ICES REPORT 17-23

September 2017

Convergence Analysis of Single Rate and Multirate FixedStress Split Iterative Coupling Schemes in Heterogeneous

Poroelastic Mediaby

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 Analysis of Single Rate andMultirate Fixed Stress Split Iterative Coupling Schemes in Heterogeneous Poroelastic Media," ICES REPORT17-23, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, September2017.

Convergence Analysis of Single Rate and Multirate Fixed

Stress Split Iterative Coupling Schemes in Heterogeneous

Poroelastic Media

T. Almani1,3, K. Kumar2, M. F. Wheeler1

1 Center for Subsurface Modeling, ICES, UT Austin, USA2 Mathematics Institute, University of Bergen, Norway

3 Saudi Arabian Oil Company, Saudi Arabiaalmanitm@utexas.edu, kundan.kumar@uib.no, mfw@ices.utexas.edu

September 12, 2017

Abstract

Recently, the accurate modeling of flow-structure interactions has gained more attentionand importance for both petroleum and environmental engineering applications. Of partic-ular interest is the coupling between subsurface flow and reservoir geomechanics. Differentsingle rate and multirate iterative and explicit coupling schemes have been proposed andanalyzed in the past. Extending the work of Mikelic and Wheeler [28], Banach contractionresults were obtained for iterative schemes, while explicit schemes were only shown to beconditionally stable. However, all previously established results consider spatially homoge-neous poroelastic media. In this work, we try to bridge this missing gap, and consider themathematical analysis of iterative coupling schemes for spatially heterogeneous poroelas-tic media. We will re-establish the contractivity of the single rate and multirate iterativecoupling schemes in the localized case. However, heterogeneities come at the expense ofimposing more restricted assumptions for the multirate iterative coupling scheme. Ourmathematical analysis will be supplemented by numerical simulations. To the best of ourknowledge, this is the first rigorous mathematical analysis of the fixed-stress split iterativecoupling scheme in heterogeneous poroelastic media.

Keywords. poroelasticity; fixed-stress split iterative coupling; heterogeneous poroelasticmedia, contraction mapping

1 Introduction

Currently, the coupling between subsurface flow and reservoir geomechanics is an active areaof research. In fact, a clear understanding of the fluid flow and the solid-phase mechanicalresponse is needed for the accurate modeling of multiscale and multiphysics phenomena

1

such as reservoir deformation, surface subsidence, well stability, sand production, wastedeposition, pore collapse, fault activation, hydraulic fracturing, CO2 sequestration, andhydrocarbon recovery [26], [27]. Traditionally, the main purpose of performing reservoirsimulation was to obtain accurate results for reservoir flow, simplifying the influence ofporous media deformations by a constant rock compressibility factor. In fact, such an influ-ence affects pore pressure which, in turn, affects the accuracy of reservoir flow models [27].By oversimplifying the rock compressibility coefficient with a constant rock compressibil-ity term, the solid phase stress and strain can never be accounted for. This poses severalconcerns on the accuracy of flow models in stress-sensitive and naturally fractured reser-voirs [27]. Therefore, it is only through the accurate coupling between subsurface flow andreservoir geomechanics that accurate and trusted results can be deduced from flow modelsin such types of reservoirs.

The coupled flow and geomechanics problem has been heavily investigated in the past. Theseed of this work can be tracked down to the work of Terzaghi [33] and Biot [10,11]. Terza-ghi was the first to propose an explanation of the soil consolidation process. He analyzedthe settlement of a column of soil under a constant load which is prevented from lateralexpansion. The success of Terzagh’s theory in predicting the settlement of different types ofsoils led to the creation of the science of soil mechanics [11]. More details about Terzaghi’stheory of consolidation can be found in [33]. Terzagh’s one dimensional work was thenextended by Biot to the three-dimensional case [11]. In subsequent work, Biot presented amore rigorous generalized theory of consolidation and continued to develop the theory ofelasticity and consolidation for isotropic and anisotropic porous media, including the theoryof deformation of porous viscoelastic anisotropic solids [9, 12,13]. A treatment of thermoe-lasticity and the mechanics of deformation and acoustic propagation in porous media canbe found in [14,15]. Several studies and interpretations based on Biot’s consolidation theorycan be found in [25,29]. To name just few, Geertsma [25] utilized Biot theory to present aunified treatment of rock mechanics problems in the field of petroleum production engineer-ing. Rice and Cleary [29] considered applications of the Biot linearized quasi-static elasticitytheory of fluid-saturated porous media. Coussy [19] presented the general theory of ther-moporoelastoplasiticy for saturated materials. A comprehensive treatment of the theory ofmechanics of porous continua and poromechanics can be found in [20,21] by Coussy. Othernonlinear extensions of the theory of poroelasticity can be found in [17,18,22,24,30,32].

There are three major approaches of coupling fluid flow with reservoir mechanics, known asthe fully implicit, the explicit, and the iterative coupling schemes. The fully implicit schemesolves the two problems simultaneously and a preconditioning technique can be employedto decouple the two problems at the linear solver level [16, 23]. In contrast, the explicitcoupling scheme decouples the two problems, and solves them in a sequential manner [1,5].The iterative coupling scheme lies in between these two approaches, decouples the two prob-lems, and imposes an iteration between the two until convergence is obtained [2–4,6–8]. Itshould be noted here that two main iterative coupling schemes, the fixed stress split and theundrained split schemes, were shown to be Banach contractive even for different time scalesfor the two coupled problems [1]. In contrast, explicit coupling schemes were only shown tobe conditionally stable [1,5]. However, the analysis for both schemes assumes homogeneous

2

flow and mechanics parameters for the whole domain of consideration. Although this isa nice theoretical assumption, it is not realistically true. In this paper, we try to bridgethis gap, and consider the analysis of iterative coupling schemes in spatially heterogeneousporo-elastic media.

Extending the work of [3] to include heterogeneities in the poroelastic parameters, we willestablish fixed point Banach contraction for both the single rate and multirate fixed stressiterative coupling schemes in heterogeneous poroelastic media. Figures 1.1a and 1.1b showthe difference between the single rate and multirate coupling schemes. In the single ratecase, the flow and mechanics problems share the same time step, while in the multiratescheme the flow takes multiple finer time steps within one coarse mechanics time step. Itshould be noted here that for the multirate scheme, heterogeneities in the poroelastic pa-rameters come at the expense of imposing an upper bound on the number of flow fine timesteps solved within one coarse mechanics time step. In addition, our localized proof outlinesa general strategy that is very likely to be useful for obtaining similar localized estimatesfor other iterative and explicit coupling schemes.

The paper is structured as follows. Model equations and associated discretizations arepresented in Section 2. Sections 3 and 4 present the formulations and analyses for the local-ized single rate and multirate iterative coupling schemes respectively. Section 4 comparesmultirate Banach contraction results established for homogeneous versus heterogeneousporoelastic media. Numerical results for a realistic reservoir model, with a heterogeneouspermeability distribution, are shown in Section 6. Conclusions and outlook are discussedin Section 7.

1.1 Preliminaries

Let Ω be an open, connected, and bounded domain of IRd, where the dimension d = 2or 3, with a Lipschitz continuous boundary ∂Ω. For the pressure unknown, we assumethat the boundary is decomposed into Dirichlet boundary ΓD, and Neumann boundaryΓN , associated with Dirichlet and Neumann boundary conditions respectively, such thatΓD ∪ ΓN = ∂Ω. In addition, Let D(Ω) be the space of all functions that are infinitelydifferentiable and with compact support in Ω, and let D′(Ω) be its dual space, i.e. thespace of distributions in Ω. As usual, we denote by H1(Ω) the classical Sobolev space

H1(Ω) = v ∈ L2(Ω) ; ∇ v ∈ L2(Ω)d,

equipped with the semi-norm and norm:

|v|H1(Ω) = ‖∇ v‖L2(Ω)d , ‖v‖H1(Ω) =(‖v‖2L2(Ω) + |v|2H1(Ω)

)1/2.

More generally, for 1 ≤ p <∞, W 1,p(Ω) is the space

W 1,p(Ω) = v ∈ Lp(Ω) ; ∇ v ∈ Lp(Ω)d,

normed by

|v|W 1,p(Ω) = ‖∇ v‖Lp(Ω)d , ‖v‖W 1,p(Ω) =(‖v‖pLp(Ω) + |v|p

W 1,p(Ω)

)1/p,

3

tflow(tf ), tmech(tm) = 0(initial time = 0)

k = 0

n = 0 (iterativecoupling index)

Fluid Flow: tflow = tflow + ∆tCompute pore pressure:

pn+1,k+1

Mechanics (Biot Model):tmech = tmech + ∆tCompute displace-

ment, un+1,k+1

Update pore volume

Converged? k = k + 1tf = tf − ∆ttm = tm − ∆tn = n + 1

No Yes

(a) Single Rate

tflow(tf ), tmech(tm) = 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∆tCompute displace-

ment, un+1,k+q

Update pore volume

Converged? k = k + qtf = tf − q∆ttm = tm − q∆tn = n + 1

No

Yes

No Yes

(b) Multirate

Figure 1.1: Flowchart for the iterative coupling algorithm using single rate and multiratetime stepping for coupled geomechanics and flow problems

with the standard modification for the case when p =∞. We also define:

H10 (Ω) = v ∈ H1(Ω) ; v|∂Ω = 0,

and for the divergence operator, we shall use the spaces

H(div; Ω)d = v ∈ L2(Ω)d ; ∇ · v ∈ L2(Ω),

andH0(div; Ω)d = v ∈ H(div; Ω)d ; v|∂Ω = 0,

4

equipped with the norm

‖v‖H(div;Ω)d =(‖v‖2L2(Ω)d + ‖∇ · v‖2L2(Ω)

)1/2.

We recall the definition of the symmetric strain tensor: ε(v) = 12(∇v+ (∇v)T ), for a vector

v in IRd. For completeness, we list below two useful inequalities that will be used in thischapter:

• Poincare’s inequality in H10 (Ω):

There exists a constant PΩ depending only on Ω such that

∀v ∈ H10 (Ω) , ‖v‖L2(Ω) ≤ PΩ|v|H1(Ω). (1.1)

• Korn’s first inequality in H10 (Ω)d:

There exists a constant Cκ depending only on Ω such that

∀v ∈ H10 (Ω)d , |v|H1(Ω)d ≤ Cκ‖ε(v)‖L2(Ω)d×d . (1.2)

2 Model Equations and Discretization

We assume a linear and elastic porous medium Ω ⊂ Rd, d = 2 or 3, in which the reservoiris saturated with a slightly compressible fluid. We start by describing the geomechanicsmodel, followed by the flow model.

2.1 Geomechanics Model

Using a quasi-static (i.e. ignoring the second order time derivative for the displacement)Biot approach to obtain the displacements (see [11]), the “geomechanics” model is as follows:

σpor(u, p) = σ(u)− αp I, (2.1)

σ(u) = λ(∇ · u)I + 2Gε(u), (2.2)

− divσpor(u, p) = f in Ω, (2.3)

where σpor is the Cauchy stress tensor, I is the identity tensor, u is the solid’s displacement,p is the fluid pressure, α > 0 is the dimensionless Biot coefficient, σ is the effective linearelastic stress tensor, λ > 0 and G > 0 are the Lame constants, f is a body force, which isusually assumed to be a gravity loading term. The last equation represents the balance oflinear momentum in the solid.

2.2 Single Phase Flow Model

Following a slightly different formulation compared to the one described in [26], we assumea linearized slightly compressible single-phase flow model for the fluid in the reservoir. Aslisted in the assumptions above, we also assume that K, the absolute permeability tensor,is bounded, symmetric, and uniformly positive definite in space and constant in time (fordiscrete time intervals). The fluid density, ρf is assumed to be a linear function of pressure:

5

ρf = ρf,r(1 + cf (p − pr)

). The porosity, or the fluid content of the medium, denoted by

ϕ∗ is related to the “mechanical” displacement and “fluid” pressure by this relation: ϕ∗ =ϕ0+α∇·u+ 1

M p, where ϕ0 is the initial porosity, and M is the Biot constant. The fluid mass

balance in the reservoir, denoted by Ω, reads: ∂∂t

(ρfϕ

∗)+∇·(ρfvD) = qs, where qs is a masssource or sink term, and vD is the velocity of the fluid in Ω, vD = − 1

µfK(∇ p − ρfg∇ η

).

Substituting the definitions of vD, ρf , and ϕ∗ into the mass balance equation, we get:

∂

∂t

(ρf,r(1 + cf (p− pr))

(ϕ0 + α∇ · u+

1

Mp))

+∇ ·(ρf,r(1 + cf (p− pr))vD

)= qs.

which can be written as (after re-arranging terms):

ρf,r

( 1

M(1 + cf (p− pr)) + cf

(ϕ0 + α∇ · u+

1

Mp)) ∂∂tp+ ρf,rα(1 + cf (p− pr))∇ ·

∂

∂tu

+∇ ·(ρf,r(1 + cf (p− pr))vD

)= qs.

For the sake of linearization, we assume that the fluid compressibility cf is small, in theorder of 10−5 or 10−6, and the term cf (p− pr) is also small as well (of the same order). Wemake the following approximations: 1

M (1 + cf (p− pr)) ≈ 1M , cf

(ϕ0 +α∇·u+ 1

M p)≈ cfϕ0,

ρf,r(1 + cf (p − pr))α ≈ ρf,rα, ρf,r(1 + cf (p − pr))vD ≈ ρf,rvD, ρf,r(1 + cf (p − pr))g∇ η ≈

ρf,rg∇ η. With such approximations, the mass balance equation now reads:

ρf,r( 1

M+ cfϕ0

) ∂∂tp+ ρf,rα∇ ·

∂

∂tu+ ρf,r∇ · vD = qs

which can be written as (after dividing by ρf,r, and submitting the expressiotn of vD):

∂

∂t

(( 1

M+ cfϕ0

)p+ α∇ · u

)−∇ ·

( 1

µfK(∇ p− ρf,rg∇ η

))= q. (2.4)

where q = qsρf,r

. This completes the derivation of the poro-elastic equations, modeling the

displacement u and pressure p in Ω.

Therefore, our quasi-static Biot model, which is quite standard in literature [11,26], reads:Find u and p satisfying the equations below for all time t ∈]0, T [:

−divσpor(u, p) = f in Ω,

σpor(u, p) = σ(u)− αp I in Ω,

σ(u) = λ(∇ · u)I + 2Gε(u) in Ω,

∂∂t

((1M

+ cfϕ0

)p+ α∇ · u

)−∇ ·

(1µfK

(∇ p− ρf,rg∇ η

))= q in Ω,

Boundary Conditions: u = 0 on ∂Ω, K(∇ p− ρf,rg∇ η) · n = 0 on ΓN , p = 0 on ΓD,

Initial Condition (t = 0) :((

1M

+ cfϕ0

)p+ α∇ · u

)(0) =

(1M

+ cfϕ0

)p0 + α∇ · u0.

where: g is the gravitational constant, η is the distance in the vertical direction (assumedto be constant in time), ρf,r > 0 is a constant reference density (relative to the referencepressure pr), ϕ0 is the initial porosity, M is the Biot constant, q = qs

ρf,rwhere qs is a mass

source or sink term taking into account injection into or out of the reservoir. We remarkthat the first three equations describe the mechanics whereas the fourth one is the flowequation. Note that the above system is linear and coupled.

6

2.3 Mixed Variational Formulation

A mixed formulation will be used for the flow equations and conformal Galerkin will be usedfor the mechanics equation. In the mixed method, the flux is defined as a separate unknown,and the flow equation is rewritten as a system of first order equations. This formulation isa standard one for flow equations as it is locally mass conservative and computes the fluxexplicitly. For time discretization, we will assume a backward-Euler scheme (for both thecontinuous and discrete in space formulations).

Accordingly, for the fully discrete formulation (discrete in time and space), let Th denotea regular family of conforming triangular elements of the domain of interest, Ω. Using thelowest order Raviart-Thomas (RT) spaces , we have the following discrete spaces (V h fordiscrete displacements, Qh for discrete pressures, and Zh for discrete velocities (fluxes)):

V h = vh ∈ H1(Ω)d

; ∀T ∈ Th,vh|T ∈ P1d,vh|∂Ω = 0 (2.5)

Qh = ph ∈ L2(Ω) ; ∀T ∈ Th, ph|T ∈ P0 (2.6)

Zh = qh ∈ H(div; Ω) ;∀T ∈ Th, qh|T ∈ P1d, qh · n = 0 on ΓN (2.7)

The space of displacements, V h, is equipped with the norm:

‖v‖Vh =( d∑i=1

‖vi‖2H1(Ω)

)1/2.

We also assume that the finer time step is given by: ∆tk = tk − tk−1. In this work, weassume uniform fine flow time steps, so for simplicity, we will drop the subscript k, anddenote the fine time step by ∆t. If we denote the total number of timesteps by N, then thetotal simulation time is given by T = ∆t N, and ti = i∆t, 0 6 i 6 N denote the discretetime points.

For the fully discrete scheme, we have chosen the Raviart-Thomas spaces for the mixedfinite element discretization. However, the proof extends to other choices for the mixedspaces (e.g. the Multipoint Flux Mixed Finite Element (MFMFE) spaces [34,35]).

Remark 2.1 Notation: Two indices will be used in this paper, one for the time step andthe other for the coupling between the mechanics and flow. The following notations will beemployed, n denotes the coupling iteration index, k denotes the coarser (mechanics) timestep iteration index, m denotes the finer (flow) time step iteration index, ∆t stands for thetime step, and q is the “fixed” number of local flow time steps per coarse mechanics timestep. A schematic showing the relations between k,m, q, and ∆t can be found in figure 1.1b.In addition, for a given time step t = tk, we define the difference between two couplingiterates as:

δξn+1,k = ξn+1,k − ξn,k,

where ξ may stand for ph, zh, or uh (the discrete pressure, flux, and displacements vari-ables).

7

2.4 Assumptions

We have the following assumptions on the model and data:

1. For mechanical modeling, the reservoir is assumed to be heterogeneous, isotropic andsaturated poro-elastic medium. The reference density of the fluid ρf > 0 is given andpositive.

2. The Lame coefficients λ > 0 and G > 0, the dimensionless Biot coefficient α, and thepore volume ϕ∗ are all positive.

3. The fluid is assumed to be slightly compressible and its density is a linear function ofpressure. The viscosity µf > 0 is assumed to be constant.

4. The absolute permeability tensor,K, is assumed to be symmetric, bounded, uniformlypositive definite in space and constant in time.

5. The parameters K, α,G,M, λ, cf , µf and ϕ0 can vary in space and time.

6. For the fully discrete formulation, we have the following additional assumptions:

(a) The spatial domain is denoted by Ω ⊂ Rd, d = 1, 2, or 3. Its external boundaryis denoted by ∂Ω, with an outward unit normal vector n.

(b) The spatial domain is discretized into NΩ conforming grid elements Ei such that:

Ω =NΩ⋃i=1

Ei.

(c) Each grid element Ei has its own, independent, set of flow and mechanics param-eters: Ki, αi, Gi,Mi, λi, cf i, µf i and ϕ0i. Moreover, we assume that the localized

permeabilities Ki include viscosities µf i (i.e. Ki = Kiµf i

).

(d) The outward normal vector for each grid element Ei is denoted by ni. In addition,for two adjacent grid elements Ei and Ei−1 sharing a common boundary interface,ni = −ni−1 across the common boundary.

3 Localized Single Rate Formulation and Analysis

3.1 Continuous in Space Global Weak Formulation

The continuous in space global weak formulation of the coupled problem reads:

8

Step (a): Find pn+1,k ∈ H1(Ω), zn+1,k ∈ H(div; Ω)d ∩ zn+1,k · n = 0 on ∂Ω such that:

∀θ ∈ L2(Ω) ,

NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)(pn+1,k − pk−1

∆t

), θ)Ei

+

NΩ∑i=1

(∇ · zn+1,k, θ

)Ei

=

NΩ∑i=1

(Li(pn,k − pk−1

∆t

)− αi∇ ·

(un,k − uk−1

∆t

), θ)Ei

+

NΩ∑i=1

(q, θ)Ei

(3.1)

∀q ∈ H(div; Ω)d ∩ q · n = 0 on ∂Ω ,NΩ∑i=1

(K−1

i zn+1,k, q

)Ei

=

NΩ∑i=1

(pn+1,k,∇ · q)Ei −NΩ∑i=1

〈pn+1,k, q · n〉∂Ei +

NΩ∑i=1

(∇(ρf,rgη), q)Ei

(3.2)

Step (b): Given pn+1,k, zn+1,k, find un+1,k ∈ H10 (Ω)

dsuch that,

∀v ∈ H10 (Ω)

d,

NΩ∑i=1

2(Giε(u

n+1,k), ε(v))Ei

+

NΩ∑i=1

(λi∇ · un+1,k,∇ · v

)Ei

−NΩ∑i=1

(αip

n+1,k,∇ · v)Ei−

NΩ∑i=1

〈σ(un+1,k)n,v〉∂Ei +

NΩ∑i=1

〈αipn+1,k I=n,v〉∂Ei =

NΩ∑i=1

(f ,v

)Ei

(3.3)

We note that at the continuum level, the Cauchy stress tensor, given by σpor(u, p) =σ(u)− αp I

=, is continuous at grid boundaries. Thus, the boundary terms in equation (3.3)

can be grouped as:

−NΩ∑i=1

〈σ(un+1,k)n,v〉∂Ei +

NΩ∑i=1

〈αipn+1,k I=n,v〉∂Ei = −

NΩ∑i=1

〈σpor(un+1,k)n,v〉∂Ei = 0

due to the continuity of σpor at grid boundaries and the fact that the normal vector hasa different sign in each two adjacent grid elements sharing a common boundary. For theouter boundary, we require that v = 0 on ∂Ω.

The boundary term in the flux equation (3.2) also vanishes due to similar reasons. Thepressure unknown is assumed to be continuous at the continuum level (otherwize ∇p is notdefined). In addition, q · n is continuous across element boundaries, as q ∈ H(div; Ω)d.This results in cancelling all inner boundary terms in equation (3.2). For outer boundaryterms, we restricted the test space such that q · n = 0 on ∂Ω. Therefore, we have:

NΩ∑i=1

〈pn+1,k, q · n〉∂Ei = 0.

The weak formulation now reads:

9

Step (a): Find pn+1,k ∈ H1(Ω), zn+1,k ∈ H(div; Ω)d ∩ zn+1,k · n = 0 on ∂Ω such that:

∀θ ∈ L2(Ω) ,

NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)(pn+1,k − pk−1

∆t

), θ)Ei

+

NΩ∑i=1

(∇ · zn+1,k, θ

)Ei

=

NΩ∑i=1

(Li(pn,k − pk−1

∆t

)− αi∇ ·

(un,k − uk−1

∆t

), θ)Ei

+

NΩ∑i=1

(q, θ)Ei

(3.4)

∀q ∈ H(div; Ω)d ∩ q · n = 0 on ∂Ω ,NΩ∑i=1

(K−1

i zn+1,k, q

)Ei

=

NΩ∑i=1

(pn+1,k,∇ · q)Ei +

NΩ∑i=1

(∇(ρf,rgη), q)Ei (3.5)

Step (b): Given pn+1,k, zn+1,k, find un+1,k ∈ H10 (Ω)

dsuch that,

∀v ∈ H10 (Ω)

d,

NΩ∑i=1

2(Giε(u

n+1,k), ε(v))Ei

+

NΩ∑i=1

(λi∇ · un+1,k,∇ · v

)Ei−

NΩ∑i=1

(αip

n+1,k,∇ · v)Ei

=

NΩ∑i=1

(f ,v

)Ei

(3.6)

3.2 Fully Discrete Weak formulation

Now, we mimic the spatially continuous weak formulation ((3.4), (3.5), and (3.6)) to obtainthe fully discrete formulation (discrete in time and space). We recall that a mixed formu-lation will be used for flow, and continuous Galerkin will be used for mechanics. Moreover,we assume no flow boundary conditions for the outer flow boundary, and zero displacementboundary conditions for mechanics. The fully-discrete weak formulation now reads:

Step (a): Find pn+1,kh ∈ Qh, zn+1,k

h ∈ Zh such that:

∀θh ∈ Qh ,NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)(pn+1,kh − pk−1

h

∆t

), θh

)Ei

+

NΩ∑i=1

(∇ · zn+1,k

h , θh)Ei

=

NΩ∑i=1

(Li(pn,kh − pk−1

h

∆t

)− αi∇ ·

(un,kh − uk−1h

∆t

), θh

)Ei

+

NΩ∑i=1

(q, θh

)Ei

(3.7)

∀qh ∈ Zh ,

NΩ∑i=1

(K−1

i zn+1,kh , qh

)Ei

=

NΩ∑i=1

(pn+1,kh ,∇ · qh)Ei +

NΩ∑i=1

(∇(ρf,rgη), qh)Ei (3.8)

10

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

h , find un+1,kh ∈ V h such that,

∀vh ∈ V h ,

NΩ∑i=1

2(Giε(uh

n+1,k), ε(vh))Ei

+

NΩ∑i=1

(λi∇ · un+1,k

h ,∇ · vh)Ei

−NΩ∑i=1

(αip

n+1,kh ,∇ · vh

)Ei

=

NΩ∑i=1

(f ,vh

)Ei

(3.9)

In terms of differences between coupling iterations, equations (3.7), (3.8), and (3.9) read:

∀θh ∈ Qh ,1

∆t

NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)δpn+1,kh , θh

)Ei

+

NΩ∑i=1

(∇ · δzn+1,k

h , θh)Ei

=1

∆t

NΩ∑i=1

(Liδp

n,kh − αi∇ · δun,kh , θh

)Ei

(3.10)

∀qh ∈ Zh ,

NΩ∑i=1

(K−1

i δzn+1,kh , qh

)Ei

=

NΩ∑i=1

(δpn+1,kh ,∇ · qh)Ei (3.11)

∀vh ∈ V h ,

NΩ∑i=1

2(Giε(δuh

n+1,k), ε(vh))Ei

+

NΩ∑i=1

(λi∇ · δun+1,k

h ,∇ · vh)Ei

−NΩ∑i=1

(αiδp

n+1,kh ,∇ · vh

)Ei

= 0 (3.12)

3.3 Proof of Contraction

• Step 1: Flow equationsFor each grid element Ei, let βi = 1

Mi+ cfiϕ0i + Li, testing (3.10) with θh = δpn+1,k

h ,and multiplying by ∆t, we obtain:

NΩ∑i=1

∥∥∥β1/2i δpn+1,k

h

∥∥∥2

Ei+ ∆t

NΩ∑i=1

(∇ · δzn+1,k

h , δpn+1,kh )Ei =

NΩ∑i=1

(Liδp

n,kh − αi∇ · δun,kh , δpn+1,k

h

)Ei.

(3.13)

Testing (3.11) with qh = δzn+1,kh , we obtain:

NΩ∑i=1

(Ki−1δzn+1,k

h , δzn+1,kh

)Ei

=

NΩ∑i=1

(δpn+1,kh ,∇ · δzn+1,k

h

)Ei. (3.14)

Substituting (3.14) into (3.13), together with Young’s inequality, we obtain:

NΩ∑i=1

∥∥∥β1/2i δpn+1,k

h

∥∥∥2

Ei+ ∆t

NΩ∑i=1

(Ki−1δzn+1,k

h , δzn+1,kh

)Ei

≤NΩ∑i=1

1

2εi

∥∥∥Liδpn,kh − αi∇ · δun,kh∥∥∥2

Ei+

NΩ∑i=1

εi2

∥∥∥δpn+1,kh

∥∥∥2

Ei.

11

Introducing a new parameter χi for each grid element Ei, we define a local quantityof contraction for each Ei as: χiδσ

n,kv = Liδp

n,kh − αi∇ · δun,kh . The choice εi = βi for

each Ei gives:

NΩ∑i=1

βi2

∥∥∥δpn+1,kh

∥∥∥2

Ei+ ∆t

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k

h

∥∥∥2

Ei≤

NΩ∑i=1

1

2βi

∥∥∥χiδσn,kv ∥∥∥2

Ei. (3.15)

• Step 2: Elasticity equationNow, test the elasticity equation (3.12) with vh = δun+1,k

h to get:

NΩ∑i=1

2Gi∥∥ε(δun+1,k

h )∥∥2

Ei+

NΩ∑i=1

λi∥∥∇ · δun+1,k

h

∥∥2

Ei−

NΩ∑i=1

αi(δpn+1,kh ,∇ · δun+1,k

h

)Ei

= 0.

(3.16)

• Step 3: Combining flow and elasticity equationsCombining flow (3.15) with elasticity (3.16), we obtain:

NΩ∑i=1

2Gi∥∥ε(δun+1,k

h )∥∥2

Ei+ ∆t

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k

h

∥∥∥2

Ei

+

NΩ∑i=1

βi2

∥∥∥δpn+1,kh

∥∥∥2

Ei− αi

(δpn+1,kh ,∇ · δun+1,k

h

)Ei

+ λi∥∥∇ · δun+1,k

h

∥∥2

Ei

≤

NΩ∑i=1

χ2i

2βi

∥∥∥δσn,kv ∥∥∥2

Ei.

(3.17)

Now, for each grid element Ei, expand the RHS to match terms on the left hand sideand form a square:∥∥∥δσn,kv ∥∥∥2

Ei=L2i

χi2

∥∥∥δpn,kh ∥∥∥2

Ei− 2αiLi

χ2i

(δpn,kh ,∇ · δun,kh

)Ei

+α2i

χ2i

∥∥∥∇ · δun,kh ∥∥∥2

Ei.

For each Ei, the following inequalities should be satisfied: βi2 ≥

L2i

χi2, 2αiLi

χ2i

= αi,

and λi ≥α2i

χ2i. The first and second inequalities give: χ2

i = 2Li, and 1Mi

+cf iϕ0i ≥ 0,

which is trivially satisfied. The third inequality gives: Li =α2i

2λi. With: Li =

α2i

2λiand

χ2i = 2Li, we have:

NΩ∑i=1

2Gi∥∥ε(δun+1,k

h )∥∥2

Ei+

NΩ∑i=1

1

2

( 1

Mi+ cf iϕ0i

)∥∥∥δpn+1,kh

∥∥∥2

Ei+ ∆t

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k

h

∥∥∥2

Ei

+

NΩ∑i=1

∥∥∥δσn+1,kv

∥∥∥2

Ei≤

NΩ∑i=1

( Li1Mi

+ cf iϕ0i + Li

)∥∥∥δσn,kv ∥∥∥2

Ei.

(3.18)

12

We finally have, for each Ei ∈ Ω, 1 ≤ i ≤ NΩ:

2

NΩ∑i=1

Gi∥∥ε(δun+1,k

h )∥∥2

Ei+

1

2

NΩ∑i=1

( 1

Mi+ cf iϕ0i

)∥∥∥δpn+1,kh

∥∥∥2

Ei+ ∆t

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k

h

∥∥∥2

Ei

+

NΩ∑i=1

∥∥∥δσn+1,kv

∥∥∥2

Ei≤ max

1≤i≤NΩ

( Li1Mi

+ cf iϕ0i + Li

) NΩ∑i=1

∥∥∥δσn,kv ∥∥∥2

Ei.

(3.19)

Theorem 3.1 [Localized Single Rate Banach Contraction Estimate] The localized multirateiterative scheme is a contraction given by

2∑NΩ

i=1Gi∥∥ε(δun+1,k

h )∥∥2

Ei+ 1

2

∑NΩi=1

(1Mi

+ cf iϕ0i

)∥∥∥δpn+1,kh

∥∥∥2

Ei+ ∆t

∑NΩi=1

∥∥∥Ki−1/2δzn+1,k

h

∥∥∥2

Ei

+∑NΩ

i=1

∥∥∥δσn+1,kv

∥∥∥2

Ei≤ max

1≤i≤NΩ

(Li

1Mi

+cf iϕ0i+Li

)∑NΩi=1

∥∥∥δσn,kv ∥∥∥2

Ei.

4 Localized Multirate Formulation and Analysis

In a similar way, we can derive a localized Banach contraction estimate for the multiratecase. We start by writing the localized spatially continuous multirate weak formulation.We note that the localized permeability tensor Ki includes the viscosity µi.

4.1 Continuous in Space Global Weak Formulation

• Step (a): For 1 ≤ m ≤ q, find pn+1,m+k ∈ H1(Ω), and zn+1,m+k ∈ H(div; Ω)d ∩zn+1,k · n = 0 on ∂Ω such that,

∀θ ∈ L2(Ω) ,1

∆t

NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)(pn+1,m+k − pn+1,m−1+k

), θ)Ei

+

NΩ∑i=1

(∇ · zn+1,m+k, θ)Ei =

1

∆t

NΩ∑i=1

(Li

(pn,m+k − pn,m−1+k

)− αi

q∇ ·(un,k+q − un,k

), θ)Ei

+

NΩ∑i=1

(q, θ)Ei,

(4.1)

∀q ∈ H(div; Ω)d ∩ q · n = 0 on ∂Ω ,NΩ∑i=1

(Ki−1zn+1,m+k, q

)Ei

=

NΩ∑i=1

(pn+1,m+k,∇ · q

)Ei

−NΩ∑i=1

〈pn+1,m+k, q · n〉∂Ei +

NΩ∑i=1

(ρf,rg∇ η, q

)Ei, (4.2)

13

• Step (b): Given pn+1,k+q and, zn+1,k+q, find un+1,k+q ∈ H10 (Ω)

dsuch that,

∀v ∈ H10 (Ω)

d,

2

NΩ∑i=1

(Giε(u

n+1,k+q), ε(v))Ei

+

NΩ∑i=1

(λi∇ · un+1,k+q,∇ · v

)Ei−

NΩ∑i=1

(αip

n+1,k+q,∇ · v)Ei

−NΩ∑i=1

〈σ(un+1,k+q)n,v〉∂Ei +

NΩ∑i=1

〈αipn+1,k+q I=n,v〉∂Ei =

NΩ∑i=1

(f ,v

)Ei. (4.3)

In a similar way, as detailed in the single rate case, all boundary terms vanish. Thecontinuous-in-space weak formulation then reads:

• Step (a): For 1 ≤ m ≤ q, find pn+1,m+k ∈ H1(Ω), and zn+1,m+k ∈ H(div; Ω)d ∩zn+1,k · n = 0 on ∂Ω such that,

∀θ ∈ L2(Ω) ,1

∆t

NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)(pn+1,m+k − pn+1,m−1+k

), θ)Ei

+

NΩ∑i=1

(∇ · zn+1,m+k, θ)Ei =

1

∆t

NΩ∑i=1

(Li

(pn,m+k − pn,m−1+k

)− αi

q∇ ·(un,k+q − un,k

), θ)Ei

+

NΩ∑i=1

(q, θ)Ei,

(4.4)

∀q ∈ H(div; Ω)d ∩ q · n = 0 on ∂Ω ,NΩ∑i=1

(Ki−1zn+1,m+k, q

)Ei

=

NΩ∑i=1

(pn+1,m+k,∇ · q

)Ei

+

NΩ∑i=1

(ρf,rg∇ η, q

)Ei, (4.5)

• Step (b): Given pn+1,k+q and, zn+1,k+q, find un+1,k+q ∈ H10 (Ω)

dsuch that,

∀v ∈ H10 (Ω)

d,

2

NΩ∑i=1

(Giε(u

n+1,k+q), ε(v))Ei

+

NΩ∑i=1

(λi∇ · un+1,k+q,∇ · v

)Ei−

NΩ∑i=1

(αip

n+1,k+q,∇ · v)Ei

=

NΩ∑i=1

(f ,v

)Ei. (4.6)

4.2 Fully Discrete Weak formulation

We mimic the spatially continuous weak formulation ((4.4), (4.5), and (4.6)) and obtain thefully discrete formulation (discrete in time and space) as follows:

14

• Step (a): For 1 ≤ m ≤ q, find pn+1,m+kh ∈ Qh, and zn+1,m+k

h ∈ Zh such that,

∀θh ∈ Qh ,1

∆t

NΩ∑i=1

(( 1

Mi+ cf iϕ0i + Li

)(pn+1,m+kh − pn+1,m−1+k

h

), θh

)Ei

+

NΩ∑i=1

(∇ · zn+1,m+k

h , θh)Ei =

1

∆t

NΩ∑i=1

(Li

(pn,m+kh − pn,m−1+k

h

)− αi

q∇ ·(un,k+qh − un,kh

), θh

)Ei

+

NΩ∑i=1

(qh, θh

)Ei,

(4.7)

∀qh ∈ Zh ,

NΩ∑i=1

(Ki−1zn+1,m+k

h , qh

)Ei

=

NΩ∑i=1

(pn+1,m+kh ,∇ · qh

)Ei

+

NΩ∑i=1

(ρf,rg∇ η, qh

)Ei,

(4.8)

• Step (b): Given pn+1,k+qh and, zn+1,k+q

h , find un+1,k+qh ∈ V h such that,

∀vh ∈ V h , 2

NΩ∑i=1

(Giε(u

n+1,k+qh ), ε(vh)

)Ei

+

NΩ∑i=1

(λi∇ · un+1,k+q

h ,∇ · vh)Ei

−NΩ∑i=1

(αip

n+1,k+qh ,∇ · vh

)Ei

=

NΩ∑i=1

(fh,vh

)Ei. (4.9)

4.3 Proof of Contraction

• Step 1: Flow equationsFor each grid element Ei, let βi = 1

Mi+cf iϕ0i+Li. . For n ≥ 1, by taking the difference

of two successive iterates of (4.7), which corresponds to one local flow iteration andits corresponding local flow iteration in the previous flow and geomechanics iterativecoupling iteration, testing with θh = δpn+1,m+k

h − δpn+1,m−1+kh , we obtain

NΩ∑i=1

βi

∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei

+ ∆t

NΩ∑i=1

(∇ · δzn+1,m+k

h , δpn+1,m+kh − δpn+1,m−1+k

h )Ei =

NΩ∑i=1

(Li(δpn,m+kh − δpn,m−1+k

h

)− αi

q∇ · δun,k+q

h , δpn+1,m+kh − δpn+1,m−1+k

h

)Ei.

(4.10)

Similarly, for the flux equation (4.8), by taking the difference of two successive iterates,followed by taking the difference at two consecutive finer time steps, t = tm+k, and

15

t = tm−1+k, and testing with qh = δzn+1,m+kh , we obtain

NΩ∑i=1

(Ki−1(δzn+1,m+k

h − δzn+1,m−1+kh

), δzn+1,m+k

h

)Ei

=

NΩ∑i=1

(δpn+1,m+kh − δpn+1,m−1+k

h ,∇ · δzn+1,m+kh

)Ei. (4.11)

We combine (4.10) with (4.11), apply Young’s inequality (for each grid Ei) to obtain

NΩ∑i=1

βi

∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei

+ ∆t

NΩ∑i=1

(Ki−1(δzn+1,m+k

h − δzn+1,m−1+kh

), δzn+1,m+k

h

)Ei

≤NΩ∑i=1

1

2εi

∥∥∥Li(δpn,m+kh − δpn,m−1+k

h

)− αi

q∇ · δun,k+q

h

∥∥∥2

Ei

+

NΩ∑i=1

εi2

∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei.

For each Ei, the choice εi = βi absorbs the pressure term on the right hand side.Together with a simple expansion of the flux product, we derive

NΩ∑i=1

βi2

∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei

+∆t

2

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,m+k

h

∥∥∥2

Ei−∥∥∥Ki

−1/2δzn+1,m−1+kh

∥∥∥2

Ei

+∥∥∥Ki

−1/2(δzn+1,m+kh − δzn+1,m−1+k

h

)∥∥∥2

Ei

≤

NΩ∑i=1

1

2βi

∥∥∥Li(δpn,m+kh − δpn,m−1+k

h

)− αi

q∇ · δun,k+q

h

∥∥∥2

Ei. (4.12)

The right hand side constitutes an expression for a quantity to be contracted on.Introducing a new parameter χi for each Ei, we define the localized volumetric meanstress for (1 ≤ m ≤ q) as

χiδσn,m+kv = Li(δp

n,m+kh − δpn,m−1+k

h )− αiq∇ · δun,k+q

h . (4.13)

The value of χi for each Ei will be chosen such that contraction can be achieved on thespatial summation of the localized norms of σn,m+k

v , summed over q flow finer timesteps, within one coarser mechanics time step. Summing up (4.12) for 1 ≤ m ≤ q,

16

substituting the new definition of the localized volumetric mean stress (4.13), and

noting that δzn+1,kh = 0, we obtain

q∑m=1

NΩ∑i=1

βi2

∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei+

∆t

2

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k+q

h

∥∥∥2

Ei

+∆t

2

q∑m=1

NΩ∑i=1

∥∥∥Ki−1/2(δzn+1,m+k

h − δzn+1,m−1+kh

)∥∥∥2

Ei≤

q∑m=1

NΩ∑i=1

1

2βi

∥∥∥χiδσn,m+kv

∥∥∥2

Ei.

(4.14)

• Step 2: Elasticity equationFor n ≥ 1, we take the difference of successive iterates of the mechanics equation (4.9),

and test with vh = δun+1,k+qh to get

2

NΩ∑i=1

Gi∥∥ε(δun+1,k+q

h )∥∥2

Ei+

NΩ∑i=1

λi∥∥∇ · δun+1,k+q

h

∥∥2

Ei

−NΩ∑i=1

αi(δpn+1,k+qh ,∇ · δun+1,k+q

h

)Ei

= 0. (4.15)

For the iterative scheme to be contractive, a quantity similar to the right hand side of(4.14), for the next iterative coupling iteration, n + 1, has to be formed. To achievethat, we introduce a term involving a summation over all flow finer time steps in (4.15)by noticing that

q∑m=1

(δpn+1,m+kh − δpn+1,m−1+k

h

)= δpn+1,k+q

h . (4.16)

Substituting (4.16) into (4.15) leads to

2

NΩ∑i=1

Gi∥∥ε(δun+1,k+q

h )∥∥2

Ei+

NΩ∑i=1

λi∥∥∇ · δun+1,k+q

h

∥∥2

Ei

−NΩ∑i=1

αi( q∑m=1

(δpn+1,m+kh − δpn+1,m−1+k

h

),∇ · δun+1,k+q

h

)Ei

= 0. (4.17)

• Step 3: Combining flow and elasticity equationsBy combining (4.17) with (4.14), and rearranging terms, we form a square term, inexpanded form, summed over flow finer time steps within one coarser mechanics time

17

step for each grid element Ei,

2

NΩ∑i=1

Gi∥∥ε(δun+1,k+q

h )∥∥2

Ei+

q∑m=1

NΩ∑i=1

βi2

∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei

+λiq

∥∥∥∇ · δun+1,k+qh

∥∥∥2

Ei− αi

(δpn+1,m+kh − δpn+1,m−1+k

h ,∇ · δun+1,k+qh

)Ei

+

∆t

2

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k+q

h

∥∥∥2

Ei+

∆t

2

q∑m=1

NΩ∑i=1

∥∥∥Ki−1/2(δzn+1,m+k

h − δzn+1,m−1+kh

)∥∥∥2

Ei

≤q∑

m=1

NΩ∑i=1

χ2i

2βi

∥∥∥δσn,m+kv

∥∥∥2

Ei. (4.18)

It remains to choose the values of our newly introduced parameters, χi and Li, suchthat the coefficients of the expanded square contributes only positive terms to the lefthand side of (4.18). Therefore, we expand the right hand side of (4.18) for each Ei as∥∥∥δσn,m+k

v

∥∥∥2

Ei=L2i

χi2

∥∥∥δpn,m+kh − δpn,m−1+k

h

∥∥∥2

Ei

− 2αiLiqχ2

i

(δpn,m+kh − δpn,m−1+k

h ,∇ · δun,k+qh

)Ei

+α2i

χ2i q

2

∥∥∥∇ · δun,k+qh

∥∥∥2

Ei. (4.19)

Now, we match the coefficients of the expansion in (4.19) to the coefficients of theexpanded square on the right hand side of (4.18), hence, deduce the values of χi andLi for each grid element Ei, respectively. For the left hand side of (4.18) to remainpositive, the following inequalities should be satisfied

βi2≥ L2

i

χi2,

2αiLiqχ2

i

= αi,λiq≥ α2

i

χ2i q

2.

The second and third inequalities give rise to the following condition

Li ≥α2i

2λifor each Ei.

The first inequality gives rise to q ≤ βiLi

. For Li =α2i

2λi, χ2

i =α2i

qλi, we derive the following

condition on the number of flow finer time steps within one coarse mechanics timestep

q ≤ 2λiαi2

( 1

Mi+ cf iϕ0i

)+ 1 for each Ei, (4.20)

which is not restrictive as typically in practice the values of λi are quite large. Now,we group the terms of the expanded square on the left hand side of (4.18) to form the

18

quantity of contraction for the next iterative coupling iteration, n+ 1, as

2

NΩ∑i=1

Gi∥∥ε(δun+1,k+q

h )∥∥2

Ei+

q∑m=1

NΩ∑i=1

(βi2− L2

i

χ2i

)∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei

+

q∑m=1

NΩ∑i=1

∥∥∥δσn+1,m+kv

∥∥∥2

Ei+

∆t

2

NΩ∑i=1

∥∥∥Ki−1/2δzn+1,k+q

h

∥∥∥2

Ei

+∆t

2

q∑m=1

NΩ∑i=1

∥∥∥Ki−1/2(δzn+1,m+k

h − δzn+1,m−1+kh

)∥∥∥2

Ei≤

q∑m=1

NΩ∑i=1

χ2i

2βi

∥∥∥δσn,m+kv

∥∥∥2

Ei.

Substituting χ2i = 2Li

q , βi = 1Mi

+ cf iϕ0i + Li for each Ei, with further algebraicsimplifications, we obtain

2

NΩ∑i=1

Gi∥∥ε(δun+1,k+q

h )∥∥2

Ei

+1

2

q∑m=1

NΩ∑i=1

( 1

Mi+ cf iϕ0i + (1− q)Li

)∥∥∥δpn+1,m+kh − δpn+1,m−1+k

h

∥∥∥2

Ei

+∆t

2

NΩ∑i=1

∥∥∥K−1/2i δzn+1,k+q

h

∥∥∥2

Ei+

∆t

2

q∑m=1

NΩ∑i=1

∥∥∥K−1/2i (δzn+1,m+k

h − δzn+1,m−1+kh )

∥∥∥2

Ei

+

q∑m=1

NΩ∑i=1

∥∥∥δσn+1,m+kv

∥∥∥2

Ei≤ max

1≤i≤NΩ

( Li

q( 1Mi

+ cf iϕ0i + Li)

) q∑m=1

NΩ∑i=1

∥∥∥δσn,m+kv

∥∥∥2

Ei.

(4.21)

The contraction coefficient: max1≤i≤NΩ

(Li

q( 1Mi

+cf iϕ0i+Li)

)< 1 for q ≥ 1. This is trivially

satisfied (at least we take one flow time step followed by one mechanics time step).

Theorem 4.1 [Localized Multirate Contraction Estimate] For q ≤ 1 + min1≤i≤NΩ

2λiαi2

(1Mi

+

cf iϕ0i

), Li =

α2i

2λiand χ2

i = 2Liq , the localized multirate iterative scheme is a contraction

given by

2∑NΩ

i=1Gi∥∥ε(δun+1,k+q

h )∥∥2

Ei

+12

∑qm=1

∑NΩi=1

(1Mi

+ cf iϕ0i + (1− q)Li)∥∥∥δpn+1,m+k

h − δpn+1,m−1+kh

∥∥∥2

Ei

+∆t2

∑NΩi=1

∥∥∥Ki−1/2δzn+1,k+q

h

∥∥∥2

Ei

+∆t2

∑qm=1

∑NΩi=1

∥∥∥Ki−1/2(δzn+1,m+k

h − δzn+1,m−1+kh )

∥∥∥2

Ei+∑q

m=1

∑NΩi=1

∥∥∥δσn+1,m+kv

∥∥∥2

Ei

≤ max1≤i≤NΩ

(Li

q( 1Mi

+cf iϕ0i+Li)

)∑qm=1

∑NΩi=1

∥∥∥δσn,m+kv

∥∥∥2

Ei.

19

Original Contraction Esti-

mates for Poroelastic Media

Localized Contraction Estimates for

Heterogeneous Poroelastic Media

Conditions on Pa-

rameters:

A degree of spatial uniformity

should be imposed as described

in remark 5.2

Parameters can be heterogeneous.

Contraction Coef-

ficient:

(L

1M

+cfϕ0+L

)2

max1≤i≤NΩ

(Li

q( 1Mi

+cf iϕ0i+Li)

)for Li =

α2i

2λifor all

Ei ∈ Ω

Condition on q: none q ≤ 1 + min1≤i≤NΩ

2λiαi

2

(1Mi

+ cf iϕ0i

).

When do Contrac-

tion Coefficients

Match?

For L = α2

2λ,(

Mα2

2(λ+Mλcfϕ0)+Mα2

)2

For q = upper limit, and Li =α2i

2λi

for all Ei ∈ Ω, contraction estimate

= max1≤i≤NΩ

(Miαi

2

2(λi+Miλicf iϕ0i)+Miα

2i

)2

. Exact

Match.

Table 1: Banach Contraction Estimates for Homogeneous vs Heterogeneous (Localized)Poro-elastic Media

5 Multirate Banach Contraction Estimates for Homogeneousvs Heterogeneous (Localized) Poroelastic Media

Table 1 compares the Banach contraction result derived for the multirate scheme in ho-mogeneous poroelastic media [3] against the one derived in this paper (for heterogeneousporo-elastic media).

Remark 5.1 Our localized Banach estimates work provides another strong justificationfor introducing the modified multirate iterative coupling scheme presented in our earlierwork [2,3]. Following a similar approach to the proof presented above, the localized modifiedmultirate iterative coupling scheme will not impose any upper bound on the number of flowfiner time steps taken within one coarse mechanics time steps. This follows immediatelyas the quantity of contraction in the modified scheme is independent of q. The details arespared.

Remark 5.2 For our earlier obtained results, the word “homogeneous” is not as restrictiveas it sounds. In fact, some degree of uniformity in the flow and mechanics parametersshould be imposed in this case. However, parameter values can change smoothly acrossthe spatial domain. The fixed stress stabilization term in this case should take the form

L = α2max

2λmin, and this value will be added to the main diagonal of the linear system in a

homogeneous manner. In fact, this leads to slower convergence rate, as the contractioncoefficient increases monotinially with L. The power of the localized contraction result isthat it allows us to add localized fixed-stress regularization terms which can vary across gridcells, yet the scheme is still contractive.

20

Injectors Wells: 10 injectors (pressure specified at 2600.0 psi)

Producers Wells: 20 producers (pressure specified at 1000.0 psi)

Total Simulation time: 192.0 days

Finer (Unit) time step: 1.0 days

Number of grids: 60048 grids (9 × 48 × 139)

Permeabilities: highly varying

kxx Range: (0.002122, 350.1372) md

kyy Range: (0.022143, 4135.124) md

kzz Range: (0.022493, 4163.053) md

Initial porosity, ϕ0 0.2

Fluid viscosity, µf 2.0 cp

Initial pressure, p0 400.0 psi

Fluid compressibility cf : 1.E-4 (1/psi)

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

Rock density: 165.44 lbm/ft3

Initial fluid density, ρf : 56.0 lbm/ft3

Young’s Modulus (E) 1.E5 psi

Possion Ratio, ν 0.4

Biot’s constant, α 0.6

Biot Modulus, M 1.0E16 psi

λ = Eν(1+ν)(1−2ν)

142857.0 psi

L (introduced fixed stress parameter) α2

2λ

Flow Boundary Conditions: no flow boundary condition on all 6 boundaries

Mechanics B.C.:

“X+” boundary (EBCXX1()) σxx = σ · nx = 5, 000psi, (overburden pressure)

“X-” - boundary (EBCXXN1()) u = 0, zero displacement

“Y+” - boundary (EBCYY1()) σyy = σ · ny = 1500psi

“Y-” - boundary (EBCYYN1()) u = 0, zero displacement

“Z+” - boundary (EBCZZ1()) u = 0, zero displacement

“Z-” - boundary (EBCZZN1()) σzz = σ · nz = 1000psi

Table 2: Input Parameters for Brugge Field Model

6 Numerical Results

6.1 Brugge Field Model

We consider the Brugge field model [1, 31] with a heterogeneous permeability distributionfor comparing the efficiency of the multirate scheme versus the single rate scheme. Themodel consists of a 9 × 48 × 139 general hexahedral elements capturing the field geometry,with 30 bottom-hole pressure specified wells, 10 of which are injectors at a pressure of2600 psi, and 20 are producers at a pressure of 1000 psi. Producers are located at a lowerelevation compared to injectors. No flow boundary condition is enforced across all externalboundaries. For the mechanics model, we apply a mixture of zero displacement and tractionboundary conditions. Gravity is not neglected in the model and detailed specifications ofthe input parameters can be found in Table 2.

21

6.2 Results

Figures 6.1 and 6.2 show the pressure profiles after 16 and 192 days of simulation forthe single rate and multirate schemes (q = 1, 2, 4 and 8). All four schemes result inalmost identical results. We also note that the upper bound for q is calculated to be:

q ≤ 1 + min1≤i≤NΩ

2λiαi2

(1Mi

+ cf iϕ0i

)= 10.52. That is, for the multirate scheme to be theoreti-

cally convergent, we should choose q ≤ 10. In other words, we should not take more than10 finer time steps within one coarse mechanics time steps.

Figure 6.3a 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 flowfiner time steps within one coarser mechanics time step (q = 2) results in 27.32% reductionin CPU run time compared to the single rate. Multirate couplings (q = 4, and q = 8) resultin 48.43%, and 51.15% reductions in CPU run times respectively. Figure 6.3b explains thereduction in CPU run time observed in the multirate case. By just solving for two flow finertime steps within one coarser mechanics time step (q = 2), the total number of mechanicslinear iterations was reduced by 51.08% with reference to the single rate case. Multiratecouplings (q = 4, and q = 8) result in 75.61%, and 86.29% reductions in the number ofmechanics linear iterations respectively, which in turn, reduce the CPU run time as well.Figure 6.3c shows the total number of flow linear iterations in the four cases. We see anincrease in the total number of flow linear iterations for multirate iterative couplings. Mul-tirate iterative coupling with two flow finer time steps (q = 2), within one coarse mechanicstime step results in 44.76% increase in the total number of flow linear iterations. Multiratecouplings (q = 2), and (q = 8) result in 49.63%, and 69.01% increase in the total numberof flow linear iterations respectively. Figure 6.3d shows the number of iterative couplingiterations per coarse mechanics time step. The increase in the number of flow linear iter-ations for multirate cases is attributed to the increase in the number of iterative couplingiterations performed for each coarse mechanics time step, as shown in figure 6.3d. However,the huge decrease in the number of mechanics linear iterations outperform the overheadintroduced by the increase in the number of flow linear iterations. This is a key factor tothe success of the iterative multirate coupling scheme in reducing the overall CPU run time.

We conclude that the applicability and efficiency of the multriate fixed-stress split iterativecoupling scheme extends to the case in which the poro-elastic media is highly heterogeneous.Its convergence is shown theoretically, and observed numerically for field-scale problems,with a heterogeneous permeability distribution.

22

Figure 6.1: Pressure Profiles after 16.0 simulation days

23

Figure 6.2: Pressure Profiles after 192.0 simulation days

24

0 50 100 150 200Simulation Period (days)

0

50

100

150

200

250

300

350

400

Accum

ula

ted C

PU

Tim

e (

min

ute

s)

Accumulated CPU Run Time vs Simulation Period

Single Rate

Multirate (q = 2)

Multirate (q = 4)

Multirate (q = 8)

(a) CPU Run Time vs Simulation Days

0 50 100 150 200Simulation Time (days)

0

10000

20000

30000

40000

50000

60000

Accum

ula

ted M

echanic

s L

inear Iteratio

ns

Accumulated # of Mechanics Linear Itrns vs Simulation Period

Single Rate

Multirate (q = 2)

Multirate (q = 4)

Multirate (q = 8)

(b) Total Number of Mechanics Linear Iterationsvs Simulation Days

0 50 100 150 200Simulation Time (days)

0

2000

4000

6000

8000

10000

12000

Accum

ula

ted F

low

Lin

ear

Itera

tions

Accumulated # of Flow Linear Itrns vs Simulation Period

Single Rate

Multirate (q = 2)

Multirate (q = 4)

Multirate (q = 8)

(c) Total Number of Flow Linear Iterationsvs Simulation Days

0 50 100 150 200Time Steps (days)

0

2

4

6

8

10

Num

ber

of

Coupling Ite

rati

ons P

er

Tim

e S

tep

Coupling Iterations per Coarse Time Step

Single Rate (q = 1)

Multirate (q = 2)

Multirate (q = 4)

Multirate (q = 8)

(d) Number of Iterative Coupling IterationsPer Coarser Time Step

Figure 6.3: Brugge Field Model Results

25

7 Conclusions and outlook

This paper considers single rate and multirate fixed stress split iterative coupling schemesin heterogeneous poroelastic media. The fixed stress split scheme allows for the sequentialcoupling of flow with mechanics and a coupling iteration is imposed until convergence isobtained for each discrete time step. The convergence of both the single rate and multiratecoupling schemes in homogeneous poroelastic media have been established in the past [3].The novelty of the work presented in this paper is that it extends these Banach contractionresults to the case in which the poroelastic media is highly heterogeneous (flow and me-chanics parameters can vary in both space and time). To the best of our knowledge, thisis the first time a Banach contraction result has been rigorously obtained for the multiratefixed stress split iterative coupling scheme in heterogeneous poroelastic media. Moreover,the localized contraction proof outlines a general strategy that is very likely to be applicablefor obtaining similar localized contraction estimates for other iterative coupling schemes,including the undrained split iterative coupling scheme. In addition, our analysis revealsan upper bound on the number of flow fine time steps that can be solved within one coarsemechanics time step in the multirate scheme. Our theoretical findings in this paper aresupplemented with numerical results for a realistic reservoir model with a heterogenouspermeability distribution. The numerical results show that the applicability and efficiencyof the multriate scheme extends to the case in which the poroelastic medium is highly het-erogeneous. In a future work, the performance of the single rate and multirate iterativecoupling schemes will be investigated for a heterogeneous fixed-stress split regularizationparameters.

Acknowledgements

TA is funded by Saudi Aramco. We thank Paulo Zunino and Ivan Yotov for helpful dis-cussions. KK would like to acknowledge the support of Statoil Akademia Grant (Bergen).The authors would like to acknowledge the CSM Industrial Affiliates program, DOE grantER25617, and ConocoPhillips grant UTA10-000444.

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