ICES REPORT 16-24

October 2016

Convergence and Error Analysis of Fully DiscreteIterative Coupling Schemes for Coupling Flow with

Geomechanicsby

Tameem Almani, Kundan Kumar, Mary F. Wheeler

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

Reference: Tameem Almani, Kundan Kumar, Mary F. Wheeler, "Convergence and Error Analysis of FullyDiscrete Iterative Coupling Schemes for Coupling Flow with Geomechanics," ICES REPORT 16-24, TheInstitute for Computational Engineering and Sciences, The University of Texas at Austin, October 2016.

Convergence and Error Analysis of Fully Discrete IterativeCoupling Schemes for Coupling Flow with Geomechanics

T. Almani1,3, K. Kumar2, M. F. Wheeler11 Center for Subsurface Modeling, ICES, UT-Austin, Austin, TX 78712, USA

2 Mathematics Institute, University of Bergen, Norway3 Saudi Arabian Oil Company (Aramco), Saudi Arabia{tameem,mfw}@ices.utexas.edu, kundan.kumar@uib.no

October 15, 2016

Abstract

We consider single rate and multirate iterative coupling schemes for the Biot system, based on thefixed-stress split coupling algorithm. In contrast to the single rate scheme, in which both flow andmechanics share the same time step, the multirate scheme exploits the different time scales of thetwo problems by allowing for multiple finer time steps for flow within one coarser mechanics timestep. For the single rate scheme considered in this work, we derive error estimates for quantifyingthe error between the solution obtained at any iterate and the true solution. Our approach is basedon studying the equations satisfied by the difference of iterates and utilizing a Banach contractionargument to show that the corresponding scheme is a fixed point iteration. Obtained contractionresults are then used to derive theoretical convergence error estimates for the single rate iterativecoupling scheme. Numerically, we first validate the efficiency of multirate schemes versus single rateschemes for field-scale problems. Second, we compare theoretically derived contraction estimatesagainst numerical computations, and conclude that theoretical estimates can predict the contractingbehavior, and subsequently, the linear rate of convergence of the iterative scheme with high accuracy.

Keywords. poroelasticity; Biot system; fixed-stress split iterative coupling; multirate scheme;

contraction mapping; a priori error estimates

1

1 Introduction

Due to oil extraction, especially in stress-sensitive reservoirs, rock compaction may occur inducinga subsidence event. Such a subsidence might not only affect the surrounding environment adversely,but can also result in a dramatic impact on reservoir production [34]. Examples of oils fields thatexperienced subsidence events in the past, due to oil extraction activities, include the Valhall field,located in the central graben of the North Sea [29], and the Willmington field, located near thesouthern edge of the Los Angeles sedimentary basin [1]. Surface subsidence is just one example of aninduced environmental phenomenon that necessitates the development of accurate, and reliable waysto model subsurface fluid flow coupled with mechanical interactions. Other examples include wellstability, sand production, waste deposition, hydraulic fracturing, and CO2 sequestration [15], [22].Classical techniques of incorporating the effects of mechanical deformations on fluid flow in porousmedia involve expressing the porosity of the reservoir as a function of a pore compressibility factor.This is found to be insufficient for structurally weak and stress-sensitive reservoirs. The accuratemodeling of such effects can not be accomplished without solving the flow and mechanics equationsin a coupled manner. This can be performed in three different ways, known as the fully implicitor simultaneous coupling, the loose or explicit coupling, and the iterative coupling methods. Thefirst approach solves the coupled equations simultaneously, and results in the most accurate, orreference solution. Although it poses several computational challenges to the underlying linearsolver, it exhibits excellent stability properties [17]. The explicit coupling approach, on the otherhand, enjoys a lower computational cost compared to the aforementioned fully implicit approach.However, it is only conditionally stable. The iterative coupling approach comes in between thesetwo extremes and iterates between the fully decoupled flow and mechanics equations until a certainconvergence criterion, with an acceptable tolerance, is achieved for a particular time step [8,21,22,35]. From an implementation point of view, this approach is easy to implement as it allows the useof existing reservoir simulators, while maintaining a relatively fast convergence rate. We shall focusour attention on studying the error analysis of the iterative coupling approach in this work.The problem of coupling flow with geomechanics has been extensively explored in the past. This canbe tracked down to the work of [33] and [6,7]. Terzaghi predicted the settlement of differesnt types ofsoils, which lead to the creation of the the science of soil mechanics, followed by Biot who extendedTerzaghi’s work to the generalized theory of consolidation [7]. Several years later, [9] presented thegeneral theory of thermoporoelastoplasiticy for saturated materials. A comprehensive treatmentof the theory of poromechanics can be found in [10]. Other nonlinear extensions of the theory ofporoelasticity can be found in [9, 12, 31]. The existence, uniqueness, and regularity of the Biotsystem have been studied by a number of authors including [32], [25], and et al [14]. Moreover, [30]presented explicit and iterative coupling schemes for coupling flow with geomechanics involvingfracture propagation.This paper addresses the error analysis and contracting behavior of iteratively coupled flow andmechanics problems. It should be noted that the rigorous mathematical analysis of the devisedcoupling schemes has received relatively less attention compared to the proposed linear and nonlinear

2

extensions currently available in literature. To the best of our knowledge, the first asymptotic errorestimates for spatially discrete Galerkin approximations of the Biot’s model were presented by [24].Few years later, [13] considered finite difference methods for the Biot’s model on staggered grids,derived stability estimates, and analyzed convergence for the discretized system. In a sequence oftwo papers, [26, 27] studied the continuous in time and fully-discrete Biot’s model in which mixedformulation is used for flow and continuous Galerkin is used for mechanics. A priori error estimatesare derived in both cases respectively. [11], on the other hand, derived a posteriori error estimatesfor the quasi-static Biot model, resulting in reliable error bounds with all constants involved in theestimates are being specified. Such error estimators can be used to perform adaptive simulations.Recently, [36] derived a priori error estimates for the quasi-static Biot model in which flow isdiscretized by the multipoint flux mixed finite element method, and elasticity uses continuouspiecewise linear Galerkin finite elements. [28] considered finite element discretizations of the Biot’smodel based on MINI and stabilized P1-P1 elements, and derived error estimates of the fully discretesystem accordingly. The work of [20] considers a formulation of the Biot’s system in four unknownsincluding pore pressure, fluid flux, stress tensor, and solid displacement, using a combination oftwo-mixed formulations for the flow and mechanics, and derived a priori error estimates of thefully coupled system accordingly. We note here that all previously derived error estimates considersimultaneous coupling of flow and mechanics.In this work, we consider iterative coupling schemes instead, and drive error estimates for the fixed-stress split iterative coupling scheme for the quasi-static Biot model. The contracting behavior ofboth schemes has been rigorously established by [23]. In addition, [17,18] utilized von Neumann sta-bility analysis techniques to study the stability and convergence of other iterative coupling schemes,including the fixed-strain and drained split techniques. We shall consider both the single rate andmultirate iterative coupling schemes in this work. Figures 1.1a and 1.1b demonstrate the couplingiteration in both schemes respectively. In the single rate case, the flow and mechanics are solvedfor the same time step during each flow-mechanics coupling iteration. In contrast, in the multiratecase, during every iterative coupling iteration, the flow is solved for a number of fine time steps,then mechanics is solved for a coarse time step. The coupling iteration is continued until a certainconvergence criterion is met, before advancing to the next coarse time step. We note here that thecontracting behavior of the multirate iterative scheme for the Biot’s system has been establishedby [2, 3] for the fixed-stress split coupling scheme, and by [5, 19] for the undrained-split couplingscheme. Moreover, the stability of the multirate explicit coupling scheme for the quasi-static Biotsystem has been established by [4] under mild assumptions on material parameters.The approach we follow in deriving our a priori error estimates utilizes previously established resultsin a clever way, under the assumption that the solution obtained by the iterative coupling schemeconverges to the solution obtained by the simultaneously coupled scheme. Under such assumption,the problem is simplified into estimating the error between the solution obtained by the iterativecoupling scheme, and the one obtained by the simultaneously coupled scheme. In fact, we showthat the former converges to the later geometrically by a Banach contraction argument. To the bestof our knowledge, this is the first rigorous derivation of a priori error estimates for the fixed-stress

3

coupling scheme for the Biot system.The rest of the paper is structured as follows. The model and discretizations are presented in Section1. Banach fixed point contraction results for the single rate and multirate schemes are presented inSection 2. A priori error estimates for the fixed-stress split single rate iterative coupling scheme arederived in 3. Numerical results are presented in Section 4. We discuss the conclusions and futurework in Section 5.

2 Model equations & discretization

Let Ω be an open and connected bounded domain of IRd, where the dimension d = 2 or 3, witha Lipschitz continuous boundary ∂Ω. Let Γ denotes the part of the boundary ∂Ω with positivemeasure. When d = 3, the boundary of Γ is also assumed to be Lipschitz continuous. Let ΓDdenotes the part of the boundary associated with Dirichlet boundary conditions, and ΓN denotesthe part associated with Neumann boundary conditions, such that ΓD ∪ ΓN = Γ. The model westudy in this work assumes a linear, homogeneous, and isotropic poro-elastic medium Ω ⊂ Rd, witha slightly compressible viscous fluid saturating the reservoir. The density of the fluid is a linearfunction of pressure, with a constant viscosity µf > 0. The Lamé coefficients λ > 0 and G > 0, theBiot coefficient α, the reference density of the fluid ρf > 0, and the pore volume ϕ

∗ are all assumedto be positive. The absolute permeability tensor, K, is assumed to be bounded, symmetric, anduniformly positive definite in space and constant in time. The quasi-static Biot model [7] we con-sider in this work reads: Find u and p satisfying the equations below for all time t ∈]0, T [:

Flow Equation: ∂∂t

((1M + cfϕ0

)p+ α∇ · u

)−∇ ·

(1µfK(∇ p− ρf,rg∇ η

))= q̃ in Ω

−divσpor(u, p) = f in Ω,Mechanics Equations: σpor(u, p) = σ(u)− αp I in Ω,

σ(u) = λ(∇ · u)I + 2Gε(u) in ΩBoundary Conditions: u = 0 on ∂Ω, K(∇ p− ρf,rg∇ η) · n = 0 on ΓN , p = 0 on ΓD,

Initial Conditions (t=0):((

1M + cfϕ0

)p+ α∇ · u

)(0) =

(1M + cfϕ0

)p0 + α∇ · u0.

where: η is the distance in the direction of gravity, which is assumed to be constant in time, ρf,r > 0is a constant reference density (with respect to a reference pressure pr), g is the gravitational ac-celeration, M is the Biot modulus, ϕ0 is the initial porosity, q̃ =

qρf,r

where q is a mass source or

sink term (injection or production wells). The above system is linear and coupled through termsinvolving the Biot coefficient α.

4

tflow, tmech = 0(initial time = 0)

k = 0

n = 0 (iterativecoupling index)

Fluid Flow:tflow = tflow + ∆t

Compute pore pressure:pn+1,k+1

Mechanics (Biot Model):tmech = tmech + ∆t

Compute displacement:un+1,k+1

Update pore volume

Converged? k = k + 1tflow = tflow − ∆ttmech = tmech − ∆t

n = n + 1

No Yes

(a) Single Rate

tflow, tmech = 0(initial time = 0)

k = 0

n = 0 (iterativecoupling index)

m = 1 (flow iteration index)

Fluid Flow:tflow = tflow + ∆t

Compute pore pressure:pn+1,k+m

m = (Maxflow

iterations:q)?

m = m + 1

Mechanics (Biot Model):tmech = tmech + q∆t

Compute displacement:un+1,k+q

Update pore volume

Converged? k = k + qtflow = tflow − q∆ttmech = tmech − q∆t

n = n + 1

No

Yes

No Yes

(b) Multirate

Figure 1.1: Flowchart for iterative algorithm using single and multirate time stepping for coupledgeomechanics and flow problems

2.1 Mixed variational formulation

For the spatial discretization, we use a mixed finite element formulation for flow and a confor-mal Galerkin formulation for mechanics. For temporal discretization, we follow a backward-Euler

5

scheme. Let Th denote a regular family of conforming elements of the domain of consideration, Ω.Using the lowest order Raviart-Thomas (RT) spaces, our discrete spaces are given as follows:

Discrete Displacements: V h = {vh ∈ H1(Ω)d

; ∀T ∈ Th,vh|T ∈ P1d,vh|∂Ω = 0},Discrete Pressures: Qh = {ph ∈ L2(Ω) ; ∀T ∈ Th, ph|T ∈ P0},Discrete Fluxes: Zh = {qh ∈ H(div; Ω)

d ;∀T ∈ Th, qh|T ∈ P1d, qh · n = 0 on ∂Ω}.

For the single rate scheme, the time step is given by ∆tk = tk − tk−1. For the multirate scheme, weassume that ∆tk denotes the fine flow time step, and q∆tk denotes the coarse mechanics time step,where q represents the number of flow fine time steps contained in one coarse mechanics time step.If we assume a uniform fine flow time step ∆t and the total number of fine time steps is denoted byN, then the total simulation time is given by T = ∆t N, and ti = i∆t, 0 6 i 6 N give the discretetime points. We note here that the proof we are about to present can be extended to other mixedformulation approaches, for instance see [37], or Conformal Galerkin discretizations.Notation: For the single rate scheme, k denotes the flow/mechanics time step index. For themultirate scheme, k denotes the coarse mechanics time step index, m denotes the fine flow timestep index, ∆t denotes the unit (fine) time step size, which is assumed to be uniform within onecoarse mechanics time step, and q denotes the number of flow time steps per coarse mechanicstime step. We note that the parameter q can vary across mechanics coarse time steps, and all ourobtained results remain valid.

3 Previous results

3.1 Standard Fixed stress split algorithm

The fixed-stress split iterative coupling scheme assumes a constant volumetric mean total stressduring the flow solve. In this scheme, the flow problem is solved first followed by the elasticityproblem. For the sake of clarity, we start by presenting the continuous strong form of the splittingscheme. We note here that n denotes the coupling iteration index between the flow and mechanics:Step (a): Given un, we solve for pn+1, zn+1 satisfying(

1M

+ cfϕ0 +α2

λ

)∂∂tpn+1 −∇ · zn+1 = α2

λ∂∂tpn − α∇ · ∂

∂tun + q̃

zn+1 = 1µfK(∇ pn+1 − ρf,rg∇ η

)Step (b): Given pn+1, zn+1, we solve for un+1 satisfying

− divσpor(un+1, pn+1) = fσpor(un+1, pn+1) = σ(un+1)− α pn+1

σ(un+1) = λ(∇ · un+1)I + 2Gε(un+1)

6

with the initial condition(( 1M

+ cfϕ0)pn+1 + α∇ · un+1

)(0) =

( 1M

+ cfϕ0)p0 + α∇ · u0. (3.1)

We first note that the initial condition is independent of the coupling iteration index “n”. Second,the regularization terms α2/λ∂tp

n+1, α2/λ∂tpn+1, added to the left and right hand sides of the flow

equation respectively, cancel each other upon convergence, and the original quasi-static Biot modelis retrieved.

3.2 Fully discrete scheme for single rate

Using a mixed formulation for the flow, continuous Galerkin for the mechanics, and the backwardEuler finite difference method in time, the weak formulation of the single-rate scheme reads asfollows.

Definition 3.1 Find pkh ∈ Qh, and zkh ∈ Zh such that,(flow equation)

∀θh ∈ Qh ,1

∆t

(( 1M

+ cfϕ0)(pkh − pk−1h

), θh

)+

1

µf

(∇ · zkh, θh) =

− αq∆t

(∇ ·(ukh − uk−1h

), θh

)+(q̃h, θh

), (3.2)

∀qh ∈ Zh ,(K−1zkh, qh

)=(pkh,∇ · qh

)+(ρf,rg∇ η, qh

), (3.3)

and (mechanics equation)find ukh ∈ V h such that,

∀vh ∈ Vh , 2G(ε(ukh), ε(vh)

)+ λ(∇ · ukh,∇ · vh

)− α

(pkh,∇ · vh

)=(f ,vh

). (3.4)

with the initial condition for the first discrete time step,

p0h = p0. (3.5)

3.3 Fully discrete scheme for multirate

Similar to the single-rate scheme, the weak formulation of the multirate scheme reads as follows.

Definition 3.2 For 1 ≤ m ≤ q, find pm+kh ∈ Qh, and zm+kh ∈ Zh such that,

7

(flow equation)

∀θh ∈ Qh ,1

∆t

(( 1M

+ cfϕ0)(pm+kh − p

m−1+kh

), θh

)+

1

µf

(∇ · zm+kh , θh) =

− αq∆t

(∇ ·(uk+qh − u

kh

), θh

)+(q̃h, θh

), (3.6)

∀qh ∈ Zh ,(K−1zm+kh , qh

)=(pm+kh ,∇ · qh

)+(ρf,rg∇ η, qh

), (3.7)

and (mechanics equation)find uk+qh ∈ V h such that,

∀vh ∈ Vh , 2G(ε(uk+qh ), ε(vh)

)+ λ(∇ · uk+qh ,∇ · vh

)− α

(pk+qh ,∇ · vh

)=(f ,vh

). (3.8)

with the initial condition for the first discrete time step,

p0h = p0. (3.9)

We note that the pressure and flux unknowns are solved at fine time steps tk+m,m = 0, . . . , q. Incontrast, the displacement unknowns are solved at coarse mechanics time steps tiq, i ∈ N. Therefore,there are q flow solves for each mechanics time step of size q∆t, which justifies the nomenclature ofmultirate.

3.4 Weak formulation of the single rate scheme:

The weak formulation of the fully discrete single rate fixed-stress split iterative coupling schemereads: Step (a): Find pn+1,kh ∈ Qh, z

n+1,kh ∈ Zh such that:

∀θh ∈ Qh ,1

∆t

( 1M

+ cfϕ0 + L)(pn+1,kh − p

k−1h , θh

)+

1

µf

(∇ · zn+1,kh , θh

)=

1

∆t

(L(pn,kh − p

k−1h

)− α∇ ·

(un,kh − u

k−1h

), θh

)+(q̃h, θh

)(3.10)

∀qh ∈ Zh ,(K−1zn+1,kh , qh

)= (pn+1,kh ,∇ · qh) +

(∇(ρf,rgη), qh) (3.11)

Step (b): Given pn+1,kh , zn+1,kh , find u

n+1,kh ∈ V h such that,

∀vh ∈ Vh , 2G(ε(un+1,kh ), ε(vh)

)+ λ(∇ · un+1,kh ,∇ · vh

)− α

(pn+1,kh ,∇ · vh

)=(f ,vh

), (3.12)

We note that in the above scheme, L is an adjustable coefficient that will be determined by math-ematical analysis. In what follows, we adopt the following notation: for a given time step t = tk,we define the difference between two iterative coupling iterates as: δξn+1,k = ξn+1,k − ξn,k. where ξmay stand for ph, zh, and uh.

8

3.5 Banach fixed-point contraction result of the single rate scheme:

Theorem 3.3 For a particular time step tk, L =α2

2λ, c0 =

4λα2

, and δσn,kv = δpn,kh − αL∇ · δu

n,kh , the

single rate fixed-stress iterative coupling scheme is a contraction given by

2∆tβµf

∥∥∥K−1/2δzn+1,kh ∥∥∥2Ω

+ 2Gc0∥∥ε(δun+1,kh )∥∥2Ω + ∥∥∥δσn+1,kv ∥∥∥2Ω ≤ ( Mα22λ+2Mλcfϕ0+Mα2)2∥∥∥δσn,kv ∥∥∥2Ω

Furthermore, the converged solution is a unique solution to the weak formulation (3.2) - (3.4).

3.6 Weak formulation of the multirate scheme:

The weak formulation of the fully discrete multirate fixed-stress split iterative coupling schemereads:

• Step (a) For 1 ≤ m ≤ q, find pn+1,m+kh ∈ Qh, and zn+1,m+kh ∈ Zh such that,

∀θh ∈ Qh ,1

∆t

(( 1M

+ cfϕ0 + L)(pn+1,m+kh − p

n+1,m−1+kh

), θh

)+

1

µf

(∇ · zn+1,m+kh , θh) =

1

∆t

(L(pn,m+kh − p

n,m−1+kh

)− αq∇ ·(un,k+qh − u

n,kh

), θh

)+(q̃h, θh

), (3.13)

∀qh ∈ Zh ,(K−1zn+1,m+kh , qh

)=(pn+1,m+kh ,∇ · qh

)+(ρf,rg∇ η, qh

), (3.14)

with the initial condition, independent of n, for the first discrete time step,

pn+1,0h = p0. (3.15)

• Step (b) Given pn+1,k+qh and, zn+1,k+qh , find u

n+1,k+qh ∈ V h such that,

∀vh ∈ Vh , 2G(ε(un+1,k+qh ), ε(vh)

)+ λ(∇ · un+1,k+qh ,∇ · vh

)− α

(pn+1,k+qh ,∇ · vh

)=(f ,vh

). (3.16)

Similar to the single rate scheme, L is an adjustable coefficient which will be determined appropri-ately by working out the contraction proof. In addition, q denotes a user-defined number of finerflow steps.

9

3.7 Banach fixed-point contraction result of the multirate scheme:

Theorem 3.4 For a coarse mechanics time step tk+q, L =α2

2λ, χ2 = L2, c0 =

2Lqχ2

, and δσn,m+kv =Lχ

(δpn,m+kh −δpn,m−1+kh )− αqχ∇·δu

n,k+qh , for 1 ≤ m ≤ q, the multirate iterative scheme is a contraction

given by

2Gc0∥∥ε(δun+1,k+qh )∥∥2 +∑qm=1 ∥∥∥δσn+1,m+kv ∥∥∥2 + ∆tβµf ∥∥∥K−1/2δzn+1,k+qh ∥∥∥2

+ ∆tβµf

∑qm=1

∥∥∥K−1/2(δzn+1,m+kh − δzn+1,m−1+kh )∥∥∥2 ≤ ( Mα22λ+2Mλcfϕ0+α2M)2∑qm=1 ∥∥∥δσn,m+kv ∥∥∥2.Furthermore, the sequences defined by this scheme converge to the unique solution of the weakformulation (3.6)–(3.8).

Remark 3.5 The contraction coefficient obtained in theorem 3.3 can be strengthened by takingadvantage of the extra terms on the left hand side of the contraction result. By triangle’s inequality,the norm of the quantity of contraction can be written as∥∥δσn+1,kv ∥∥ ≤ αL‖∇ · δun+1,kh ‖+ ‖δpn+1,kh ‖.By standard mixed method techniques of estimating the pressure by the flux (using Poincare inequal-ity), and bounding the volumetric strain term by Korn’s inequality, we obtain∥∥δσn+1,kv ∥∥ ≤ C( ∥∥∥ε(δun+1,kh )∥∥∥+ ∥∥∥δzn+1,kh ∥∥∥ ). (3.17)for a constant C > 0. Now, denote the first two terms on the left hand side of the result of theorem3.3 by In+1,k:

In+1,k = 2∆tβµf

∥∥∥K−1/2δzn+1,kh ∥∥∥2Ω

+ 2Gc0∥∥ε(δun+1,kh )∥∥2Ω.

For a generic constant C > 0, inequality (3.17) can be written as:∥∥δσn+1,kv ∥∥2 ≤ CIn+1,k. (3.18)The contraction result in theorem 3.3 takes the form:( 1

C+ 1)∥∥δσn+1,kv ∥∥2 ≤ ( Mα22λ+ 2Mλcfϕ0 +Mα2

)2 ∥∥δσn,kv ∥∥2resulting in the improved contraction constant,∥∥δσn+1,kv ∥∥2 ≤ ( CC + 1)( Mα22λ+ 2Mλcfϕ0 +Mα2

)2 ∥∥δσn,kv ∥∥2 . (3.19)10

It is not difficult to see that the constant C increases monotonically as the values of the Lamécoefficients, λ and G, increase (assuming the Poisson ratio, ν, is fixed). However, estimating itsexact value is difficult in practice. The above computations show that the contraction coefficientobtained earlier is damped by a factor strictly less than one: C

C+1< 1. Therefore, for C � 1,

the contraction estimate obtained in theorem 3.3 is relatively sharp. However, for small values ofC, the contraction coefficient obtained earlier can be severely affected by the damping factor C

C+1.

We will validate these observations numerically by varying the values of the Lamé coefficients andcomputing contraction estimates numerically. We will show that for larger Lamé coefficients’ values,contraction estimates computed numerically approach the value of the theoretical estimate obtainedin theorem 3.3. In other words, our theoretical contraction estimates are sharper for larger Lamécoefficients. In fact, these computations show the impact of the extra positive terms on the left handside of the result in theorem 3.3 on the sharpness of the contraction coefficient.

Remark 3.6 As in the single rate case, the contraction coefficient in the multirate case, derived intheorem 3.4, can be improved as well. The improved contraction result reads:

q∑m=1

∥∥δσn+1,m+kv ∥∥2 ≤ ( C1 + C)( Mα22λ+ 2Mλcfϕ0 + α2M)2 q∑

m=1

∥∥δσn+1,m+kv ∥∥2 . (3.20)For a constant C > 0.

Remark 3.7 We can theoretically derive the value of the constant C given in the previous remark.For simplicity, consider the single rate case q = 1. For L = α

2

2λ, χ = L, and by Poincare’s and

Korn’s inequalities, we write:∥∥δσn+1,kv ∥∥ ≤ αL‖∇ · δun+1,kh ‖L2(Ω) + ‖δpn+1,kh ‖L2(Ω)≤ 2λ

α|δun+1,kh |H1(Ω) + ‖δp

n+1,kh ‖L2(Ω)

≤ 2λCκα

∥∥ε(δun+1,kh )∥∥L2(Ω) + PΩ‖∇δpn+1,kh ‖L2(Ω)≤ 2λCκ

α

∥∥ε(δun+1,kh )∥∥L2(Ω) + PΩ‖K−1δzn+1,kh ‖L2(Ω)Thus, we have:∥∥δσn+1,kv ∥∥2 ≤ 4λ2Cκ2α2 ∥∥ε(δun+1,kh )∥∥2L2(Ω) + P2Ω‖K−1δzn+1,kh ‖2L2(Ω)

+4λCκPΩ

α

∥∥ε(δun+1,kh )∥∥L2(Ω)‖K−1δzn+1,kh ‖L2(Ω)≤(4λ2Cκ2

α2+

2λCκPΩ�α

)∥∥ε(δun+1,kh )∥∥2L2(Ω) + (P2Ω + 2λCκPΩα� )‖K−1δzn+1,kh ‖2L2(Ω)(3.21)

11

For � > 0. Assuming that the permeability tensor K is uniformly bounded and uniformly elliptic.There exits positive constants λmin, and λmax, such that

λmin‖ξ‖2 ≤ ξtK(x)ξ ≤ λmax‖ξ‖2. (3.22)We write:∥∥δσn+1,kv ∥∥2 ≤ (4λ2Cκ2α2 + 2λCκPΩ�α )∥∥ε(δun+1,kh )∥∥2L2(Ω) + 1λ2min

(P2Ω +

2λCκPΩα�

)‖δzn+1,kh ‖

2L2(Ω)

(3.23)

Therefore, we have:∥∥δσn+1,kv ∥∥2 ≤Max

((4λ2Cκ2α2

+2λCκPΩ�

α

),

1

λ2min

(P2Ω +

2λCκPΩα�

))(∥∥ε(δun+1,kh )∥∥2L2(Ω) + ‖δzn+1,kh ‖2L2(Ω))(3.24)

For the single rate case (recall: c0 =4λα2

), In+1,kq takes the form:

In+1,kq = 2Gc0∥∥ε(δun+1,kh )∥∥2L2(Ω) + 2∆tβµf

∥∥∥K−1/2δzn+1,kh ∥∥∥2L2(Ω)

≥ 8Gλα2∥∥ε(δun+1,kh )∥∥2L2(Ω) + 1λmax 2∆tβµf

∥∥∥δzn+1,kh ∥∥∥2L2(Ω)

≥Min(8Gλα2

,1

λmax

2∆t

βµf

)(∥∥ε(δun+1,kh )∥∥2L2(Ω) + ∥∥∥δzn+1,kh ∥∥∥2L2(Ω)) (3.25)Combining (3.24) and (3.25), we have:

∥∥δσn+1,kv ∥∥2 ≤(Max((4λ2Cκ2

α2+ 2λCκPΩ�

α

), 1λ2min

(P2Ω + 2λCκPΩα�

))Min

(8Gλα2, 1λmax

2∆tβµf

) )In+1,kqwith the constant C given by:

C =

Max

((4λ2Cκ2

α2+ 2λCκPΩ�

α

), 1λ2min

(P2Ω + 2λCκPΩα�

))Min

(8Gλα2, 1λmax

2∆tβµf

) (3.26)Clearly, C scales monotonically with the values of Lame’s parameters (G and λ). Therefore, forlarger Lame parameters, the value of the constant C increases, and in turn, the damping factorC

1+Cstart approaching the value of one. This results in reducing the gap between the theoretical

value of the contraction coefficient(

Mα2

2λ+2Mλcfϕ0+α2M

)2, and the ratio of

‖δσn+1,kv ‖2

‖δσn,kv ‖2. This is validated

numerically for the Frio field model in figure 5.3.

12

4 Error analysis of the single rate iterative scheme

For a given time step t = tk, and a given iterative coupling iteration n ≥ 0, we need to estimate∥∥ξn,kh − ξ(tk)∥∥, where ξ may stand for ph, zh, and uh. By the triangle inequality, we can write:∥∥ξn,kh − ξ(tk)∥∥ ≤ ∥∥ξn,kh − ξkh∥∥+ ∥∥ξkh − ξ(tk)∥∥where ξkh is the solution obtained by solving the coupled flow and mechanics equations simultane-ously. Error estimates for the second term on the right hand side have been derived in the workof [26, 27]. It only remains to estimate the first term

∥∥ξn,kh − ξkh∥∥. This can be done in two steps:first we derive a Banach contraction argument on the difference between the solution obtained at aparticular iterative coupling iteration ξn,kh , and the solution obtained by solving the coupled systemsimultaneously (fully implicit scheme, ξkh). Then, we derive stability estimates for the fully implicitscheme, and combine the two to bound the term

∥∥ξn,k − ξkh∥∥. The two steps are detailed below.• Step 1: Banach Contraction estimate on

∥∥ξn,k − ξkh∥∥:We first note that the weak formulation of the fully discrete single-rate fixed-stress split iter-ative coupling scheme is given in equations (3.10) - (3.12). In contrast, the weak formulationof the fully discrete implicit scheme reads:Find pkh ∈ Qh, zkh ∈ Zh, and ukh ∈ V h such that,

∀θh ∈ Qh ,1

∆t

(( 1M

+ cfϕ0)(pkh − pk−1h

), θh

)+

1

µf

(∇ · zkh, θh

)= − α

∆t

(∇ ·(ukh − uk−1h

), θh

)+(q̃h, θh

), (4.1)

∀qh ∈ Zh ,(K−1zkh, qh

)=(pkh,∇ · qh

)+(ρf,rg∇ η, qh

), (4.2)

∀vh ∈ Vh , 2G(ε(ukh), ε(vh)

)+ λ(∇ · ukh,∇ · vh

)− α

(pkh,∇ · vh

)=(fkh,vh

). (4.3)

Subtracting equations (4.3), (4.2), and (4.1), from (3.12), (3.11), and (3.10) respectively, andnoting that fn+1,kh = f

kh, we get:

∀θh ∈ Qh ,1

∆t

(( 1M

+ cfϕ0)(pn+1,kh − p

kh

), θh

)+

1

µf

(∇ · (zn+1,kh − z

kh), θh

)= − α

∆t

(∇ ·(un,kh − u

kh

), θh

)− L

∆t

(pn+1,kh − p

n,kh , θh

),

(4.4)

∀qh ∈ Zh ,(K−1(zn+1,kh − z

kh), qh

)=(pn+1,kh − p

kh,∇ · qh

), (4.5)

∀vh ∈ Vh , 2G(ε(un+1,kh − u

kh), ε(vh)

)+ λ(∇ · (un+1,kh − u

kh),∇ · vh

)− α

(pn+1,kh − p

kh,∇ · vh

).

(4.6)

13

Define en+1p = pn+1,kh − pkh, en+1u = u

n+1,kh −ukh, and en+1z = z

n+1,kh −zkh. Equations (4.4), (4.5),

and (4.6) can be written as:

∀θh ∈ Qh ,1

∆t

( 1M

+ cfϕ0 + L)(en+1p , θh

)+

1

µf

(∇ · en+1z , θh

)=

1

∆t

(− α∇ · enu + Lenp , θh

),

(4.7)

∀qh ∈ Zh ,(K−1en+1z , qh

)=(en+1p ,∇ · qh

), (4.8)

∀vh ∈ Vh , 2G(ε(en+1u ), ε(vh)

)+ λ(∇ · en+1u ,∇ · vh

)− α

(en+1p ,∇ · vh

)= 0. (4.9)

Let β = 1M

+ cfϕ0 + L. Testing (4.7) with θh = en+1p , and (4.8) with qh = e

n+1z , we obtain:

β∥∥∥en+1p ∥∥∥2

Ω+

∆t

µf

(∇ · en+1z , en+1p )Ω =

(− α∇ · enu + Lenp , en+1p

)Ω. (4.10)(

K−1en+1z , en+1z

)Ω

=(en+1p ,∇ · en+1z

)Ω. (4.11)

Substituting (4.11) into (4.10), defining enσ as χenσ = Le

np − α∇ · enu, where χ is an adjustable

parameter, and applying Young’s inequality, we obtain:

β∥∥∥en+1p ∥∥∥2

Ω+

∆t

µf

∥∥∥K−1/2en+1z ∥∥∥Ω≤ 1

2�

∥∥∥enσ∥∥∥2Ω

+�

2

∥∥∥en+1p ∥∥∥2Ω.

The choice � = β gives (after multiplying by 2β):∥∥∥en+1p ∥∥∥2

Ω+

2∆t

βµf

∥∥∥K−1/2en+1z ∥∥∥2Ω≤ 1β2

∥∥∥χenσ∥∥∥2Ω. (4.12)

Multiplying the elasticity equation (4.9) by a free parameter c0, and testing with vh = en+1u ,

we get:

2Gc0∥∥ε(en+1u )∥∥2Ω + λc0∥∥∇ · en+1u ∥∥2Ω − αc0(en+1p ,∇ · en+1u )Ω = 0. (4.13)

Combining flow (4.12) with elasticity (4.13), we obtain:

2Gc0∥∥ε(en+1u )∥∥2Ω + {∥∥∥en+1p ∥∥∥2Ω − αc0(en+1p ,∇ · en+1u )Ω + λc0∥∥∇ · en+1u ∥∥2Ω}

+2∆t

βµf

∥∥∥K−1/2en+1z ∥∥∥2Ω≤ χ

2

β2

∥∥∥enσ∥∥∥2Ω. (4.14)

Expanding the right hand side to match terms on the left hand side (to form a completesquare): ∥∥∥enσ∥∥∥2 = L2χ2 ∥∥∥enp∥∥∥2Ω − 2αLχ2 (enp ,∇ · enu)Ω + α2χ2∥∥∥∇ · δenu∥∥∥2Ω.

14

The following inequalities should be satisfied: 1 > L2

χ2, 2αL

χ2= αc0, and λc0 =

α2

χ2. The

second and third equalities lead to the following parameter assignments: c0 =2Lχ2

, and L = α2

2λ.

The first inequality leads to the condition: χ > α2

2λ. Now, (4.14) can be written as:

2Gc0∥∥ε(en+1u )∥∥2Ω + 2∆tβµf

∥∥∥K−1/2en+1z ∥∥∥2Ω

+(

1− L2

χ2

)∥∥∥en+1p ∥∥∥2Ω

+∥∥∥en+1σ ∥∥∥2

Ω≤(χβ

)2∥∥∥enσ∥∥∥2Ω.

(4.15)

For contraction to hold, we require χβ< 1. Together with the previous condition χ > α

2

2λ, the

value of χ should be chosen such that

α2

2λ< χ <

1

M+ cfϕ0 +

α2

2λ

This imposes the following condition on our given parameters (which corresponds to thecondition on the constrained specific storage coefficient in the work of [26,27]:

1

M+ cfϕ0 ≥ γ0 > 0. for some positive constant γ0. (4.16)

In general, for n ≥ 0, we can write:∥∥∥en+1σ ∥∥∥2Ω≤(χβ

)2∥∥∥enσ∥∥∥2Ω

≤(χβ

)2(n+1)∥∥∥e0σ∥∥∥2Ω.

≤(χβ

)2(n+1)∥∥∥Le0p − α∇ · e0u∥∥∥2Ω. (4.17)

Combining (4.15) with (4.17), together with Young’s inequality, we can write:(1− L

2

χ2

)∥∥∥en+1p ∥∥∥2Ω≤(χβ

)2(n+1)∥∥∥Le0p − α∇ · e0u∥∥∥2Ω

≤(χβ

)2(n+1)(L2∥∥∥e0p∥∥∥2

Ω+ α2

∥∥∥∇ · e0u∥∥∥2Ω− 2Lα(e0p,∇ · e0u)

)≤(χβ

)2(n+1)(L2∥∥∥e0p∥∥∥2

Ω+ α2

∥∥∥∇ · e0u∥∥∥2Ω

+ 2Lα(1

2�

∥∥∥e0p∥∥∥2Ω

+�

2

∥∥∥∇ · e0u∥∥∥2Ω

))

≤(χβ

)2(n+1)((L2 +

Lα

�)∥∥∥e0p∥∥∥2

Ω+ (α2 + Lα�)

∥∥∥∇ · e0u∥∥∥2Ω

)for � > 0.

(4.18)

15

Similarly, we can write:

2Gc0∥∥ε(en+1u )∥∥2Ω ≤ (χβ)2(n+1)((L2 + Lα� )∥∥∥e0p∥∥∥2Ω + (α2 + Lα�)∥∥∥∇ · e0u∥∥∥2Ω). (4.19)

2∆t

βµf

∥∥∥K−1/2en+1z ∥∥∥2Ω≤(χβ

)2(n+1)((L2 +

Lα

�)∥∥∥e0p∥∥∥2

Ω+ (α2 + Lα�)

∥∥∥∇ · e0u∥∥∥2Ω

). (4.20)

Combining (4.18), (4.19), and (4.20), we have:∥∥∥en+1p ∥∥∥2Ω

+∥∥ε(en+1u )∥∥2Ω + ∥∥∥K−1/2en+1z ∥∥∥2Ω ≤(χ

β

)2(n+1)C1

((L2 +

Lα

�)∥∥∥e0p∥∥∥2

Ω+ (α2 + Lα�)

∥∥∥∇ · e0u∥∥∥2Ω

). (4.21)

where C1 =

[χ2

χ2−L2 +1

2Gc0+

βµf2∆t

]. Noting that: e0p = p

0,kh −pkh = p

k−1h −pkh, and e0u = u

0,kh −ukh =

uk−1h − ukh, (4.21) can be written as:∥∥∥pn+1,kh − pkh∥∥∥2Ω

+∥∥ε(un+1,kh − ukh)∥∥2Ω + ∥∥∥K−1/2zn+1,kh − zkh∥∥∥2Ω ≤(χ

β

)2(n+1)C1

(L2 +

Lα

�

)∥∥∥pkh − pk−1h ∥∥∥2Ω

+(χβ

)2(n+1)C1(α

2 + Lα�)∥∥∥∇ · ukh − uk−1h ∥∥∥2

Ω

(4.22)

Let η̃1 = C1(L2 + Lα

�), and η̃2 = C1(α

2 + Lα�) for � > 0, (4.22) reduces to:∥∥∥pn+1,kh − pkh∥∥∥2Ω

+∥∥ε(un+1,kh − ukh)∥∥2Ω + ∥∥∥K−1/2(zn+1,kh − zkh)∥∥∥2Ω

≤(χβ

)2(n+1)(η̃1

∥∥∥pkh − pk−1h ∥∥∥2Ω

+ η̃2

∥∥∥∇ · ukh − uk−1h ∥∥∥2Ω

)(4.23)

• Step 2: Stability estimate on∥∥ξkh− ξk−1h ∥∥: We recall that the weak formulation of the implicit

scheme is given by equations (4.1) - (4.3). The derivation of the stability estimate for theimplicit scheme is carried out in three steps: by first considering the flow equations, followedby the mechanics equation and then combining the two to derive the final estimate. Forsimplicity, we define c̃f =

1M

+ cfϕ0.

16

4.1 Flow equations

Testing (4.1) with θh = pkh − pk−1h , and multiplying the whole equation by ∆t, we obtain

c̃f

∥∥∥pkh − pk−1h ∥∥∥2 + ∆tµf(∇ · zkh, pkh − pk−1h

)= α

(∇ · (ukh − uk−1h ), p

kh − pk−1h

)+(q̃h, p

kh − pk−1h

)(4.24)

Next, we consider the flux equation (4.2). Taking the difference of two consecutive time stepst = tk and t = tk−1 and testing with qh = z

kh, we obtain:(

K−1(zkh − zk−1h ), zkh

)=(pkh − pk−1h ,∇ · z

kh

)(4.25)

Substituting (4.25) into (4.24), with some algebraic manipulations of the resulting term (usingthe identity: a(a− b) = 1

2(a2 − b2 + (a− b)2)), we derive

c̃f

∥∥∥pkh − pk−1h ∥∥∥2 + ∆t2µf(∥∥∥K−1/2zkh∥∥∥2 − ∥∥∥K−1/2zk−1h ∥∥∥2 + ∥∥∥K−1/2(zkh − zk−1h )∥∥∥2)

= −α(∇ · (ukh − uk−1h ), p

kh − pk−1h

)+(q̃h, p

kh − pk−1h

)(4.26)

4.2 Elasticity equation

Considering (4.3) for the difference of two consecutive time steps, t = tk and t = tk−1, andtesting with vh = u

kh − uk−1h , we obtain

2G∥∥∥ε(ukh − uk−1h )∥∥∥2 + λ∥∥∥∇ · (ukh − uk−1h )∥∥∥2 − α(pkh − pk−1h ,∇ · (ukh − uk−1h ))

=(fkh − fk−1h ,ukh − uk−1h

)(4.27)

4.3 Combining flow and elasticity equations

Combining (4.26) with (4.27) yields

c̃f

∥∥∥pkh − pk−1h ∥∥∥2 + ∆t2µf(∥∥∥K−1/2zkh∥∥∥2 − ∥∥∥K−1/2zk−1h ∥∥∥2 + ∥∥∥K−1/2(zkh − zk−1h )∥∥∥2)

+2G∥∥∥ε(ukh − uk−1h )∥∥∥2 + λ∥∥∥∇ · (ukh − uk−1h )∥∥∥2 = (q̃h, pkh − pk−1h )︸ ︷︷ ︸

R1

+(fkh − fk−1h ,ukh − uk−1h

)︸ ︷︷ ︸

R2

(4.28)

17

To bound the terms (R1 and R2), we will use Poincaré’s and Korn’s inequalities. Poincaré’sinequality in H10 (Ω) reads: there exists a constant PΩ depending only on Ω such that

∀v ∈ H10 (Ω) , ‖v‖L2(Ω) ≤ PΩ|v|H1(Ω). (4.29)Korn’s first inequality in H10 (Ω)

d reads: there exists a constant Cκ depending only on Ω suchthat

∀v ∈ H10 (Ω)d , |v|H1(Ω) ≤ Cκ‖ε(v)‖L2(Ω). (4.30)By Poincaré, Korn, and Young inequalities, we bound R1 & R2 as:

|R1| ≤1

2�1

∥∥∥q̃h∥∥∥2 + �12

∥∥∥pkh − pk−1h ∥∥∥2|R2| ≤

1

2�2

∥∥∥fkh − fk−1h ∥∥∥2 + �22 ∥∥∥ukh − uk−1h ∥∥∥2≤ 1

2�2

∥∥∥fkh − fk−1h ∥∥∥2 + �2P2ΩC2κ2 ‖ε(ukh − uk−1h )‖2.for �1, and �2 > 0. Choosing �1 = c̃f , and �2 =

2GP2ΩC2κ

, and summing for 1 ≤ k ≤ N , where Ndenotes the total number of time steps (note telescopic cancellations), we derive

c̃f2

N∑k=1

∥∥∥pkh − pk−1h ∥∥∥2 + ∆t2µf(∥∥∥K−1/2zNh ∥∥∥2 + N∑

k=1

∥∥∥K−1/2(zkh − zk−1h )∥∥∥2)+G N∑k=1

∥∥∥ε(ukh − uk−1h )∥∥∥2+λ

N∑k=1

∥∥∥∇ · (ukh − uk−1h )∥∥∥2 ≤ ∆t2µf∥∥∥K−1/2z0h∥∥∥2 + 12c̃f

N∑k=1

∥∥∥q̃h∥∥∥2 + P2ΩC2κ4G

N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2,(4.31)

Therefore, we can write:

N∑k=1

∥∥∥pkh − pk−1h ∥∥∥2 ≤ ∆tµf c̃f∥∥∥K−1/2z0h∥∥∥2 + 1c̃2f

N∑k=1

∥∥∥q̃h∥∥∥2 + P2ΩC2κ2Gc̃f

N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2,(4.32)

N∑k=1

∥∥∥∇ · (ukh − uk−1h )∥∥∥2 ≤ ∆t2µfλ∥∥∥K−1/2z0h∥∥∥2 + 12c̃fλ

N∑k=1

∥∥∥q̃h∥∥∥2 + P2ΩC2κ4Gλ

N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2.(4.33)

Combining (4.32) with (4.33), we have:

N∑k=1

∥∥∥pkh − pk−1h ∥∥∥2 + N∑k=1

∥∥∥∇ · (ukh − uk−1h )∥∥∥2 ≤ ∆tη̃3∥∥∥K−1/2z0h∥∥∥2 + η̃4 N∑k=1

∥∥∥q̃h∥∥∥2+ η̃5

N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2 (4.34)18

where η̃3 =1µfC2, η̃4 =

1c̃fC2, η̃5 =

P2ΩC2κ

2GC2, and C2 =

(1c̃f

+ 12λ

). Combining (4.23) with

(4.34), for a generic constant C3 > 0 (which will be revealed by the end of the derivation butwe suppress its value now for the sake of simplicity), we can derive:∥∥∥pn+1,kh − pkh∥∥∥2

Ω+∥∥ε(un+1,kh − ukh)∥∥2Ω + ∥∥∥K−1/2(zn+1,kh − zkh)∥∥∥2Ω≤(χβ

)2(n+1)(η̃1

∥∥∥pkh − pk−1h ∥∥∥2Ω

+ η̃2

∥∥∥∇ · ukh − uk−1h ∥∥∥2Ω

)≤(χβ

)2(n+1)C3

[∥∥∥pkh − pk−1h ∥∥∥2Ω

+∥∥∥∇ · ukh − uk−1h ∥∥∥2

Ω

]≤(χβ

)2(n+1)C3

[ N∑k=1

∥∥∥pkh − pk−1h ∥∥∥2Ω

+N∑k=1

∥∥∥∇ · ukh − uk−1h ∥∥∥2Ω

]≤(χβ

)2(n+1)C3

[∆tη̃3

∥∥∥K−1/2z0h∥∥∥2 + η̃4 N∑k=1

∥∥∥q̃h∥∥∥2 + η̃5 N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2]≤(χβ

)2(n+1)C3

[∆t∥∥∥K−1/2z0h∥∥∥2 + N∑

k=1

∥∥∥q̃h∥∥∥2 + N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2]Therefore, we can write:∥∥∥pn+1,kh − pkh∥∥∥

L2(Ω)+∥∥ε(un+1,kh − ukh)∥∥L2(Ω) + ∥∥∥K−1/2(zn+1,kh − zkh)∥∥∥L2(Ω)≤ C3

(χβ

)(n+1)[∆t∥∥∥K−1/2z0h∥∥∥2

L2(Ω)+

N∑k=1

∥∥∥q̃h∥∥∥2L2(Ω)

+N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2L2(Ω)

]1/2(4.35)

Now, we assume that the permeability tensor K is uniformly bounded and uniformly elliptic. Thereexits positive constants λmin, and λmax, such that

λmin‖ξ‖2 ≤ ξtK(x)ξ ≤ λmax‖ξ‖2. (4.36)

We can write

‖K−1/2(zn+1,kh − zkh)‖L2(Ω) ≥

1

λ1/2max

‖zn+1,kh − zkh‖L2(Ω).

In addition, by Poincaré’s inequality and Korn’s first inequality, we have (for Ck1 > 0):

‖ε(un+1,kh − ukh)‖L2(Ω) ≥

1

Ck1‖un+1,kh − u

kh‖H1(Ω).

19

Therefore, (4.37) can be written as:∥∥∥pn+1,kh − pkh∥∥∥L2(Ω)

+∥∥∥un+1,kh − ukh∥∥∥

H1(Ω)+∥∥∥zn+1,kh − zkh∥∥∥

L2(Ω)

≤ C3(χβ

)(n+1)[∆t∥∥∥K−1/2z0h∥∥∥2

L2(Ω)+

N∑k=1

∥∥∥q̃h∥∥∥2L2(Ω)

+N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2L2(Ω)

]1/2(4.37)

We conclude that for every coupling iteration n ≥ 0,∥∥∥pn+1,kh − p(tk)∥∥∥`∞(L2)

+∥∥∥un+1,kh − u(tk)∥∥∥

`∞(H1)+∥∥∥zn+1,kh − z(tk)∥∥∥

`∞(L2)

≤∥∥∥pn+1,kh − pkh∥∥∥

`∞(L2)+∥∥∥un+1,kh − ukh∥∥∥

`∞(H1)+∥∥∥zn+1,kh − zkh∥∥∥

`∞(L2)

+∥∥∥pkh − p(tk)∥∥∥

`∞(L2)+∥∥∥ukh − u(tk)∥∥∥

`∞(H1)+∥∥∥zkh − z(tk)∥∥∥

`∞(L2)

≤ C3(χβ

)(n+1)[∆t∥∥∥K−1/2z0h∥∥∥2

L2(Ω)+

N∑k=1

∥∥∥q̃h∥∥∥2L2(Ω)

+N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2L2(Ω)

]1/2+∥∥∥pkh − p(tk)∥∥∥

`∞(L2)+∥∥∥ukh − u(tk)∥∥∥

`∞(H1)+∥∥∥zkh − z(tk)∥∥∥

`2(L2)

By [26,27], we have:∥∥∥pkh − p(tk)∥∥∥2`∞(L2)

+∥∥∥ukh − u(tk)∥∥∥2

`∞(H1)+∥∥∥zkh − z(tk)∥∥∥2

`2(L2)≤ C(h2r1+2 + h2r2) +O(∆t2)

for a positive constant C > 0 and mesh size h. We note that r1 denotes the degree of the polyno-mials used in the mixed space (Qh,Zh), and r2 denotes the degree of the polynomials used in thedisplacement space V h. In our case, r1 = 0, and r2 = 1. Therefore, our final estimate takes theform:∥∥∥pn+1,kh − p(tk)∥∥∥

`∞(L2)+∥∥∥un+1,kh − u(tk)∥∥∥

`∞(H1)+∥∥∥zn+1,kh − z(tk)∥∥∥

`∞(L2)

≤ C3(χβ

)(n+1)[∆t∥∥∥K−1/2z0h∥∥∥2

L2(Ω)+

N∑k=1

∥∥∥q̃h∥∥∥2L2(Ω)

+N∑k=1

∥∥∥fkh − fk−1h ∥∥∥2L2(Ω)

]1/2+ 3(

2Ch2 +O(∆t2))1/2

where C3 = 3(

1 +Ck1 +λ1/2max

)(Max(η̃1, η̃2)×Max(η̃3, η̃4, η̃5)

)2. The above discussions are summa-

rized in the following theorem:

Theorem 4.1 For a particular time step tk, and a particular flow-mechanics coupling iterationn ≥ 1, and assuming the lowest order Raviart-Thomas spaces for flow, and continuous piecewiselinear approximations for mechanics, and assuming equations (4.16) and (4.36), and sufficient

20

regularity in the true solution, the following finite element error estimate, to the leading order intime, for the single rate fixed-stress split iterative coupling scheme holds:∥∥∥pn,kh − p(tk)∥∥∥

`∞(L2)+∥∥∥un,kh − u(tk)∥∥∥

`∞(H1)+∥∥∥zn,kh − z(tk)∥∥∥

`∞(L2)

≤ C1(χβ

)n[∆t∥∥∥K−1/2z0h∥∥∥2

L2(Ω)+∑N

k=1

∥∥∥q̃h∥∥∥2L2(Ω)

+∑N

k=1

∥∥∥fkh − fk−1h ∥∥∥2L2(Ω)

]1/2+(C2h

2 +O(∆t2))1/2

where

C1 = 3(

1 + Ck1 + λ1/2max

)(Max(η̃1, η̃2)×Max(η̃3, η̃4, η̃5)

)2C2 = C2(T,K,M, cf , ϕ0, Ck1 , p

kh, p

kh,t, z

kh,u

kh,t).

Remark 4.2 We note that the contracting behavior of the undrained split iterative coupling schemehas been established in [23] when Continuous Galerkin (CG) is used to discretize both flow andmechanics. The work of [5, 19] extends this result to the case when a mixed form is used for flowand CG is used for mechanics for both the single rate and multirate schemes. Based on that, apriori error estimates for the single rate undrained split scheme can be derived in the same wayas in the fixed stress split scheme. A Banach contraction estimate on the difference

∥∥ξn,k − ξkh∥∥can be obtained in a similar way as described in [5, 19]. This complete the first step of the erroranalysis. The second step, involving the derivation of stability estimates on the difference

∥∥ξk−ξk−1h ∥∥remains unchanged, as both the fixed-stress split and undrained split iterative schemes converge tothe solution obtained by the simultaneously coupled scheme.

5 Numerical Results

The single rate and multirate fixed-stress split iterative coupling schemes have been implementedin the Integrated Parallel Accurate Reservoir Simulator (IPARS) for single-phase and two-phaseflow models coupled with a linear poroelasticity model. Conformal Galerkin is used for elasticitydiscretization and the Multipoint Flux Mixed Finite Element Method (MFMFE) is used for flowdiscretization. The two-phase model assumes an IMPES (implicit pressure explicit saturation)scheme. We first highlight efficiency gains of the multirate scheme over the single rate scheme byconsidering the Frio field model (a field-scale problem) run as a two-phase (oil and water) flowproblem coupled with mechanics. Then, we the investigate the effects of Lame parameters on thesharpness of the predicted theoretical estimate for the Frio field model run as a single-phase flowproblem coupled with mechanics.

5.1 Frio Field Model

The Frio field model is a field-scale problem representing a reservoir model located at South Libertyoil field on the Gulf Coast, near Dayton, Texas. The field is curved in the depth direction, and

21

contains several thin curved faults, with a geometrically challenging geological formation [16]. Inthis work, we consider the challenging geometry of the field, and its permeability distribution.Gravity effects are included in the simulation model. Other input parameters are listed in Table 1.We recall that q denotes the number of fine flow time steps of size ∆t within one coarse mechanicstime step of size q∆t. We run the simulation for q = 1, which corresponds to the single rate case,and for q = 2, 4, and 8, which correspond to three different multirate schemes, in which we taketwo, four, and eight flow fine time steps of size ∆t within one coarse mechanics time step of size2∆t, 4∆t, and 8∆t, respectively.

5.2 Results

Figures 5.2a, 5.2b, 5.2c, and 5.2d show water pressure profile and mechanical displacements in thex,y, and z directions respectively for the frio field model after 128.0 simulation days. We clearlysee that the results for both single rate and multirate implementations are identical. It shouldbe noted that the pore pressure change in this case is relatively small compared to the value ofthe Young’s modulus of the porous medium, which leads to small variations in the displacementvectors. Figure 5.1a shows the accumulated CPU run time for the single rate case (q = 1), and formultirate cases: q = 2, 4, and 8. The multirate iterative coupling algorithm with two flow finertime steps within one coarser mechanics time step (q = 2) results in 12.25% reduction in CPU runtime compared to the single rate case. Multirate couplings (q = 4 and q = 8) result in 18.18% and20.05% reductions in CPU run times respectively. Figure 5.1b explains the reduction in CPU runtime observed in the multirate case. By just solving for two flow finer time steps within one coarsemechanics time step (q = 2), the total number of mechanics linear iterations was reduced by 47.78%with reference to the single rate case. Multirate couplings (q = 4 and q = 8) result in 73.07% and85.75% reductions in the number of mechanics linear iterations respectively, which in turn, reducethe CPU run time as well. Figure 5.1c shows the total number of flow linear iterations in the fourcases. We see a slight increase in the total number of flow linear iterations for multirate iterativecoupling schemes. The case (q = 2) results in 1.25% increase in the total number of flow lineariterations. Multirate couplings (q = 4) and (q = 8) result in 2.89% and 4.82% increase in the totalnumber of flow linear iterations respectively. We conclude that the huge reduction in the numberof mechanics linear iterations outperform the overhead introduced by the increase in the number offlow linear iterations. This is a key factor to the success of the iterative multirate coupling schemein reducing the overall CPU run time. We note here that the observed increase in the number offlow linear iterations in the multirate case is a direct consequence of the increase in the number ofiterative flow coupling-mechanics iterations in the multirate case, compared to the single rate case,as shown in figure 5.1d.

22

Wells: 3 production wells, 6 injection wellInjection well (1): Pressure specified, 4000.0 psiInjection well (2): Pressure specified, 3300.0 psiInjection well (3): Pressure specified, 4000.0 psiInjection well (4): Pressure specified, 4400.0 psiInjection well (5): Pressure specified, 3700.0 psiInjection well (6): Pressure specified, 4400.0 psi

Production well (1): Pressure specified, 2000.0 psiProduction well (2): Pressure specified, 2000.0 psiProduction well (3): Pressure specified, 2000.0 psi

Total Simulation time: 128.0 daysFiner (Unit) time step: 0.05 days

Number of grids: 891 grids (33 × 9 × 1)Absolute Permeabilities: kxx, kyy, kzz highly varying, range: (5.27E-10, 3.10E+3) md

Initial porosity, ϕ0 0.2Water viscosity, µw 1.0 cp

Oil viscosity, µo 2.0 cp

Initial oil concentration, co 44.8 lbm/ft3

Initial oil pressure, po 400.0 psiWater compressibility cfw : 1.E-6 (1/psi)

Oil compressibility cfo : 1.E-4 (1/psi)Rock compressibility: 1.E-6 (1/psi)

Rock density: 165.44 lbm/ft3

Initial water density, ρw: 56.0 lbm/ft3

Initial oil density, ρo 62.34 lbm/ft3

Young’s Modulus (E) 1.2E6 psiPossion Ratio, ν 0.35

Biot’s constant, α 1.0Biot Modulus, M 1.E8 psi

λ = Eν(1+ν)(1−2ν) 1.037E6 psi

L (introduced fixed stress parameter) α2

2λ

Flow Boundary Conditions: no flow boundary condition on all 6 boundariesMechanics B.C.:

“X+” boundary (EBCXX1()) σxx = σ · nx = 10, 000psi, (overburden pressure)“X-” - boundary (EBCXXN1()) u = 0, zero displacement“Y+” - boundary (EBCYY1()) u = 0, zero displacement“Y-” - boundary (EBCYYN1()) σyy = σ · ny = 2000psi“Z+” - boundary (EBCZZ1()) u = 0, zero displacement“Z-” - boundary (EBCZZN1()) σzz = σ · nz = 1000psi

Table 1: Input Parameters for Frio Field Model

23

0 20 40 60 80 100 120 140Simulation Period (days)

0

5

10

15

20

25

30

35

40A

ccum

ula

ted C

PU

Tim

e (

min

ute

s)Accumulated CPU Run Time vs Simulation Period

Single RateMultirate (q = 2)Multirate (q = 4)Multirate (q = 8)

(a) CPU Run Time vs Simulation Days

0 20 40 60 80 100 120 140Simulation Time (days)

0

10000

20000

30000

40000

50000

60000

Accum

ula

ted M

echanic

s L

inear

Itera

tions

Accumulated # of Mechanics Linear Itrns vs Simulation PeriodSingle RateMultirate (q = 2)Multirate (q = 4)Multirate (q = 8)

(b) Total Number of Mechanics Linear Iterationsvs Simulation Days

0 20 40 60 80 100 120 140Simulation Time (days)

0

10000

20000

30000

40000

50000

60000

Accum

ula

ted F

low

Lin

ear

Itera

tions

Accumulated # of Flow Linear Itrns vs Simulation PeriodSingle RateMultirate (q = 2)Multirate (q = 4)Multirate (q = 8)

(c) Total Number of Flow Linear Iterationsvs Simulation Days

0 20 40 60 80 100 120Time Steps (days)

0

1

2

3

4

5

Num

ber

of

Couplin

g Ite

rati

ons

Per

Tim

e S

tep

Coupling Iterations per Coarse Time StepSingle Rate (q = 1)Multirate (q = 2)Multirate (q = 4)Multirate (q = 8)

(d) Number of Iterative Coupling IterationsPer Coarser Time Step

Figure 5.1: Frio Field Model Simulation Results: the top left plot shows the CPU time savings ofdifferent multirate schemes (q = 2, 4, and 8) over the single rate scheme (q = 1). The top right

plot shows the reduction in the number of mechanics linear iterations observed for thecorresponding multirate scheme. The bottom left plot shows the increase in the number of flow

linear iterations induced by the slight increase in the number of flow-mechanics couplingiterations, as shown in the bottom right plot, for multirate schemes (q = 2, 4, and 8) compared to

the single rate scheme (q = 1).

24

(a) Pressure Profiles after 128.0 simulation days (b) Displacement in (x) direction after 128.0simulation days

(c) Displacement in (y) direction after 128.0simulation days

(d) Displacement in (z) direction after 128.0simulation days

Figure 5.2: Frio Field Model Pressure and Displacement Fields at The End of The Simulation

25

5.3 Theoretical Vs. Numerical Contraction Coefficients

In this section, we compare our theoretically derived contraction estimate against numerically ob-served values for the single rate case. We consider the Frio field model run as a single phase flowproblem coupled with mechanics. We assume the same permeability distribution as in the pre-vious case. The values of the Biot Modulus (M), Biot’s constant (α), Poisson Ratio (ν), fluidcompressibility (cf ), and initial porosity (ϕ0) are 10

8 psi , 0.9, 0.35, 10−4 (1/psi), and 0.2 respec-tively. We varied the value of the Young’s Modulus (E), and computed the numerical contractionestimates for first 12 simulation days in each case. Numerical contraction estimates are computedby taking the ratio of the quantity of contraction between two consecutive iterative coupling it-

erations

(∥∥∥δσn+1,m+kv ∥∥∥2/∥∥∥δσn,m+kv ∥∥∥2

). For each time step, the maximum value of this ratio across

all coupling iterations is considered. Theoretical contraction estimates are given by the expression(Mα2

2λ+2Mλcfϕ0+Mα2

)2. Results are shown in figure 5.3. We clearly see that theoretical estimates act

as upper bounds for numerically computed contraction estimates. Moreover, numerical contractionestimates are larger for early time steps, which is expected, as the coupled problem has not reacheda steady-state yet. We also recall that the improved contraction estimate for the single rate case

(derived in remark 3.5), is given by∥∥δσn+1,kv ∥∥2 ≤ ( CC+1)( Mα22λ+2Mλcfϕ0+Mα2)2 ∥∥δσn,kv ∥∥2 for a constant

C > 0, which is difficult to compute in practice. However, we anticipate that it scales monotoni-cally with the values of Lamé’s first parameter λ, and Young’s modulus E (when fixing the value

of the Poisson ratio). Therefore, theoretically, we expect the value of the damping factor(

CC+1

)to

approach one, as the value of Young’s modulus increases (assuming ν is fixed), which means thatour derived contraction estimates are sharper for larger Young’s modulus values. The results shownin figure 5.3 validate this theoretical discussion numerically. We see that as the value of the Young’smodulus increases, the gap between theoretical contraction estimates, and numerically computedvalues shrinks. For E = 106, theoretical estimates and numerical ones are both of the order of 10−4.

6 Conclusions

This paper considers the contracting behavior of the single rate and multirate fixed-stress splititerative coupling schemes both theoretically and numerically. In the single rate case, the flow andmechanics problems share the same time step, while in the multirate case, the flow takes multiplefine time steps within each coarse mechanics time step. A priori error estimates for the single ratefixed-stress split iterative coupling scheme are derived in this work. The novelty of the approachused in deriving our error estimates lies in its the ability to utilize previously established results forthe simultaneously coupled scheme. We anticipate that this approach can be used to derive errorestimates for other contractive iterative schemes, including the undrained-split coupling scheme.

26

Figure 5.3: Numerical contraction estimates for different values of Young’s modulus (E) for thesingle rate scheme for the first 12 simulation days. As the value of Young’s modulus increases, the

gap between theoretically predicted contraction estimates and numerically observed valuesshrinks, validating our theoretical derivations shown in remarks 3.5 and 3.6.

0 2 4 6 8 10 12Simulation Period (days)

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

Num

eric

al C

ontr

actio

n Es

timat

es

Numerical Contraction Estimates For Different Values of Young's ModulusYoung's Modulus (E) = 1.E+4, Theoretical. Est. = 0.4921Young's Modulus (E) = 5.E+4, Theoretical. Est. = 0.1018Young's Modulus (E) = 1.E+5, Theoretical. Est. = 0.0360Young's Modulus (E) = 5.E+5, Theoretical. Est. = 0.0020Young's Modulus (E) = 1.E+6, Theoretical. Est. = 5.2E-4

Our numerical results highlight the ability of the multirate scheme in reducing the number ofmechanics linear iterations, and in turn, the CPU run time, efficiently while maintaining the samelevel of accuracy compared to the results obtained by the single rate scheme. In addition, subjectto the value of the damping factor obtained in remarks 3.5, and 3.6, by comparing our theoreticalcontraction estimates against numerical computations, we conclude that the theoretical estimatescan predict the contracting behavior, and subsequently, the rate of convergence of the correspondingiterative scheme with high accuracy.

Acknowledgements

TA is funded by Saudi Aramco. We thank Paulo Zunino and Ivan Yotov for helpful discussions. KKwould like to acknowledge the support of StatOil Akademia Grant (Bergen). The authors would like

27

to acknowledge the CSM Industrial Affiliates program, DOE grant ER25617, and ConocoPhillipsgrant UTA10-000444. Moreover, we thank Gurpreet Singh for his tremendous help and supportwith IPARS.

References

[1] D. R. Allen. Physical changes of reservoir properties caused by subsidence and repressurizingoperations. Society of Petroleum Engineers, 1968. SPE 1811.

[2] T. Almani, A. H. Dogru, K. Kumar, G. Singh, and M. F. Wheeler. Convergence of multirateiterative coupling of geomechanics with flow in a poroelastic medium. Saudi Aramco Journalof Technology, Spring 2016, 2016.

[3] T. Almani, K. Kumar, A. H. Dogru, G. Singh, and M. F. Wheeler. Convergence analysis ofmultirate fixed-stress split iterative schemes for coupling flow with geomechanics. Ices report 16-07, Institute for Computational Engineering and Sciences, The University of Texas at Austin,Austin, Texas, 2016.

[4] T. Almani, K. Kumar, G. Singh, and M. F. Wheeler. Stability of multirate explicit coupled ofgeomechanics with flow in a poroelastic medium. Ices report 16-12, Institute for ComputationalEngineering and Sciences, The University of Texas at Austin, Austin, Texas, 2016.

[5] T. Almani, K. Kumar, and M. F. Wheeler. Multirate undrained splitting for coupled flow andgeomechanics in porous media. Ices report 16-13, Institute for Computational Engineering andSciences, The University of Texas at Austin, Austin, Texas, 2016.

[6] M. A. Biot. Consolidation settlement under a rectangular load distribution. J. Appl. Phys.,12(5):426–430, 1941.

[7] M. A. Biot. General theory of three-dimensional consolidation. J. Appl. Phys., 12(2):155–164,1941.

[8] L. Y. Chin, L. K. Thomas, J. E. Sylte, and R. G. Pierson. Iterative coupled analysis ofgeomechanics and fluid flow for rock compaction in reservoir simulation. Oil and Gas Scienceand Technology, 57(5):485–497, 2002.

[9] O. Coussy. A general theory of thermoporoelastoplasticity for saturated porous materials.Transport in Porous Media, 4:281–293, June 1989.

[10] O. Coussy. Mechanics of Porous Continua. Wiley, West Sussex PO19 1UD, England, 1995.

28

[11] A. Ern and S. Meunier. A posteriori error analysis of euler-galerkin approximations to cou-pled elliptic-parabolic problems. ESAIM: Mathematical Modelling and Numerical Analysis,43(2):353–375, 12 2008.

[12] X. Gai, R. H. Dean, M. F. Wheeler, and R. Liu. Coupled geomechanical and reservoir modelingon parallel computers. In The SPE Reservoir Simulation Symposium, Houston, Texas, 2003.

[13] F.J. Gaspar, F.J. Lisbona, and P.N. Vabishchevich. A finite difference analysis of biot’s con-solidation model. Applied Numerical Mathematics, 44(4):487 – 506, 2003.

[14] V. Girault, G Pencheva, M. F. Wheeler, and T. Wildey. Domain decomposition for poroelas-ticity and elasticity with dg jumps and mortars. Mathematical Models and Methods in AppliedSciences, 21(1):169–213, 2011.

[15] V. Girault, M. F. Wheeler, B. Ganis, and M. Mear. A lubrication fracture model in a poro-elastic medium. Technical report, The Institute for Computational Engineering and Sciences,The University of Texas at Austin, 2013.

[16] Mika Juntunen and Mary Wheeler. Two-phase flow in complicated geometries - modeling thefrio data using improved computational meshes. Computational Geosciences, 17:239–247, 2013.

[17] J. Kim, H. A. Tchelepi, and R. Juanes. Stability, accuracy, and efficiency of sequential methodsfor coupled flow and geomechanics. In The SPE Reservoir Simulation Symposium, Houston,Texas, 2009. SPE119084.

[18] J. Kim, H. A. Tchelepi, and R. Juanes. Stability and convergence of sequential methods forcoupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl.Mech. Engrg., 200(13-16):1591–1606, 2011.

[19] K. Kumar, T. Almani, G. Singh, and M. F. Wheeler. Multirate undrained splitting for coupledflow and geomechanics in porous media. In ENUMATH 2015 Proceedings. European Conferenceon Numerical Mathematics and Advanced Applications, 2015. submitted.

[20] J. J. Lee. Robust error analysis of coupled mixed methods for biot’s consolidation model.Journal of Scientific Computing, pages 1–23, 2016.

[21] M. Mainguy and P. Longuemare. Coupling fluid flow and rock mechanics: formulations ofthe partial coupling between reservoir and geomechanics simulators. Oil and Gas Science andTechnology - Rev. IFP, 57(4):355–367, 2002.

[22] A. Mikelić, B. Wang, and M. F. Wheeler. Numerical convergence study of iterative couplingfor coupled flow and geomechanics. Computational Geosciences, 18:325–341, 2014.

29

[23] A. Mikelić and M. F. Wheeler. Convergence of iterative coupling for coupled flow and geome-chanics. Computational Geosciences, 17:455–461, 2013.

[24] M. A. Murad, V. Thomée, and A. F. D. Loula. Asymptotic behavior of semidiscrete finite-element approximations of biot?s consolidation problem. SIAM Journal of Numerical Analysis,33:1065–1083, 1996.

[25] P. J. Phillips and M. F. Wheeler. A coupling of mixed and continuous Galerkin finite elementmethods for poroelasticity. I. The continuous in time case. Comput. Geosci., 11(2):131–144,2007.

[26] Phillip Joseph Phillips and Mary F. Wheeler. A coupling of mixed and continuous galerkin finiteelement methods for poroelasticity i: the continuous in time case. Computational Geosciences,11(2):131–144, 2007.

[27] Phillip Joseph Phillips and Mary F. Wheeler. A coupling of mixed and continuous galerkin finiteelement methods for poroelasticity ii: the discrete-in-time case. Computational Geosciences,11(2):145–158, 2007.

[28] C. Rodrigo, F. J. Gaspar, X. Hu, and L. T. Zikatanov. Stability and monotonicity for somediscretizations of the biot?s consolidation model. Computer Methods in Applied Mechanics andEngineering, 298:183 – 204, 2016.

[29] I. Ruddy, Mark A. Andersen, P.D. Pattillo, M. Bishlawl, and N. Foged. Rock compressibility,compaction, and subsidence in a high-porosity chalk reservoir: A case study of valhall field.Society of Petroleum Engineers, 1989. SPE 18278.

[30] A. Settari and F. Mourits. A coupled reservoir and geomechanical simulation system. SPEJournal, 1998.

[31] A. Settari and F. M. Mourits. Coupling of geomechanics and reservoir simulation models. InSiriwardane and Zema, editors, Comp. Methods and Advances in Geomech., pages 2151–2158,Balkema, Rotterdam, 1994.

[32] R. E. Showalter. Diffusion in poro-elastic media. J. Math. Anal. Appl., 251(1):310–340, 2000.

[33] K. von Terzaghi. Theoretical Soil Mechanics. Wiley, New York, 1943.

[34] Jing Wan. Stabilized Finite Element Methods for Coupled Geomechanics and Multiphase Flow.PhD thesis, Stanford University, Stanford, California, 2003.

[35] M. F. Wheeler, G. Xue, and I. Yotov. A family of multipoint flux mixed finite element methodsfor elliptic problems on general grids. In Procedia Computer Science, volume 4, pages 918–927.International Conference on Computational Science, ICCS 2011, 2011.

30

[36] M. F. Wheeler, G. Xue, and I. Yotov. Coupling multipoint flux mixed finite element meth-odswith continuous galerkin methods for poroelasticity. Computational Geosciences, 18(1):57–75, 2014.

[37] M. F. Wheeler and I. Yotov. A multipoint flux mixed finite element method. SIAM Journalof Numerical Analysis, 44:2082–2106, 2006.

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