrated Workflow Results and Discussion Tuning Results Field Development Optimization Results Marcellus Example Without Geomechanics viii

9 3.3.1 Integrated Workflow Results Sensitivity Results Marcellus Example With Geomechanics Tuning Results Integrated Workflow Results Concluding Remarks Conclusions Suggestions for Future Work ix

10 List of Tables 2.1 Reservoir Properties for the Full-Physics Marcellus Model Field Development Economic Parameters Field Development Decision Variables and Constraints Marcellus and Barnett Reservoir Parameters Initial Guess and Resulting Optimal Tuning Parameters for Six-well and Four-well Configurations Summary of Tuning Results Summary of Barnett Field Development Optimization Optimal Tuning Parameters for the Integrated Workflow Example Marcellus Geomechanical Properties for Coupled Flow-Geomechanics Simulations Optimal Tuning Parameters for Six-well Configuration with and without Geomechanics Summary of Tuning Results with Geomechanics Optimal Tuning Parameters for the Integrated Workflow Example with Geomechanics x

11 List of Figures 1.1 Projected shale gas growth to 2040 (from [12]) Illustration of full-physics horizontal well model. The wellbore is shown in red with the hydraulic fracture in each stage shown in blue Natural fracture network in a Marcellus shale outcrop (from Engelder et al. [13]) Effect of desorption on gas production in Marcellus shale (from Heller and Zoback [20]) Fracture conductivity as a function of effective stress in Marcellus shale (from McGinley et al. [28]) Normalized fracture permeability as a function of effective stress for fractures in Marcellus shale (from McGinley et al. [28] and Cipolla et al. [9]) Normalized permeability as a function of pore pressure (adapted from McGinley et al. [28]) xi

12 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 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]) Flowchart of PSO-MADS hybrid algorithm (from [22]) Schematic of the integrated workflow Tuning results for a six-well development plan Progress of the tuning optimization for the six-well configuration Results using initial guess for the six-well configuration Tuning results for a four-well development plan NPVs for optimal configurations at each well count NPVs for optimal configurations at each well count (circles). The star indicates the NPV of the best variable well count case Permeability and pressure map of the best optimum from the variable well count case Progress of the optimization during the variable well count case Progress of the optimization during the integrated workflow. Stars indicate tuning/re-tuning Permeability map of the base-case configuration for the integrated workflow without geomechanics Permeability map for optimal configuration from integrated workflow without geomechanics xii

13 3.12 Final pressure map for optimal configuration from integrated workflow without geomechanics Permeability maps for sensitivity cases Tuning results for a six-well development plan with geomechanics Progress of the optimization during the integrated workflow with geomechanics. Stars indicate tuning/re-tuning Permeability map for optimal field development plan with geomechanics 57 xiii

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15 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 unrecoverable. 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

16 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

17 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 dualpermeability method with local grid refinement [9, 44]. Mayerhofer et al. [27] employed a stimulated reservoir volume technique with natural fractures explicitly modeled. 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 dualporosity, 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

18 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 offset 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 comprehensive optimization workflow that considered well spacing and completion design using an analytical model to predict well production. This method should be applicable 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 simple (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-towell 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

19 value (NPV) of the field was also not considered. Wilson [41] and Wilson and Durlofsky [42] combined a number of the ideas presented above into a general workflow. They applied this simulation-based optimization 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 development 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 optimizations were typically considered individually, but are more appropriately viewed jointly. Specifically, Zandvliet et al. [46] and Forouzanfar and Reynolds [15] demonstrated 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 producer), their locations, the drilling sequence, and time-varying well BHP profiles. This work utilized a hybrid global-local optimization algorithm and demonstrated its efficiency 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 Direct Search), and a global method, PSO (Particle Swarm Optimization), to search 5

20 through complex solution spaces that may contain many local optima. This thesis aims to advance shale field optimization by incorporating recent methods used in conventional field development. Here, we will combine aspects of previously 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, dualporosity, 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 parameters, SRV porosity and permeability multipliers, are determined through a history matching process that minimizes the difference in gas production between the original 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 development plan. During the field development optimization we consider 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. Previously, the development plan was constrained to include a specified number of wells, but here 6

21 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 example 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

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23 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 Gridding and Model Properties One of the plays modeled in this work is the Marcellus shale, and we describe our modeling 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

24 5300 ft 162 ft. This domain is represented on a Cartesian grid of dimensions 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 permeability value that maintains the actual fracture conductivity (F c = w f k f, where F c is fracture conductivity, w f is the fracture aperture and k f is the fracture permeability). Here, we employ a simple bi-wing planar hydraulic fracture representation where the fracture half length, x f, is 500 ft, and each hydraulic fracture is surrounded by a logrithmically spaced, locally refined grid, as shown in Fig

25 Fig. 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

26 The grid blocks around each hydraulic fracture and along the wellbore are logarithmically 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 perforation 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 k m v + βρv 2, (2.1) where P is pressure, µ is viscosity, v is Darcy velocity, k m 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 = , (2.2) φ m km

27 Fig. 2.2: Natural fracture network in a Marcellus shale outcrop (from Engelder et 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 equation is given as: ωc = ωc,max Bc yc,g P P, 1 + P Bj yj,g (2.3) j 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

28 Fig. 2.3: Effect of desorption on gas production in Marcellus shale (from Heller and Zoback [20]) In this work, however, we only consider a single methane component, so the summation is not needed. Here B c and ω c,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 fractures), additional physics (Knudsen diffusion, non-darcy flow in natural fractures) and additional completion design effects (type of proppant, proppant loading levels) 14

29 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 Grid Cell Dimension ft Reservoir Depth 8593 ft Initial Reservoir Pressure 4726 psi Matrix Porosity, φ m 6% Matrix Permeability, k m md Fracture Half Length, x f 500 ft 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 particularly 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

30 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 simulator, 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 calculations. In the isothermal case, it treats porosity as a function of pressure and total mean stress (σ m ) using the following formula developed by Tran et al. [38]: φ n+1 = φ n + (c 1 + c 2 a 1 )(P P n ), (2.4) where: c 1 = 1 Vb 0 ( dv p dp + V dσ m bαc b ), (2.5) dp c 2 = V p αc b, (2.6) Vb 0 a 1 = 2 E 9 (1 ν) αc b, (2.7) where φ is porosity, c b is bulk compressibility (1/psi), V p is pore volume (m 3 ), E is Young s modulus (psi), V b is bulk volume (m 3 ), V 0 b Biot number, and ν is Poisson s ratio. is initial bulk volume, α is the Additionally, GEM allows for the input of a table describing permeability as a 16

31 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 fractures 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 vertical 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 normalized conductivity chart for the hydraulic and natural fractures. Our treatment is 17

32 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 order 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

33 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 fractures 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 = w f k f ). Thus the normalized fracture conductivity plot is identical to the normalized fracture permeability plot. This logic is also applicable to the natural fracture data set where the initial fracture permeability is 10 md with an aperture of 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 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 w f. This process was applied to the hydraulic fracture conductivity data from McGinley 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 hydraulic and natural fractures are shown in Fig 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

34 Fig. 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]) a coupled flow-geomechanics simulation is the increase in runtime. Our coupledgeomechanics 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

35 2.2.1 Surrogate Model Description The surrogate model is a computationally inexpensive representation of the fullphysics 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, desorption 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, M k and M φ respectively, for the original matrix parameters, k m and φ m : φ s = M φ φ m, (2.8) k s = M k k m. (2.9) The stimulated reservoir parameters, k s and φ s, 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 x f. 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, k m and φ m, 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

36 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 pressuredependent permeability multiplier, M g (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 k s is k s (P ) = M g (P ) M k k m. (2.10) Fig. 2.6: Normalized permeability as a function of pore pressure (adapted from McGinley 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 This is accomplished by assuming a Biot coefficient value of 1 and using 22

37 the relationship: σ = σ T P, (2.11) where σ T 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) Tuning Process In order to use the surrogate model in place of the computationally expensive fullphysics model, the surrogate model production must match, to within a small tolerance, 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, M k and M φ, are adjusted to achieve satisfactory agreement with the full-physics model. To systematically search through the M k -M φ 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 N w i=1 k=1 ( ) 2 Q fp ik Qsur ik e rt i, (2.12) where Q fp ik and Qsur ik are respectively the full-physics and surrogate model gas production over time step i for well k. Here, N w 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

38 function R to the quantity of primary interest, NPV, we include a discount factor, e rt i, which essentially places a higher weighting on matching early gas production. Here, t i 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 M k and M φ. 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

39 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

40 (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 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

41 PSO-MADS combines the individual advantages of the MADS and PSO techniques 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 improvement 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 improving, 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 violations 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 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, M k and M φ, we can now use the surrogate 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

42 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, L k, determines the length of the horizontal well in the y- direction. The variable N f,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 x f 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 See [23, 24] for discussion of the treatments for integer and binary variables. 28

43 The objective function we now seek to maximize is the NPV of the field: NP V = T DCF i C cap, (2.13) i=1 where DCF indicates discounted cash flow and the capital expenditure is represented by C cap. The equation for DCF is based on the work of Williams-Kovacs and Clarkson [39]. Discounted cash flow is given as: DCF i = (1 X) [Q i (G T )(1 R p ) C LOE ] e rt i, (2.14) where Q i 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, R p is the royalty percentage, and C LOE 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 encompasses drilling, completion and leasing costs: C cap = C drill + C frac + C lease. (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, C lease 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

44 Table 2.2: Field Development Economic Parameters Economic Parameter Value Gas Price $4.00/mscf Drilling $500/ft Completion $150,000/stage Transportation & Gathering $0.70/mscf Operating $30/well/day Royalty 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 = [D 1, x 1, N f,1, L 1, B 1, D N, x N, N f,n, L N, B N ]. (2.16) Each well is described by a set of five variables. The variable D k is binary and determines whether or not well k is drilled. The variables x k, N f,k, L k are all integerbased (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 B k 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

45 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, D k N k=1 D k 10 Well Location, x k x k x k 1 2x f Fracture Stage Count, N f,k 2 50 Lateral Length, L k ft Minimum BHP, B k 535 psi 2.5 Integrated Workflow In Sections 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: M k and M φ. The second optimization, a field development optimization, uses M k and M φ 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

46 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

47 Fig. 2.9: Schematic of the integrated workflow 33

48 34

49 Chapter 3 Results and Discussion In this chapter we present results using the methods and workflow discussed in Chapter 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

50 Table 3.1: Marcellus and Barnett Reservoir Parameters Reservoir Property Marcellus Barnett Pressure 4726 psi 3800 psi Temperature 175 F 180 F Matrix Porosity, φ m 6% 4% Matrix Permeability, k m md md Langmuir Volume 28.3 scf/ton 88 scf/ton Langmuir Pressure psi 440 psi Minimum Well BHP 535 psi 1000 psi Simulation Time 10 years 10 years 3.1 Tuning Results Tuning the surrogate model is a critical part of this work. As discussed in Section 2.2.3, the tuning process utilizes PSO-MADS to find optimal tuning parameters, M k and M φ, which minimize R in Eq 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 configuration and a four-well configuration. The six-well configuration is based on a result shown later in Section 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 Comparisons of monthly gas rates and cumulative production are shown in Fig. 3.1(a) and (b), 36

51 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 and Four-well Configurations Tuning Parameter M φ M k Initial Guess 1 17 N w = N w = The progress of the tuning optimization for the six-well configuration is shown in Fig 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 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 M φ and M k for this case are shown in Table 3.2. Note that M φ 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, M k, 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

52 (a) Gas rate comparison (b) Cumulative production comparison Fig. 3.1: Tuning results for a six-well development plan 38

53 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 M φ and M k values for the two configurations are similar. 39

54 Table 3.3: Summary of Tuning Results Model, Configuration NPV ($MM) Cum. Production (bcf) Full-Physics, N w = Surrogate, N w = Full-Physics, N w = Surrogate, N w = Fig. 3.3: Results using initial guess for the six-well configuration 40

55 (a) Gas rate comparison (b) Cumulative production comparison Fig. 3.4: Tuning results for a four-well development plan 41

56 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 N w = 6, where N w is the number of wells. The optimizer then provides x k, N f,k, L k and B k 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 This process was repeated for N w = 7, 8, 9 and 10. Each of these four cases was again run three times, and the resulting NPVs are plotted in Fig 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

57 The maximum NPV in these results ($35.7 MM) occurs with N w = 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 N w 10). Results for this case are shown in Table 3.4. In each of the three runs, PSO-MADS found an optimum N w 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 N w set to 8) is a satisfactory result. Fig. 3.6: NPVs for optimal configurations at each well count (circles). indicates the NPV of the best variable well count case The star 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, M k and M φ, and blue areas are the unstimulated region of the reservoir. 43

58 Table 3.4: Summary of Barnett Field Development Optimization N w =7 N w =8 N w =9 Variable NPV Run 1 ($MM) (N w =8) NPV Run 2 ($MM) (N w =8) NPV Run 3 ($MM) (N w =8) Average ($MM) 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 optimization, as the well count increases from the initial value of N w = 2. After 500 function evaluations the optimal configuration contains ten wells, and after 1800 function evaluations the well count remains constant at eight wells. Further NPV improvements are the result of optimizing other decision variables, such as N f,k and L k. 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

59 (a) Permeability map (b) Final pressure map Fig. 3.7: Permeability and pressure map of the best optimum from the variable well count case 45

60 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 Integrated Workflow Results The reservoir parameters for this case are shown in Table 3.1. As discussed in Section 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 development 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

61 start the field development optimization, as shown in the schematic in Fig 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 reaching 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 development 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 The final pressure map is displayed in Fig Function evaluations (simulations) during field development optimization are run in parallel using 100 nodes. The integrated workflow required just under 23,000 function 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 required 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

62 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 indicate tuning/re-tuning Table 3.5: Optimal Tuning Parameters for the Integrated Workflow Example Case M φ M k Base-case Re-tuning # Re-tuning # Re-tuning #

63 Fig. 3.10: Permeability map of the base-case configuration for the integrated workflow without geomechanics 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 alongside the original optimal configuration from the example shown in Section 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 Here, we see that the optimizer 49

64 Fig. 3.11: Permeability map for optimal configuration from integrated workflow without geomechanics Fig. 3.12: Final pressure map for optimal configuration from integrated workflow without geomechanics 50

65 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 production 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 reservoir 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 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 Simulations Geomechanical Property Value Young s Modulus psi Biot Coefficient 1.0 Poisson s Ratio

66 (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

67 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 fullphysics production profile, the surrogate model needed to include desorption and a pressure-dependent permeability multiplier, M g (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 surrogate 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 (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 M g (P )) is able to accurately capture the production profile of the full-physics model with geomechanics. Fig 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 M g (P ) 53

68 (a) Gas rate comparison (b) Cumulative production comparison Fig. 3.14: Tuning results for a six-well development plan with geomechanics 54

69 as given in Eq 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 without Geomechanics Case M φ M k Six-well (Geom.) Six-well (No Geom.) Table 3.8: Summary of Tuning Results with Geomechanics Model, Configuration NPV ($MM) Cum. Production (bcf) Full-Physics, N w = Surrogate, N w = 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 development optimization is the optimal configuration found for the non-geomechanics case shown in Fig 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 The workflow completes just over 12,500 function evaluations, which requires around 16 minutes of CPU time. Similar to the non-geomechanics integrated workflow example, the tuning 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

70 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 with Geomechanics Case M φ M k Base-case Re-tuning # Re-tuning # 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 This example indicates that geomechanics can have an impact on the optimal field development plan and should be considered during optimization. 56

71 Fig. 3.15: Progress of the optimization during the integrated workflow with geomechanics. Stars indicate tuning/re-tuning Fig. 3.16: Permeability map for optimal field development plan with geomechanics 57