ICES REPORT 13-17

June 2013

Modeling hydraulic fractures with a reservoir simulatorcoupled to a boundary element method

by

Benjamin Ganis, Mark Mear, A. Sakhaee-Pour, Mary F. Wheeler, Thomas Wick

The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustin, Texas 78712

Reference: Benjamin Ganis, Mark Mear, A. Sakhaee-Pour, Mary F. Wheeler, Thomas Wick, Modeling hydraulicfractures with a reservoir simulator coupled to a boundary element method, ICES REPORT 13-17, The Institutefor Computational Engineering and Sciences, The University of Texas at Austin, June 2013.

Modeling Hydraulic Fractures with a Reservoir Simulator

Coupled to a Boundary Element Method

Benjamin Ganis∗† Mark E. Mear∗ A. Sakhaee-Pour∗ Mary F. Wheeler∗

Thomas Wick∗

July 16, 2013

Abstract

We describe an algorithm for modeling stationary hydraulic fractures in a poroelas-tic domain in which the reservoir simulator is coupled with a boundary element method.A fixed stress splitting is used on the underlying fractured Biot system to iterativelycouple fluid and solid mechanics systems. The fluid system consists of Darcy’s lawin the reservoir with a multipoint flux mixed finite element method, and a Reynolds’lubrication equation in the fracture with a mimetic finite difference method. The me-chanics system consists of linear elasticity in the reservoir with a continuous Galerkinmethod, and linear elasticity in the fracture with a weakly-singular symmetric Galerkinboundary element method. This algorithm is able to compute both unknown fracturewidth and unknown fluid leakage rate. An interesting numerical example is presentedwith an injection well inside of a circular fracture.

Keywords. hydraulic fracture, poroelasticity, multipoint flux mixed finite element,mimetic finite difference, Galerkin finite element, boundary element

1 Introduction

A better understanding of the interaction between the fluid flow inside fractures coupledwith fluid flow in a surrounding poroelastic medium is of great interest to the petroleumindustry. This becomes more apparent for unconventional resources such as tight gas sand-stones and shales [31], as production at an economic rate is achieved only when theseformations are heavily fractured. With more accurate mathematical and computationalmodels, it would be possible to produce these resources in a greater quantity, and also toinvestigate environmental impacts with greater confidence.

The interaction of the fluid flow inside the fracture and porous media is a multiphysicsand multiscale problem [11]. The fluid flow and solid mechanics are inherently coupled,

∗Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The Uni-versity of Texas at Austin, Austin, TX 78712, Email: bganis,sakhaee,mfw,twick@ices.utexas.edu,mear@mail.utexas.edu†Corresponding author

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whereby the fluid pressure influences both the fracture width and the reservoir displace-ment. The effective porosity in the reservoir is also a function of both fluid pressure andsolid displacement. In addition, flow properties may differ by several orders of magnitudein different parts of fractured reservoirs, and the fracture width can be many orders ofmagnitude smaller than its length. This combination of factors makes realistic large scalefracture simulations challenging to realize, and has spurred the development of many solu-tion techniques over many decades [1].

The literature on fractures is extensive, and these models can be categorized in manyways. Fractures may be stationary objects for production problems [19, 2, 12], they may beallowed to propagate [27, 11], and they may be allowed to nucleate or branch [4, 14]. Thereis a range of geometrical representations for fractures, including lines [25, 19], rectangles[16], internal element boundaries [3], portions of external boundaries [11], partitions ofunity [14], characteristic functions [4], and non-planar surfaces [22, 27]. Many studiesinclude poroelastic effects in the surrounding rock matrix, while another class of studiesis primarily focused on coupling fluid equations alone [15, 2]. Reservoir domains rangefrom 2D to pseudo-3D to full-3D models and number of fractures range from one to severalhundred. In this work, we model a stationary circular fracture represented by a 2D surfaceinside a 3D poroelastic domain.

There are many studies in which fluid flow in a fracture is considered without couplingto the surrounding reservoir fluid [27, 28, 20, 4, 3, 7, 14]. This is achieved by assuming thesurrounding rock matrix is completely impermeable, or it is assumed that fluid leak-off fromfracture to reservoir is explicitly approximated using phenomenological models such as theCarter model [8]. While these assumptions may be valid under certain circumstances, theeffect of capturing an accurate fluid leak-off can be a very important factor in the hydraulicfracturing process [1]. Computing an unknown leak-off can be achieved by a full couplingof reservoir flow to fracture flow using internal boundary conditions [11, 15, 19, 12, 2], andwe utilize this approach in this work.

To compliment the realm of fracture models with varying physical characteristics, thereexists a multitude of numerical methods. This work uses a boundary element method, whichapplies finite element techniques to boundary integral equations, to solve for a quantity ofinterest on a specific manifold. Certain weakly-singular, weak-form integral equations wereutilized to develop a symmetric Galerkin boundary element method (SGBEM) in [21], andit was used in hydraulic fracture problems in [27] with the Carter leak-off model. ThisSGBEM was coupled with a finite element method in [26], and other examples of SGBEM-FEM couplings include [17, 24].

In this study, we modify the approach used in [12] to couple a reservoir simulator with theboundary element method in [21]. This model captures the interaction between the fluid flowinside a stationary fracture and fluid flow inside the reservoir with poroelastic effects underquasi-static assumptions. The solution algorithm uses iterative coupling based on the fixedstress splitting for the Biot system [23]. The two fluid models are also iteratively coupled inorder to compute an accurate fluid leak-off. The fluid system consists of Darcy’s law in thereservoir with a multipoint flux mixed finite element (MFMFE) method, and a Reynolds’lubrication equation in the fracture with a mimetic finite difference (MFD) method. Themechanics system consists of linear elasticity in the reservoir with a continuous Galerkinfinite element method (FEM), and linear elasticity in the fracture with a weakly-singular

2

symmetric Galerkin boundary element method.The remainder of this paper is organized as follows. The physical problem is formulated

in Section 2, the numerical algorithms are described in Section 3, an interesting numericalexample is presented with an injection well inside a circular fracture in Section 4, andconclusions and future work are described in Section 5.

2 Problem Formulation

C

!

Figure 1: Illustration of reservoir and fracture domains.

In this section, we describe the physical equations, their boundary and initial conditionsand their relation to each other through right hand side terms and interface conditions.We are concerned with four different types of equations. Let Ω ⊂ R3 be a spatial domain(the reservoir) and C ⊂ R2 a smooth lower-dimensional manifold (the fracture), as shownin Figure 1. In the present work, the fracture surface is assumed to be stationary, but itswidth is allowed to change in time. Technically, the fracture has an upper surface C+ and alower surface C−, which are geometrically coincident and their unit normals (directed intothe crack) satisfy n+ = −n−. Our fracture problems are posed on the upper surface and thenormal will be used without superscript. In this study we assume there is a single fracturefor notational convenience, but this framework is naturally extended to the case of multiplefractures. The time t belongs to the interval (0, T ), where T is the end time value.

We start by describing the fluid problems. Let the fluid density ρ depend on the fluidpressure p by the slightly compressible equation of state

ρ(p) = ρ0 exp(cp), (1)

where ρ0 is a reference density and c is the fluid compressibility constant. The effective

3

porosity φ∗ in the poroelastic medium is defined by

φ∗ = φ0 + α(∇ · u) +1

M(p− p0) in Ω \ C, (2)

where φ0 is the initial porosity, α ∈ [0, 1] is the Biot coefficient, u is the solid displacement,M is the Biot’s modulus, and p0 is the initial pressure. With these definitions, we pose thereservoir flow problem.

Problem 2.1 (Reservoir flow). Find the reservoir pressure pres and mass flux vres such that

vres = − K

µρ(∇pres − ρg∇η) in Ω \ C × (0, T ), (3)

∂t(ρφ∗) +∇ · vres = qres in Ω \ C × (0, T ), (4)

vres · n = 0 on ∂Ω× (0, T ), (5)

pres = pfrac on C × (0, T ), (6)

−JvresK · n = qL on C × (0, T ), (7)

pres = p0 at Ω \ C × t = 0. (8)

Here, K denotes the permeability tensor, µ the fluid viscosity, g is the gravitation constant,η is (positive downward) vertical coordinate, and qres denotes sources and sink terms inthe reservoir. No-flow external boundary conditions are assumed, the fluid pressures arecontinuous across the fracture, and the jump in reservoir fluxes is equal to the fluid leak-offrate qL from fracture to reservoir.

Let ∇ represent the surface gradient operator on C. In the fracture, we solve theReynolds’ lubrication model.

Problem 2.2 (Fracture flow). Find the fracture pressure pfrac and mass flux vfrac such that

vfrac = − w3

12µρ(∇pfrac − ρg∇η) on C × (0, T ), (9)

∂t(ρw) +∇ · vfrac = qI − qL on C × (0, T ), (10)

vfrac · n = 0 on ∂C × (0, T ), (11)

pfrac = p0 at C × t = 0. (12)

Here, w = −JufracK · n denotes the width of the fracture computed as the jump of normaldisplacements, qI is a known fluid injection rate, and no flow boundary conditions areassumed on the boundary of the fracture (but fluid leak-off is handled through source terms).

In particular, on the fracture surface C, we have pfrac = pres, so that the overall fluid pressurep in the entire domain Ω is defined by

p =

pres, in Ω \ Cpfrac, on C.

(13)

We now turn to the description of the fractured reservoir poro-mechanics problem. Here,the poroelastic stress tensor σpor depends on the displacement u and the pore pressure p by

σpor(u, p) = σ(u)− αpI, (14)

4

where σ(u) is the solid stress. In a homogeneous isotropic material, the linear relationshipbetween strain and stress reads

σ(u) = λ(∇ · u)I + µε(u), (15)

where λ and G denote the Lame parameters, I is the second order identity tensor, and thestrain is the symmetric gradient

ε(u) =1

2(∇u+∇uT ). (16)

In the following, let f denote the body force due to gravitation loading, and we defineDirichlet and Neumann boundaries, with ∂Ω = ΓD ∪ ΓN , assuming |ΓD| > 0 for uniquesolvability. (If this condition is relaxed, then rigid modes may be constrained.) Underquasi-static assumptions, the boundary value problem for linear elasticity is to find thedisplacement field upor such that

−∇ · σpor = f in Ω \ C, (17)

upor = 0 on ΓD, (18)

σporn = tN on ΓN , (19)

σporn = −pn on C. (20)

Since (17) is linear, by superposition we break the solution into two pieces, i.e.

upor = ures + ufrac, and (21)

σpor = σres + σfrac. (22)

This decomposition is necessary since we would like to use different methods for the solutionof reservoir mechanics and fracture mechanics problems. The solution to the latter will beapproximated. Therefore, we pose the following two problems.

Problem 2.3 (Reservoir mechanics). Find the reservoir displacement ures such that

−∇ · σres(ures) = f in Ω, (23)

σres = σ(ures)− αpI in Ω, (24)

ures = 0 on ΓD, (25)

σresn = tN on ΓN . (26)

Problem 2.4 (Fracture mechanics). Find the fracture displacement ufrac such that

−∇ · σfrac(ufrac) = 0 in R3 \ C, (27)

σfracn = − σresn− pn on C. (28)

Remark 2.1. The fracture mechanics problem above is solved using a boundary elementmethod where conditions on ∂Ω are dropped, and the problem is posed on the unboundeddomain R3 \ C. For this reason, the decomposition upor = ures + ufrac is an approximation.If the domain size is large and fractures are not close to the boundary, then this error shouldbe small. The effective porosity φ∗ in the reservoir is also approximated by using ∇ · ures in(2). We could account for the effect of the finite domain within the context of the boundaryelement formulation, see [22].

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3 Numerical Algorithm

In the present section, we account for discretization details and the solution algorithm.Specifically, the order in which the four problems are solved is explained. Finally, sincethe two mechanics problems are solved on different grids, we discuss interpolation rules totransfer quantities from one grid to the other.

3.1 Discrete Formulations

Since the mechanics problems are quasi-statically coupled to the fluid problems, time dis-cretization is only necessary in the latter. The spatial discretizations for the fluid problemsuse MFMFE and MFD methods, providing local mass conservation and the ability to repre-sent general geometries. The interface conditions between fluid problems are implementedusing internal boundary conditions, so that a pressure condition can be imposed in thereservoir flow and a jump in fluxes can be provided to the fracture flow.

Let us introduce some notation. The time step is defined as δt = tn − tn−1, for n =1, . . . , NT . Whenever possible the superscript n for time index will be suppressed. Let Thbe a decomposition of the domain Ω into structured hexahedral grid cells, which conformto the fracture surface C. Let Fh be the union of quadrilateral faces of the hexahedral gridconforming to the fracture surface C. We use (·, ·) to denote the L2(Ω) inner product incase of the MFMFE spaces, and 〈·, ·〉C to denote the inner product on L2(C).

For the reservoir fluid discretization, let Vh and Wh denote the MFMFE discrete massflux and pressure spaces, respectively, on Th. This method reduces to cell-centered finitedifferences for full tensors on irregular geometries. The space for fluxes is based on thelowest order Brezzi-Douglas-Marini mixed finite elements [5]. A special quadrature ruleis applied to the mass matrix, which transforms the saddle point problem via eliminationinto a positive definite algebraic system for the pressure variable. For more details on theMFMFE method, see [13, 30]. With these preliminaries, we pose the following discretereservoir flow problem.

Problem 3.1 (MFMFE reservoir fluid discretization). Find (vres, pres) ∈ Vh×Wh such thatfor n = 1, . . . , NT ,

(µρ−1K−1vres, ψ)− (pres,∇ · ψ)− (ρg∇η, ψ) + 〈pfrac,∇ · ψ〉C = 0 ∀ψ ∈ Vh, (29)

−(φ∗ρ, χ)− δt(∇ · vres, χ)− (φ∗ρn−1 − δtqres, χ) = 0 ∀χ ∈Wh. (30)

For the fracture flow problem, we use a MFD discretization. Let Zh and Θh denote theMFD discrete mass flux and pressure spaces, respectively, on Fh. We use [·, ·]Z and [·, ·]Θto denote special discrete scalar products for these spaces, respectively, which are chosento satisfy the stability and consistency assumptions given in [6]. The MFD method can beviewed as a generalization of the MFMFE method, which can be used on completely generalpolygonal and polyhedral meshes. However in our case, our MFD grid Fh is composed ofquadrilateral cells which are a union of faces from the MFMFE hexahedral grid. For moredetails on the MFD method, see the appendix in [2].

We pose the following discrete fracture flow problem.

6

Problem 3.2 (MFD fracture flow discretization). Find (vfrac, pfrac) ∈ Zh × Θh such thatfor n = 1, . . . , NT ,

[µρ−1w−3vfrac, ψ]Z − [p,DIV ψ]Θ − [ρg GRAD η, ψ]Z = 0 ∀ψ ∈ Zh, (31)

−[wρ, χ]Θ − δt[DIV vfrac, χ]Θ − [wρn−1 − δt(qI − qL), χ]Θ = 0 ∀χ ∈ Θh. (32)

Here, DIV and GRAD denote the discrete divergence and gradient operator, such that thefollowing relations hold. On a cell E with edge ∂E,

(DIV uh)E =1

|E|∑e∈∂E

ueE |e|.

For two discrete pressure functions p1,frac, p2,frac ∈ Θh, we define

[p1,frac, p2,frac]Θ =∑E∈Fh

p1,Ep2,E |E|.

For two discrete velocity functions v1,frac, v2,frac ∈ Zh, we define

[v1,frac, v2,frac]Z =∑E∈Fh

[v1,frac, v2,frac]E ,

with[v1,frac, v2,frac]E = (v1,frac)

TEME(u2,frac)E ,

where the matrix ME is the crucial part in designing a stable discretization.Note that in our method, the fracture flow problem will be linearized using a constantdensity. Therefore to avoid solving a pure Neumann problem, a stabilization term β[pfrac−pold

frac, χ]Θ is added for some small β > 0 for uniqueness, where poldfrac is the solution from the

previous Newton step.Turning to the mechanics problems, we discretize these problems using well-known stan-

dard techniques. A continuous Galerkin FEM [9] is employed for the reservoir mechanics.Let Xh denote the Galerkin FE space over the hexahedral grid Th. Note here that we arenot meshing in the fracture as was performed in [12], so that the space Xh will be continuousover C. Then, the formulation reads as follows.

Problem 3.3 (FEM reservoir mechanics discretization). Find the reservoir displacementures ∈ Xh such that∫

Ωσ(ures) : ε(ψ) dV =

∫Ω

(f − α∇p)ψ dV +

∫ΓN

tNψ dS ∀ψ ∈ Xh, (33)

where σres is decomposed by (24) and σ is defined in (15).

For the fracture mechanics discretization, a weakly-singular SGBEM is employed [21,22]. Let Gh be a decomposition of the fracture surface C into unstructured quadrilaterals,which do not conform to the decomposition Fh, see figure 3. Let Bh denote the SGBEMspace on Gh. We compute a relative fracture face displacement, denoted by JufracK ∈ Bh.

7

We make the following definitions using summation notation. Let

Dm = niεism∂

∂ξs

denote a surface integral operator in which εism denotes the Civita-Levi symbol and nithe i-th component of the outward unit normal to the surface (directed ‘into’ the crack).Furthermore, we denote the weakly-singular kernel (a fourth order tensor)

Ctkmj(ξ − y) =G

4π(1− ν)r

[(1− ν)δktδjm + 2νδkmδjt − δkjδtm −

(ξj − yj)(ξk − yk)

r2δtm

],

where δij is the Kronecker delta symbol, G is the shear modulus, ν is Poisson’s ratio, y andξ denote a source point and field point, respectively, and r = ‖ξ−y‖. For more information,we refer the reader to [21]. The discrete fracture mechanics problem is as follows.

Problem 3.4 (SGBEM fracture mechanics discretization). Find the relative fracture facedisplacement JufracK ∈ Bh such that∫CDtJψk(y)K

∫CCtkmj(ξ − y)DmJuj(ξ)KdS(ξ)dS(y) = −

∫CJψk(y)Ktk(y)dS(y) ∀ψ ∈ Bh.

(34)

Here, tk is the k-th Cartesian component of σfracn.Summarizing, we performed temporal and spatial discretization for the four physical sub-

problems. In the next section, we discuss the solution algorithm and explain the iterativeprocedure in which the problems are coupled.

3.2 Coupling algorithm

The main difficulty in solving this multiscale and multiphysics problem is designing thesolution algorithm that couples the four sub-problems together. One possibility is to solvethe entire discrete problem simultaneously in a monolithic fashion, although this wouldbe very challenging to implement. On the other end of the spectrum, a loosely coupledapproach is possible, although this does not lead to accurate results. Our solution algorithmuses an iteratively coupled approach, which balances the simplicity in implementation ofthe loosely coupled approach with the accuracy of the monolithic scheme. To separate theflow and mechanics, we employ a fixed stress splitting [23].

Consequently, we propose the following iterative solution algorithm as sketched in Figure2. At each new time step, we first solve the non-linear fluid problems using a Newtoniteration. Within each Newton step, we iterate between reservoir and fracture flow problems.In the reservoir flow Problem 3.1, the effective porosity φ∗, internal boundary condition forfracture pressure pfrac, and fluid leak-off qL are known data using the most recent solutions.Using the solution from the reservoir flow problem, a jump in flux is computed and becomesa new fluid leak-off term qL given to the fracture flow Problem 3.2. The width w in thisproblem is also known data from the most recent solution to the fracture mechanics. Thesolution the fracture flow problem gives an updated solution to the next reservoir flowproblem, until the change the fracture pressure δpfrac is below a given tolerance.

8

Reservoir Flow

(MFMFE)

Fracture Flow

(Mimetic FD)

Reservoir

Mechanics

(Galerkin FEM)

if !pf > TOL

Fracture

Mechanics

(SGBEM)

if !ϕ* > TOL

n = n+1

Begin time

step n Data Unknowns

∗frac res res L

qL, w pfrac, vfracp u

L frac frac

pres uresσ , p u

σres, pfrac ufrac

φ∗, pfrac pres, vres

Figure 2: Flow chart of the iteratively coupled algorithm used in this work.

After the two fluid sub-problems are converged, we solve the two mechanics sub-problems.These two problems are performed in sequence since they are decoupled. The most recentreservoir pressure is known data for the reservoir mechanics Problem 3.3. Unlike the pre-vious work in [12], there is no internal boundary condition in reservoir mechanics. Usingthe most recent solid stress from reservoir mechanics and the most recent fracture pressurefrom fracture flow, traction data is given to the fracture mechanics Problem 3.2. At thispoint, both fracture width w and effective porosity φ∗ are updated, and convergence isachieved between mechanics and flow when the change in effective porosity δφ∗ is below agiven tolerance.

3.3 Interpolation between grids

The MFD grid Fh conforms to the Galerkin FEM and MFMFE grid Th, so it is trivial totransfer data such as solid stress σres, fluid leak-off qL, and fracture pressure pfrac betweenFh and Th. However, the underlying SGBEM grid Gh does not conform to Fh, so a crucialissue is to interpolate certain functional values between different grids on the surface C.These grids are distinguished from each other in both geometry and structure, as shown inFigure 3. The fracture width w must be interpolated from Gh to Fh, and the traction σfrac

must be interpolated from Fh to Gh. We describe these processes in Algorithms 1 and 2,respectively.

9

X

Y

Z

Figure 3: Example of structured MFD grid Fh (thin line) and unstructured SGBEM gridGh (bold line) on a circular fracture surface C. Note these grids are nonconforming.

Algorithm 1 Projection of width w from Gh to Fh.

for each quadrature point x(k) ∈ Xh dofor each element Em ∈ Gh do

xm(k) = closest point projection of x(k) onto Em.d(m) = distance from xm(k) to center of mass of cell Em.

end form = min argd(m), ∀m.Interpolate the normal component of JufracK at xm(k) to set the value of width w at

quadrature point x(k).end for

Algorithm 2 Projection of traction t from Fh to Gh.

for each quadrature point x(k) ∈ Bh dofor each element Em ∈ Fh do

xm(k) = closest point projection of x(k) onto Em.d(m) = distance from xm(k) to center of mass of cell Em.

end form = min argd(m), ∀m.Interpolate the solid stress σres and fracture pressure pfrac at xm(k) to set the value

of traction t at quadrature point x(k).end for

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4 Numerical Results

In this section, we present an interesting numerical example with an injection well insidea circular fracture to present a realistic example and illustrate the phenomena which themodel is able to capture. Following the flowchart in Figure 2, we have implemented theReservoir-BEM algorithm by coupling the boundary element code described in [22] withthe reservoir simulator IPARS [18, 29].

The domain is taken to be the cube Ω = (−7.14, 7.14)3 m, with a circular fracture Ccentered on the plane x = 0 m of radius 2.5 m, as shown in Figure 1. Using the meshgeneration software GridPro [10], the domain is discretized into 32 × 32 × 32 = 32, 768logically structured hexahedral elements, meaning each interior hexahedral element hasexactly 6 neighbors. The nodes on this grid conform to the fracture in such a way that thecenter 16× 16 = 256 quadrilateral faces on each vertical layer form a circle of radius 2.5 m,and make up a cylinder throughout the entire mesh, as shown in Figure 4. The boundaryelement discretization is made up of 56 unstructured quadrilateral elements, which can beseen in Figures 3, 5, and 6.

X Y

Z X

Y

Z

Figure 4: Top view of hexahedral reservoir mesh, side view of hexahedral reservoir mesh(middle), and side view of hexahedral reservoir mesh scaled by reservoir displacements onfinal time step (right).

The initial fluid pressure in both the reservoir and fracture is taken to be p0 = 3.5 ×106 Pa. The external fluid boundary conditions are no flow, i.e. vres · n = 0 on ∂Ω. Theexternal mechanical boundary conditions are such that the bottom face is completely pinned,ures = 0, and the five remaining faces are traction free, tN = 0. The tolerances for theNewton method, linear solver, fixed stress splitting, and reservoir-fracture flow iteration are10−6, 10−9, 10−6, and 10−6, respectively, and the MFD stabilization parameter is taken tobe β = 10−4. Using these tolerances, our numerical simulation takes 2 fixed stress iterationson the first time step, followed by 1 iteration on subsequent time steps. The number ofNewton iterations is 2 on the first time step, followed by 1 iteration on subsequent timesteps. The average number of reservoir-fracture flow iterations per time step is 48.

Fluid is injected into the 4 fracture cells located at the center of the fracture with aconstant rate of qI = 25 kg · s−1 into each cell. There are no sources and sinks in thereservoir domain, so qres = 0. The remaining input parameters are summarized in Table 1.Note that the Lame constants are given by λ = Eν/[(1 + ν)(1− 2ν)] and G = E/[2(1 + ν)],

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where E is Young’s modulus and ν is Poisson’s ratio.

Parameter Quantity Value

K Reservoir Permeability diag(5, 20, 20)× 10−15 m2

φ0 Initial Porosity 0.2µ Fluid Viscocity 1× 10−3 Pa sc Fluid Compressibility 5.8× 10−7 Pa−1

ρ0 Reference Fluid Density 897 kg m−3

E Young’s Modulus 7.0× 1010 Paν Poisson’s Ratio 0.3α Biot’s Coefficient 1.0M Biot’s Modulus 2.0× 108 PaT Total Simulation Time 100 sδt Time Step 1 s

Table 1: Input parameters for the numerical example in Si units.

Each fixed stress mechanics iteration involves one reservoir mechanics solve followed byone boundary element solve. As the boundary element method has traction as input andreturns relative fracture face displacement (for which width is the normal component) asoutput, this procedure involves calling projection Algorithms 1 and 2. Figures 5 and 6 showthe accuracy which is resolved on each type of grid, for traction and width, respectively, onthe first and final time steps. The fracture mechanics solve is performed after the reservoirmechanics solve so that the most recent value of solid stress and fracture pressure can supplyan accurate traction to the boundary element method.

The reservoir displacement ures is shown in Figure 7 and the swelling effect inducedby the pressure increase in the Biot system can be seen in Figure 4. Although ures is acontinuous field, the discontinuity across the fracture surface is obtained with the solutionto the boundary element method JufracK. Therefore, the true poroelastic displacementupor = ures + ufrac is computed only on the fracture surface, and is an approximationas discussed in Remark 2.1.

One of the main benefits of coupling fracture flow with the surrounding reservoir fluid isthat the fluid leak-off qL can be computed without phenomenological approximations suchas the Carter formula [8]. Figure 8 shows the fluid leak-off for this numerical example. Inour experience, this quantity is especially sensitive to the geometry of the fracture grid.Even in unconventional reservoirs with an extremely low permeability, fluid leak-off is avery important quantity to capture accurately, as it can affect phenomena such as proppanttransport and the saturations of multiphase fluids in realistic hydraulic fracture applications.

The computed fluid pressure in both the fracture and the surrounding reservoir areshown in Figure 9. A high rate of fluid injection into the fracture causes a maximum pressureof 8.15 MPa at time t = 1 s, and this maximum fluctuates between 6.0 to 6.2 MPa in thefracture until the end of the simulation at t = 100 s. The reservoir pressure in the center ofthe cells adjacent to the fracture slowly increases from the initial value of 3.50 MPa to thefinal value of 4.28 MPa at t = 100 s, as the total fluid mass of the closed system increasesand the fluid constantly leaks off from the fracture into the reservoir. The pressure buildupin the reservoir causes a gradual increase in solid stress. Combining solid stress with fracture

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X

Y

Z

Trac

7.80E+06

7.40E+06

7.00E+06

6.60E+06

6.20E+06

5.80E+06

5.40E+06

5.00E+06

4.60E+06

4.20E+06

X

Y

Z

Trac

7.80E+06

7.40E+06

7.00E+06

6.60E+06

6.20E+06

5.80E+06

5.40E+06

5.00E+06

4.60E+06

4.20E+06

Figure 5: Traction on mimetic grid at first time step (top left), projected traction onboundary element grid at first time step (top right), traction on mimetic grid at final timestep (bottom left), projected traction on boundary element grid at final time step (bottomright).

X

Y

Z

W

5.00E-04

4.50E-04

4.00E-04

3.50E-04

3.00E-04

2.50E-04

2.00E-04

1.50E-04

1.00E-04

X

Y

Z

W

5.00E-04

4.50E-04

4.00E-04

3.50E-04

3.00E-04

2.50E-04

2.00E-04

1.50E-04

1.00E-04

Figure 6: Width on boundary element grid at first time step (top left), projected width onmimetic grid at first time step (top right), width on boundary element gird at final time step(bottom left), projected width on mimetic element grid at final time step (bottom right).

13

X

Y

Z

UX

2.00E-06

1.50E-06

1.00E-06

5.00E-07

0.00E+00

-5.00E-07

-1.00E-06

-1.50E-06

-2.00E-06

-2.50E-06

X

Y

Z

UY

1.50E-06

1.00E-06

5.00E-07

0.00E+00

-5.00E-07

-1.00E-06

-1.50E-06

Figure 7: Reservoir displacements at final time step, shown on the plane z = 0: displace-ment in x-direction (top left), displacement in y-direction (top right), and displacementvectors with magnitude (bottom).

qL

-2.00E-01

-3.00E-01

-4.00E-01

-5.00E-01

-6.00E-01

-7.00E-01

-8.00E-01

-9.00E-01

-1.00E+00

Figure 8: Fluid leak-off rate from fracture into reservoir at first time step (left) and finaltime step (right).

14

pressure provides the traction to the boundary element method, and returns the width thatbecomes the “fracture permeability”. The feedback in this multiphysics simulation makesquantities especially hard to predict a priori, and this coupling can cause the system tobehave in complex ways.

P

8.00E+06

7.60E+06

7.20E+06

6.80E+06

6.40E+06

6.00E+06

5.60E+06

5.20E+06

4.80E+06

4.40E+06

X

Y

Z

P

4.25E+06

4.15E+06

4.05E+06

3.95E+06

3.85E+06

3.75E+06

3.65E+06

3.55E+06

Figure 9: Fracture pressure at first time step (top left) and final time step (top right);computed reservoir pressure at final time step on plane z = 0 (bottom).

5 Conclusions and Future Work

In this work, we successfully coupled a reservoir simulator with a boundary element methodin order to model a stationary hydraulic fracture in a poroelastic domain. The overall prob-lem consisted of four different equations: reservoir fluid, fracture fluid, reservoir mechanics,fracture mechanics. These systems were solved with MFMFE, MFD, FEM, and SGBEMdiscretizations, respectively. They were combined into a single framework with an itera-tively coupled solution algorithm based on a fixed stress splitting. Interpolation algorithmswere described in detail in order to transfer functional values between different grids. Ouralgorithm was demonstrated with a realistic three-dimensional numerical example with fluidinjection into a circular fracture. In particular, we were able to compute accurate leak-offwithout using a phenomenological leak-off law such as that put forth by Carter [8], and wewere able to compute accurate fracture widths by accounting for geomechanical effects.

One of the main benefits of using the SGBEM is the ability to model growing fractures,by computing stress intensity values and remeshing the boundary element grid. We planto extend this work to the case of a growing fracture in a future work. This algorithmcould also be modified to solve both reservoir and fracture fluid problems simultaneously toreduce computational cost, as this was the most expensive part of the solution algorithm.

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Acknowledgements

This research was funded by ConocoPhillips grant UTA10-000444, DOE grant ER25617,and NSF CDI grant DMS 0835745. The authors would like to thank Drs. Rick Dean andJoe Schmidt for many valuable discussions. The authors would also like to thank PeterEiseman for providing the GridPro software that was used for hexahedral mesh generation.

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