ices report 13-09 modeling fractures in a poro-elastic medium · reference: benjamin ganis, vivette...
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ICES REPORT 13-09
April 2013
Modeling fractures in a poro-elastic mediumby
Benjamin Ganis, Vivette Girault, Mark Mear, Gurpreet Singh, Mary F. Wheeler
The Institute for Computational Engineering and SciencesThe University of Texas at AustinAustin, Texas 78712
Reference: Benjamin Ganis, Vivette Girault, Mark Mear, Gurpreet Singh, Mary F. Wheeler, Modeling fracturesin a poro-elastic medium, ICES REPORT 13-09, The Institute for Computational Engineering and Sciences, TheUniversity of Texas at Austin, April 2013.
Modeling fractures in a poro-elastic medium
Benjamin Ganis1, Vivette Girault2*, Mark Mear1, Gurpreet Singh1 and Mary Wheeler1
1 Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712
2 Laboratoire Jacques-Louis Lions, Université Paris 6, France. Department of Mathematics, TAMU, College Station, TX 77843
e-mail: bganis,mfw,[email protected],[email protected] — [email protected]
*Corresponding author
Résumé — Nous présentons un modèle de fracture dans un milieu poro-élastique. Il décrit la fracturecomme une courbe ou une surface, selon la dimension, l’épaisseur étant incorporée dans l’équationde l’écoulement dans la fracture. La discrétisation utilise des éléments finis mixtes pour le fluide etdes éléments finis continus pour le déplacement du milieu. Le schéma est résolu par un algorithmequi découple d’une part le calcul du déplacement de celui de l’écoulement, et d’autre part le calcul del’écoulement dans le réservoir de celui dans la fracture. Le modèle est illustré par un essai numérique.
Abstract — We present a fracture model in a poro-elastic medium. The model describes the fracture asa curve or surface according to the dimension, the width of the crack being included into the equationof flow in the fracture. The discretization uses mixed finite elements for the fluid and continuous finiteelements for the porous medium’s displacement. The numerical scheme is solved by an algorithm thatdecouples on one hand the computation of the mechanics from that of the fluid, and on the other handthe computation of the flow in the reservoir from that in the fracture. The model is illustrated by anumerical experiment.
1 INTRODUCTION
In petroleum and environmental engineering, studies of mul-tiscale and multiphysics phenomena such as reservoir defor-mation, surface subsidence, well stability, sand production,waste deposition, hydraulic fracturing, and CO2 sequestra-tion ([10]) require a clear understanding of both the fluidflow and solid phase mechanical response.Traditionally, fluid flow problems were focused on reservoirflow whereas the influence of porous media deformation onpore pressure was usually simplified and approximated asa constant rock compressibility that could not account forrock response especially in naturally fractured and/or stress-sensitive reservoirs ([7]). In this setting, models for narrowfractures can be derived by treating the width as a small pa-
rameter ε and letting this parameter tend to zero. One ap-proach in this direction can be found in the theoretical anal-ysis of Morales and Showalter [19, 20], where only flowis considered, the crack has a flat basis and vertical heightof the order of ε, and the pressure is continuous at the in-terfaces. The former reference deals with the divergenceform of Darcy’s law and the latter with the mixed form. An-other approach consists in treating the fracture as a thin do-main in the framework of domain decomposition. We referthe reader to the extensive work of Jaffré, Roberts and co-authors on Darcy flow, see [1],[16].
To overcome the limitations of decoupled flow simulation,solving the coupled fluid flow and mechanics model inporous media setting has become more feasible with emerg-ing computational machine power.
2
In 1941, Biot ([3]) proposed the first three-dimensional con-solidation theory. Later, poro-elasticity and poro-plasticitymodels for single phase, black oil, and compositional flowmodels were developed following Biot’s work ([7, 9, 13, 15,21, 24]).There are three approaches that are currently employed inthe numerical coupling of fluid flow and the mechanical re-sponse of the reservoir solid structure ([13, 24]). They havebeen referred to in the literature as fully implicit, loose orexplicit coupling, and iterative coupling. The fully implicitinvolves solving all of the governing equations simultane-ously and involves careful implementation with substantiallocal memory and complex solvers. The loosely or explic-itly coupled is less accurate and requires estimates of whento update the mechanic response. Iterative coupling is a se-quential procedure where either the flow or the mechanicsis solved first, followed by incorporating the latest solutioninformation. At each step the procedure is iterated until thesolution converges within an acceptable tolerance. This lastapproach appears to be more scalable on emerging exascalecomputer platforms; it can also be used as a preconditionerfor the fully implicit coupling. But of course, iterative cou-pling schemes must be carefully designed. For instance,two iterative coupling schemes, the undrained split and thefixed stress split, are of particular interest; they converge forslightly compressible linearized flow, see [17]. In contrast,a Von Neumann stability analysis shows (cf. [13]) that thedrained split and the fixed strain split iterative methods arenot stable for the same model problem.Hydraulic fracking is one of the primary ways manufactur-ers retrieve natural gas. Hence, modeling fractures in porousmedia, that are either naturally present or are created bystimulation processes, is a high profile topic, but it is alsoa source of additional and challenging complexities. Thiswork focuses on the simulation of the time-dependent flowof a fluid in a deformable porous medium that contains acrack. The medium in which the crack is embedded is gov-erned by the standard equations of linear poro-elasticity andthe flow of the fluid within the crack is governed by a spe-cific channel flow relation. The two key assumptions of thischannel flow equation are:
1. The width of the crack is small compared to its otherrelevant dimensions, and is such that the crack can berepresented as a single surface. The relevant kinemati-cal data is the jump in the displacement of the mediumacross the crack, i.e. the crack’s width.
2. The permeability in the crack is much larger than in thereservoir.
One advantage of treating the crack as a single surface is thatit alleviates the need for meshing a thin region, thus avoidingall the complexities related to anisotropic elements.In this paper we formulate a discretization that is suitablefor irregular and rough grids and discontinuous full tensor
permeabilities that are often encountered in modeling sub-surface flows. To this end, we develop a multiphysics al-gorithm that couples multipoint flux mixed finite element(MFMFE) methods for the fluid, both in the medium and inthe crack, with continuous Galerkin finite element methods(CG) for the elastic displacement. The MFMFE method wasdeveloped for Darcy flow in [12, 25, 26]. It is locally con-servative with continuous fluxes and can be viewed withina variational framework as a mixed finite element methodwith special approximating spaces and quadrature rules. Itallows for an accurate and efficient treatment of irregular ge-ometries and heterogeneities such as faults, layers, and pin-chouts that require highly distorted grids and discontinuouscoefficients. The resulting discretizations are cell-centeredwith convergent pressures and velocities on general hexahe-dral and simplicial grids. The MFMFE method is extendedto poro-elasticity in [27]; the analysis is based on the tech-nique developed by Phillips and Wheeler in [22].We solve the numerical scheme by an extension to discretefractures of the fixed stress splitting algorithm. Its salientfeatures at each time step are:
1. We decouple the computation of the displacement fromthat of the fluid flow until convergence.
2. We decouple the computation of the fluid flow in thereservoir from that of the flow in the crack until con-vergence.
The model described here does not include crack propaga-tion, i.e. the crack front is stationary. However, the workpresented here can be viewed as a starting point toward hy-draulic fracture modeling in which non planar crack prop-agation is simulated. Our future plans include couplingthe present software with the HYFRAC boundary elementmethod developed by Mear and discussed in [23], as well astreating multiple cracks or multiphase flow in the reservoirand non Newtonian flow in the fracture.This article is organized as follows. The modeling equationsare defined in Section 2. Section 3 summarizes theoreticalresults on the linear model. The numerical scheme and al-gorithm are described in Section 4. A numerical experimentillustrating the model is presented in Section 5. We end withsome conclusions in Section 6.
2 PROBLEM FORMULATION
Let the reservoir Ω be a bounded domain of IRd d = 2 or 3,with piecewise smooth Lipschitz boundary ∂Ω and exteriornormal n. Let the fracture C b Ω be a simple piecewisesmooth curve with endpoints a and b when d = 2 or a simplepiecewise smooth surface with piecewise smooth Lipschitzboundary ∂C when d = 3. To simplify, we denote partialderivatives with respect to time by the index t.
3
2.1 Equations in Ω \ C
The displacement of the solid is modeled in Ω \ C by thequasi-static Biot equations for a linear elastic, homoge-neous, isotropic, porous solid saturated with a slightly com-pressible viscous fluid. The constitutive equation for theCauchy stress tensor σpor is
σpor(u, p) = σ(u) − α p I, (2.1)
where I is the identity tensor, u is the solid’s displacement,p is the fluid pressure, σ is the effective linear elastic stresstensor:
σ(u) = λ(∇ · u)I + 2 Gε(u). (2.2)
Here λ > 0 and G > 0 are the Lamé constants, α > 0 is thedimensionless Biot coefficient, and ε(u) = (∇u + ∇uT )/2 isthe strain tensor. Then the balance of linear momentum inthe solid reads
−∇ · σpor(u, p) = f in Ω \ C, (2.3)
where f is a body force. For the fluid, we use a linearizedslightly compressible single-phase model. Let pr be a ref-erence pressure, ρ f > 0 the fluid phase density, ρ f ,r > 0a constant reference density relative to pr, and c f the fluidcompressibility. We consider the simplified case when ρ f isa linear function of pressure:
ρ f = ρ f ,r(1 + c f (p − pr)
). (2.4)
Next, let ϕ∗ denote the fluid content of the medium; it isrelated to the displacement and pressure by
ϕ∗ = ϕ0 + α∇ · u +1M
p, (2.5)
where ϕ0 is the initial porosity, and M a Biot constant. Thevelocity of the fluid vD in Ω \ C obeys Darcy’s Law:
vD = −1µ f
K(∇ p − ρ f g∇ η
), (2.6)
where K is the absolute permeability tensor, assumed to besymmetric, bounded, uniformly positive definite in spaceand constant in time, µ f > 0 is the constant fluid viscosity,g is the gravitation constant, and η is the distance in the ver-tical direction, variable in space, but constant in time. Thefluid mass balance in Ω \ C reads(
ρ fϕ∗)
t + ∇ · (ρ f vD) = q, (2.7)
where q is a mass source or sink term taking into accountinjection into or out of the reservoir. The equation obtainedby substituting (2.4) and (2.5) into (2.7) is linearized throughthe following considerations. The fluid compressibility c f isof the order of 10−5 or 10−6, i.e. it is small. The fraction p/Mis small and the divergence of u is also small. Therefore
these terms can be neglected. With these approximations,when dividing (2.7) by ρ f ,r, substituting (2.6), and settingq =
qρ f ,r
, we obtain
( 1M
+c fϕ0)pt +α∇·ut−∇·
( 1µ f
K(∇ p−ρ f ,rg∇ η
))= q. (2.8)
Thus the poro-elastic system we are considering for model-ing the displacement u and pressure p in Ω \ C is governedby (2.1), (2.3) and (2.8).
2.2 Equation in C
The trace of p on C is denoted by pc and ∇ is the surfacegradient operator on C, it is the tangential trace of the gra-dient. These quantities are well defined for our problem.The width of the fracture is represented by a non-negativefunction w defined on C; it is the jump of the displacementu in the normal direction. Since the medium is elastic andthe energy is finite, w must be bounded and must vanish onthe boundary of the fracture. We adopt a channel flow rela-tion for the crack in which the volumetric flow rate Q on C
satisfies (e.g. [10])
Q = −w3
12µ f
(∇ pc − ρ f g∇ η
),
and the conservation of mass in the fracture satisfies
(ρ f w)t = −∇ · (ρ fQ) + qI − qL.
Here, qI is a known injection term into the fracture, and qL isan unknown leakoff term from the fracture into the reservoirthat guarantees the conservation of mass in the system. Ap-proximating ρ f by ρ f ,r in the time derivative, linearizing thediffusion term as in the previous section, dividing by ρ f ,r,and setting qI =
qIρ f ,r
, qL =qLρ f ,r
, yield the equation in C:
wt − ∇ ·( w3
12µ f(∇ pc − ρ f ,rg∇ η)
)= qI − qL. (2.9)
In order to specify the relation between the displacement uof the medium and the width w of the fracture, let us dis-tinguish the two sides (or faces) of C by the superscripts +
and −; A specific choice must be selected but is arbitrary.To simplify the discussion, we use a superscript ? to denoteeither + or −. Let Ω? denote the part of Ω adjacent to C?
and let n? denote the unit normal vector to C exterior to Ω?,? = +,−. As the fracture is represented geometrically bya figure with no width, then n− = −n+. For any function gdefined in Ω \ C that has a trace, let g? denote the trace of gon C?, ? = +,−. Then we define the jump of g on C in thedirection of n+ by
[g]C = g+ − g−.
4
The width w is the jump of u · n− on C:
w = −[u]C · n+.
Therefore the only unknown in (2.9) is the leakoff term qL.Summarizing, the equations in Ω \C are (2.3) and (2.8), andthe equation in C is (2.9); the corresponding unknowns areu, p and qL. These equations are complemented in the nextsection by interface, boundary and initial conditions.
2.3 Interface, boundary, and initial conditions
The balance of the normal traction vector and the conserva-tion of mass yield the interface conditions on each side (orface) of C:
(σpor(u, p))?n? = −pcn? , ? = +,−. (2.10)
Then the continuity of pc through C yields
[σpor(u, p)]Cn+ = 0.
The conservation of mass at the interface gives
1µ f
[K(∇ p − ρ f ,rg∇ η)]C · n+ = qL. (2.11)
General conditions on the exterior boundary ∂Ω of Ω can beprescribed for the poro-elastic system, but to simplify, weassume that the displacement u vanishes as well as the fluxK(∇ p − ρ f ,rg∇ η) · n. According to the above hypotheseson the energy and medium, we assume that w is bounded inC and vanishes on ∂C. Finally, the only initial data that weneed are the initial pressure and initial porosity, p0 and ϕ0.Therefore the complete problem statement, called Problem(Q), is: Find u, p, and qL satisfying (2.1), (2.3), (2.8) inΩ \ C and (2.9) in C, for all time t ∈]0,T [, with the interfaceconditions (2.10) and (2.11) on C:
−∇ · σpor(u, p) = f ,
σpor(u, p) = σ(u) − α p I,( 1M
+ c fϕ0)pt + α∇ · ut − ∇ ·
( 1µ f
K(∇ p − ρ f ,rg∇ η
))= q,
wt − ∇ ·( w3
12µ f(∇ pc − ρ f ,rg η)
)= qI − qL,
σpor(u, p))?n? = −pcn?, ? = +,− on C,
1µ f
[K(∇ p − ρ f ,rg∇ η)]C · n+ = qL on C,
where pc = p|C and
w = −[u]C · n+, (2.12)
with the boundary conditions
u = 0 , K(∇ p − ρ f ,rg∇ η) · n = 0 on ∂Ω, (2.13)
w = 0 on ∂C, (2.14)
and the initial condition at time t = 0:
p(0) = p0.
3 THEORETICAL RESULTS
Problem (Q) is highly non linear and its analysis is outsidethe scope of this work. Therefore we present here some the-oretical results on a linearized problem where (2.12) is sub-stituted into the first term of (2.9) while the factor w3 in thesecond term is assumed to be known. Knowing w3 amountsto linearizing the nonlinear term with respect to w, such ascould be encountered in a time-stepping algorithm.
3.1 Variational formulation
It is convenient (but not fundamental) to generalize the no-tation of Section 2.2 by introducing an auxiliary partitionof Ω into two non-overlapping subdomains Ω+ and Ω− withLipschitz interface Γ containing C, Ω? being adjacent to C?,? = +,−. The precise shape of Γ is not important as longas Ω+ and Ω− are both Lipschitz. Let Γ? = ∂Ω? \ Γ; forany funtion g defined in Ω, we set g? = g|Ω? , ? = +,−. LetW = H1(Ω+ ∪Ω−), i.e.
W = v ∈ L2(Ω) ; v? ∈ H1(Ω?), ? = +,−,
normed by the graph norm:
‖v‖W =(‖v+‖2H1(Ω+) + ‖v−‖2H1(Ω−)
)1/2.
The space for the displacement is
V = v ∈ Wd ; [v]Γ\C = 0, v?|Γ? = 0, ? = +,−, (3.1)
with the norm of Wd:
‖v‖V =( d∑
i=1
‖vi‖2W)1/2
. (3.2)
The space for the pressure is more subtle: It is H1 in Ω, butin C it is
H1w(C) = z ∈ H1/2(C) ; w3/2∇ z ∈ L2(C)d−1, (3.3)
equipped with the norm
‖z‖H1w(C) =
(‖z‖2H1/2(C) + ‖w3/2∇ z‖2L2(C)
)1/2, (3.4)
where H1/2(C) is the space of traces on C of H1(Ω) func-tions. Thus p belongs to the space:
Q = q ∈ H1(Ω) ; qc ∈ H1w(C) , qc := q|C, (3.5)
equipped with the graph norm
‖q‖Q =(‖q‖2H1(Ω) + ‖qc‖
2H1
w(C)
)1/2, (3.6)
Finally, the space for the leakoff variable qL is H1w(C)′, the
dual space of H1w(C), equipped with the dual norm. Then, by
5
multiplying (2.3), (2.8) and (2.9) with adequate test func-tions, applying Green’s formula, using the interface con-ditions (2.10) and (2.11), the boundary conditions (2.13)and (2.14), substituting (2.12) into the first term of (2.9),and assuming sufficiently smooth data, we obtain the varia-tional formulation: Find p ∈ H1(0,T ; L2(Ω)) ∩ L∞(0,T ; Q),u ∈ H1(0,T ; V), and qL ∈ L2(0,T ; H1
w(C)′) solution a.e. in]0,T [ of
2G(ε(u), ε(v)
)+ λ
(∇ · u,∇ · v
)− α
(p,∇ · v
)+
(pc, [v]C · n+)
C =(f , v
), ∀v ∈ V,
(3.7)
∀θ ∈ Q ,( 1
M+ c fϕ0
)(pt, θ
)+ α
(∇ · ut, θ
)+
1µ f
(K(∇ p − ρ f ,rg∇ η
),∇ θ
)=
(q, θ
)+ 〈qL, θc〉C,
(3.8)
where the scalar products are taken in Ω \ C,
−([ut]C · n+, θc
)C +
( w3
12µ f
(∇ pc − ρ f ,rg∇ η
),∇θc
)C
= 〈qI − qL, θc〉C , ∀θ ∈ Q,(3.9)
with the initial condition at time t = 0
p(0) = p0. (3.10)
3.2 Two reduced formulations
A first reduced formulation is obtained by summing(3.8) and (3.9), thereby eliminating qL: Find p ∈
H1(0,T ; L2(Ω)) ∩ L2(0,T ; Q) and u ∈ H1(0,T ; V), solutionof (3.10) and a.e. in ]0,T [ of (3.7) and for all θ ∈ Q
( 1M
+ c fϕ0)(
pt, θ)
+ α(∇ · ut, θ
)−
([ut]C · n+, θc
)C
+1µ f
(K(∇ p − ρ f ,rg∇ η
),∇ θ
)+
( w3
12µ f
(∇ pc − ρ f ,rg∇ η
),∇θc
)C = (q, θ) + (qI , θc)C.
(3.11)
Next, (3.7) can be viewed as an equation for the displace-ment u as an affine function of p. More precisely, we split uinto two parts:
u = u + u(p),
where, for f given in L2(Ω)d, u ∈ V solves for all v ∈ V
2G(ε(u), ε(v)
)+ λ
(∇ · u,∇ · v
)=
(f , v
), (3.12)
and for p given in H1(Ω), u(p) ∈ V solves for all v ∈ V
2G(ε(u(p)), ε(v)
)+ λ
(∇ · u(p),∇ · v
)= α
(p,∇ · v
)−
(pc, [v]C · n+)
C.(3.13)
It is easy to prove that each of these two systems has aunique solution. This leads to a second reduced problem:Find p ∈ H1(0,T ; L2(Ω)) ∩ L2(0,T ; Q) solution of (3.10)and
∀θ ∈ Q ,( 1
M+ c fϕ0
)(pt, θ
)+ α
(∇ · (ut + u(pt)), θ
)−
([ut + u(pt)]C · n+, θc
)C +
1µ f
(K(∇ p − ρ f ,rg∇ η
),∇ θ
)+
( w3
12µ(∇ pc − ρ f ,rg∇ η
),∇θc
)C =
(q, θ
)+ (qI , θc)C.
(3.14)
where u ∈ H1(0,T ; V) and u(p) ∈ H1(0,T ; V) are theunique solutions of (3.12) and (3.13) respectively. The ad-vantage of (3.14) is that its only unknown is p.
3.3 Existence and uniqueness
The analysis of Problem (3.7) – (3.10) relies on suitableproperties of H1
w(C), that in turn depend on the regularityof w. For the practical applications we have in mind, w hasthe following properties when d = 3; the statement easilyextends to d = 2.
Hypothesis 3.1. The non negative function w is H1 in timeand is smooth in space away from the crack front, i.e. theboundary ∂C. It vanishes on ∂C and in a neighborhood ofany point of ∂C, w is asymptotically of the form:
w(x, y) ' x1/2 f (y), (3.15)
where y is locally parallel to the crack front, x is the distanceto the crack, and f is smooth.
With this hypothesis, we can validate the integrations byparts required to pass from Problem (Q) to its variational for-mulation (3.7) – (3.9) and we recover the leakoff unknownqL from (3.11), thus showing equivalence between all threeformulations. This equivalence allows to “construct" a so-lution of Problem (Q) by discretizing (3.14) with a Galerkinmethod, more precisely by approximating p in a truncatedbasis, say of dimension m, of the space Q. This leads to asquare system of linear Ordinary Differential Equations oforder one and of dimension m, with an initial condition. Aclose examination of this system’s matrices shows that it hasa unique solution. Then reasonable regularity assumptionson the data, Hypothesis 3.1, and the theory developed by[22], yield a set of basic a priori estimates on the discretepressure, say pm, and displacement u(pm). These are suffi-cient to pass to the limit in the discrete version of (3.14), but,as is the case of a poro-elastic system, they are not sufficientto recover the initial condition on p. This initial conditioncan be recovered by means of a suitable a priori estimate onthe time derivative p′m, but it requires a slightly stronger reg-ularity assumption on the data (still fairly reasonable) and
6
an assumption on the time growth of w. It allows the timeevolution of a crack, but it cannot be used in the formationof a crack, since it forbids an opening of a region that isoriginally closed. More precisely, we make the followinghypothesis.
Hypothesis 3.2. There exists a constant C such that
a.e. inC×]0,T [ , w′ ≤ C w. (3.16)
With this additional assumption, we can prove existence anduniqueness of a solution of Problem (Q).
Theorem 3.1. If f is in H2(0,T ; L2(Ω)d), q in L2(Ω×]0,T [),qI in H1(0,T ; L2(C)), p0 in Q, and w satisfies Hypotheseses3.1 and 3.2, then the linearized problem (3.7)–(3.10) hasone and only one solution p in H1(0,T ; L2(Ω))∩L2(0,T ; Q),u in H1(0,T ; V), and qL in L2(0,T ; H1
w(C)′). This solutiondepends continuously on the data.
3.4 Heuristic mixed formulation
A mixed formulation for the flow is useful in view of dis-cretization because it leads to locally conservative schemes.The mixed formulation we present in this short section is de-rived heuristically in the sense that it is written in spaces ofsufficiently smooth functions in the fracture. As the discretefunctions are always locally smooth, all terms in the discreteformulation will make sense.For the pressure in Ω \ C, we define the auxiliary velocity zby
z = −K∇(p − ρ f ,rgη), (3.17)
and for the pressure in C, we define a surface velocity ζ by
ζ = −∇(pc − ρ f ,rgη). (3.18)
Let
Z = q ∈ H(div; Ω+ ∪Ω−) ; [q] · n+ = 0 on Γ \ C
q · n = 0 on ∂Ω,
ZC = µ ∈ L2(C)d−1 ; ∇ · (w3µ) ∈ L2(C).
In view of (2.13), (2.11), and the regularity in Theorem 3.1,we have z ∈ Z and we see that it must satisfy the essentialjump condition:
1µ f
[z]C · n+ = −qL on C. (3.19)
We use the same space V for u, but we reduce the regularityof p and set
Q = L2(Ω).
As the trace of functions in Q are no longer meaningful, weintroduce an additional variable pc in the space
ΘC = L2(C),
and we assume that the leakoff term is in ΘC. Then themixed variational formulation reads (to simplify, we do notspecify the dependence of the spaces on time): Find u ∈ V,p ∈ Q, pc ∈ ΘC, qL ∈ ΘC, z ∈ Z, and ζ ∈ ZC such that(3.7), (3.10), and (3.19) hold:
2G(ε(u), ε(v)
)+ λ
(∇ · u,∇ · v
)− α
(p,∇ · v
)+
(pc, [v]C · n+)
C =(f , v
), ∀v ∈ V,
p(0) = p0,
1µ f
[z]C · n+ = −qL on C,
and for all θ in Q
( 1M
+c fϕ0)(
pt, θ)+α
(∇·ut, θ
)+
1µ f
(div z, θ
)=
(q, θ
), (3.20)
for all θc in ΘC
−([ut]C · n+, θc
)C +
112µ f
(∇ · (w3ζ), θc
)C =
(qI − qL, θc
)C,
(3.21)∀q ∈ Z ,
(K−1 z, q
)=(p,∇ · q) −
(pc, [q]C · n+)
C
+(∇(ρ f ,rgη), q),
(3.22)
∀µ ∈ ZC ,(w3ζ,µ
)C =
(pc,∇ · (w3µ)
)C
+(∇(ρ f ,rgη),w3µ
)C.
(3.23)
It follows from (3.22) that p belongs to H1(Ω) and pc = p|C.Hence pc acts like a Lagrange multiplier that enforces thecontinuity of the traces on C. Moreover, when the solutionsare sufficiently smooth, the mixed and primitive formula-tions are equivalent. In addition, qL can be immediatelyeliminated by substituting (3.19) into (3.21); this yields
∀θc ∈ ΘC , −([ut]C · n+, θc
)C +
112µ f
(∇ · (w3ζ), θc
)C
−1µ f
([z]C · n+, θc
)C =
(qI , θc
)C.
(3.24)
4 DISCRETIZATION
In this section, we present a space–time discretization ofthe linearized problem (3.7), (3.10), (3.20), (3.22), (3.23),(3.24) with a backward Euler scheme in time and Galerkinfinite elements in space for the displacement, except in aneighborhood of the crack. The flow equation is discretizedby a multipoint flux mixed finite element method (MFMFE).In order to avoid handling curved elements or approximat-ing curved surfaces, we assume that both ∂Ω and the crack C
are polygonal or polyhedral surfaces. We shall refer to thisscheme as the MFMFE-CG scheme. A convergence anal-ysis of the MFE-CG scheme for the poro-elasticity system
7
without a crack is done in [27], where two versions of themethod: one with a symmetric quadrature rule on simplicialgrids or on smooth quadrilateral and hexahedral grids, andone with a non-symmetric quadrature rule on rough quadri-lateral and hexahedral grids are treated. Theoretical and nu-merical results demonstrated first order convergence in timeand space for the fluid pressure and velocity as well as forsolid displacement. In the formulation below we restrict ourattention to the symmetric quadrature rule for the flow.
4.1 MFMFE spaces
We first describe the interior discretization. Let Ω be ex-actly partitioned into a conforming union of finite elementsof characteristic size h, i.e. without hanging nodes. For con-venience we assume that the elements are quadrilaterals in2D and hexahedra in 3D. Let us denote the partition by Th
and assume that it is shape-regular [8]. For convenience, wealso assume that Th triangulates exactly Ω+ and Ω−; in par-ticular, no element crosses C. The displacement, velocityand pressure finite element spaces on any physical elementE are defined, respectively, via the vector transformation
v↔ v : v = v F−1E ,
via the Piola transformation
z↔ z : z =1JE
DFE z F−1E , (4.1)
and via the scalar transformation
w↔ w : w = w F−1E ,
where FE denotes a mapping from the reference element Eto the physical element E, DFE is the Jacobian of FE , andJE is its determinant. The advantage of the Piola transfor-mation is that it preserves the divergence and the normalcomponents of the velocity vectors on the faces (edges) [11,Ch.III,4.4], [4]:
(∇·v,w)E = (∇·v, w)E and (v·ne,w)e = (v·ne, w)e, (4.2)
which is needed for an H(div; Ω)-conforming velocity spaceas required by (4.3).The finite element spaces Vh, Zh and Qh on Th are given by
Vh =v ∈ V ; v|E ↔ v, v ∈ V(E), ∀E ∈ Th
,
Zh =z ∈ Z ; z|E ↔ z, z ∈ Z(E), ∀E ∈ Th
,
Qh =q ∈ Q ; q|E ↔ q, q ∈ Q(E), ∀E ∈ Th
,
(4.3)
where V(E), Z(E) and Q(E) are finite element spaces on thereference element E. Note that since C is an open set and Vh
is contained in V, the functions of Vh are continuous on ∂C,i.e. they have no jump on ∂C.Let Pk denote the space of polynomials of degree k in twoor three variables, and let Q1 denote the bilinear or trilinearpolynomial spaces in two or three variables, according to thedimension.
Convex quadrilaterals
In the case of convex quadrilaterals, E is the unit squarewith vertices r1 = (0, 0)T , r2 = (1, 0)T , r3 = (1, 1)T , andr4 = (0, 1)T . Denote by ri, i = 1, . . . , 4 the correspondingvertices of E. In this case, FE is the bilinear mapping givenas
FE(x, y) = r1(1 − x)(1 − y) + r2 x(1 − y) + r3 xy + r4(1 − x)y;
the space for the displacement is
V(E) = Q1(E)2,
and the space for the flow is the lowest order BDM1 [5]space
Z(E) = P1(E)2 + r curl(x2y) + s curl(xy2),
Q(E) = P0(E),
where r and s are real constants.
Hexahedra
In the case of hexahedra, E is the unit cube with verticesr1 = (0, 0, 0)T , r2 = (1, 0, 0)T , r3 = (1, 1, 0)T , r4 = (0, 1, 0)T ,r5 = (0, 0, 1)T , r6 = (1, 0, 1)T , r7 = (1, 1, 1)T , and r8 =
(0, 1, 1)T . Denote by ri = (xi, yi, zi)T , i = 1, . . . , 8, the eightcorresponding vertices of E. We note that the element canhave non-planar faces. In this case FE is a trilinear mappinggiven by
FE(x, y, z) = r1(1 − x)(1 − y)(1 − z) + r2 x(1 − y)(1 − z)+ r3 xy(1 − z) + r4(1 − x)y(1 − z) + r5(1 − x)(1 − y)z+ r6 x(1 − y)z + r7 xyz + r8(1 − x)yz,
and the spaces are defined by
V(E) = Q1(E)3,
for the displacement, and by enhancing the BDDF1 spaces[12] for the flow :
Z(E) = BDDF1(E)
+ s2curl(0, 0, x2z)T + s3curl(0, 0, x2yz)T
+ t2curl(xy2, 0, 0)T + t3curl(xy2z, 0, 0)T
+ w2curl(0, yz2, 0)T + w3curl(0, xyz2, 0)T ,
Q(E) = P0(E),
(4.4)
where the BDDF1(E) space is defined as [6]:
BDDF1(E) = P1(E)3 + s0curl(0, 0, xyz)T
+ s1curl(0, 0, xy2)T + t0curl(xyz, 0, 0)T
+ t1curl(yz2, 0, 0)T + w0curl(0, xyz, 0)T
+ w1curl(0, x2z, 0)T .
8
In the above equations, si, ti,wi (i = 0, . . . , 3) are real con-stants. In all cases the degrees of freedom (DOF) for thedisplacements are chosen as Lagrangian nodal point values.The velocity DOF are chosen to be the normal componentsat the d vertices on each face. The dimension of the space isdnv, where d is the dimension and nv is the number of ver-tices in E. Note that, although the original BDDF1 spaceshave only three DOF on square faces, these spaces have beenenhanced in [12] to have four DOF on square faces. Thisspecial choice is needed in the reduction to a cell-centeredpressure stencil in a pure Darcy flow problem as describedlater in this section.
4.2 A quadrature rule
The integration on a physical element is performed by map-ping to the reference element and choosing a quadrature ruleon E. Using the Piola transformation, we write (K−1·, ·) in(3.22) as
(K−1 z, q)E =( 1
JEDFT
E K−1(FE(x))DFE z, q)
E
≡ (ME z, q)E ,
where
ME =1JE
DFTE K−1(FE(x))DFE . (4.5)
Then, we denote the trapezoidal rule on E by Trap(·, ·)E :
Trap( z, q)E ≡|E|k
k∑i=1
z(ri) · q(ri), (4.6)
where riki=1 are the vertices of E. The quadrature rule on
an element E is defined as
(K−1 z, q)Q,E ≡ Trap(ME z, q)E
=|E|k
k∑i=1
ME(ri) z(ri) · q(ri).(4.7)
Mapping back to the physical element E, we have thequadrature rule on E as
(K−1 z, q)Q,E =|E|k
k∑i=1
JE(ri)K−1(ri)z(ri) · q(ri), (4.8)
and we define (K−1 z, q)Q by summing (4.8) on all elementsE of Th. As the trapezoidal rule induces a scalar product onthe above finite element spaces that yields a norm uniformlyequivalent to the L2 norm (see [12, 25]), and the tensor Kis symmetric, bounded, and uniformly positive definite, thisquadrature rule also yields a norm on these spaces uniformlyequivalent to the L2 norm.
4.3 Discretization in the fracture
Since C is assumed to be polygonal or polyhedral, we canmap each line segment or plane face of C onto a segment inthe x1 line (when d = 2) or a polygon in the x1 − x2 plane(when d = 3) by a rigid-body motion that preserves bothsurface gradient and divergence, maps the normal n+ into aunit vector along x3, for exemple −e3, and whose Jacobian isone. After this change in variable, all operations on this linesegment or plane face can be treated as the same operationson the x1 axis or x1 − x2 plane. To simplify, we do not use aparticular notation for this change in variable, and work as ifthe line segments or plane faces of C lie on the x1 line or x1−
x2 plane. Let Si, 1 ≤ i ≤ I, denote the line segments or planefaces of C; to simplify, we drop the index i. Let TS,h be ashape regular partition of S, for instance the trace of Th on S
(but it can be something else) let e denote a generic elementof TS,h, with reference element e, and let the scalar and Piolatransforms be defined by the same formula as above, butwith respect to e instead of E. Then we define the finiteelement spaces on C by:
ZC,h =z ∈ ZC ; z|Si ∈ ZSi,h, 1 ≤ i ≤ I
,
ΘC,h =q ∈ ΘC ; q|Si ∈ ΘSi,h, 1 ≤ i ≤ I
,
with
ZS,h =z ∈ ZC ; z|e ↔ z, z ∈ ZC(e), ∀e ∈ TS,h
,
ΘS,h =q ∈ ΘC ; q|e ↔ q, q ∈ ΘC(e), ∀e ∈ TS,h
,
where ZC(e) and ΘC(e) are finite element spaces on the ref-erence element e.We approximate (w3ζ,µ)S in (3.23) by first writing
(w3ζ,µ)e =( 1
Jew(Fe(x))3DFT
e DFeζ, µ)
e≡ (Meζ, µ)e,
where
Me =1Je
w(Fe(x))3DFTe DFe, (4.9)
and next applying the trapezoidal rule Trap(·, ·)e on e:
Trap(ζ, µ)e ≡|e|k
k∑i=1
ζ(ri) · µ(ri), (4.10)
where riki=1 are the vertices of e. Hence mapping back to
the physical element e, we approximate (w3ζ,µ)e by
(w3ζ,µ)Q,e =|e|k
k∑i=1
Je(ri)w(ri)3ζ(ri) · µ(ri), (4.11)
and we obtain the approximation (w3ζ,µ)Q by summingover all elements of TS,h and over all S of C.
9
Let Ih denote the standard nodal Lagrange interpolant on P1or Q1. Then
(w3ζ,µ)Q = (Ih(w3)ζ,µ)Q,
and the properties of the quadrature formula imply that
∀µ ∈ ZS,h , (w3µ,µ)1/2Q ' ‖Ih(w3/2)µ‖L2(C).
4.4 The scheme
Let N ≥ 1 be a fixed integer, ∆t = T/N the time step, andti = i∆t, 0 ≤ i ≤ N, the discrete time points. We define theapproximations f n of f (tn) and qn of q(tn) for almost everyx ∈ Ω+ ∪Ω− or Ω by
f n(x) =1∆t
∫ tn
tn−1
f (x, t)dt , qn(x) =1∆t
∫ tn
tn−1
q(x, t)dt.
(4.12)Similarly, we define the approximation qn
I of qI(tn) for al-most every s ∈ C by
qnI (s) =
1∆t
∫ tn
tn−1
qI(s, t)dt. (4.13)
On the other hand, we use point values for w in time, i.e.define
∀s ∈ C , wn(s) = w(s, tn).
Then we propose an iterative algorithm for solving thefully discrete version of (3.7), (3.10), (3.20), (3.22), (3.23),(3.24) that uses the quadrature rule of Sections 4.2 and4.3. A flowchart of the scheme is given in Figure 1. Ituses the Fixed Stress split iterative algorithm of Mikelic &Wheeler [18]. The mean stress is given by σ = 1
3∑3
i=1 σporii ,
where σpor is defined by (2.1). Thus the discrete mean stressreads
σh =α2
cr∇ · uh − αph,
where
cr =3α2
3λ + 2G.
For a given displacement and mean stress, the algorithm de-couples sequentially the computation of the flow in the reser-voir from that in the fracture.Recall that the given initial pressure p0 is sufficientlysmooth and has a trace, say p0
c , on C; this is the initialpressure in C.
• At time t = 0, let p0h = Ph(p0) and p0
c,h = Ph(p0c) where Ph
and Ph are suitable approximation operators from Q into Qh
and ΘC into ΘC,h respectively. Then the initial displacementis approximated by discretizing the elasticity equation (3.13)in Ω \ C: Find u0
h ∈ Vh solution for all vh ∈ Vh of
2G(ε(u0
h), ε(vh))
+ λ(∇ · u0
h,∇ · vh)
= α(p0
h,∇ · vh)− (p0
c,h, [vh]C · n+)C + ( f 0, vh).(4.14)
Time step
Fixed stress iter.
Solve Res.-Frac. flow with fixed stress
Start
Fixed Stress Iterative Coupling of Poroelasticity with Fracture
Solve mechanics system (4.15)
End
Yes No
!
n + 1
!"#n+1,!! < TOL ?
Reservoir-Fracture Flow with fixed stress
Start
End
Res.-Frac. iter j
Yes
No
Solve reservoir flow system (4.16)-(4.17)
Solve fracture flow system (4.18)-(4.19)
!"pn+1,!,j! < TOL ?
Figure 1
Flowcharts for outer iteration (left) and inner iteration (right)with fixed stress iterative coupling of poroelasticity with frac-ture.
This gives the initial mean stress
σ0h =
α2
cr∇ · u0
h − αp0h.
•Marching in timeFor any n, 0 ≤ n ≤ N − 1, knowing un
h, pnh, σn
h and pnc,h, the
discrete functions at time tn+1 are computed iteratively bytwo nested loops:
• Starting functions
pn+1,0h = pn
h , un+1,0h = un
h , pn+1,0c,h = pn
c,h , σn+1,0h = σn
h.
• Outer loopFor ` = 0, 1, . . . , knowing un+1,`
h , σn+1,`h , and pn+1,`
c,h compute
iteratively intermediate functions pn+1, jh , zn+1, j
h , ζn+1, jh , and
pn+1, jc,h , for j = 1, 2, . . . , jm by the inner loop written below.
Setpn+1,`+1
h := pn+1, jmh , pn+1,`+1
c,h := pn+1, jmc,h ,
and compute the displacement un+1,`+1h ∈ Vh by solving for
all vh ∈ Vh
2G(ε(un+1,`+1
h ), ε(vh))
+ λ(∇ · un+1,`+1
h ,∇ · vh)
= α(pn+1,`+1
h ,∇ · vh)− (pn+1,`+1
c,h , [vh]C · n+)C
+ ( f n+1, vh).
(4.15)
Update the mean stress by
σn+1,`+1h =
α2
cr∇ · un+1,`+1
h − αpn+1,`+1h ,
10
and test the difference in porosities (see (2.5)):
α∇ ·(un+1,`+1
h − un+1,`h
)+
1M
(pn+1,`+1
h − pn+1,`h
).
If it is larger than the tolerance, increment ` and return to theouter loop. Otherwise, set
un+1h := un+1,`+1
h , pn+1h = pn+1,`+1
h , pn+1c,h := pn+1,`+1
c,h ,
σn+1h = σn+1,`+1
h ,
increment n and return to the marching in time.• • Inner loopSet
pn+1,0c,h := pn+1,`
c,h .
• • For j = 0, 1, . . . , knowing σn+1,`h , un+1,`
h , and pn+1, jc,h , com-
pute pn+1, j+1h ∈ Qh and zn+1, j+1
h ∈ Zh solution of
( 1M
+ c fϕ0) 1∆t
(pn+1, j+1
h − pnh, θh
)+
1µ f
(div zn+1, j+1
h , θh)
=(qn+1, θh
)−
cr
α
1∆t
(σn+1,`
h − σnh, θh
), ∀θh ∈ Qh,
(4.16)
∀qh ∈ Zh ,(K−1 zn+1, j+1
h , qh)
Q = (pn+1, j+1h ,∇ · qh)
− (pn+1, jc,h , [qh]C · n+)C +
(∇(ρ f ,rgη), qh
).
(4.17)
Compute ζn+1, j+1h ∈ ZC,h and pn+1, j+1
c,h ∈ ΘC,h solution for allθc,h in ΘC,h of
−1∆t
([un+1,`
h − unh]C · n+, θc,h
)C
−1µ f
([zn+1, j+1
h ]C · n+, θc,h)C
+1
12µ f
(∇ · (w(tn+1)3ζn+1, j+1
h ), θc,h)C
+ β(pn+1, j+1
c,h − pn+1, jc,h , θc,h
)C = (qn+1
I , θc,h)C,
(4.18)
(w(tn+1)3ζn+1, j+1
h ,µh)
Q =(pn+1, j+1
c,h ,∇ · (w(tn+1)3µh))C
+(∇(ρ f ,rgη),w(tn+1)3µh
)C , ∀µh ∈ ZC,h,
(4.19)
where β > 0 is a stabilizing factor to be adjusted. Test thedifference pn+1, j+1
c,h − pn+1, jc,h . If it is less than the tolerance, set
jm := j + 1 , pn+1, jmh := pn+1, j+1
h , pn+1, jmc,h := pn+1, j+1
c,h ,
and return to the outer loop.For given pressures, (4.14) and (4.15) have a uniquesolution. It is easy to prove that (4.16)–(4.17) deter-mine pn+1, j+1
h and zn+1, j+1h , and that (4.18)–(4.19) determine
Ih(w(tn+1)3/2)ζn+1, j+1h and pn+1, j+1
c,h , owing in particular to thestabilizing term with factor β. Two remarks are in order re-garding the algorithm. Equation (4.15) utilizes discontinu-ous spaces of functions on C and weakly imposes a traction
boundary condition: pn+1,`+1c,h . Computing the solution re-
lies on the work of Liu in [14]. The system (4.18)–(4.19)is solved in Section 5 below by means of a mimetic formu-lation (see [2]), which is closely related to a mixed finiteelement method when using quadrilateral elements.
5 NUMERICAL EXPERIMENT
We provide the following numerical example as a simpleillustration of the model and its predictive capability. Fol-lowing the flowchart in Figure 1, we have implemented theFixed Stress iterative coupling scheme with the inner itera-tion between reservoir flow and fracture flow into the IPARSreservoir simulation code.The domain is taken to be the cube Ω = (0, 10)3 m, witha square fracture C centered on the plane y = 5 m ofsize 7.5 × 7.5 m. The domain is discretized into 5 × 6 × 5structured hexahedral elements. There are uniform meshwidths of 2 m in the x and z directions, and mesh widths of2.475, 2.475, 0.05, 0.05, 2.475, 2.475 m in the y direction.The initial fluid pressure in both the reservoir and fractureis taken to be p0 = 3.5 × 106 Pa. The external fluid bound-ary conditions are set to be p = p0 on ∂Ω, which will allowfluid to escape as pressure rises. The external mechanicalboundary conditions are such that the bottom face is com-pletely pinned: u = 0 m; the four lateral faces are tractionfree: σporn = 0 psi; and the top face has an overburdentraction of σporn = (−3.8, 0, 0) × 106 psi. The stabilizationparameter is taken to be β = 1.0×10−4. The remaining inputparameters are summarized in Table 1. The Lamé constantsare given by λ = Eν/[(1 + ν)(1− 2ν)] and G = E/[2(1 + ν)].Fluid is injected into the center of the fracture with a con-stant rate of qI = 1000 kg · s−1, which will induce a pressuregradient in the fracture and leak off into the domain. Boththe leakoff rate qL and the fracture pressure pc are unknownsdetermined by the coupling of the reservoir and fracture flowmodels. The fluid pressure computed in the fracture plane isshown in Figure 2. The pressurized fracture also induces atraction on C, which creates a discontinuity in the displace-ment field, allowing the calculation of a dynamic width,which is shown in Figure 3.
6 SOME CONCLUSIONS
We have studied the numerical approximation of a fracturemodel in a poroelastic medium, where the fracture is rep-resented as a curve or surface and its width is incorporatedinto the flow equation in the fracture. We have used a Mul-tipoint Flux Mixed Finite Element Method for the flow inthe reservoir and a Mimetic Finite Difference Method in thefracture, and a Continuous Galerkin Method for the geome-chanics. The scheme was solved by a fixed stress split it-erative coupling for the flow and mechanics and an iterative
11
Parameter Quantity ValueK 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
ρ f ,r Reference Fluid Density 897 kg m−3
g Gravitational Acceleration 0 m s−2
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 20 s∆t Time Step 1 s
TABLE 1
Input parameters for the numerical example in Si units.
algorithm coupling the flow in the reservoir and in the crack.A numerical experiment illustrated the relevance of the me-chanical model and the efficiency of the numerical methodand algorithm.
ACKNOWLEDGMENTS
The first, fourth and fifth authors were funded by theDOE grant DE-GGO2-04ER25617. The second author wasfunded by the Center For Subsurface Modeling IndustrialAffiliates and by a J.T. Oden Faculty Fellowship. The thirdauthor was funded by Conoco Phillips. The authors wouldlike to thank Drs. Rick Dean, Joe Schmidt, and Horacio Flo-rez for their assistance in this paper. We also wish to thankOmar Al Hinai for his assistance with the fracture flow code.
REFERENCES
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9 O. Coussy. Mechanics of Porous Continua. Wiley, New York,1995.
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11 V. Girault and P.A. Raviart. Finite element methods for Navier-Stokes equations: Theory and algorithms. Springer Series inComputational Mathematics, 5. Springer-Verlag, Berlin, 1986.
12 R. Ingram, M.F. Wheeler, and I. Yotov. A multipoint flux mixedfinite element method on hexahedra. SIAM J. Numer. Anal., 48:1281–1312, 2010.
13 J. Kim, H.A. Tchelepi, and R. Juanes. Stability, accuracy, andefficiency of sequential methods for coupled flow and geome-chanics. SPE Journal, pages 249–262, June 2011.
14 R. Liu. Discontinuous Galerkin for Mechanics. PhD thesis,The University of Texas at Austin, Austin, Texas, 2004.
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12
X
YZ
P
5.20E+06
5.08E+06
4.95E+06
4.83E+06
4.71E+06
4.58E+06
4.46E+06
4.34E+06
4.22E+06
4.09E+06
3.97E+06
3.85E+06
3.72E+06
3.60E+06
X
YZ
P
5.20E+06
5.08E+06
4.95E+06
4.83E+06
4.71E+06
4.58E+06
4.46E+06
4.34E+06
4.22E+06
4.09E+06
3.97E+06
3.85E+06
3.72E+06
3.60E+06
Figure 2
Computed fluid pressure in the fracture plane and a perpendic-ular plane on the first time step (top), and the final time step(bottom).
X
Y Z
W
7E06
6.5E06
6E06
5.5E06
5E06
4.5E06
4E06
3.5E06
3E06
2.5E06
2E06
1.5E06
1E06
5E07
X
Y Z
W
7E06
6.5E06
6E06
5.5E06
5E06
4.5E06
4E06
3.5E06
3E06
2.5E06
2E06
1.5E06
1E06
5E07
Figure 3
Computed fracture width on the first time step (top), and the finaltime step (bottom).
13
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