Flow and transport in fractured poroelastic media Ilona Ambartsumyan * Eldar Khattatov * Truong Nguyen * Ivan Yotov * October 6, 2018 Abstract We study flow and transport in fractured poroelastic media using Stokes flow in the fractures and the Biot model in the porous media. The Stokes-Biot model is coupled with an advection-diffusion equation for modeling transport of chemical species within the fluid. The continuity of flux on the fracture-matrix interfaces is imposed via a Lagrange multiplier. The coupled system is discretized by a finite element method using Stokes elements, mixed Darcy elements, conforming displacement elements, and discontinuous Galerkin for transport. The stability and convergence of the coupled scheme are analyzed. Computational results verifying the theory as well as simulations of flow and transport in fractured poroelastic media are presented. 1 Introduction Flow and transport in fractured poroelastic media occur in many applications, including enhanced oil and gas recovery, hydraulic fracturing, groundwater hydrology, and subsurface waste repositories. These are challenging multiphysics processes involving interaction between a free fluid in the fractures with a fluid within the porous medium. The fluid flow may cause and be affected by solid deformation. For example, geomechanics effects are critical in hydraulic fracturing, as well as in modeling phenomena such as subsidence and compaction. Furthermore, the flow process may be coupled with transport phenomena, with the substance of interest propagating both through the fracture network and the porous matrix. Typical examples include tracking and cleaning up groundwater contaminants, leakage of subsurface radioactive waste, and proppant injection in hydraulic fracturing. We use the Stokes equations to model the free fluid in the fractures and the Biot poroelasticity model [9] for the fluid in the poroelastic region. The latter is based on a linear stress-strain constitutive relationship for the porous solid, and Darcy’s law, which describes the average velocity of the fluid in the pores. The interaction across the fracture-matrix interfaces exhibits features of both Stokes-Darcy coupling [24, 34, 40, 54, 61] and fluid-structure interaction (FSI) [7, 17, 26, 31, 52]. We refer to the Stokes- Biot coupling considered in this paper as fluid-poroelastic structure interaction (FPSI). There has been growing interest in such models in the literature. The well-posedness of the mathematical model was studied in [57]. Numerical studies include variational multiscale methods for the monolithic system and iterative partitioned scheme [6], a non-iterative operator-splitting method [15], a partitioned method based on Nitsche’s coupling [14], and a Lagrange multiplier formulation for the continuity of flux [4]. To simplify the presentation we consider a fixed domain in time. As presented, the model is suitable for deformations that are small relative to the width of the fractures. This is valid for scales that are * Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA; [email protected], [email protected], [email protected], [email protected]; partially supported by DOE grant DE-FG02-04ER25618 and NSF grant DMS 1418947. 1
Transcript
Flow and transport in fractured poroelastic media
Ilona Ambartsumyan∗ Eldar Khattatov∗ Truong Nguyen∗ Ivan Yotov∗
October 6, 2018
Abstract
We study flow and transport in fractured poroelastic media using Stokes flow in the fractures andthe Biot model in the porous media. The Stokes-Biot model is coupled with an advection-diffusionequation for modeling transport of chemical species within the fluid. The continuity of flux on thefracture-matrix interfaces is imposed via a Lagrange multiplier. The coupled system is discretizedby a finite element method using Stokes elements, mixed Darcy elements, conforming displacementelements, and discontinuous Galerkin for transport. The stability and convergence of the coupledscheme are analyzed. Computational results verifying the theory as well as simulations of flow andtransport in fractured poroelastic media are presented.
1 Introduction
Flow and transport in fractured poroelastic media occur in many applications, including enhanced oiland gas recovery, hydraulic fracturing, groundwater hydrology, and subsurface waste repositories. Theseare challenging multiphysics processes involving interaction between a free fluid in the fractures with afluid within the porous medium. The fluid flow may cause and be affected by solid deformation. Forexample, geomechanics effects are critical in hydraulic fracturing, as well as in modeling phenomenasuch as subsidence and compaction. Furthermore, the flow process may be coupled with transportphenomena, with the substance of interest propagating both through the fracture network and theporous matrix. Typical examples include tracking and cleaning up groundwater contaminants, leakageof subsurface radioactive waste, and proppant injection in hydraulic fracturing.
We use the Stokes equations to model the free fluid in the fractures and the Biot poroelasticitymodel [9] for the fluid in the poroelastic region. The latter is based on a linear stress-strain constitutiverelationship for the porous solid, and Darcy’s law, which describes the average velocity of the fluid inthe pores. The interaction across the fracture-matrix interfaces exhibits features of both Stokes-Darcycoupling [24,34,40,54,61] and fluid-structure interaction (FSI) [7,17,26,31,52]. We refer to the Stokes-Biot coupling considered in this paper as fluid-poroelastic structure interaction (FPSI). There has beengrowing interest in such models in the literature. The well-posedness of the mathematical model wasstudied in [57]. Numerical studies include variational multiscale methods for the monolithic system anditerative partitioned scheme [6], a non-iterative operator-splitting method [15], a partitioned methodbased on Nitsche’s coupling [14], and a Lagrange multiplier formulation for the continuity of flux [4].
To simplify the presentation we consider a fixed domain in time. As presented, the model is suitablefor deformations that are small relative to the width of the fractures. This is valid for scales that are
zoomed-in on the fractures or for meso-scale inclusions such as cavities. The model can be extended toaccount for the motion of the fluid domain by using the Arbitrary Lagrangian-Eulerian (ALE) approach,which has been done in [6, 15].
In this work we employ a monolithic scheme for the full-dimensional Stokes-Biot problem to modelflow in fractured poroelastic media. We note that an alternative approach is based on a reduced-dimension fracture model, including the Reynolds lubrication equation [33, 35, 42, 46] and an averagedBrinkman equation [16]. Works that do not account for elastic deformation of the media include averagedDarcy models [11,21,25,27,29,44,47], Forchheimer models [28], Brinkman models [43], and an averagedStokes model that results in a Brinkman model for the fracture flow [48].
For the discretization of the full-dimensional Stokes-Biot problem we consider the mixed formulationfor Darcy flow in the Biot system, which provides a locally mass conservative flow approximation and anaccurate Darcy velocity. This formulation results in the continuity of normal velocity condition beingof essential type, which is enforced through a Lagrange multiplier [4]. The discretization allows for theuse of any stable Stokes spaces in the fracture region and any stable mixed Darcy spaces [10]. For theelasticity equation we employ a displacement formulation with continuous Lagrange elements.
The Stokes-Biot system is coupled with an advection diffusion equation for modeling transport ofchemical species within the fluid. The transport equation is discretized by a discontinuous Galerkin(DG) method. DG methods [5,20,49,53,60] exhibit local mass conservation, reduced numerical diffusion,variable degrees of approximation, and accurate approximations for problems with discontinuous coef-ficients. Due to their low numerical diffusion, DG methods are especially suited for advection-diffusionproblems [2, 19, 20, 23, 60, 63]. Coupled Darcy flow and transport problems utilizing DG for transporthave been studied in [22,58,59,63]. Coupling of Stokes-Darcy flow with transport using a local discontin-uous Galerkin scheme was developed in [62]. A coupled phase field-transport model for proppant-filledfractures is studied in [41]. A flow-transport reduced fracture model using Darcy flow in the fractureand the matrix is developed in [30]. To the best of our knowledge, the coupled Stokes-Biot-transportproblem has not been studied in the literature. Here we follow the approach from [58] for miscibledisplacement in porous media and employ the non-symmetric interior penalty Galerkin (NIPG) methodfor the transport problem. We note that the dispersion tensor in the transport equation is a nonlinearfunction of the velocity. The work in [58] handles this difficulty by utilizing a cut-off operator. Herewe avoid the need for the cut-off operator by establishing an L∞-bound for the computed Stokes-Biotvelocity. As a result, the velocity is directly incorporated into the transport scheme. We present astability bound and an error estimate for the solution of the transport equation. The analysis in thispaper is presented for saturated flow and linear transport. Extensions to unsaturated flow in poroelasticmedia and nonlinear transport can also be studied, using for example techniques developed in [12,51].
The rest of the paper is organized as follows. The coupled Stokes-Darcy-transport problem and itsvariational formulation are presented in Section 2. The semi-discrete continuous-in-time approximationis developed in Section 3 and analyzed in Section 4. Computational experiments confirming the con-vergence of the method and illustrating its performance for a range of applications of flow in fracturedporoelastic media are presented in Section 5.
2 Model problem
We consider a simulation domain Ω ⊂ Rd, d = 2, 3 which is a union of non-overlapping and possiblynon-connected regions Ωf and Ωp, where Ωf is a fracture region and Ωp is a poroelasticity region, seeFigure 1. We denote by Γfp = ∂Ωf ∩ ∂Ωp the interface between Ωf and Ωp. We further denote by(u?, p?) the velocity-pressure pairs in Ω?, ? = f , p, by ηp the displacement in Ωp and by (f?, q?) the
2
Figure 1: Schematic representation of the physical domain.
body force and the external source or sink terms. The flow in the fracture region Ωf is governed by theStokes equations
−∇ · σf (uf , pf ) = ff in Ωf × (0, T ], (2.1)
∇ · uf = qf in Ωf × (0, T ], (2.2)
where the deformation and stress tensors, ε(u) and σf (uf , pf ), are given by
ε(u) =1
2
(∇uf +∇uTf
), σf (uf , pf ) = −pfI + 2νε(uf ),
and ν denotes the fluid viscosity.Let σe(η) and σp(η, p) be the elasticity and poroelasticity stress tensors, respectively:
where αp is the Biot-Willis constant and λp, µp are the Lame coefficients.The poroelasticity region Ωp is governed by the quasi-static Biot system [9]
−∇ · σp(ηp, pp) = fp, νK−1up +∇pp = 0 in Ωp × (0, T ], (2.3)
∂
∂t(s0pp + α∇ · ηp) +∇ · up = qp in Ωp × (0, T ], (2.4)
where s0 is a storage coefficient and K is a symmetric and uniformly positive definite permeabilitytensor.
Following [4, 6, 57], on the fluid-poroelasticity interface Γfp we prescribe the following interfaceconditions: mass conservation, balance of normal stress, conservation of momentum, and the Beavers-Joseph-Saffman (BJS) condition modeling slip with friction [8, 55]:
uf · nf +
(∂ηp∂t
+ up
)· np = 0 on Γfp × (0, T ], (2.5)
− (σfnf ) · nf = pp, σfnf + σpnp = 0 on Γfp × (0, T ], (2.6)
− (σfnf ) · τ f,j = ναBJS
√K−1j
(uf −
∂ηp∂t
)· τ f,j on Γfp × (0, T ], (2.7)
where nf and np are the outward unit normal vectors to ∂Ωf and ∂Ωp, respectively, τ f,j , 1 ≤ j ≤ d−1,is an orthogonal system of unit tangent vectors on Γfp, Kj = (Kτ f,j) · τ f,j and αBJS > 0 is anexperimentally determined friction coefficient.
3
The above system of equations is complemented by a set of boundary and initial conditions. LetΓf = ∂Ωf \Γfp, Γp = ∂Ωp\Γfp = ΓNp ∪ΓDp . For simplicity we assume homogeneous boundary conditions
uf = 0 on Γf × (0, T ], up · np = 0 on ΓNp × (0, T ], pp = 0 on ΓDp × (0, T ], ηp = 0 on Γp × (0, T ].
We further set the initial conditions
pp(x, 0) = pp,0(x), ηp(x, 0) = ηp,0(x) in Ωp.
Throughout the paper we will use the following standard notation. For a domain G ⊂ Rd, theL2(G) inner product and norm for scalar and vector valued functions are denoted by (·, ·)G and ‖ · ‖G,respectively. The norms and seminorms of the Sobolev spaces W k,p(G), k ∈ R, p > 0 are denoted by‖·‖k,p,G and |·|k,p,G, respectively. Conventionally, the norms and seminorms of Hilbert spaces Hk(G) aredenoted by ‖ · ‖k,G and | · |k,G, respectively. For a section of the domain or element boundary S ⊂ Rd−1
we write 〈·, ·〉S and ‖ · ‖S for the L2(S) inner product (or duality pairing) and norm, respectively. Wewill also use the space
H(div;G) = v ∈ L2(G) : ∇ · v ∈ L2(G)
equipped with the norm
‖v‖div,G =(‖v‖2 + ‖∇ · v‖2
)1/2.
For the weak formulation of the coupled Stokes-Biot equations we introduce the following functionspaces:
The weak formulation is obtained by multiplying the equations in each region by the corresponding testfunctions, integrating by parts the second order terms in space, and utilizing the interface and boundaryconditions. The integration by parts in (2.1) and (2.3) leads to the bilinear forms, corresponding to theStokes, Darcy and the elasticity operators:
af (·, ·) : Vf ×Vf −→ R, af (uf ,vf ) := (2νε(uf ), ε(vf ))Ωf,
We note that for the well-posedness of bΓ term, we require λ ∈ (Vp ·np∣∣Γfp
)′. The normal trace theorem
for vp ∈ Vp ⊂ H(div; Ωp) implies that vp · np ∈ H−1/2(∂Ωp). With our choice of boundary conditions,one can verify that vp · np ∈ H−1/2(Γfp), see [4]. Therefore, we take Λ = H1/2(Γfp).
Stokes-Biot variational formulation: given pp(0) = pp,0 ∈ Wp, ηp(0) = ηp,0 ∈ Xp, find, fort ∈ (0, T ], uf (t) ∈ Vf , pf (t) ∈ Wf , up(t) ∈ Vp, pp(t) ∈ Wp, ηp(t) ∈ Xp, and λ(t) ∈ Λ such that for allvf ∈ Vf , wf ∈Wf , vp ∈ Vp, wp ∈Wp, ξp ∈ Xp, and µ ∈ Λ,
The well-posedness of the above problem has been established in [3].
Theorem 2.1. For each ff ∈ W 1,1(0, T ; V′f ), fp ∈ W 1,1(0, T ; X′p), qf ∈ W 1,1(0, T ;W ′f ),
qp ∈ W 1,1(0, T ;W ′p), and pp(0) = pp,0 ∈ Wp, ηp(0) = ηp,0 ∈ Xp, there exists a unique solution(uf (t), pf (t),up(t), pp(t),ηp(t), λ(t)) ∈ L∞(0, T ; Vf )×L∞(0, T ;Wf )×L∞(0, T ; Vp)×W 1,∞(0, T ;Wp)×W 1,∞(0, T ; Xp)× L∞(0, T ; Λ) of (2.11)–(2.13).
The Stokes-Biot problem is coupled with the transport equation in Ω:
φct +∇ · (cu−D∇c) = qc, in Ω× (0, T ], (2.14)
where c(x, t) is the concentration of some chemical component, 0 < φ∗ ≤ φ(x) ≤ φ∗ is the porosityof the medium in Ωp (it is set to 1 in Ωf ), u is the velocity field over Ω = Ωf ∪ Ωp, defined asu|Ωf
= uf ,u|Ωp = up, q is the source term given by q∣∣Ωf
= qf and q∣∣Ωp
= qp, and
c =
injected concentration cw, q > 0,
resident concentration c, q < 0.
The diffusion/dispersion tensor D, which combines the effects of molecular diffusion and mechanicaldispersion, is a nonlinear function of the velocity, given by [50]
D(u) = dmI + |u|αlE(u) + αt(I−E(u)), (2.15)
5
where dm = φτDm, τ is the tortuosity coefficient, Dm is the molecular diffusivity, E(u) is the tensorthat projects onto the u direction with (E(u))ij =
uiuj|u|2 , and αl, αt are the longitudinal and transverse
dispersion, respectively.The model is complemented by the initial condition
c(x, 0) = c0(x) in Ω, (2.16)
and the boundary conditions
(cu−D∇c) · n = (cinu) · n on Γin × (0, T ], (2.17)
(D∇c) · n = 0 on Γout × (0, T ], (2.18)
where Γin := x ∈ ∂Ω : u ·n < 0, Γout := x ∈ ∂Ω : u ·n ≥ 0 and n is the unit outward normal vectorto ∂Ω.
Remark 2.1. We note that the coupling between the flow and transport problems is one way. Inparticular, the transport equation uses the Stokes-Biot velocity, but the flow problem does not depend onthe concentration.
3 Semi-discrete continuous-in-time formulation
Let T fh and T ph be shape-regular and quasi-uniform affine element partitions of Ωf and Ωp [18], re-spectively, both consisting of elements with maximal element diameter h. The two partitions may benon-matching at the interface Γfp. We also consider a shape-regular and quasi-uniform affine element
partition of Ω, denoted by Th. We note that Th may be different from T fh and T ph . We denote by Eh theset of all interior facets of Th and on each facet we arbitrarily fix a unit normal vector ne. We furtherdenote Eouth and Einh the set of facets on Γout and Γin, for which ne coincides with the outward unitnormal vector.
For the discretization of the fluid velocity and pressure we choose finite element spaces Vf,h ⊂ Vf andWf,h ⊂Wf , which are assumed to be inf-sup stable. Examples of such spaces include the MINI elements,the Taylor-Hood elements and the conforming Crouzeix-Raviart elements [10]. For the discretization ofthe porous medium problem we choose Vp,h ⊂ Vp and Wp,h ⊂Wp to be any inf-sup stable mixed finiteelement spaces, such as the Raviart-Thomas or the Brezzi-Douglas-Marini spaces, [10]. We employ aconforming Lagrange finite element space Xp,h ⊂ Xp to approximate the structure displacement. Forthe discretization of the Lagrange multiplier variable we set
Λh = Vp,h · np|Γfp,
which allows for optimal order approximation on non-matching grids [40]. We note that this is a non-conforming choice, since Λh 6⊂ H1/2(Γfp). The space is equipped with the discrete H1/2-norm analogue,‖µh‖2Λh
= ‖µh‖2Γfp+ |µh|2Λh
[4, 32] with the semi-norm
|µh|2Λh= adp(u
∗p,h(µh),u∗p,h(µh)), (3.1)
where (u∗p,h(µh), p∗p,h(µh)) ∈ Vp,h×Wp,h is the mixed finite element solution to the Darcy problem withDirichlet data µh on Γfp:
adp(u∗p,h(µh),vp,h) + bp(vp,h, p
∗h(µh)) = −〈vp,h · np, µh〉Γfp
, ∀vp,h ∈ Vp,h,
6
bp(u∗p,h(µh), wp,h) = 0, ∀wp,h ∈Wp,h.
We denote by kf ≥ 1 and sf ≥ 1 the degrees of polynomials in the spaces Vf,h and Wf,h respectively.Let kp ≥ 0 and sp ≥ 0 be the degrees of polynomials in the spaces Vp,h and Wp,h respectively. Finally,let ks ≥ 1 be the polynomial degree in Xp,h.
Semi-discrete Stokes-Biot problem: given pp,h(0) and ηp,h(0), for t ∈ (0, T ], find uf,h(t) ∈ Vf,h,pf,h(t) ∈ Wf,h, up,h(t) ∈ Vp,h, pp,h(t) ∈ Wp,h, ηp,h(t) ∈ Xp,h, and λh(t) ∈ Λh such that for allvf,h ∈ Vf,h, wf,h ∈Wf,h, vp,h ∈ Vp,h, wp,h ∈Wp,h, ξp,h ∈ Xp,h, and µh ∈ Λh,
We note that the result was derived under the assumption |ΓDp | 6= 0, but can be extended to the caseof full Neumann boundary condition on Γp by restricting the mean value of the pressure over the entiredomain to be zero.
Next, we derive the numerical method for the transport problem. Following [58], we adopt the DGscheme known as the non-symmetric interior penalty Galerkin (NIPG) [53].
For s ≥ 0, we define the space
Hs(Th) = φ ∈ L2(Ω) : φ ∈ Hs(E), E ∈ Th.
7
The jump and average for φ ∈ Hs(Th), s > 1/2 are defined as follows. Let Ei, Ej ∈ Th and e =∂Ei ∩ ∂Ej ∈ Eh, with ne exterior to Ei. Let
where Pr(E) denotes the space of polynomials of degree less than or equal to r on E.Let the bilinear form Buh
(ch, ψh) and the linear functional Lh(ψh) be defined as follows:
Buh(ch, ψh) =
∑E∈Th
∫E
(D(uh)∇ch − chuh) · ∇ψh −∑e∈Eh
∫eD(uh)∇ch · ne[ψh]
+∑e∈Eh
∫eD(uh)∇ψh · ne[ch] +
∑e∈Eh
∫ec∗huh · ne[ψh]
+∑
e∈Eouth
∫echuh · neψh −
∫Ωchq−ψh + Jσ0 (ch, ψh),
Lh(ψh) =
∫Ωcwq
+ψh −∑e∈Ein
h
∫ecinuh · neψh. (3.6)
Here q+ = max(q, 0) is the injection part of the source term and q− = min(q, 0) is the extraction part,c∗h|e is the upwind value of concentration, defined as
c∗h|e =
ch|E1 if uh · ne > 0,
ch|E2 if uh · ne < 0,(3.7)
and Jσ0 (ch, ψh) is the interior penalty term
Jσ0 (ch, ψh) =∑e∈Eh
σehe
∫e[ch][ψh], (3.8)
where, σ is a discrete positive function that takes constant value σe on the edge and is bounded belowby σ∗ > 0 and above by σ∗, and he is the diameter of side of facet e.
Semi-discrete DG transport problem: find ch(t) ∈ Dr(Th) such that ∀ψh ∈ Dr(Th),
(φ∂tch, ψh) +Buh(ch, ψh) = Lh(ψh), (3.9)
with initial condition ch(0) a suitable approximation of c0.It is easy to verify that, if the solution to (2.14) is sufficiently regular, it satisfies (3.9) with Buh
replaced by Bu.
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4 Analysis of the semi-discrete problem
In this section we discuss the stability and error estimates for the transport problem (3.9). We note thata similar scheme has been used and analyzed in details in [58]. The main difference and improvementin this work is the fact that the numerically computed velocity field uh is directly incorporated into thescheme for transport (3.9), while in [58] the authors used a cut-off operator in order to ensure stability oftheir method. We avoid the need for a cut-off operator by establishing pointwise stability of the velocitysolution in space and time, which is done in the next two lemmas. We first establish an error estimatefor the fluid velocity in L∞(0, T ). The result requires control of uf,h(0) and up,h(0). To simplify theanalysis, we assume that the initial pressure pp,0 and displacement ηp,0 are constants.
Lemma 4.1. Assume that pp,0 and ηp,0 are constants. If the solution of (2.11)–(2.13) is sufficientlyregular, there exists a positive constant C independent of h such that
To obtain a velocity bound in L∞(0, T ), we differentiate (2.11) and (3.2) in time, and then subtract(3.2)–(3.3) from (2.11)–(2.12) to form the error equation
where we also used that Λh = Vp,h · np|Γfpfor the last equality. We also have
bΓ(φf,h,φp,h, ∂tφs,h; ∂tφλ,h
)= 0, bΓ
(φf,h,φp,h, ∂tφs,h; ∂tχλ
)=⟨φf,h · nf + ∂tφs,h · np, ∂tχλ
⟩Γfp
,
where we have used (4.3) and (3.4) for the first equality and the last equality in (4.16) for the secondequality. Using these results, the error equation (4.15) becomes
1
2∂t
(af (φf,h,φf,h) + adp(φp,h,φp,h) +
∣∣φf,h − ∂tφs,h∣∣2aBJS
)+ aep(∂tφs,h, ∂tφs,h) + s0‖∂tφpp,h‖2L2(Ωp)
10
= af (∂tχf ,φf,h) + adp(∂tχp,φp,h) + aep(∂tχs, ∂tφs,h
where we used the Cauchy-Schwartz, Young’s and trace inequalities. Using the coercivity of the bilinearforms af (·, ·), adp(·, ·), and aep(·, ·), choosing ε small enough, and integrating (4.17) in time from 0 to anarbitrary t ∈ (0, T ] gives
Next, using an inf-sup condition for the Stokes-Darcy problem [4, 32] and the error equation obtainedby subtracting (3.2) from (2.11) and taking ξp,h = 0, we obtain
Finally, to control the error at t = 0, we note that the assumed solution regularity on the right handside of (4.1) implies that (2.11)–(2.13) and (3.2)–(3.4) hold at t = 0. We subtract (3.2)–(3.3) from (2.11)–(2.12) at t = 0, sum the two equations, and take vf,h = φf,h,vp,h = φp,h, ξp,h = ∂tφs,h, wf,h = φfp,h,and wp,h = φpp,h, to obtain
af (φf,h(0),φf,h(0)) + adp(φp,h(0),φp,h(0)) +∣∣φf,h(0)− ∂tφs,h(0)
Since pp,h(0) = Qp,hpp,0 and ηp,h(0) = Is,hηp,0, we have that φpp,h(0) = 0 and φs,h(0) = 0. Since pp,0and ηp,0 are constants, we also have that χs = 0, χpp = 0, and χλ = 0. It is then easy to see that
The assertion of the lemma follows from combining (4.18)–(4.21) and using Gronwall’s inequality,the triangle inequality, and the approximation properties (4.9)–(4.14).
Lemma 4.2. Under the assumptions of Lemma 4.1, for any choice of stable spaces when d = 2, andfor kf ≥ 2, kp ≥ 1, sp ≥ 1, and ks ≥ 2 when d = 3, there exists a positive constant M = M(uf , pf ,up, pp, ηp, λ), such that, for t ∈ (0, T ], the solution uh of (3.2)–(3.4) satisfies
‖uh‖L∞(Ω) ≤M. (4.22)
Proof. We recall that by definition
uh =
uf,h in Ωf ,
up,h in Ωp.
Therefore, we prove (4.22) separately for uf,h in the fluid domain and for up,h in the poroelastic domain.Let Sf,h be the Scott-Zhang interpolant onto Vf,h [56], satisfying
The proof is completed by combining (4.26), (4.29), (4.23)–(4.24), (4.27)–(4.28), and (4.1).
Remark 4.1. We note that the above result assumes sufficient regularity of the solution, as indicatedby (4.23)–(4.24), (4.27)–(4.28), and (4.1).
We will utilize the following positive definite property of the dispersion tensor, proved in [58].
Lemma 4.3. Assume that for D(u) defined in (2.15), dm(x) ≥ dm,∗ > 0, αl(x) ≥ 0 and αt(x) ≥ 0uniformly in Ω. Then D(u) is uniformly positive definite and for all x ∈ Ω,
D(u)∇c · ∇c ≥ dm,∗|∇c|2. (4.30)
We next prove a Garding’s inequality for the bilinear form Buh(·, ·). To simplify the analysis we
assume velocity boundary condition for the Darcy problem.
Lemma 4.4. Under the assumptions of Lemma 4.2, and if ΓDp = ∅, then the bilinear form Buh(·, ·)
defined in (3.6) satisfies, ∀t ∈ (0, T ],
Buh(ψh, ψh) ≥ C
(|||∇ψh|||2Ω + Jσ(ψh, ψh)− ‖ψh‖2Ω
), ∀ψh ∈ Df (Th). (4.31)
Proof. For any ψh ∈ Df (Th) we have
Buh(ψh, ψh) =
∑E∈Th
∫E
(D(uh)∇ψh − ψhuh) · ∇ψh −∑e∈Eh
∫eD(uh)∇ψh · ne[ψh]
13
+∑e∈Eh
∫eD(uh)∇ψh · ne[ψh] +
∑e∈Eh
∫eψ∗huh · ne[ψh]
+
∫Γout
ψhuh · nψh −∫
Ωψhq
−ψh + Jσ0 (ψh, ψh). (4.32)
Next we introduce the notation
J1 :=∑E∈Th
∫E
(D(uh)∇ψh − ψhuh) · ∇ψh, J2 :=∑e∈Eh
∫eψ∗h[ψh]uh · ne,
J3 :=
∫Γout
ψ2h uh · n−
∫Ωq−ψ2
h + Jσ0 (ψh, ψh),
and rewrite (4.32) as
Buh(ψh, ψh) = J1 + J2 + J3. (4.33)
Using (4.22) and (4.30), we bound J1 as
J1 =∑E∈Th
∫E
D(uh)∇ψh · ∇ψh −∑E∈Th
∫Eψhuh · ∇ψh ≥ dm,∗|||∇ψh|||2Ω −M
∑E∈Th
‖ψh‖E‖∇ψh‖E
≥ dm,∗|||∇ψh|||2Ω − Cε−1‖ψh‖2Ω − ε|||∇ψh|||
2Ω. (4.34)
For J2 we have
J2 =∑e∈Eh
∫eψ∗h[ψh]uh · ne ≥ −M
∣∣∣∣∣∣∑e∈Eh
∫eψ∗h[ψh]
∣∣∣∣∣∣ ≥ −M∑e∈Eh
‖ψ∗h‖e‖[ψh]‖e
≥ −∑e∈Eh
(εσehe‖[ψh]‖2e +
Cheε‖ψ∗h‖2e
)≥ −εJσ(ψh, ψh)− Ch
ε
∑E∈Th
h−1‖ψh‖2E
≥ −εJσ(ψh, ψh)− Cε−1‖ψh‖2Ω. (4.35)
We bound J3, using that uh · n = u · n ≥ 0 on Γout and that q− ≤ 0,
J3 =
∫Γout
ψ2hu · n−
∫Ωq−ψ2
h + Jσ0 (ψh, ψh) ≥ Jσ0 (ψh, ψh). (4.36)
Combining (4.33)–(4.36) we obtain
Buh(ψh, ψh) ≥ (dm,∗ − ε)|||∇ψh|||2Ω − Cε
−1‖ψh‖2Ω + (1− ε)Jσ(ψh, ψh).
Choosing ε small enough completes the proof.
We are now ready to prove a stability bound for the solution of (3.9).
Theorem 4.1. Under the assumptions of Lemma 4.4, there exists a positive constant C independentof h such that the solution ch(t) of (3.9) satisfies, ∀t ∈ (0, T ],
‖ch(t)‖2Ω +
∫ t
0|||∇ch(s)|||2Ω ds ≤ C
(‖ch(0)‖2Ω +
∫ t
0
(‖cw(s)q+(s)‖2Ω + ‖cin(s)‖2Γin
)ds
).
14
Proof. With the choice ψh = ch, (3.9) reads∫Ωφch∂tch +Buh
(ch, ch) = Lh(ch).
Using (4.31), the definition (3.6) of Lh, and that uh · n = u · n, we obtain
φ1
2∂t‖ch‖2Ω + C
(|||∇ch|||2Ω + Jσ(ch, ch)
)≤ C‖ch‖2Ω +
∫Ωcwq
+ch −∫
Γin
cinu · n ch.
Integrating in time from s = 0 to s = t for 0 < t ≤ T and using Cauchy-Schwarz and Young’sinequalities, we obtain
‖ch(t)‖2Ω +
∫ t
0
(|||∇ch(s)|||2Ω + Jσ(ch, ch)
)ds
≤ C(‖ch(0)‖2Ω +
∫ t
0
(‖ch(s)‖2Ω + ‖cw(s)q+(s)‖2Ω + ε−1‖cin(s)‖2Γin
+ ε‖ch(s)‖2Γin
)ds
). (4.37)
The last term above is bounded as
‖ch‖2Γin≤ C
(‖ch‖2Ω + |||∇ch|||2Ω + Jσ(ch, ch)
), (4.38)
which can be shown following the argument presented in [13]. The theorem follows from (4.37)–(4.38),taking ε small enough, and using Gronwall’s inequality.
In the next theorem we state the error estimate for the transport problem (3.9). Derivation of thebound follows the steps in the proof of Theorem 4.1 in [58], using the estimate (4.22), rather than aboundedness property of the ”cut-off” operator. For the sake of space, we omit the proof and the readeris referred to [58] for the details.
Theorem 4.2. Under the assumptions of Lemma 4.4, and assuming further that the solution of (2.14)–(2.18) satisfies c ∈ L∞(0, T ;W 1,∞(Ω)) ∩ L2(0, T ;Hr+1(Ω)), there exists a positive constant C indepen-dent of h such that, ∀t ∈ (0, T ],
‖c(t)− ch(t)‖Ω +
(∫ t
0|||∇(c(s)− ch(s))|||2Ω ds
)1/2
≤ Chminkf ,sf+1,kp+1,sp+1,ks,r.
5 Numerical results
In this section, we present results from several computational experiments in two dimensions. Thecomputations are performed using a fully discrete scheme with Backward Euler time discretization.The method is implemented using the finite element package FreeFem++ [36]. We use a monolithicsolver for the Stokes-Biot system at each time step. It is possible to design a non-overlapping domaindecomposition algorithm, similar to the Stokes-Darcy problem, see e.g. [61]. One can also use varioussplitting schemes for the Biot system [37, 38, 45]. We first present a numerical test that confirmsthe theoretical convergence rates for the Biot-Stokes-transport problem using an analytical solution,followed by five examples with simulations of fluid flow in a fractured reservoir with physically realisticparameters.
15
5.1 Convergence test
In this test we study the convergence of the spatial discretization using an analytical solution. Thedomain is a rectangle Ω = [0, 1] × [−1, 1]. We associate the upper half with the Stokes flow, while thelower half represents the flow in the poroelastic structure governed by the Biot system. The appropriateinterface conditions are enforced along the interface y = 0. Following the example from [4], the solutionin the Stokes region is
uf = π cos(πt)
(−3x+ cos(y)
y + 1
), pf = et sin(πx) cos(
πy
2) + 2π cos(πt).
The Biot solution is chosen accordingly to satisfy the interface conditions (2.5)–(2.7):
up = πet
(cos(πx) cos(πy2 )12 sin(πx) sin(πy2 )
), pp = et sin(πx) cos(
πy
2), ηp = sin(πt)
(−3x+ cos(y)
y + 1
).
The right hand side functions ff , qf , fp and qp are computed from (2.1)–(2.4) using the above solution.The model problem is then complemented with the appropriate Dirichlet boundary conditions and initialdata. The total simulation time for this test case is T = 10−3 and the time step is ∆t = 10−4. The timestep is chosen sufficiently small, so that the time discretization error does not affect the convergencerates. The transport solution is set to
c = t (cos(πx) + cos(πy)) /π,
with the diffusion tensor D = 10−3I and porosity φ = 1.For the spatial discretization we use the MINI elements Pb1 − P1 for Stokes, the Raviart-Thomas
RT 0 − P0 for Darcy, continuous Lagrangian P1 elements for the displacement, and piecewise constantLagrange multiplier P0. The transport equation is discretized using discontinuous piecewise linears, Pdc1 .For simplicity the Stokes and Biot meshes are made matching along the interface and the transportmesh is the same as the flow mesh. Theorem 4.2 predicts first order convergence for all variables,which is confirmed by the results reported in Table 1. We also observe second order convergence forthe concentration in the l∞(L2(Ω)) norm. Here the notation l2(·) and l∞(·) refers to discrete-in-timenorms.
Table 1: Relative numerical errors and convergence rates.
16
(a) Example 1 (b) Example 2
(c) Example 3 (d) Example 4 (e) Example 5
Figure 2: Computational domains.
5.2 Applications to coupled flow and transport through fractured poroelastic media
We present five examples with simulations of fluid flow in a fractured reservoir with physically realisticparameters. The examples are designed to illustrate the robustness of the method with respect toreservoir and fracture geometry, rock heterogeneities, and various flow and transport scenarios. Thecomputational domains for the five examples are shown in Figure 2. Examples 1 and 2 are motivated byhydraulic fracturing. Example 1 features irregularly shaped fracture and reservoir, with fluid injected inthe center of the fracture. Example 2 has heterogeneous permeability, porosity, and Young’s modulus,with fluid injected into the fracture via inflow boundary condition. Examples 3-5 model flow andtransport through vuggy or naturally fractured poroelastic media. The flow is induced by a pressuredrop between the left and right boundaries. The transport equation models the concentration of atracer, which enters the domain with the fluid along the inflow boundary. The reservoir in Example 3has a large irregularly shaped cavity. Examples 4 and 5 consider a network of channels and fractures,respectively. The latter is the computationally most challenging example, due to the small fracturethickness and sharp angles at the fracture intersections. For all examples in this section the physicalunits are meters for length, seconds for time, and kPa for pressure.
5.2.1 Example 1: fluid and tracer injection into a fracture
For this example, we first introduce the reference domain Ω given by a square [−1, 1] × [−1, 1]. Afracture, representing the reference fluid domain Ωf , is then described by its top and bottom boundaries,as follows
y2 = 82(x− 0.35)2(x+ 0.35)2, x ∈ [−0.35, 0.35].
17
The physical domain, see Figure 2a, is obtained from the reference one Ω via the mapping[xy
]=
[x
8(
cos(πx+y100 ) + y
4
)] .This example models the interaction between a fracture filled with fluid and a surrounding poroelasticreservoir. The physical parameters are given in the Table 2. They are taken from [35] and are realisticfor hydraulic fracturing.
Parameter Symbol Units Values
Young’s modulus E (kPa) 107
Poisson’s ratio ν 0.2
Lame coefficient λp (kPa) 5/18× 107
Lame coefficient µp (kPa) 5/12× 107
Dynamic viscosity µ (kPa s) 10−6
Permeability K (m2) diag(200, 50)× 10−12
Mass storativity s0 (kPa−1) 6.89× 10−2
Biot-Willis constant α 1.0
Beavers-Joseph-Saffman coefficient αBJS 1.0
Table 2: Poroelasticity and fluid parameters in Example 1.
The Lame coefficients are determined from the Young’s modulus E and the Poisson’s ratio ν via thewell-known relationships
λp =Eν
(1 + ν)(1− 2ν), µp =
E
2(1 + ν).
The boundary conditions are
pp = 1000, ηp = 0 on Γp.
The initial conditions are set accordingly to ηp(0) = 0 and pp(0) = 1000. The initial concentration isc(0) = 0. The total simulation time is T = 100 s with a time step of size ∆t = 1s.
The flow is driven by the injection of the fluid into the fracture with the constant rate 5 · 10−3 kg/s.The fluid is injected into a region obtained from mapping a disk of radius 0.017m at the center of thereference fracture Ωf . A tracer is injected in this same region, continuously over the entire simulationperiod, i.e. cw = 1 in the region specified above.
Recall, see (2.15), that the diffusion tensor is given as
D(u) = dmI + |u|αlE + αt(I−E).
For all examples in this section, in Ωf we set dm = 10−6 m/s, αl = αt = 0, i.e., D = 10−6I m/s. In Ωp
we set dm = 10−4 m/s, αl = αt = 10−4. The porosity is set to φ = 0.4.In all examples we use the Taylor-Hood P2 − P1 elements for the fluid velocity and pressure in the
fracture region, the Raviart-ThomasRT 1−Pdc1 elements for the Darcy velocity and pressure, continuousLagrangian P1 elements for the displacement, and the Pdc1 elements for the Lagrange multiplier. Weuse discontinuous piecewise linears Pdc1 for the concentration.
18
Figure 3 shows the computed velocity and pressure in the reservoir, the displacement, and thevelocity in the fracture at the final time T = 100s. We observe channel-like flow in the fracture region,from the center to the tips. The leak-off into the reservoir is highest at the fracture tips, but there isalso a noticeable leak-off along the fracture length. The structure displacement is small, but it is highestin the vicinity of the fracture and indicates a slight opening of the fracture, as expected.
Figure 4 shows the solution obtained for the concentration at various time moments. At early times,the tracer propagates in accordance with the Stokes velocity field, moving preferentially in horizontaldirections towards the tips of the fracture. At later times, despite the small permeability, the tracerpenetrates into the reservoir and it is further transported/diffused in it, following the Darcy velocity.We note that, due to the singular shape of the fracture tips, the Darcy velocity at the tips is slightlyhigher than the Stokes velocity inside the fracture near the tips. This acceleration effect leads to slightlylower concentration values at the fracture tips. This example demonstrates the ability of the method tohandle irregularly shaped domains and fractures with a computationally challenging set of parameters.
(a) Darcy velocity (m/s) over pres-sure (kPa)
(b) Displacement (m) (c) Stokes velocity (m/s)
Figure 3: Example 1, computed Stokes-Biot solution at t = 100s.
5.2.2 Example 2: flow and transport through a fractured heterogeneous reservoir
As in the previous example, this example is motivated by hydraulic fracturing, while we illustrate theability of the method to handle heterogeneous permeability, porosity, and Young’s modulus. The domainΩ is given by the rectangle [0, 1]m× [−1, 1]m. A fracture, which represents the fluid domain Ωf is thenpositioned in the middle of the rectangle, with the boundaries defined by
x2 = 200(0.05− y)(0.05 + y), y ∈ [−0.05, 0.05],
see Figure 2b. Fluid is injected into the opening of the fracture on the left boundary. The externalboundary of Ωp is split into Γp,?, where ? ∈ left, right, top, bottom. The boundary conditions are
uf · nf = 10, uf · τ f = 0 on Γf,inflow,
up · np = 0 on Γp,left,
pp = 1000 on Γp,top ∪ Γp,right ∪ Γp,bottom,
ηp · np = 0 on Γp,top ∪ Γp,right ∪ Γp,bottom,
(σpnp) · τ p = 0 on Γp,top ∪ Γp,right ∪ Γp,bottom,
σpnp = 0 on Γp,left,
(cu−D∇c) · n = (cinu) · n, cin = 1 on Γf,inflow,
(D∇c) · n = 0 on ∂Ω \ Γf,inflow.
19
(a) Concentration at t = 10s. (b) Concentration at t = 25s.
(c) Concentration at t = 50s. (d) Concentration at t = 100s.
Figure 4: Example 1, computed concentration solution.
The initial conditions are ηp(0) = 0 and pp(0) = 1000. The same physical parameters as in Example1 from Table 2 are used, except for the porosity φ, the permeability K, and the Young’s modulus E.The permeability and porosity data is taken from a two-dimensional cross-section of the data providedby the Society of Petroleum Engineers (SPE) Comparative Solution Project 101. The SPE data, whichis given on a rectangular 60 × 220 grid is projected onto the triangular grid on the domain Ω, andvisualized in Figures 5a–5c. We note that the permeability tensor is isotropic in this example. Givenporosity φ, the Young’s modulus is determined from the relationship [39]
E = E0
(1− φ
β
)2.1
,
where E0 = 107 is the Young’s modulus for the non-porous material and the constant β = 0.5 representsthe porosity at which the effective Young’s modulus becomes zero.
The computed Darcy velocity, pressure, and displacement at the final time are shown in Figure5d and 5e, respectively. We observe that most of the leak-off is through the fracture tip and theDarcy velocity is largest in a channel-like high permeability region near the tip of the fracture. Thedisplacement field indicates that the fracture opens up as fluid is injected. We also note that theheterogeneities featuring higher permeability and porosity, and correspondingly less stiff material, resultin overall larger displacement compared to the previous homogeneous example. Five snapshots of theconcentration solution at various time steps are given in Figure 6. At the early times the tracerpropagates along the fracture following the Stokes velocity and penetrates into the high permeabilityreservoir regions near the middle top, middle bottom, and tip of the fracture. At later times, the
1http://www.spe.org/csp
20
tracer is diffused in the poroelastic region; however the overall profile of the concentration front roughlyresembles the underlying permeability field.
Figure 5: Example 2, material properties and computed Stokes-Biot solution at t = 100s.
5.2.3 Example 3: irregularly shaped fluid-filled cavity
The next two examples feature highly irregularly shaped fractures and grids that conform to the fracturegeometries. They are motivated by modeling flow and transport through vuggy or naturally fractured
21
(a) Concentration at t = 0.5s. (b) Concentration at t = 1s. (c) Concentration at t = 2.5s.
(d) Concentration at t = 5s. (e) Concentration at t = 10s.
Figure 6: Example 2, computed concentration solution.
reservoirs or aquifers. The domain in this example has a large fluid-filled cavity, see Figure 2c. Theflow is driven from left to right via a pressure drop of 1 kPa. We take the Darcy pressure boundarycondition values to be 1 and 0, which can be considered as an offset from a reference pressure. We notethat including the reference pressure in the boundary conditions for Darcy and Stokes, the latter beingset through (σf nf ) · nf , produces similar results. The boundary conditions are
pp = 1 on Γp,left,
pp = 0 on Γp,right,
22
up · np = 0 on Γp,top ∪ Γp,bottom,
σp np = 0 on Γp,left,
ηp = 0 on Γp,right,
(σp np) · np = 0, ηp · τ p = 0 on Γp,top ∪ Γp,bottom,
(σf nf ) · nf = 0, uf · τ f = 0 on Γf,right,
(cu−D∇c) · n = (cinu) · n, cin = 1 on Γp,left,
(D∇c) · n = 0 on ∂Ω \ Γp,left.
The physical parameters for this test case are chosen as in the previous example, except for the perme-ability, which is K = 10−8I m2. The total simulation time is 10s, with time step size ∆t = 0.1s.
The velocity fields in the poroelastic and fracture regions are shown in Figures 7a and 7c, respectively,while the rock displacement is given in Figure 7b. The Darcy velocity is largest in the region betweenthe left inflow boundary and the cavity. This results in a larger displacement in this region. The Stokesvelocity in the cavity is an order of magnitude larger than in the poroelastic region. A channel-likeflow profile is clearly visible within the cavity, with the largest velocity along a central path away fromthe cavity walls. Four snapshots of the concentration solution at different time moments are shown inFigure 8. As expected, the tracer follows the flow, and tends to get into the free fluid region throughthe nearest fracture tip. After that, it is transported quickly toward the opening in the right boundary,following the Stokes velocity profile and with very little diffusion. In particular, the tracer follows anarrow central path within the cavity away from the walls. This behavior agrees qualitatively with theparameters in the transport equation.
(a) Darcy velocity (m/s) overpressure (kPa)
(b) Displacement (m) (c) Stokes velocity (m/s)
Figure 7: Example 3, computed velocity, pressure, and displacement fields.
5.2.4 Example 4: flow through poroelastic media with channel network
The domain for this example is given in Figure 2d. It features an irregularly shaped network of channels.The physical parameters and boundary and initial conditions for both flow and transport are as inExample 3 except for the boundaries of the Stokes region. Since the channel network has openings atboth the left and right boundaries, we set
uf · nf = 0.2, uf · τf = 0 on Γf,left,
23
(a) Concentration at t = 1s. (b) Concentration at t = 2.5s. (c) Concentration at t = 5s. (d) Concentration at t = 10s.
Figure 8: Example 3, computed concentration solution.
We present the computed velocity fields in the poroelastic and fracture regions in Figures 9a, 9c and thestructure displacement in Figure 9b. Four snapshots of the concentration solution at different times areshown in Figure 10. The qualitative behavior of the flow and transport solution is similar to Example3, with channel-like flow profile and higher velocity within the channel network. The tracer propagatesmuch faster through the channel network, following the widest channel as a preferential path to reachthe outflow boundary. Another interesting feature is that in some channels, the tracer enters bothfrom the channel network and from the porous media. Since the diffusion in the channels is very low,this results in two coexisting tracer streams in close proximity to each other, but not mixing, beingtransported by the free fluid, see the upper outflow at time t = 5s.
(a) Darcy velocity (m/s) overpressure (kPa)
(b) Displacement (m) (c) Stokes velocity (m/s)
Figure 9: Example 4, computed velocity, pressure, and displacement fields.
5.2.5 Example 5: flow through poroelastic media with fracture network
The final example is the most challenging, since it involves a network of thin fractures that intersect atsharp angles, see Figure 2e. The setting for this test case matches the one from Example 4, including
24
(a) Concentration at t = 1s. (b) Concentration at t = 2.5s. (c) Concentration at t = 5s.
(d) Concentration at t = 10s. (e) Concentration at t = 15s. (f) Concentration at t = 25s.
Figure 10: Example 4, computed concentration solution.
physical parameters, initial and boundary conditions, The computed velocity fields in the poroelasticregion and the fracture network are visualized in Figures 11a and 11c, while the displacement of theporous media skeleton is shown in Figure 11b. We note that the velocity in the fractures is higher thanthe velocity in the channels in the previous example, due to the smaller fracture thickness. Also, thevelocity is highest in branches of the network where fluid enters from two different branches and thathave connection to the outflow boundary. As seen in the plots of the concentration solution in Figure 12,the tracer is transported quickly through the fractures toward the outflow boundary. Initially it followswell the fracture network geometry, despite the sharp angles between the branches, However, due torelatively small size of the fracture outflow boundaries, the concentration eventually builds up in thefracture region and the tracer starts to penetrate and diffuse into the porous matrix. This can be seenat the later times near the right boundary.
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(a) Darcy velocity (m/s) overpressure (kPa)
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Figure 11: Example 5, computed velocity, pressure, and displacement fields.
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