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Accepted at Computational Geosciencesdoi:10.1007/s10596-018-9728-6

Optimal design of hydraulic fracturing in porous media usingthe phase field fracture model coupled with genetic algorithm

Sanghyun Lee · Baehyun Min∗ · Mary F. Wheeler

Received: date / Accepted: date

Abstract We present a framework for the coupling of fluid-filled fracture propagation anda genetic inverse algorithm for optimizing hydraulic fracturing scenarios in porous media.Fracture propagations are described by employing a phase field approach, which treats frac-ture surfaces as diffusive zones rather than of interfaces. Performance of the coupled ap-proach is provided with applications to numerical experiments related to maximizing pro-duction or reservoir history matching for emphasizing the capability of the framework.

Keywords Hydraulic fracturing · Phase field · Genetic algorithm · Production maximiza-tion · History matching

1 Introduction

Multistage hydraulic fracturing (i.e., fracking) has been considered the most pragmatic tech-nology to improve the productivity of hydrocarbon-bearing shale formations since hydraulicfractures play the role of conductive conduits in tight porous media. For fracking, slickwatercontaining additives (e.g., proppants and chemicals) are injected at sufficiently high rates andpressures above well fracture pressure down into a well. The created fractures are kept openby the proppants, and provide increased contact between the wellbore and the shale forma-tion, resulting in increased oil and gas production. Recent advances in hydraulic fracturingwith horizontal drilling have increased the area of drainage in contact with the wellbore bysix orders of magnitude [62].

Sanghyun LeeDepartment of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL 32306-4510,United States E-mail: [email protected]

Baehyun Min∗

Department of Climate and Energy Systems Engineering, Division of Sustainable Systems Engineering,Ewha Womans University, Ewhayeodae-gil 52, Seodaemun-gu, Seoul 03760, Republic of Korea. E-mail:[email protected]

Mary F. WheelerCenter for Subsurface Modeling, The Institute of Computational Engineering and Sciences, The Universityof Texas at Austin, 201 East 24th Street, Austin TX 78712, USA

2 S.Lee B.Min M.F. Wheeler

Productivity of a fractured hydrocarbon-bearing formation depends on reservoir proper-ties (e.g., permeability) and operating parameters (e.g., well locations, the number of frac-turing stages, and stage spacing). A study of an optimal number of hydraulic fractures foreach horizontal well at Bakken shale model was presented in [58]. It was found that in-creasing the density of hydraulic fractures and decreasing the fracturing spacing enhancedthe productivity, in general; however, the incremental production resulting from additionalfracking decreased [23]. The longer the fracture length was, the more profitable the wellwas; however, the revenue increment decreased due to the increase in treatment volume[61, 10]. Maximizing the net present value of a tight gas reservoir was also studied in [37].

Fig. 1: Schematic of wellbore and near-wellbore of a horizontal well with multiple hydraulicfractures.

Fractured zones are often mimicked as thin planes in which enhanced permeability val-ues are allocated. This simplified approach might ignore interference effects among hy-draulic fractures called stress-shadowing. The stress shadowing effect refers to suppressionof hydraulic fractures resulting from compressive stresses exerted on them by neighboringhydraulic fractures [1, 17, 39]. If the distance between the hydraulic fractures along the wellis not far enough, the propagation of those fractures is affected by the interaction betweenthe stress around each fracture. The interferences may result a less effective fracture propa-gation followed by the deterioration of hydrocarbon production. In Bakken and Utica shalewells, for example, Hess Corporation [15] deployed downhole geophones to monitor micro-seismic events associated with multistage hydraulic fracturing and observed the stress shad-owing effect. The observation showed that closely spaced fracturing stages were interferedwith one another and bounced cyclically in and out of the fractured zone. For maximizingthe performance of fractured horizontal wells, it is necessary to develop a model that cansimulate the propagation of multiple fractures while taking the interaction between adjacentfractures into account [54, 70, 36, 15, 12, 66, 67, 68]. Recently, Cheng et al. [12] proposedapplying the C2Frac model for simulating the growth of hydraulic fractures from multipleperforation intervals in a single stage with low computational costs. The effectiveness of theC2Frac model was verified by comparing numerical solutions provided by a fully-coupledplanar 3D simulator ILSA [56, 55].

In this paper, we investigate the hydraulic fracturing simulation by using the variationalmethods and energy minimization that has been widely developed in [9, 18]. The numericalapproaches for the studies are based on Ambrosio-Tortorelli elliptic functionals [4, 5]. An

Optimal design of hydraulic fracturing 3

important numerical extension of [18] towards a thermodynamically consistent phase fieldfracture model has been studied in [40, 42]. In the phase field fracture approach, disconti-nuities in the displacement field u across the lower-dimensional crack surface are approx-imated by an indicator scalar function ϕ ∈ [0,1]. This phase field function ϕ introduces adiffusive transition zone, which has a half bandwidth ε , between the fracture (ϕ = 0) andthe un-fractured (ϕ = 1) material. This approach becomes as a variational inequality sincethe fracture propagation is required to satisfy a crack irreversibility constraint.

A crucial step to applying variational phase field fracture techniques in subsurface mod-eling (Biot system) has been achieved in [44]. Here, the phase field fracture has been ex-tended to poro-elasticity in which the pressure is included. The concepts outlined in [44]have been further substantiated both theoretically and numerically in [47, 45, 48, 33]. Fur-ther studies using phase-field fracture techniques treating multiphysics include [41, 38, 21,30, 35, 65, 31, 29]. Advantages of the phase-field method are that joining and branching ofcurvilinear paths are automatically included in the energy minimization; that is, calculatingstress intensity factors and re-meshing along the crack path are embedded in the model [25].Because the phase field fracture approach provides detailed and robust fracture propagationin heterogeneous reservoir and geomechanical parameters, and can be extended to multiphysics problems [29, 30], it is important to investigate coupling techniques for optimiza-tion and history matching.

The objective of this study is to establish a numerical simulation framework by com-bining a phase field fracture model and an evolutionary optimization algorithm to tacklerealistic scenarios. For example, the framework can be used for maximizing the productiv-ity of multistage hydraulic fracturing at a tight shale formation by avoiding stress shadowingeffect. In particular, here for convenience we adopt a non intrusive approach such as geneticalgorithm as the optimization module in the framework. The performance of the coupledmodel is demonstrated with applications to several numerical experiments. In addition, wenote that the phase-field model is being integrated with a reservoir simulator for predictingflow rate. Here, as an alternative, the total volume (or total area for two dimension) of hy-draulic fractures is employed as the metric of well productivity. The relation between theproduction and the area of the fractures is discussed in [69].

2 Methods

2.1 Modeling of hydraulic fractures with phase field

In this section, we briefly recapitulate essential phase field fracture propagation concepts,which have been modeled and implemented in [33, 45, 47, 48].

Fig. 2: An example of a fracture defined with the phase field function ϕ ∈ [0.1] .


Here, Λ := Λ(t) ∈ Rd (d = 2,3) is a smooth open and bounded computational domainwith Lipschitz boundary ∂Λ := ∂Λ(t). Let [0,T ] be the computational time interval withT > 0 and time is denoted by t ∈ [0,T ]. As discussed in [9, 18], the fracture C (t) is con-tained compactly in Λ(t). The displacement of the solid and diffusive flow in an un-fracturedporous medium (reservoir) are modeled in ΩR := ΩR(t) = Λ\C (t) by the classical quasi-static elliptic-parabolic Biot system for a porous solid saturated with a slightly compressibleviscous fluid.

2.1.1 Geomechanics and phase-field fracture equations

The constitutive equation for the Cauchy stress tensor is given as

σpor(u, pR)−σ0 = σR(u)−α(pR− p0)I, in ΩR× (0,T ], (1)

where u(·, t) : ΩR× [0,T ]→ Rd is the solid’s displacement, pR(·, t) : ΩR× [0,T ]→ R is thepressure in the reservoir with an initial pressure p0, α ∈ [0,1] is the Biot coefficient, I is theidentity tensor, and σ0 is the initial stress value.

Given the effective linear elastic stress tensor as

σR := σR(u) = λ (∇ ·u)I +2µe(u), (2)

the balance of linear momentum in the solid reads

−∇ ·σpor(u, pR) = ρsg in ΩR× (0,T ], (3)

where λ ,µ > 0 are the Lame coefficients, the linear elastic strain tensor is given as e(u) :=12 (∇u+∇uT ), ρs is the density of the solid, and g is the gravitational acceleration term.For simplicity, we prescribe homogeneous Dirichlet boundary conditions on ∂Λ for thedisplacement u.

To discuss the phase field fracture, we first introduce the Francfort-Marigo functional[18], which describes the energy with a fracture in an elastic medium as

E(u,C ) =12

∫ΩR

σR(u) : e(u)+GcHd−1(C ) dx, (4)

where the Hausdorff measure Hd−1(C ) denotes the length of the fracture and is multipliedby Gc, the fracture toughness value (critical energy release rate). Extension to the energywith a fracture in a poro-elastic medium based on Biot system was derived and discussed in[33, 45, 47, 48], which is

E(u, p,C ) =12

∫ΩR

σR(u) : e(u) dx−∫

ΩR

α pR∇ ·u dx+GcHd−1(C ). (5)

Next, we consider the global constitutive dissipation functional of Ambrosio-Tortorellitype [4, 5] to describe the Hd−1(C ) with the phase field function for a rate independentfracture process. The latter was initially introduced for linear elasticity in [9, 18, 42]. Weemphasize that here the fracture C (t) is regarded as a thin three-dimensional volume ΩF(t),where the thickness of the crack is much larger than the pore size of porous media. Thescalar-valued phase field function ϕ(·, t) : Λ × [0,T ]→ [0,1] can be viewed as an indicatorfunction, which introduces a diffusive transition zone (0 < ϕ < 1) between the fracturedregion (ΩF ) having ϕ = 0 and the un-fractured porous media (ΩR) having ϕ = 1. The diffu-sive zone has length ε > 0 on each side of the fracture. The intermediate values constitute a


smooth transition zone dependent upon the regularization parameter (length) ε . See Figure2 for more details. The boundary of the fracture is denoted by ΓF(t) := ΩF(t)∩ ΩR(t).

We assume that the effective fluid stress is given by σF := σ |ΩF =−pF I and continuityof normal stresses:

(σR(u)−α pRI) ·nF =−pF nF on ΓF , (6)

where nF is the outward pointing normal vector of the interface ΓF and pF = p|ΩF is thepressure in the fracture. The leading order of the fracture fluid stress is derived from lubri-cation theory in [45, 48]. In addition, we use the continuity of pressures

pR = pF on ΓF , (7)

and obtain:(σR(u)−α pI) ·nF =−pnF on ΓF , (8)

by p = pR = pF . We do not distinguish between σR, pR and σF , pF in the global dissipationform by employing the definition of ϕ and the interface conditions (7)-(8). We rewrite (5)as the global formation where all definitions are extended to Λ . Then, we obtain

Eε(u, p,ϕ) =∫

Λ

12((1−κ)ϕ2 +κ)σ(u) : e(u) dx−

∫Λ

(α−1)ϕ2 p∇ ·u dx

+∫

Λ

(ϕ2∇p)u dx+Gc

∫Λ

(1

2ε(1−ϕ)2 +

ε

2(∇ϕ)2

)dx, (9)

where ε is the length of the diffusive zone presented in Figure 2 and κ ε is a small positiveregularization parameter which is set to 10−10.

Moreover, fracture evolutions satisfy a crack irreversibility condition that is an inequal-ity condition in time: ∂tϕ ≤ 0. Consequently, the resulting system can be characterizedas a quasi-stationary (time-dependent) variational inequality, in which the continuous ir-reversibility constraint is approximated by ϕ ≤ ϕold, where ϕold is the phase field solutionat the previous time step and ϕ is the current solution. The phase field function is subject tohomogeneous Neumann conditions on ∂Λ . The initial domains, ΩF(·,0) and ΩR(·,0), aredefined by a given phase field value ϕ(·,0).

Finally, we introduce a strong form of the Euler-Lagrangian formulation for the dis-placement and phase field: let p and ϕ be given, then find u such that

−∇ ·((

(1−κ)ϕ2 +κ)σ++σ

−)+(α−1)∇ · (ϕ2 p)+ϕ

2∇p = 0 in Λ ,

u = 0 on ∂Λ ,

(α−1)ϕ2 p ·n = 0 on ∂Λ ,

where n is the outward normal vector of ∂Λ . Similarly, let u and p be given, then find ϕ

such that((1−κ)σ+ : e(u)ϕ−Gcε∆ϕ− Gc

ε(1−ϕ)

−2(α−1)ϕ p∇ ·u+2ϕ∇pu)·∂tϕ = 0 in Λ ,

∂nϕ = 0 on ∂Λ ,


with a compatibility condition (which is called the Rice condition in the presence of fractures[57]),

(1−κ)σ+ : e(u)ϕ−Gcε∆ϕ− Gc

ε(1−ϕ)−2(α−1)ϕ p∇ ·u+2ϕ∇pu≤ 0 in Λ ,

∂tϕ ≤ 0 in Λ .

In the discretized system, we assume h ε , where h is the local discretization parameter.We follow [6], where the stress tensor is additively decomposed into a tensile part σ+(u)

and a compressive part σ−(u) by:

σ+ := σ

+(u) = (2d

µ +λ )tr+(e(u))I +2µ(e(u)− 1d

tr(e(u))I), (10)

σ− := σ

−(u) = (2d

µ +λ )tr−(e(u))I, (11)

respectively. Here tr denotes the trace of the matrix and

tr+(e(u)) = max(tr(e(u)),0), tr−(e(u)) = tr(e(u))− tr+(e(u)). (12)

Note that the energy degradation only acts on the tensile part σ+(u).

2.1.2 A pressure diffraction system for modeling reservoir and fracture flow

In this section, the flow (pressure) is formulated for both reservoir and fracture as a pressurediffraction system following [33, 47]. To derive the pressure system for each sub-domain,first we consider the two separate mass conservation equations for the fluid in the reservoirand the fracture:

∂t(ρiφ?i )+∇ · (ρivi) = ρiqi in Λ × (0,T ], (13)

where i = F indicates the fracture domain ΩF and i = R indicates for the reservoir domainΩR. Here φ ?

i is the fluid fraction, where we assume φ ?F = 1 (porosity of the fracture is one)

and the reservoir fluid fraction is given as

φ?R := φ

?0 +α∇ ·u+

1M(p− p0),

where M is the Biot modulus coefficient. The velocity is given by Darcy’s law

vi =−Ki

ηi(∇pi−ρig), (14)

with the permeability Ki, fluid viscosity ηi for the fracture (i = F) and for the reservoir(i = R), respectively. We assume that the fluid in the reservoir and the fracture is slightlycompressible: thus, we define each fluid density as

ρi := ρ0i exp(ci(pi− p0

i ))≈ ρ0i [1+ ci(pi− p0

i )], (15)

where ρ0i is the reference density and ci is the fluid compressibility. In addition, qi is a

source/sink term including leak-off [47]. The leak-off term derived in [47](equation (2.10))depends on the fracture opening displacement values (aperture), which affects the total vol-ume of the fractures. Following the general reservoir approximation with the assumptionthat cR and cF are sufficiently small, we use ρR = ρ0

R and ρF = ρ0F .


Next, in the fracture flow equation, the fracture permeability is assumed to be isotropicand has the form

KF =112

w(u)2, (16)

as shown in [47]. Here, w(u) = [u · n] denotes the aperture (width) of the fracture, whichrequires the computation of the jump [·] of the normal displacements. The correspondingdetails of approximating the aperture of the non-planar fractures using a level set method andinterpolating these values inside the fracture are described in [34]. Non-isotropic lubricationlaws have been specifically derived for fluid-filled phase-field fractures in [47].

Finally, we formulate the pressure diffraction system by treating ϕ as an indicator func-tion. Let ϕ and u be given, find p(·, t) such that

ρ0∂tφ?+∇ · (ρ0Keff(∇p−ρ0g)) = ρ0q, in Λ × (0,T ], (17)

where the coefficient functions are defined as

ρ0 = χΩR ρ0R +χΩF ρ

0F , (18)

φ? = φ

?(·, t) := χΩR

( 1M

pR +α∇ ·u)+χΩF (cF pF), (19)

q = q(·, t) := χΩR qR +χΩF qF , (20)

Keff = Keff(u) := χΩR

KR

ηR+χΩF

KF

ηF, (21)

where χΩR ,χΩR are linear interpolation functions satisfying χΩR = 1 and χΩF = 0 in ΩR,and χΩR = 0 and χΩF = 1 in ΩF . In particular, once the phase field value ϕ is computed, wedefine,

χΩF :=1−H(ϕ)

2, χΩR :=

1+H(ϕ)

2(22)

with the Heaviside function H(·) given as

H(s) =

1 if s > 0.5+Dε ,−1 if s < 0.5−Dε ,

s−0.5Dε

otherwise.(23)

For simplicity, we set Dε = 0.1 throughout this paper.We prescribe the boundary and interface conditions for pressure as

KR(∇pR−ρ0Rg) ·n = 0 on ∂Λ × (0,T ], (24)

[p] = 0 on ΓF × (0,T ], (25)

KRρ0R

ηR(∇pR−ρ

0Rg) ·nF =

KF ρ0F

ηF(∇pF −ρ

0F g) ·nF on ΓF × (0,T ], (26)

Extensions to proppant transport with non-Newtonian flow and two phase flow in the frac-ture have been developed in [30, 29].


Eachtime step n

Solve pn

with un−1 and ϕn−1Solve un andϕn with pn

RefineMesh

n⇒ n+1

Fixed-stress iterations

Mesh adaptivity (predictor-corrector)

Fig. 3: Flow chart of global solution algorithm for the phase-field model.

2.1.3 Global solution algorithm for phase field fracture

Figure 3, we present the overall algorithms for the phase field fracture propagation. First,the geomechanics-phase-field system are discretized by classical continuous Galerkin linearfinite element methods. Flow (pressure) equation is modeled using a locally conservativeenriched Galerkin approximations approach [28, 32]. This Biot system is coupled in timeby employing the fixed stress iterative scheme which has been widely applied in [13, 26,27, 43, 46, 59], for fractured poro-elastic media in [19, 34, 47, 60], and including multi-ratetime stepping algorithms in [2, 3].

To reduce the computational cost, we employ a predictor-corrector dynamic mesh re-finement strategy developed in [22] and which was extended to three dimension in [33, 64]with parallel high performance computing using MPI (message passing interface). Nonlinearproblems are solved using Newton’s method and an analytical evaluation of the Jacobian.The constraint minimization problem is treated with a semi-smooth Newton method (i.e.,a primal-dual active set method) [22]. The linear equations are solved with parallel MPI-based iterative solvers (GMRES) with block-diagonal pre-conditioning. For more details,the reader may refer to [22, 33, 64, 34]. The programming code is developed based on theopen-source finite element library deal.II [7].

2.2 Optimization of hydraulic fracturing design

Most real-world engineering problems consist of multiple uncertain parameters as:

f (x) = f (x1,x2, · · · ,xN), (27)

where f (x) is the scalarized objective function, x is the vector of uncertain parameters, andN is the number of uncertain parameters. For a subsurface modeling problem, x may consistof reservoir properties (e.g., permeability, Poisson’s ratio, and Young’s modulus) or decisionvariables (e.g., well trajectory and number of hydraulic fractures).

Optimal design of hydraulic fracturing treatment can be defined as a maximization prob-lem as:

arg maxx∈RN

f (x) = arg maxx∈RN

dsim (28)

where dsim is the vector of simulated subsurface responses. In this paper, dsim is the totalvolume (area for two dimensional case) of hydraulic fractures Vf , which is computed byintegrating the fractured zones ΩFv obtained after running the phase-field model as:

Vf =∫

ΩFv

1 dA, (29)


ΩFv := x | ϕ < cv,∀x ∈Ω, (30)

where cv is the threshold used for distinguishing between fractured mesh and un-fracturedmesh in the given domain. In this study, cv is set to 0.1.

Meanwhile, subsurface characterization of a fractured reservoir can be regarded as ahistory-matching problem that minimizes the discrepancy between dsim and observed (i.e.,actual) subsurface responses dobs as:

arg minx∈RN

f (x) = arg minx∈RN‖dobs−dsim‖. (31)

In real-world problems, field-measurements (e.g., oil production rate, water-cut, and bot-tom hole pressure) would compose dobs while its corresponding dsim would be evaluated bysolving governing equations by use of a petroleum reservoir simulator. In the latter case, theobjective function Vf would be replaced with hydrocarbon production rates. This couplingproduction reservoir simulator with the phase-field model will be treated in future works.

2.3 Genetic algorithm

Genetic algorithm (GA), a stochastic optimization approach based on the evolutionary pro-cess of natural selection [24], is adopted to search for the global optimal solution of ahydraulic fracturing problem. The advantage of this gradient-free method is to require noassumption on the form of the objective function. GA is a population-based process [20].Qualities of populations in subsequent generations are expected to be more improved thanthat of the population in the current generation.

1 : INITIALIZE the parent population Xpt=0 with Npop solutions

generated based on prior knowledge or at random.The initial offspring population Xq

t=0 = ∅.2 : EVALUATE each parent solution in Xp

0 .3 : while termination condition is not achieved do4 : t = t +1.5 : SELECT Npop superior solutions from the mating pool Xp

t−1 ∪Xqt−1.

6 : UPDATE the parent population Xpt with the selected solutions.

7 : CROSSOVER pairs of the parent solutions for creating Npop-sized Xqt .

8 : MUTATE the resulting offspring population Xqt .

9 : EVALUATE each offspring solution in Xqt .

10 : end

Table 1: Pseudo code of the evolutionary process in genetic algorithm

Pseudo code of the evolutionary process in genetic algorithm is described in Table 1.This iterative process performs different genetic operations on a pool of solutions. Let Xp

and Xo be the parent population and the offspring population, respectively: xp ∈ Xp andxo ∈ Xo. The initial parent population Xp

t=0 consists of Npop solutions that are generated atrandom or based on prior knowledge. Here, Npop is the population size per generation. Eachsolution (i.e., individual) encodes its own vector of uncertain parameters x by real or binarynumber. In GA, the individual x and each element of individual x are called chromosomeand gene, respectively. At each generation, Xo is created by recombining qualified solutionsin the mating pool Xp ∪Xo. The qualified solutions of the mating pool are selected usinga tournament or a roulette wheel method [20]. Note that the fitness (i.e., objective function


1 : i = 12 : while i < Npop do3 : GENERATE the random probability Pr ∈ [0,1]4 : SELECT two parent members xp

i and xpi+1

5 : if Pr < Pc then6 : GENERATE the crossover point rc ∈ [1,N]7 : xq

i = xpi,1, · · · ,x

pi,rc

,xpi+1,rc+1, · · · ,x

pi+1,N

8 : xqi+1 = x

pi+1,1, · · · ,x

pi+1,rc

,xpi,rc+1, · · · ,x

pi,N

9 : else10 : xq

i = xpi

11 : xqi+1 = xp

i+112 : end13 : i = i+214 : end

Table 2: Pseudo code of crossover to create new solutions in genetic algorithm

1 : i = 12 : while i < Npop do3 : GENERATE the random probability Pr ∈ [0,1]4 : if Pr < Pm then5 : GENERATE the mutation point rm ∈ [1,N].6 : GENERATE the random variable xq,∗

i,rm∈ [xq,min

rm ,xq,maxrm ] at the mutation point rm

7 : xqi = x

qi,1, · · · ,x

qi,rm−1

,xq,∗i,rm

, · · · ,xqi,N

8 : end9 : i = i+110 : end

Table 3: Pseudo code of mutation to create new solutions in genetic algorithm

value) of each solution in the mating pool is evaluated by running the phase field model inthis study.

Recombination is sequentially invoked using two genetic operators: crossover (see Ta-ble 2) and mutation (see Table 3). Let Pr, Pc, and Pm be the random probability at eachgenetic operation, the probability of crossover, and the probability of mutation, respectively.The evolutionary process activates crossover if Pr < Pc and then does mutation if Pr < Pm.Both Pc and Pm are specified a priori, in general. Through the paper, Pc and Pm are 0.9 and0.1, respectively. Here, rc ∈ [1,N] is the crossover point that switches all genes after thatpoint. Here, rm ∈ [1,N] is the mutation point at which the random variable is altered in therange of its lower and upper limits. Note that both rc and rm are picked at random at eachgenetic operation. More details on crossover and mutation can be found in [20].

The termination condition of the evolutionary process is either the achievement of apre-specified tolerance level or the arrival on the maximum number of generations (i.e.,iterations). If the maximum number of generations is Ngen, the maximum number of eval-uations Neval = Ngen ×Npop. Note that the genetic algorithm tends to explore the singleglobal-optimum having the smallest objective-sum of a minimization problem or the largestobjective-sum of a maximization problem because of the one-dimension of the global-objective function.


2.4 Coupling of phase field model with assisted optimization program

Phase field fracture model discussed in Section 2.1 was integrated with an optimization tool-box, namely UT-OPT (University of Texas Austin-Multiple Realization Optimizer) [53], inorder to realize, run, and optimize geomechanics simulations of hydraulic fracturing. Fig-ure 4 shows a framework of UT-OPT. UT-OPT consists of three modules: builder, geneticalgorithm (GA) [20] as a global-objective optimizer, and non-dominated sorting geneticalgorithm-II (NSGA-II) [14] as a multi-objective optimizer. Builder generates ASCII inputdata files that are imported to one of the optimization algorithms to be executed. The inputfiles are categorized as algorithm parameters (e.g., generation number and population size);uncertain parameters (e.g., permeability, Young’s modulus, and fracture spacing); responseparameters that are targeted for quality check of reservoir models (e.g., volume of frac-ture); and watch parameters that are un-targeted for quality check (e.g., runtime). Dynamicgoal programming and successive linear-objective-reduction modules are advanced optionsthat can be coupled with NSGA-II to improve the rate of convergence with a small loss indiversity-preservation for solving a multi-objective problem in case M > 4 [49, 50, 51]. Forreducing the computational cost associated with the generational algorithms, UT-OPT hasthe capability to run multiple simulations concurrently. Each simulation can be executed inhigh-performance parallel computing environments. Besides the phase-field model, CSM’sin-house simulator, namely Integrated Parallel Accurate Reservoir Simulator (IPARS) [63],and commercial reservoir simulators are linked to UT-OPT. This general-purpose optimiza-tion program can be utilized for performing sensitivity analysis, history matching, produc-tion forecasts, economic analysis, and uncertainty analysis. UT-OPT was written in C++programming language.

Fig. 4: Framework of the assisted optimization tool, UT-OPT.

3 Numerical experiments

In this section, we present three numerical examples to illustrate capabilities of the coupledalgorithm. The first two examples provide maximization problems and the last demonstratesthe minimization. We clarify that the domain size of each example herein is designed aslab-scale for saving computational costs accompanied with multiple phase field runs during


optimization. All the examples run 20 experiments concurrently in each generation, andevery phase field fracture simulation is executed with 4 parallel processors.

3.1 Example 1: Maximization of fracture area with a well in a homogeneous reservoir

Fig. 5: Example 1. Schematic of wellbore and near-wellbore of a horizontal well with three-stage hydraulic fracturing.

In this example, given three fractures in a fixed domain Ω , find the optimal spacing thatprovides the largest fracture area in Ω . Let Ω = (0m,4m)2 with H = L = 4, we prescribeparallel initial cracks with half lengths l0 = 0.2m as illustrated in Figure 5. The initial meshis six times locally refined around the fracture. The smallest mesh size is hmin = 0.0013mand the regularization parameters are chosen as ε = 2hmin. The fracture in the center is fixedat (2− hmin,2+ hmin)× (2− l0,2+ l0), but the position of left and right fractures dependon two spacings, left and right, i.e., x1 and x2, respectively, in each fracturing scenario (seeFigure 5). In the homogeneous media, the critical energy release rate is chosen as Gc =1Nm−1, Young’s modulus is E = 108 Pa, and Poisson ratio is set to ν = 0.2. The relationshipto the Lame coefficients µ and λ is given by:

µ =E

2(1+ν), λ =

νE(1+ν)(1−2ν)

.

For the fluid parameters, Biot’s coefficient is set as α = 1, M = 1×108 Pa, ηR = ηF =1×10−3 Ns/m2, the reservoir permeability is KR = 1D≈ 1×10−12 m2, qF = 60, cF =1×10−12 Pa−1, and the density is ρ0

F = 1000kg/m3.Variable vector x consists of fracture spacing between the inner fracture and the outer

fractures: x = (x1,x2) where x1,x2 ∈ [0.5m,1.5m]. Thus, the number of variables N = 2 inthis example. The objective function f (x) of this example to be maximized takes the formof Equation (28) as:

arg maxx∈RN

f (x) = arg maxx∈R2

Nh f

∑i=1

Vf ,i , (32)

where the number of hydraulic fractures Nh f = 3.


(a) (b)

Fig. 6: Example 1. Evolution of objective function values in the homogeneous medium: (a)convergence of objective function values and (b) objective function values in variable space.

Figure 6(a) is a boxplot to show the evolution of objective function values from the firstto the seventh generations. The box in each generation consists of 20 solutions. Thus, the to-tal number of evaluations is 140. The initial generation was created at random. The height ofthe box indicates the interquartile ranges (IQR) determined by the 25th and 75th percentileof the posterior solutions. The solid red line inside the box is the median of the posterior so-lutions. Whiskers are vertical lines above and below the box that extend to the most extremesolutions not considered outliers, while any solution regarded as an outlier is plotted as a redplus. The solutions evolve in the direction of maximizing the total area of three fractures. Itappears that all solutions in the seventh generation stably converge to the global maximum,(x1, x2) = (0.75 m, 0.75 m). Figure 6(b) demonstrates the relationship between variablesand objective values. Kriging was used to interpolate 140 objective function values evalu-ated during optimization. Homogeneity of Young’s modulus in the domain delivers radialgradients of objective function values from top right corner in a two-dimensional variablespace. The ideal case of Figure 6(b) must be symmetric because x1 and x2 are interchange-able. A slight asymmetry shown in Figure 6(b) is due to the interpolation using Krigingfrom 140 evaluations. Interpolated regions where their objective values are inferred fromthe evaluations cause the asymmetry. We expect increasing the number of evaluations im-proves the degree of the symmetry. For example, adding and simulating (x1, x2) = (1.0 m,0.5 m), which are not created during this optimization example, would improve the degree ofthe symmetry shown in Figure 6(b) significantly. Note that the initial 20 evaluations are gen-erated at random and the subsequent evaluations are done by use of crossover and mutationof their preceding solutions in GA.

In Figures 7 and 8, the bent outer fractures and the constrained growth of the innerfracture indicate stress shadowing effects as shown in [11] and references cited therein.Figure 7 illustrates multiple parallel fracture propagation of the global optimal solution.The optimized fracture spacing (x1, x2) = (0.75 m, 0.75 m) yields after 140 evaluations,the largest hydraulic fractured total area Vf = 0.0212261m2. The stress shadowing effectsbecome noticeable after n = 100 in Figure 7. The propagated fractures of each fracturingscenario appear to be fairly symmetric due to the spatial homogeneity of Young’s modulus.Symmetry of fracture propagation is also observed in the non-optimal solutions.


(a) n = 0 (b) n = 50 (c) n = 100

(d) n = 150 (e) n = 200 (f) n = 250

Fig. 7: Example 1. Fracture propagation of the optimal solution. Each figure illustrates therefined mesh near the fractures (which is better for visualization) for each time step. Itsfracture spacing (x1, x2) = (0.75 m, 0.75 m) yields Vf = 0.0212261m2 in the homogeneousmedium.

(a) (x1, x2) = (0.5 m, 0.5 m) (b) (x1, x2) = (1.0 m, 1.0 m) (c) x1, x2) = (1.5 m, 1.5 m)

Fig. 8: Example 1. Fractures of three non-optimal solutions at timestep n=250 in thehomogeneous medium: (a) the closest fracture spacing (x1, x2) = (0.5 m, 0.5 m) yieldsVf = 0.0168475m2, (b) (x1, x2) = (1.0 m, 1.0 m) yields Vf = 0.0147084m2, and (c) thefarthest fracture spacing (x1, x2) = (1.5 m, 1.5 m) yields Vf = 0.0128308m2.

Figure 8 illustrates the shape of fractures at the last time step n = 250. The non-optimalsolutions result in smaller Vf values than that of the global-optimum. It seems that the closestfracture spacing reveals strong stress shadowing effects resulting in the inhibition of fracturegrowth (see Figure 8 (a)), while the farthest fracture spacing reveals little stress shadowingeffect (see Figure 8(c)). Note that Vf of Figure 8(c) is smaller than that of Figure 8(a) because


the three propagated fractures in Figure 8(c) are longer but thinner than those in Figure 8(a).This result arises since the fractures in Figure 8(c) are isolated due to larger spacing.

3.2 Example 2: Maximization of fracture area in a heterogeneous reservoir

In the computational domain Ω = (0m,4m)2, we set prescribed parallel initial cracks withhalf lengths l0 = 0.2m as described in Figure 9(a). The boundary conditions, initial meshgeneration, fixed stress iteration tolerance, variable vector, objective function, and mechan-ical parameters except Young’s modulus are the same as in the previous Example 1. Figure9(b) shows the spatial variation of Young’s modulus taking the value E ∈ [106 Pa,108 Pa].Note that a bar-shaped zone (denoted by A in Figure 9(b)) having the highest Young’s mod-ulus is located in the upper middle region of the domain.

(a) Initial Setup (b) Young’s modulus

Fig. 9: Example 2. Setup: (a) parallel fractures and (b) heterogeneous E values.

(a) (b)

Fig. 10: Example 2. Evolution of objective function values in the heterogeneous medium:(a) convergence of objective function values and (b) objective function values in variablespace.


(a) n = 0 (b) n = 50 (c) n = 100

(d) n = 150 (e) n = 200 (f) n = 250

Fig. 11: Example 2. Fracture propagation of the optimal solution. Its fracture spacing (x1,x2) = (1.0 m, 1.0625 m) yields Vf = 0.0192650m2 in the heterogeneous medium.

Similar to Figure 6(a), Figure 10(a) shows the convergence of objective function valuesin the heterogeneous medium during global optimization. All solutions following the sixthgeneration converge on the global maximum, which appears in the second generation. Theobjective value of each variable vector is depicted in Figure 10(b). Because of the hetero-geneity of Young’s modulus, the contour lines of Figure 10(b) are irregular. The optimizedfracture spacing (x1, x2) = (1.0 m, 1.0625 m) yields after 140 evaluations, the largest hy-draulic fractured total area Vf = 0.0192650m2.

Figure 11 illustrates the multiple parallel fracture propagation of the optimal solution.Compared to Figure 7, Figure 11 demonstrates that reservoir heterogeneity has a significantinfluence on fracture propagation. Each fracture tends to grow towards regions having lowerYoung’s modulus by detouring regions having higher Young’s modulus. It seems that thegrowth of the inner fracture to the north is suppressed by the existence of the bar-shapedzone with the highest Young’s modulus in the upper middle region of the domain. Thesuppression is also observed in the growth of the left outer fracture to the south of thedomain. Meanwhile, the bent inner fracture propagated to the southeast hinders the growthof the right outer fracture to the south. Fracture propagation results of three non-optimalsolutions at the last time step n = 250 are illustrated in Figure 12. Compared to Figures11(f), 12(b), and 12(c), Figure 12(a) having the closest perforating spacing deteriorates thefracture propagation due to the stress shadowing effects. Figure 12(c) yields a similar Vf tothe optimal solution.

Distinguishing the stress shadowing effect from heterogeneity of reservoir propertiesremains a challenging issue. Clearly, heterogeneity would have the greatest impact on theperformance of fracturing design. Nevertheless, it is necessary to consider the stress shad-


owing effect is to be considered carefully for estimating a propagation path of each hydraulicfracture in heterogeneous media if operating parameters are to be determined.

(a) (x1, x2) = (0.5 m, 0.5 m) (b) (x1, x2) = (1.0 m, 1.0 m) (c) (x1, x2) = (1.5 m, 1.375 m)

Fig. 12: Example 3. Fractures of three non-optimal solutions at time step n = 250 inthe heterogeneous medium: (a) the fracture spacing (x1, x2) = (0.5 m, 0.5 m) yields Vf =0.0116455m2, (b) (x1, x2) = (1.0 m, 1.0 m) yields Vf = 0.0182794m2, and (c)(x1, x2) =(1.5 m, 1.375 m) yields Vf = 0.0189270m2.

3.3 Example 3: History matching

Fig. 13: Example 3. Propagated fractures at time step n = 250 with heterogeneous E valuesof the reference model.

This example solves a minimization problem to explore the global minimum having thesmallest discrepancy between actual (reference) and simulated area of fractures. The actualareas of fractures was computed using the simulation results of the reference field modelshown in Figure 13. As shown in Example 1, three parallel fractures are positioned in thecomputational domain Ω = (0m,4m)2. All the other numerical and physical parameters areidentical to those of Example 1, with an exception that a rectangle region ([2.0m,3.5m]×[2.5m,3.0m]) of which Young’s modulus is largest.


The variable vector x of this minimization problem in Equation (31) is set as:

x = x1,b,x2,b,y1,b,y2,b,Ein,Eout, (33)

where x1,b is x-coordinate of the bottom-left corner of the bar, x2,b is x-coordinate of theupper-right corner of the bar, y1,b is y-coordinate of the bottom-left corner of the bar, y2,bis y-coordinate of the upper-right corner of the bar, Ein is Young’s modulus of inside thebar, and Eout is Young’s modulus of outside the bar. The bottom-left corner of the domainis taken to be the origin of the domain. The range of each variable is as follows: x1,b,x2,b ∈[0m,4m]; y1,b,y2,b ∈ [0m,4m]; Ein ∈ [106 Pa, 1010 Pa]; and Eout ∈ [107 Pa, 109 Pa]. Weintentionally made genetic algorithm more difficult to converge by giving a wide range forYoung’s modulus.

The objective function for this example is the absolute average percent error derivedfrom the form of Equation (31) as:

arg minx∈RN

f (x) = arg minx∈R6

1Nobs

Nobs

∑i=1|V re f

f ,i −Vf ,i

V re ff ,i

|×100(%), (34)

where V re ff is the fracture area of the reference solution that corresponds to dobs in Equation

(31) and Nobs is the number of observations. In this example, Nobs = 1 as V re ff measured at

the last time step is the only observation data used to calculate the objective function.Operating parameters of hydraulic fracturing are fixed in all phase-field runs: the number

of fractures is three, injection rate is constant at each fracture interval, and fracture spacingbetween the inner fracture and each outer fracture is constant 1 m. At the last time stepn = 250, the reference model yields the total area of fractures V re f

f = 0.00974681m2, whichresults from x1,b = 2.0m, x2,b = 3.5m, y1,b = 2.5m, y2,b = 3.0m, Ein = 1010 Pa, and Eout =108 Pa. Because the Example 3 set up the single bar, it might be reasonable if the solution setconverges not only the reference solution but also the following three solutions: (x1,b,y1,b)= (0.5m, 2.0m), (x2,b,y2,b) = (2.5m, 3.0m); (x1,b,y1,b) = (2.0m, 3.5m), (x2,b,y2,b) = (1.0m,1.5m); and (x1,b,y1,b) = (0.5m, 2.0m), (x2,b,y2,b) = (1.0m, 1.5m).

Figure 14 illustrates the numerical results for selected generations and experiments.Each generation had 20 experiments and 10 generations were tested. Since no conditions be-tween Ein and Eout were provided, we observed some cases with Ein < Eout that yielded un-favorable simulation results during earlier generations. After the fourth generation, it seemsthat most experiments get closer to the reference solution in terms of the position of the bar.

As a consequence, Figure 15(a) provides the optimized solution similar to the referencesolution shown in Figure 13. The derived fractures are somewhat symmetric despite theexistence of the single bar with the high Young’s modulus on the upper part of the domain,but the degree of fracture asymmetry depends on the characteristics of the bar (e.g., sizeand position). We expect a larger bar enhances the asymmetry more remarkably as seen inFigure 14(d). Also, that kind of asymmetric fractures can be more pronounced if increasingthe injection time step. The variance of each variable decreased during the generationalprocess. Interestingly, the thickness of the block y2,b− y1,b was a less crucial factor to theperformance of genetic algorithm because the phase field model did not allow fractures topenetrate the bar under the given operating condition. Note that the fracture-growth mecha-nisms in a complex sedimentary basin are dominated by not only heterogeneities of reservoirproperties such as material property contrast but also the operating condition [16]. A higher


(a) Gen. 1 Exp. 02 (b) Gen. 1 Exp. 02 (c) Gen. 1 Exp. 04

(d) Gen. 2 Exp. 06 (e) Gen. 2 Exp. 14 (f) Gen. 2 Exp. 20

(g) Gen. 4 Exp. 04 (h) Gen. 4 Exp. 07 (i) Gen. 4 Exp. 14

(j) Gen. 7 Exp. 03 (k) Gen. 7 Exp. 06 (l) Gen. 7 Exp. 19

Fig. 14: Example 3. Illustrates the numerical results for selected generations (Gen=1,2,4,and 7) and experiments (Exp.). Each row indicates different generations. We observe a con-vergence of the solution to the reference model.


(a) (b)

Fig. 15: Example 3. Evolution of objective function values for the history matching problem:(a) the optimal solution and (b) convergence of objective function values.

fluid injection rate that can grow fractures with penetrating the bar would reduce the un-certainty related to the thickness of the block by providing more information beyond thebar.

Figure 15(b) shows the stable decrease in the objective function values from the first tothe seventh generations. As most experiments were assimilated in the seventh generation, theresults from the eighth to the tenth generations are omitted in this figure. Notwithstanding thesmall positive absolute average percent error, it seems that all solutions arrive near the globalminimum (i.e., reference model) in the seventh generation. Note that this six-dimensionalproblem requires more expensive computational costs for achieving the convergence to-wards the global optimal solution than the two-dimensional problem described in Examples1 and 2. Employing faster inverse algorithms [8] or coupling a surrogate model with theforward model [52] would contribute to saving computational costs for solving more high-dimensional complex problems. It is also anticipated that reflecting reservoir uncertainty onthe parameters adjusted in these examples would deliver more realistic outcomes in futureworks.

Conclusions

This work integrated the phase field fracture propagation model with an optimization tool(UT-OPT) for accomplishing optimal hydraulic fracturing design at unconventional shaleoil or gas reservoirs. Genetic algorithm implemented in UT-OPT adjusted fracture intervalsfor three-stage hydraulic fracturing problems with a single well in a heterogeneous porousmedia. The operating parameters derived from GA were assigned to the phase field modelthat estimated the volume (area) of propagated fractures as a performance indicator of hy-draulic fracturing. This indicator will be replaced with oil production or net present valueafter completing the integration of the phase field fracture propagation model with a reser-voir simulator. The optimization process was iterated until no further improvements in theobjective function were achieved or when the maximum number of iteration was reached.


Computational effects of heterogeneity and stress-shadowing effects due to fracture spac-ing were analyzed with applications to both homogeneous and heterogeneous media. Allnumerical simulations herein were executed concurrently using a high-performance parallelcomputing. for reducing the computational cost. These results illustrate the capabilities ofthis integrated algorithm to further apply for more realistic scenarios in petroleum engineer-ing. Coupling the phase field fracture algorithms to the field scale reservoir simulator is ourongoing work.

Acknowledgements

The research by S. Lee, B. Min, and M. F. Wheeler were funded by U.S. Department of En-ergy, National Energy Technology Laboratory grant DOE FG02-04ER25617. S. Lee and M.F. Wheeler were partially supported by NSF grant NSF 1546553. B. Min was partially sup-ported by the National Research Foundation of Korea (NRF) grant funded by the Ministryof Education, Science and Technology (No. 2016R1A6A3A03012796) and the Ministry ofScience, ICT & Future Planning (No. 2017R1C1B5017767).

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