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A heterogeneous modeling method for porous media flows

Georgette Hlepas1,2, Timothy Truster3,4 and Arif Masud4,*,†

1Chicago District, US Army Corps of Engineers, Chicago, IL 60606, USA2Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60612, USA

3Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN 37996-2313, USA4Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL

61801-8039, USA

SUMMARY

This paper presents a new heterogeneous multiscale modeling method for porous media flows. Physics at theglobal level is governed by one set of PDEs, while features in the solution that are beyond the resolutioncapacity of the global model are accounted for by the next refined set of governing equations. In this method,the global or coarse model is given by the Darcy equation, while the local or refined model is given by theDarcy–Stokes equation. Concurrent domain decomposition where global and local models are applied toadjacent subdomains, as well as overlapping domain decomposition where global and local models coexiston overlapping domains, is considered. An interface operator is developed for the case where global and localmodels commute along the common interface. For the overlapping decomposition, a residual-based couplingtechnique is developed that consistently facilitates bottom-up embedding of scale effects from the localDarcy–Stokes model into the global Darcy model. Numerical results are presented for nonoverlapping andoverlapping domain decompositions for various benchmark problems. Computed results show that thehierarchically coupled models accurately account for the heterogeneity of the medium and efficientlyincorporate local features into the global response. Copyright © 2014 John Wiley & Sons, Ltd.

Received 29 December 2011; Revised 25 December 2013; Accepted 26 February 2014

KEY WORDS: multiscale methods; variational methods; heterogeneous methods; Darcy flow;Darcy–Stokes flow

1. INTRODUCTION

Flow through porous media is an important physical problem when considered in the largergeotechnical and geoenvironmental context. Seepage flow has been of primary importance inthe field of groundwater flow, drainage studies, and reservoir engineering. Most availablenumerical formulations for porous media flows are based on the Darcy equation that modelsflows under pressure gradients through porous media with low permeability, and wherein dragforce alone is the significant and dominant factor. As permeability increases, which is the casefor fractured rocks or for flow through larger-sized granular materials, flow rate also increases,and internal frictional forces in the fluid start playing an important role. Accordingly, regionsthat have higher permeability and are associated with higher flow rates are modeled viaDarcy–Stokes equations. Subsurface geologic formations are naturally heterogeneous, and wheninspected at a fine level, they reveal differing soil types with varying permeability values.Modeling of such systems calls for methods that can account for the heterogeneity of themedium and multiscale nature of the resulting flow field.

*Correspondence to: Arif Masud, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-8039, USA.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd.

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2014; 75:487–518Published online 3 April 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.3904

In this paper, we present a new method for the linking of hierarchical physical models (Rajagopal[1]) in a mathematically consistent fashion. Hierarchical modeling of physical phenomena has beenan area of active research, and a number of approaches have been proposed to address issues relatedto accurate representation of scale effects and interscale couplings between the different models(Engquist [2], Wienan [3]). Hierarchical models are also used for material and spatial randomness(Ostoja-Starzewski [4]) and to generate missing data in macroscopic models via stochasticupscaling of microscopic models (Ganapathysubrmanian [5]). The notion of linking hierarchicalmodels has been the basis for the development of bridging scale methods in micromechanics andnanomechanics (Masud and Kannan [6], Shenoy [7]) and in the coupling of atomistic andcontinuum models (Curtin [8], Hou [9]). Fracture mechanics is another area where different modelsare employed at different scale levels to capture localized effects in the process zones around cracks(Shilkrot [10], Wagner [11]).Modeling of multiscale phenomena in the class of problems enumerated previously poses a major

challenge because of an important mathematical question: How should the information obtainedfrom a model at one level be incorporated into a model at a different level? A literature reviewreveals that the common features of the various heterogeneous multiscale (HMS) methods are asfollows: (i) a global model (GM) that adequately represents the overall behavior of the system underinvestigation and invariably resolves only the coarse scales of the physics involved and (ii) ascheme for appropriately representing the physical features that are beyond the modeling capabilityof the GMs, usually via refined local models (LMs). Computational expediency suggests employingLMs in the smaller or localized regions of the computational domain where detailed physics isrequired, while using GMs or the coarse-scale models in the rest of the domain. An importantconsideration in the coupling of different models is the exchange of information across theinterfaces between the domains for LM and GM.To simplify ideas, consider the example of an oil reservoir with an overall low permeability

where Darcy equations can serve as the GM. In the presence of stratified layers with higherpermeability where flow takes place at a higher rate and therefore shearing effects becomesignificant, Darcy–Stokes equations can be employed as the preferred LM. From the modelingperspective, this hierarchical intricacy in flow physics that is a consequence of the hierarchy inlength scales associated with the porous medium calls for a framework that can effectively couplethe increasingly sophisticated mathematical equations in a consistent fashion. In this work, wefollow the heterogeneous modeling framework presented in Masud and Scovazzi [12] and callthe GM as the ‘coarse’ model and the LM as the ‘fine’ model. We consider two scenarios: Onein which GM/LMs (coarse/fine scales) augment each other and are spatially separated by aninterface, and the other in which GM/LMs co-occupy the domain of interest, and one is embeddedin the other.It is well known that an application of the standard mixed FEM to the Darcy equation or the

Darcy–Stokes equation is required to satisfy the Babuska–Brezzi inf-sup condition. With theobjective to develop a method that is easy to implement and works with equal-order interpolationsfor velocity and pressure, we employ stabilized methods for Darcy [13, 14] and Darcy–Stokes [15]equations. We embed these stabilized formulations in the proposed hierarchical framework [12] anddevelop the heterogeneous modeling method for porous media flows.An outline of the paper is as follows. Section 2 presents a brief overview of consistent upscaling

via the HMS method. Section 3 presents the Darcy model, Section 4 presents the Darcy–Stokesmodel, and Section 5 presents the heterogeneous model that provides upscaling of solution fromLM to GM. The coupling operator at the interface boundary between GM and LM is presented inSection 6. Section 7 presents numerical examples for Darcy, Darcy–Stokes, and the heterogeneousmodels. Conclusions are drawn in Section 8.

2. THE MULTIMODEL MODELING PARADIGM

This section presents the hierarchical modeling framework that employs two systems of PDEs.Figure 1 presents an abstract representation of the global and local domains where ΩG⊂ℝnsd

represents the global spatial domain and is associated with GM, and ΩL⊂ℝnsd represents the local

488 G. HLEPAS, T. TRUSTER AND A. MASUD

Copyright © 2014 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2014; 75:487–518DOI: 10.1002/fld

region wherein flow physics possesses finer features that are described by the LM. Specifically,Figure 1(a) depicts the case in which GM and LM (coarse and fine scales) are spatiallyseparated by an interface, and Figure 1(b) depicts the case where GM and LM co-occupythe domain of interest with an overlapping decomposition of scales. For the current system,GM is facilitated by the Darcy equations, while the LM is given by the Darcy–Stokesequations. It is important to realize that there can be multiple subdomains ΩLi with associatedLMs coexisting in ΩG.

2.1. Consistent upscaling via the heterogeneous modeling method

We first present a synopsis of the upscaling technique proposed in [12] that is applied to upscalingfrom LM (Darcy–Stokes) to GM (Darcy). For easy comprehension of ideas, let us assume for nowthat the coarse solution uG has been obtained from the GM. We term it as the global solutionuG ∈ SG where SG is the admissible space of functions for the trial solutions for GM over ΩG.The solution from the subdomain ΩL will now be obtained via the LM, that is, uL∈ SL whereSL is the admissible space of functions for the trial solutions for LM over ΩL. At the interfaceboundary between ΩL and ΩG, indicated as ΓI, a ‘natural requirement’ for the interface conditionis that uLjΓI

¼ uGjΓI. Therefore, appropriate boundary conditions may be posed as some average

or projection of uLjΓI¼ uGjΓI

along the boundary ΓI. Accordingly, we will define an interfacecoupling operator LI (uL;uG) that is designed such that information is exchanged betweenGM and LM across the interface ΓI in a mathematically consistent fashion. In the discussionto follow in the rest of the section, we adopt the following notational convention. Solutionscales that are considered ‘large’ are indicated with superposed bar ·ð Þ; those considered ‘small’are indicated with superposed hat ·ð Þ. Scales considered coarse appear with superposed tilde e·ð Þ,and scales considered fine have superposed prime (· ′)Let LL be the differential operator of the PDEs for LM governing physics over subdomain ΩL

where we want to find the solution with high precision. The governing equations are as follows:

LL uL ¼ f L in ΩL (1)

together with the condition that uL= uG on ΓI, where uL; uGf g∈ SL because SG ⊆ SL, and f ∈ L2 (Ω).Let w be the set of admissible weighting functions such that w∈VL , where VL is the admissiblespace of weighting functions. The weak form is as follows:

w;LLuLð Þ þ w;LI uL; uGð Þh iΓI¼ w; f Lð Þ (2)

where (�,�) = ∫Ω(·) dΩ is the L2-inner product, LI (� ; �) is the interface coupling operator at ΓI, andthe brackets indicate the duality pairing.

Figure 1. Abstract representation of global and local models: (a) nonoverlapping subdomains and (b)overlapping subdomains.

489A HETEROGENEOUS MODELING METHOD FOR POROUS MEDIA FLOWS


At this point, we consider two scenarios to develop the coupling method:

1. Nonoverlapping solution decomposition in which case the solution of GM communicates withthe solution of LM only at ΓI, and

2. Overlapping solution decomposition wherein the solution of GM not only matches the solutionof LM at ΓI but also satisfies the LM in ΩL in a weak sense.

2.1.1. Nonoverlapping solution decomposition. For the case where solution of GM communicateswith the solution of LM only at the boundary ΓI, the variational form given in (2) becomes themodified weak form. Further discussion on this nonoverlapping coupling is presented in Section 6,and the interface coupling operator is developed for Darcy, Stokes, and Stokes–Darcy models.

2.1.2. Overlapping solution decomposition. A more intricate situation arises when the solution ofGM not only matches the solution of LM at ΓI but also satisfies the LM in ΩL in a weak sense.Consequently, overlapping solution decomposition provides a tighter coupling of scales betweenGM and LM. We consider that LM assumes a unique additive decomposition of the local solutionas follows:

uL ¼ uL|{z}large scales

þ uL|{z}small scales

(3)

This decomposition can be made precise via specification of the appropriate spaces of functions.Furthermore, we assume that large scales uL are on the order of the scales in uG. Substituting (3) in(2), we obtain the following:

w;LL uL þ uLð Þð Þ þ w;LI uL þ uL; uGð Þh iΓI¼ w; fð Þ (4)

For overlapping solution decomposition between GM and LM, we require that the solution ofGM exactly matches the large-scale solution of LM at the interface, that is, uLjΓI

¼ uG on ΓI. Inaddition, the global solution uG satisfies the governing model LM inside ΩL only in a weak sense.Substituting uG for uL in (4) and writing in a residual form, we obtain the following:

w;LL uLð Þ þ w;LI uG þ uL;uGð Þh iΓI¼ w; f � LL uGð Þ (5)

where f�LL uG represents the residual generated by the GM solution when inserted in the LM modelequations. Computational model (5) yields a local solution where the global solution is fully embed-ded. Furthermore, this modeling framework can be applied even if there are multiple subdomainsΩLi with different LMs in the various subregions of ΩG.

2.1.3. The variational multiscale method applied to the local model over ΩL. We can apply varia-tional multiscale (VMS) ideas to the LM to develop multiscale formulations for LM over ΩL. TheVMS method is based on an overlapping additive decomposition of the solution into coarse and finescales. The rationale behind this scale decomposition is that fine-scale features in the solution maybe beyond the resolution capacity of a given mesh [16]. However, the influence of the fine-scalefeatures on the coarse-scale physics may not be negligible, and therefore, fine scales need to beconsidered for an accurate modeling of the problem.Accordingly, we assume an additive decomposition of the small features in the LM solution

uL into coarse scales eu and fine scales u′, respectively.

uL xð Þ ¼ eu xð Þ|ffl{zffl}coarse scale

þ u′ xð Þ|ffl{zffl}fine scale

(6)



Likewise, we consider a corresponding decomposition of the weighting functions w xð Þ ¼ew xð Þ þ w′ xð Þ. Consistency of the method requires that the spaces of functions for the coarse and finescales are linearly independent. This can be achieved by specifying a direct sum decomposition on

these spaces SL ¼ eSL⊕ S′L , where eSL and S′

L are the spaces of functions for the coarse-scale andfine-scale fields, respectively. Substituting the additively decomposed form of trial solution andweighting function in (5) and employing the linearity of the weighting function slot lead to two setsof problems:

Coarse-scale problem for LM¯

: ew;LL euþ u′� �� þ ew;LI uG þ eu;uGð Þh iΓI

¼ ew; f � LL uGð Þ (7)

Fine-scale problem for LM¯

: w′;LL euþ u′� �� ¼ w ′; f � LL uGð Þ (8)

The key idea at this point is to solve (8) and extract an expression for u′ that can then besubstituted in (7). This step leads to the multiscale form for the LM over ΩL. With suitable assump-tions on the fine scales, we can make (8) a finite dimensional problem that can then be solved overelement interiors or over patches of elements. Variationally embedding the fine-scale solution intothe coarse-scale problem leads to a hierarchically multiscale form for the LM that governs ΩL.

ew;LL euð Þ þ L�Lew; τLL eu� �þ ew;LI uG þ eu;uGð Þh iΓI

¼ ew; f � LL uGð Þ þ L�Lew; τ f � LL uGð Þ� �

(9)

where L�L is the adjoint operator for LM. Equation (9) is the multiscale LM where the effect of the

GM has been accounted for via residual-based forcing terms that emanate because of the insertionof GM solution into the LM equations. An application of VMS to LM for the modeling of porousmedia flows is made precise in Section 5.

Remark 1: It is important to note that because LL is defined over ΩL alone, the contribution fromLL uG is nonzero only in ΩL⊆ΩG.

3. DARCY EQUATIONS (GLOBAL MODEL)

Darcy equations model flow through porous media wherein permeabilities are low, and frictionalforce between the fluid and the porous medium is the dominant factor, that is, flows through claysand other dense geologic materials. In our framework, Darcy equations constitute the GM.

3.1. Mixed velocity–pressure formulation

LetΩG⊆ℝnsd be an open bounded region with piecewise smooth boundary Γ. The number of spacedimensions, nsd, is equal to 2 or 3. The Darcy law and conservation of mass are given by thefollowing equations:

v ¼ � κμ

∇pþ ρgcg

� �in Ω Darcy Lawð Þ (10)

div v ¼ φ in Ω Conservation of Massð Þ (11)

v�n ¼ ψ on Γ (12)

where v is the Darcy velocity vector, p is the pressure, g is the gravity vector, φ is the volumetricflow rate source or sink, ψ is the normal component of the velocity field on the boundary, μ> 0 is



the viscosity, κ> 0 is the permeability, ρ> 0 is the density, gc is a conversion constant, and n isthe unit outward normal vector to Γ. It is apparent from (11) and (12) that the prescribed data φand ψ must satisfy the constraint ∫Ω φ dΩ= ∫Γψ dΓ.

3.2. The standard weak form

Let

V ¼ H div;Ωð Þ¼def ν ν∈ L2 Ωð Þð Þnsd ; div ν ∈ L2 Ωð Þ; trace ν� nð Þ ¼ ψ ∈H�12 Γð Þg��

(13)

P ¼ L2 Ωð Þ\ℝ ¼def p p ∈ L2 Ωð Þ; ∫Ω pdΩ ¼ 0g��(14)

V0 ¼ H0 div;Ωð Þ ¼def ν ν ∈ L2 Ωð Þð Þnsd ; div ν ∈ L2 Ωð Þ; trace ν �nð Þ ¼ 0gjf (15)

Further elaboration on these spaces is available in the work of Brezzi and Fortin [17]. It isassumed that κ,μ, ρ, gc, g, φ and ψ are given. Thus, the standard weak form of (10–12) is as follows:Find ν ∈ V; p ∈P, such that, for all w ∈ V0; q ∈P,

w;μκv

� div w; pð Þ þ q; div vð Þ ¼ � w

ρgc

g

� �þ q; φð Þ (16)

where (�,�) is the L2(Ω) inner product. A unique solution to the weak form exists for sufficientlyregular data. For simplicity, it is convenient to rewrite (16) as follows: Let Y ¼ V � P; Y0 ¼ V0 � P;V ¼ ν; pf g and W= {w, q}. Find V∈Y, such that for all W∈Y0,

BD W;Vð Þ ¼ LD Wð Þ (17)

where

BD W;Vð Þ ¼ w;μκv

� div w; pð Þ þ q; div vð Þ (18)

LD Wð Þ ¼ � w;ρgc

g

� �þ q; φð Þ (19)

Remark 2:The Galerkin FEM is based on (17). Stability is achieved for only certain combinations ofvelocity and pressure interpolations [17–19].

3.3. The stabilized weak form

Employing the VMS framework, Masud and Hughes [13] proposed a stabilized form that issummarized as follows: Let

Q ¼ H1 Ωð Þ\ℝ¼def q q ∈H1 Ωð Þ; ∫Ω qdΩ ¼ 0g��(20)

Z ¼ V �Q; Z0 ¼ V0 �Q (21)



The stabilized weak form is as follows: Find V∈Z, such that for all W∈Z0,

BDstab W;Vð Þ ¼ LDstab Wð Þ (22)

where

BDstab W;Vð Þ ¼ BD W;Vð Þ þ 1

2�μκwþ ∇q

;κμ

μκνþ ∇p

� �(23)

LDstab Wð Þ ¼ LD Wð Þ � 12

�μκwþ ∇q

;κμ

ρgc

g

� �� (24)

4. DARCY–STOKES EQUATIONS (LOCAL MODEL)

The local changes in the pore length scales in the subsurface geologic formations, for example, sandand gravel layers stratified within clay, can have significant impact on the local flow features. Flowsin these stratifications take place at a higher rate, and therefore, accounting for the shear viscosity ofthe fluid becomes important. Furthermore, these flows possess finer features that cannot be modeledby the Darcy equations alone. Consequently, for the LM, we employ the Darcy–Stokes equations aspresented below.

4.1. Darcy–Stokes mixed velocity–pressure form

Let ΩG ⊆ℝnsd be an open bounded region with piecewise smooth boundary Γ. The number ofspace dimensions, nsd, is equal to 2 or 3. Darcy–Stokes equations and conservation of massequations are as follows:

μκνþ ∇p� μΔν ¼ f inΩ Darcy-Stokesð Þ (25)

div v ¼ φ in Ω Conservation of Massð Þ (26)

where v is the velocity vector, p is the pressure, f is the body force vector, φ is the volumetric flowrate at source or sink, μ is the viscosity, and κ is the permeability. As boundary conditions, we willconsider ν¼ g on Γ, which is not to be confused with the gravity vector for Darcy flow.

4.2. The standard weak form

The appropriate spaces for velocity and pressure are ν ∈W; p ∈P.

W ¼ H1 Ωð Þ¼def ν ν ∈ L2 Ωð Þð Þnsd ;∇ν ∈ L2 Ωð Þð Þnsd�nsd ; and ν ¼ g in H�12 Γð Þg��

(27)

It is assumed that κ,μ and φ are the given data. The classical weak form of (25) and (26) is asfollows:

w;μκv

� div w; pð Þ þ q; div vð Þ þ ∇w; μ∇vð Þ ¼ w; fð Þ þ q; φð Þ (28)



For sufficiently regular data, the weak formulation is known to possess a unique solution. It isconvenient to rewrite (28) in an abstract form as follows: Let X ¼ W �Q; X 0 ¼ W0 �Q;V ¼ v; pf g and W= {w, q}. Find V¼X , such that for all W∈X0,

BDST W;Vð Þ ¼ LDST Wð Þ (29)

where

BDST W;Vð Þ ¼ w;μκv

� div w; pð Þ þ q; div vð Þ þ ∇w; μ∇vð Þ (30)

LDST Wð Þ ¼ w; fð Þ þ q; φð Þ (31)

4.3. The stabilized weak form

We employ the stabilized/multiscale method proposed in Masud [15] in the present work. We firstsummarize the important aspects of [15] within the context of the heterogeneous framework. Thestabilized weak formulation is as follows: Find V¼X , such that for all W∈X0,

BDSTstab W;Vð Þ ¼ LDSTstab Wð Þ (32)

where

BDSTstab W;Vð Þ ¼ BDST W;Vð Þ þ �μ

κwþ ∇qþ μΔw

; τ

μκνþ ∇p� μΔν

(33)

LDSTstab Wð Þ ¼ LDST Wð Þ þ �μκwþ ∇qþ μΔw

; τ f

(34)

and BDST(W,V) and LDST(W) are given by (30) and (31), respectively. As reported in Section 3.4 of[15], the stabilization tensor τ in terms of the bubble function representation of the fine scales is givenby the following expression:

τ ¼ be ∫Ωe be dΩ ∫Ωe beμκbe dΩ

I þ ∫Ωe μ∇be� ∇be dΩ

� �I þ ∫Ωe μ∇be⊗∇be dΩ

h i�1(35)

where be(x) is the bubble function supported over element Ωe that vanishes on the element bound-ary Γe. Previous studies [20, 21] have indicated that the choice of simple polynomial functions issufficient to provide the necessary stability. The specific form of the bubble function for eachelement type is provided in Section 7.

Remark 3: The derivations for the stabilized Darcy equations [13, 14, 22] and the stabilizedDarcy–Stokes equations [15] were based on the VMS method that yields a two-level descriptionof the problem: a coarse-scale problem and a fine-scale problem. The fine-scale problem is solvedvia a bubble function approach [23–25], and the fine-scale fields are variationally projected ontothe coarse-scale space.

Remark 4: The bubble functions only reside in the definition of the stability tensor τ in (33) and(34). Consequently, the choice of the bubble functions only affects the value of the stability tensor.

Remark 5: In the Darcy limit, the stabilization tensor τ contained in (35) is O (κ/μ), while in theStokes limit τ is O (h2/μ). These asymptotic behaviors are in agreement with the correspondingstabilization methods for the underlying formulations [13, 20].



5. THE HETEROGENEOUS MULTISCALE MODELING METHOD

Following along the lines of Section 2, we integrate LM from Section 4 into GM from Section 3. Inthe heterogeneous framework, the GM (Darcy) can be expressed in an abstract form as follows:

LG uG ¼ fG in ΩG (36)

The corresponding weak form can be written as follows:

wG;LGuGð Þ ¼ wG; fGð Þ (37)

The corresponding stabilized form is as follows:

wG;LGuGð Þ þ L�GuG; τLGuG

� � ¼ wG; fG � τ fGð Þ (38)

For the GM, the stabilized form corresponding to (36) is given by (22). Solution of (22) yields theglobal velocity and pressure fields ν; pf gTG.Following (3), we split the solution of the LM into local-large ν; pf gTL and local-small ν; pf gTL

components.

νp

� �L

¼ νp

� �L|fflfflffl{zfflfflffl}

local-large

þ νp

� �L|fflfflffl{zfflfflffl}

local-small

(39)

⇒ Vf gL ¼ V�

Lþ V�

L(40)

We set the solution of the GM (Darcy) solution equal to the local-large LM solution(Darcy–Stokes) at ΓI.

ν

p

( )G

¼ν

p

( )L|fflfflffl{zfflfflffl}

local-large

at ΓI(41)

⇒ Vf gG ¼ V�

Lat ΓI (42)

This prescription implies that the local-small solution VL is taken to vanish on ΓI. Now consider, thestabilized LM given in (9). The equivalent Heterogeneous Multiscale (HMS) version for the one-wayupscaling from LM (Darcy–Stokes) to GM (Darcy) can be developed from (32) through (34).

BHMSstab W;Vð Þ ¼ LHMS

stab Wð Þ (43)

BHMSstab W;Vð Þ ¼ BDST

stab WL;VLð Þ þ WDG;LI VG þ VL;VG

� �� (44)

LHMSstab Wð Þ ¼ LDSTstab WLð Þ � WL;LLVGð Þ � L�

LWL; τ LL VGð Þð Þ (45)

where BDSTstab W;Vð Þ is explicitly defined in (33), and LDSTstab Wð Þ is defined in (34).



An algorithm for solving the one-way coupled heterogeneous system is provided in Table I.

Remark 6: The unique contribution of the present approach is that, contrary to the usual approachin the VMS method, the residual (45) of the coarser mathematical model is used to drive thefine-scale solution and hence the name Heterogeneous Multiscale (HMS) modeling method.

Remark 7: The Dirichlet boundary conditions are applied to VL∈X so that the local solutioncan be superimposed on the global solution within the region of overlap. Also, a projectionoperator may be required in the discrete setting to ensure that the global Darcy solutionVG ∈Z is a well-defined argument for the Darcy–Stokes operator LL on the right-hand sideof (45). A discussion of the implementation is given in Section 7.5 of the numerical results.

Remark 8: In the abstract framework for the hierarchically refined models presented in [12], thefunction space of global solutions is assumed to be a subset of the admissible space for the localsolutions, that is, SG⊆SL. While the LM is expected to capture the more refined physics of thesystem, the mathematical function spaces must satisfy the regularity requirements of theparticular governing PDEs. In the present porous media flow problem, we have the reverseembedding SL⊆SG: the proper space for Darcy–Stokes flow SL ¼ H1 Ωð Þ is a subset of the spacefor Darcy flow SG ¼ H div;Ωð Þ (see [17] for a discussion on the properties of these spaces). Inthis context, embedding of VG ∈ SG in LM should be interpreted in the sense of projection of VG

onto SL. For other applications, the conditions SG ¼ SL or SG ⊆ SL may hold.

6. DESIGN OF THE INTERFACE COUPLING OPERATOR FOR NONOVERLAPPINGDECOMPOSITION OF MODELS

In this section, we consider the coupling of Stokes and Darcy flow as a model problem to developthe interface coupling operator in the case of nonoverlapping solution decomposition. Whilesimilar developments could be performed for combining Darcy and Darcy–Stokes flow regimesas presented in the preceding sections, we instead focus on a related topic that has recently beenan active area of research. A sampling of applications for combined Stokes and Darcy models in-clude the modeling of contaminant transfer between rivers and groundwater, coupled models ofnutrient transfer between biological tissue and the bloodstream, and design of industrial filtrationsystems. A common approach to couple the flow regions is through domain decomposition (see,for example, the work by Vassilev and Yotov [26] in which the transport equation is also incorpo-rated). A summary of numerical techniques for coupling flow regimes is contained in [27]. Herein,we adopt the procedure proposed by Truster and Masud [21, 28] to derive a primal interfaceoperator with the character of a discontinuous Galerkin (DG) method by starting from a Lagrangemultiplier interface formulation. This approach relies crucially on applying concepts from the VMSmethod [16] locally at the interface between the LG and GM to derive the numerical flux terms forthe DG method, which allows for different element types and jumps in material properties betweenthe two models. First, we discuss the procedure for the case when both models are represented bythe same governing equations, and then we present the generalization to the Stokes–Darcy systemin Section 6.3.

Table I. Algorithm for formation and solution of overlapping heterogeneous system.

Obtain solution VG by solving (22) over the Darcy subdomain ΩG

Initialize local model: assign homogeneous Dirichlet boundary conditions to VL on ΓIFor elements Ωe

L⊂ΩL:○ Assemble contributions to the stiffness matrix from (44):BDSTstab WL;VLð Þ (a)

○ Assemble contributions to the force vector from (45):LDSTstab WLð Þ � BDST

stab WL;VGð Þ (b)Solve for VL from (43) and obtain V =VG+VL



6.1. Interface operator for Darcy equation

The launching point for the derivation in the context of Darcy flow is the standard Lagrangemultiplier method for weakly enforcing continuity of the normal component of velocity at theinterface, which is stated using the functionals from (17) and (18) as follows:

BDG WG;VGð Þ þ BD

L WL;VLð Þ þ wG�n; λh iΓI� wL�n; λh iΓI

¼ LDG WGð Þ þ LDL WLð Þ (46)

μ; vG�nh iΓI� μ; vL�nh iΓI

¼ 0 (47)

where λ;μ ∈H�12 ΓIð Þ are the Lagrange multipliers, the subscripts indicate the restriction of the

associated fields and functional forms to the respective subdomains ΩG or ΩL, and n is the unitoutward normal vector to ΩG.Now, the key idea is to derive an expression for the Lagrange multiplier field λ in terms of the

primary fields VG and VL at the interface in order to remove the explicit appearance of λ in (46)to arrive at a form analogous to (2). This is accomplished by following the general frameworkdescribed in [21] whereby a multiscale decomposition is applied to the velocity field locally atthe interface. By applying modeling assumptions to the fine-scales v′G and v′L on either side of the

interface, we arrive at analytical expressions for these scales in terms of the coarse-scales eVG andeVL and the multiplier λ. The reader is referred to [21] for a complete discussion of the modelingprocedures. The major assumption is that the fine scales are considered to be localized to the

elements from regions ΩL and ΩG that border the interface ΓI. Specifically, let γjn onseg

j¼1be a partition

of ΓI into a series of segments γj defined by adjacent pairs of elements ΩeG and Ωe

L such that∂Ωe

L ∩ ∂ΩeG ¼ γj for each j. Throughout, we use the subscript α to denote the model region (GM

or LM). In the vicinity of each segment γj, we represent the fine scales by edge bubble functionsbjα that are supported over sectors ωj

α ⊆Ωjα as follows:

v′α��ωjα¼

Xnsdk¼1

βαkbjα xð Þeαk; w′

α

��ωjα¼

Xnsdl¼1

ηαlbjα xð Þeαl (48)

where eαkf gnsegk¼1 is a set of linearly independent unit vectors spanningℝnsd, and βαk; ηαl are undetermined

coefficients. The concept of segments and sectors associated with a discretized interface is illustrated inFigure 2, which is adapted from [21]. The specific form of the edge bubble functions for variouselement types is provided in Section 7.Carrying through the derivation along the lines of [21] by employing the representation (48)

within the fine-scale problem associated with (46) leads to the following analytical expression forv′α at the interface:

v′α�n��γj¼ �1ð Þα�1τ jα �λþ pαð Þ (49)

τjα ¼ meas γj h i�1

∫ω jαb jα

� �2μα=κα dΩ

�1∫γj b

jα dΓ

2(50)

where τjα are stabilization parameters accounting for the element geometry and material parameters.Note that the fine scales are driven by the boundary residual, since for the exact solution of (46) and(47) we have λ= pα.



Embedding the representation of the fine scales (49) into the coarse-scale problem associatedwith (46) and (47) results in the following stabilized mixed weak form:

BDG

fWG; eVG

þ BD

LfWL; eVL

þ ewG�n; λh iΓI

� ewL�n; λh iΓI

þ qG; τjG �λþ pGð Þ� �

ΓIþ qL; τ

jL �λþ pLð Þ� �

ΓI¼ LDG fWG

þ LDL fWL

(51)

μ;evG�nh iΓI� μ;evL�nh iΓI

þ μ; τ jG �λþ pGð Þ� �ΓIþ μ; τ jL �λþ pLð Þ� �

ΓI¼ 0 (52)

Because the formulation is stabilized by the terms arising consistently from the fine scales, we arefree to select the interpolation space of the multipliers without recourse to the Babuska–Brezzi con-dition. By adopting a discontinuous approximation of the multipliers as L2 functions along eachsegment, we can solve the continuity equation (52) to obtain an analytical expression for λ:

λjγj ¼ pf gjγj þ τ j〚ev〛��γj (53)

in which the numerical flux is defined as pf g ¼ δ jGpG þ δ j

LpL, and the velocity jump is denoted by〚v〛= vG � nG+ vL � nL. The weighting coefficients δ j

α and the velocity penalty parameter τ j areevaluated from the fine-scale stabilization parameters τ jα along each interface segment as follows:

δ jα ¼ τ jτ jα and τ j ¼ τ jG þ τ jL

� ��1, respectively. In this manner, possible heterogeneity in the element

geometry or material properties is accounted for within the expression for the numerical flux.

Observe that by definition, we have that δ jG þ δ j

L ¼ 1.Substituting (53) into (51) and regrouping terms, we arrive at the stabilized interface formulation

for Darcy flow:

BDG

fWG; eVG

þ BD

LfWL; eVL

� qf g; 〚ev〛h iΓI

þ 〚ew〛; pf gh iΓI

þ 〚q〛; δ j〚p〛

� �ΓIþ 〚ew〛; τ j〚ev〛h iΓ I

¼ LDG fWG

þ LDL fWL

(54)

Figure 2. Interface segment γj.



where the pressure (flux) jump and penalty parameter are defined as 〚 p 〛¼pG nG + pL nL and δ j ¼τ jG� ��1 þ τ jL

� ��1h i�1

, respectively. This formulation is form equivalent to standard DG methods for

the Darcy equation (see, e.g., Hughes et al. [14]) except for the enhanced definitions of the numer-ical flux and penalty parameters. Comparing (54) and (2), we conclude that the flux and jump termsplay the role of the interface operator LI uL; uGð Þ.As remarked in Section 3.2, only certain combinations of velocity–pressure interpolations yield

stable results for the standard Galerkin form, and this observation is also relevant for the presentinterface problem. Therefore, we elect to replace the functional forms on the domain interiors ΩG

and ΩL with their stabilized counterparts from Section 3.3, resulting in the final form of the methodfor nonoverlapping solution decomposition: Find VG;VLf g∈ZG � ZL such that for allWG;WLf g∈Z0;G � Z0;L:

BDstab;G WG;VGð Þ þ BD

stab;L WL;VLð Þ � qf g; 〚v〛h iΓIþ 〚w〛; pf gh iΓI

þ 〚q〛; δj〚p〛� �

ΓIþ 〚w〛; τj〚v〛h iΓI ¼ LDstab;G WGð Þ þ LDstab;L WLð Þ (55)

While the modeling approach for overlapping domains leads naturally to a staggered solutionstrategy, the present approach is more amenable to solving for VG and VL in a concurrent fashion.

6.2. Interface operator for Stokes equation

The preceding developments can be easily extended to treat the situation where both GM and LMare governed by the Stokes equation. Let ΩG ⊂ℝnsd be an open bounded region with piecewisesmooth boundary Γ. The number of space dimensions, nsd, is equal to 2 or 3. The governingequations for Stokes flow are given by the following equations:

∇p� div 2με vð Þð Þ ¼ f in Ω Stokes Equationð Þ (56)

divv ¼ 0 in Ω Conservation of Massð Þ (57)

v ¼ g on Γ (58)

where v is the velocity vector, p is the pressure, ε vð Þ ¼ 12 ∇vþ ∇vð ÞT� �

is the deformation ratetensor, μ is the viscosity, f is a source term, and g is the prescribed velocity on the boundary. We alsodefine the stress tensor combining volumetric and viscous effects as σ (u, p)¼� pI + 2μ ε (u).Because our main focus is on developing the interfacial coupling operator, we suppress the

discussion of the weak form for the domain interior and simply adopt the stabilized formulationof Masud and Khurram [20] to accommodate equal-order interpolation spaces. To reduce the for-mulation from the Navier–Stokes equations to the present Stokes equations, we drop the effectsof the time-dependent and nonlinear advection terms. Another relevant note is that the properfunction spaces for the velocity field are H1(Ω) rather than H(div,Ω) in the case of Darcy flow.In regards to the interface, the major differences compared with Section 6.1 are that the viscous effectsare incorporated into the numerical flux, and full continuity of the velocity is imposed across ΓI.Sparing the details, for which the reader is referred to [21] for an analogous discussion of mixedelasticity, the resulting interfacial weak form is stated as follows:



BSTstab;G WG;VGð Þ þ BST

stab;L WL;VLð Þ � σw w; qð Þnf g; 〚v〛h iΓI

� 〚w〛; σ v; pð Þnf gh iΓI� 〚σw w; qð Þn〛; δ j〚σ v; pð Þn〛� �

ΓI

þ 〚w〛; τ j〚v〛h iΓI ¼ LSTstab;G WGð Þ þ LSTstab;L WLð Þ(59)

Starting with the interfacial integrals, the jump and flux terms are defined as follows: 〚v〛= vG� vL,σnf g ¼ δ j

G σGnGð Þ � δ jL σLnLð Þ , and 〚σn〛=σG nG +σLnL, where the arguments to the stress

tensor (either (v, p) or (w, q)) have been suppressed to make the notation compact. Also, in themodified stress tensor σw for the weighting function, the sign of the pressure variable q is reversed:

σw(w, q) = qI + 2με (w). The stabilization tensors are evaluated as δ jα ¼ τ jτ j

α, τj ¼ τ j

G þ τ jL

� ��1, and

δ j ¼ τ jG

� ��1 þ τ jL

� ��1h i�1

, where the individual fine-scale tensors τjα are obtained as follows:

τ jα ¼ meas γj

h i�1∫γj b

jα dΓ

2∫ωj

αμ∇b j

α�∇b jα dΩ

I þ ∫ω j

αμ∇b j

α⊗∇b jα dΩ

h i�1(60)

The domain interior contributions from the bilinear form and linear form BSTstab W;Vð Þ and

LSTstab Wð Þ, respectively, are determined as follows:

BSTstab W;Vð Þ ¼ BST W;Vð Þ þ ∇qþ 2μdiv ε wð Þð Þ; τ ∇p� 2μdiv ε vð Þð Þð Þ (61)

LSTstab Wð Þ ¼ LST Wð Þ þ ∇qþ 2μdivε wð Þð Þ; τ f Þð (62)

in which the standard Galerkin terms are as follows:

BST W;Vð Þ ¼ ε wð Þ; 2μ ε vð Þð Þ � divw; pð Þ þ q; divvð Þ (63)

LST Wð Þ ¼ w; fð Þ (64)

and the Stokes stabilization tensor τ is defined in terms of an element interior bubble function be

according to the formula presented in Section 4.1.1 of [20]:

τ ¼ be∫Ωe be dΩ ∫Ωe μ∇be�∇be dΩ� �

I þ ∫Ωe μ∇be⊗∇be dΩ� ��1

(65)

6.3. Interface operator for combined Stokes–Darcy system

In the following developments, the Darcy regime is indicated as GM and the Stokes regime as LM.When distinct Stokes and Darcy flow regimes are combined within the same modeling domain, theinterfacial conditions between the two regions play a key role in properly modeling the physics. Addi-tionally, the conditions must be mathematically consistent with the different regularity requirementsfor functions in H(div,ΩG) compared with H1(ΩL) that are associated with the Darcy equation andStokes equation, respectively. Herein, we impose the following conditions for the velocity and flux fields:

vG � nG þ vL � nL ¼ 0 on ΓI (66)

nL � σL vL; pLð ÞnL ¼ pG on ΓI (67)

� I � nL⊗nL½ �σL vL; pLð ÞnL ¼ 2μαoffiffiffiκ

p I � nL⊗nL½ �vL on ΓI (68)



Equations (66) and (67) represent the continuity of the normal component of the velocity and fluxfield, respectively. The third condition (68) imposes the so-called Beavers–Joseph–Saffman law[29] to account for the experimentally observed slip at the surface of the porous medium, whereαo is an experimentally determined material parameter. The combination of (66–68) implies thatboth normal and tangential boundary conditions are assigned to the Stokes region, while only thenormal direction conditions are assigned to the Darcy region.The standard Lagrange multiplier formulation obtained by combining the governing equations

for Darcy (10–12) and Stokes (56–58) with weak imposition of the interface conditions (66–68)is as follows:

BDG WG;VGð Þ þ BST

L WL;VLð Þ þ wG�n; λh iΓI� wL�n; λh iΓI

þ wL;2αoμLffiffiffiffiffiffi

KIp I � n⊗n½ �vL

� �ΓI

¼ LDG WGð Þ þ LSTL WLð Þ (69)

μ; vG�nh iΓ I� μ; vL�nh iΓ I

¼ 0 (70)

To derive the interfacial operator for this model system, we combine the developments in thepreceding sections and utilize the local fine-scale modeling procedure on either side of the interface.Presently, the enforcement of normal direction continuity alone implies that the Stokes stabilizationtensor from (60) must be converted into a scalar quantity. Modifying the corresponding develop-ments in Section 6.2 leads to the scalar stabilization parameter: τjL ¼ n�τjLn.Carrying out the steps for the interface modeling and incorporating the domain interior

stabilization for both models, we arrive at the composite interface formulation for Stokes andDarcy flow:

BDstab;G WG;VGð Þ þ BST

stab;L WL;VLð Þ � n�σw w; qð Þnf g; 〚v〛h iΓI � 〚w〛; n�σ v; pð Þnf gh iΓI� 〚σw w; qð Þn〛; δ j

〚σ v; pð Þn〛� �

ΓIþ 〚w〛; τ j〚v〛h iΓI

þ wL;2αoμLffiffiffiffiffiffi

KIp I � n⊗n½ �vL

� �ΓI

¼ LDstab;G WGð Þ þ LSTstab;L WLð Þ(71)

The interface terms in (71) are defined using composite formulas including both the Darcy andStokes contributions:

〚v〛 ¼ vG�nG þ vL�nL (72)

n�σnf g ¼ δ jG �pGð Þ þ δ j

LnL�σLnL; n�σwnf g ¼ δ jG qGð Þ þ δ j

LnL�σwLnL (73)

〚σn〛 ¼ �pGð ÞnG þ σL vL; pLð ÞnL; 〚σwn〛 ¼ qGð ÞnG þ σwL wL; qLð ÞnL (74)

and recall that the modified stress tensor σw(w, q) is defined in Section 6.2.An algorithm for solving the discrete counterpart of the coupled system (71) is provided in Table II.

Remark 9: The stabilization parameters τj, δj and the weighting coefficients δjα encapsulate the infor-mation concerning the material properties, element geometry, and differential operators through thefine-scale models (50) and (60). Additionally, the flux and jump terms are such that the functions fromH(div,ΩG) andH

1(ΩL) mathematically commute across the interface. Thus, these DG terms provide arich underlying mathematical structure to the coupled problem (71).



7. NUMERICAL RESULTS

Figure 3 shows the equal-order elements employed in the numerical studies. In all the test cases, velocityand pressure fields are assumed continuous across elements within individual modeling regions. The fol-lowing quadrature rules were used throughout: linear quadrilaterals, 2� 2 Gauss quadrature; linear trian-gles, four-point quadrature; and higher-order elements, appropriate extensions of the linear rules [30].The bubble functions employed for the domain-based and interface-based stabilization terms are given

in terms of element natural coordinates (ξ, η) by the expressions shown in Tables III and IV, respectively.These particular forms of the bubble functions were employed in the DGmethod and themixed elasticityformulation presented in [21]. The element type abbreviations designate the shape of the element, eithertriangular (T) or quadrilateral (Q), and the number of nodes per element, varying between 3 and 9.

7.1. Convergence rate study

This section presents numerical convergence study for Darcy–Stokes equations that constitute theLM in the proposed heterogeneous model presented in Section 4. The domain under consideration

Figure 3. Equal-order triangular and quadrilateral elements.

Table III. Interior bubble functions employed for fine-scale fields.

Element Bubble function

T3 ξη(1� ξ � η)Q4 (1� ξ2)(1� η2)

Table II. Algorithm for formation and solution of nonoverlapping Stokes–Darcy system.

Assemble contributions from elements ΩeG in Darcy subdomain ΩG:

BDstab;G WG;VGð Þ; LDstab;G WGð Þ from (23), (24) (a)

Assemble contributions from elements ΩeL in Stokes subdomain ΩL:

BSTstab;L WL;VLð Þ; LSTstab;L WLð Þ from (61), (62) (b)

Assemble contributions from interface segments γj corresponding to element pairs ΩeG;Ω

eL

� such that

∂ΩeG∩∂Ω

eL≠∅:

� n�σwnf g; 〚v〛h iΓI� 〚w〛; n�σnf gh iΓI � 〚σwn〛; δj〚σn〛

� �ΓI

þ 〚w〛; τj〚v〛h iΓIþ wL;

2αoμLffiffiffiffiffiffiKI

p I � n⊗n½ �vL� �

ΓI

(c)

Solve the fully coupled system (71) for VG and VL



is a bi-unit square, and the exact solution for the velocity and pressure fields is given as follows(see Appendix A for details):

vx x; yð Þ ¼ � κμ2π cos 2πxð Þ sin 2πyð Þ (75)

vy x; yð Þ ¼ � κμ2π sin 2πxð Þ cos 2πyð Þ (76)

p ¼ 1þ 16κπ2� �

sin 2πxð Þ sin 2πyð Þ (77)

The material properties are taken as κ,μ = 1, the volumetric flow rate φ is calculated from (26) bytaking the divergence of the velocity field, and the prescribed velocity g is evaluated from the exactsolution (75) and (76). In specifying the boundary value problem, φ is prescribed over Ω, while g isprescribed at the boundary. Uniform meshes were employed in obtaining the results presented inthis section.For linear quadrilateral elements, the meshes employed consisted of 400, 1600, and 6400

elements. The linear triangular element meshes consisted of exactly twice as many elements. Theelement mesh parameter, h, is taken to be the edge length of the quadrilaterals and the short-edgelength for triangles.Figures 4 and 5 show plots of the numerical error with respect to the exact solution for the linear

triangles and quadrilaterals, respectively, measured in the L2-norm for the velocity field and L2-norm and H1-seminorm of the pressure field. Theoretical rates are achieved for the L2-norm ofvelocity. Nearly optimal rates for both error measures of the pressure field are also attained forthese cases.

Table IV. Edge bubble functions employed for fine-scale fields.

Element Bubble function

T3 4ξ(1� ξ � η)Q4 1

2 1� ξ2� �

1� ηð ÞT6 4ξ2(1� ξ � η)2

Q9 14 1� ξ4� �

1� ηð Þ2 þ 14 1� ξ2� �

1� ηð Þ

Figure 4. Convergence rates for equal-order linear triangles.



7.2. Five-spot problem

7.2.1. Darcy–Stokes equations. This section presents numerical results for a quarter of the five-spot problem. The bi-unit square domain shown in Figure 6 has prescribed velocity at the sourceand the sink. Because of symmetry of the problem, zero normal flow is prescribed along theboundaries. We assumed the divergence of the velocity field, φ, was a Dirac delta function actingat source and sink, with strengthþ1

4 and�14, respectively. We calculated an equivalent distribution

of normal velocity, φ, and drove the problem with φ, setting φ¼ 0.For linear elements, we assumed a linear distribution of ψ along the external edges of the corner

elements. This uniquely determines the distribution of ψ on the edges, as depicted in Figure 7 for

Figure 5. Convergence rates for equal-order bilinear quadrilaterals.

Figure 6. Schematic diagram of a quarter of the five-spot problem.

Figure 7. Distribution of ψ along corner elements at sink for linear elements.



the sink well. Distribution of ψ at source well is equal but opposite in direction. Parameter valuesare set as μ = 1 and κ = 0.5.For the stabilized Darcy model presented in (22), the results for the five-spot problem are given in

Masud and Hughes [13]. In this section, we present the five-spot problem for Darcy–Stokes model.Figure 8 shows the pressure contour in 3D for the 1600 element mesh using four-node linearquadrilaterals.Figures 9 and 10 show the pressure profile along the diagonal from the source to the sink for the

four-node quadrilaterals and three-node triangles, respectively. Figure 11 shows the comparison ofthe pressure values along the diagonal for the four-node and three-node elements.

7.2.2. Patched Darcy domain. This test case considers the five-spot problem for the Darcy flowwhere the domain is composed of subregions that have been patched together via the interfaceoperator presented in Section 6.1. We also compare the performance of the proposed interfaceoperator with the operator that could have been developed using the standard Nitsche approach[31] in the context of DG methods. Symmetry boundary conditions for the normal componentof the velocity field are prescribed along each edge.In this test case, we focus on the checkerboard version of the problem where the four zones in

Figure 12 have a large discrepancy in permeability; namely, we set κ/μ = 1 in zones I and IV and

Figure 8. 3D pressure profile for 1600 four-node linear quadrilateral elements.

Figure 9. Pressure along the diagonal for four-node elements.



κ/μ= 0.01 in zones II and III. Additionally, we treat the boundaries of these zones to be interfacesalong which continuity is weakly imposed, and nonconforming meshes are permitted. Weinvestigate the performance of the proposed interface method and the standard Nitsche methodfor this rather difficult problem. For the Nitsche method, we take the penalty parameters for thevelocity and pressure jumps to be τ j ¼ max μα=καð Þϵh and δ j= (τ j)� 1, respectively, with ϵ ¼ 1

4.

Figure 11. Pressure along the diagonal for refined meshes of three-node and four-node elements.

Figure 10. Pressure along the diagonal for three-node elements.

Figure 12. Problem description for patched domain for five-spot problem.



We first present results for a conforming mesh of Q4 elements with 10� 10 elements in eachzone. The pressure field p and velocity component vy obtained from the proposed interface methodare plotted in Figure 13. The checkerboard pattern for the permeability induces a sharp front in thecenter of the bi-unit square as evidenced by the discontinuous pressure field and the spikes in thevelocity field. The pressure contour plot agrees closely with the results from the discontinuouspressure nine-node quadrilateral elements in [13].In Figure 14, we compare the interfacial pressure variation along the boundary between zones I

and III obtained from the stabilized interface formulation and the Nitsche method. Similar to [21],we evaluate the quality of the computed solution using two measures of the flux at the interface,

termed the ‘total flux’ λ= {ph} + τ j〚vh〛 and the ‘gradient flux’ eλ ¼ ph�

. In both cases, the resultsdo not exhibit fluctuations, although the VMS-based interface coefficients produce results that aresmoother. Also, the gap between the gradient flux and total flux for the Nitsche method results islarger near the singular point x ¼ y ¼ 1

2. This can be partially attributed to the higher penalty termfor the Nitsche method, which is τ j= 1.25 compared with τ j= 0.005 obtained from (50) and (53).Next, we make the problem more challenging by discretizing each of the zones using different

element types: T3 in zone I, Q9 in zone II, T6 in zone III, and Q4 in zone IV. The same numberof nodes (11� 11) is employed for each zone, and we compare again the results from the twointerface methods. The discretization for this example is shown in Figure 15 in the backgroundof the velocity contour plots. These results are comparable with the field obtained for the all-quadrilateral mesh, although the velocity field in the center of the domain from the T3 elementsis slightly higher in Figure 15(a). Plots are given in Figure 16 for the interfacial pressure along eachof the four boundaries between the pairs of zones. In all cases, we observe that the profiles from the

Figure 13. Solution contours for conforming mesh: (a) pressure p and (b) velocity vy.

Figure 14. Interface pressure along zone I–III interface.



proposed interface method are smoother than those from the Nitsche method. Particularly, thepressure field from the Nitsche method along the interface between zones II and IV is quiteoscillatory. Also, the gradient flux and the total flux as obtained from the stabilized interface methodexhibit closer agreement along the interfaces. From these results, we conclude that the proposedinterface method is more effective than the standard Nitsche method at coupling dissimilar elementtypes across material interfaces.As a final example, we solve the same problem again using distorted meshes in each of the zones.

The contour plot for the velocity field on these distorted meshes obtained from both methods ispresented in Figure 17. We observe that oscillations appear for the proposed method in the velocityvy field along the interface between zones I and III where five T3 elements border a single T6element. These oscillations are not visible in the results from the Nitsche method, and in all othercases, the fields appear rather smooth.

Figure 15. Velocity vy contour plots: (a) stabilized interface method and (b) Nitsche method.

Figure 16. Interface pressure: (a) zone I–III interface, (b) zone I–II interface, (c) zone III–IV interface, and(d) zone II–IV interface.



However, the interfacial pressure curves in Figure 18 tell a completely different story. While thepressure field along the interfaces bounding zone III in Figure 18(a) and (c) exhibits mildoscillations for the VMS results, the curves from the Nitsche method contain sharp oscillationsalong all four interfaces. Also, the gradient flux and total flux curves for the Nitsche results oftenexhibit significant discrepancies. These results for this difficult numerical example illustrate theadvantages offered by the consistent derivation of the penalty and flux terms provided by theproposed interface method.

7.3. Stokes–Darcy coupled problem

As another example of the case of nonoverlapping solution decomposition, we consider a coupledStokes–Darcy problem posed over a bi-unit square domain. The Darcy model occupies the regionΩG ¼ 0; 1ð Þ � 0; 12ð Þ, and the Stokes model occupies the region ΩL ¼ 0; 1ð Þ � 1

2; 1ð Þ. We devise an

Figure 17. Velocity vy contour plots: (a) stabilized interface method and (b) Nitsche method.

Figure 18. Interface pressure: (a) zone I–III interface, (b) zone I–II interface, (c) zone III–IV interface, and(d) zone II–IV interface.



exact solution similar to those proposed in [26] that satisfies the interfacial conditions between thetwo regions:

vGx ¼ sinx

Gþ ω

ey=G þ ω sin ωxð Þ (78)

vGy ¼ � cosx

Gþ ω

ey=G (79)

pG ¼ μGκ

cosx

Gþ ω

ey=G þ μ

κcos ωxð Þ (80)

vLx ¼ sinx

Gþ ω

ey=G (81)

vLy ¼ � cosx

Gþ ω

ey=G (82)

pL ¼ yþ e1=2GμGκ

� 2μG

� �cos

x

Gþ ω

þ 2μy

κcos ωxð Þ � 1

2(83)

where the parameter G ¼ ffiffiffiκ

p=αo . The associated body force terms ρgG/gc, φG, fL are obtained by

substituting these analytical expressions into the governing equations (10), (11), and (56) (note thatgG = 0 by design). Notice that both the tangential component of velocity and the pressure field arediscontinuous across ΓI;LG ¼ 0; 1ð Þ � 1

2

� , while the normal component of velocity and the stress

are continuous. Additionally, careful attention is paid to satisfying the Beavers–Joseph–Saffman law(68) at the lower boundary of the Stokes region. For the numerical simulations that follow, DGinterfaces are also inserted to partition the two model regions to allow nonconforming meshes, namely,ΓI;GG ¼ 1

2

� � 0; 12ð Þ and ΓI;LL ¼ 12

� � 12; 1ð Þ (Figure 19). As boundary conditions, the normal velocity

vG � n is prescribed on the left edge of the Darcy region along with flux conditions consistent with thepressure pG on the lower and right edges. The velocity field vL is prescribed on the left edge of the

Figure 19. Stokes–Darcy coupled domain problem description.



Stokes region along with flux conditions on the upper and right edges. Lastly, the material propertiesare taken as κ = 1, μ = 1, αo= 0.5, and ω = 1.05.Two convergence rate studies employing linear elements are presented. The first involves linear

triangular and quadrilateral elements on uniform structured meshes, where the coarsest meshcontaining 87 elements is depicted in Figure 20(a). A contour plot of the pressure field obtainedon the mesh with 348 elements is provided in Figure 20(b). We remark that the interfaces betweenthe triangular and quadrilateral elements of each modeled region are nearly invisible, while thediscontinuity in the pressure between the Darcy and Stokes regions is accurately resolved.The convergence of the numerical error between the computed and exact solutions is provided in

Figure 21(a) and (b) separately for the Darcy region ΩG and the Stokes region ΩL, respectively. Forthe Darcy region, the convergence rates of the pressure error norms achieve the theoreticallypredicted rates of 2 and 1 for equal-order elements as analyzed in [13], while the velocity L2-normexhibits a consistent trend of 1.67. By comparison, the velocity error norms in the Stokes regionattain the theoretical rates of 2 and 1 according to the analysis conducted in [32] for a similarformulation, and the pressure error rates of 1.5 and 0.5 also conform to the behavior reported inthe aforementioned reference.Next, a convergence study is performed on intentionally distorted meshes, where the coarsest

discretization is depicted in Figure 22(a). This study highlights the benefit of having computableexpressions for the stabilization parameters and flux weighting coefficients for highlynonconforming interfaces. Contour plots of the solution obtained on the mesh with 384 elementsare presented in Figure 22(b)–(d). The major features of the solution field are well resolved; namely,the vertical component of velocity is nearly continuous between the two modeling regions, and thejumps in the field are controlled across the vertical interfaces.The convergence of the error measured in the various norms for each subdomain is reported in

Figure 23. Even in the presence of significant mesh distortion and nonconformity at the discreteinterfaces, the optimal convergence rates are obtained by the pressure field in the Darcy region

Figure 20. Uniform structured mesh: (a) coarse mesh and (b) pressure field.

Figure 21. Convergence rates for structured meshes: (a) Darcy region and (b) Stokes region.



and by the velocity field in the Stokes region. Some degradation of the rates is present for thevelocity field in the Darcy region and for the pressure field in the Stokes region.

7.4. Patched domains for heterogeneous flows with overlapping global and local models

The five-spot problem is modified for the heterogeneous case presented in Section 5, and a patcheddomain is created to further test the heterogeneous flow model. Figure 24 represents the configura-tion, where zone I, II, and IV are governed by Darcy equations, and zone III is governed by theheterogeneous model. For zone III, τ ¼ κ

μ. In this problem, low permeability values are associatedwith Darcy flow, and high permeability values are associated with the heterogeneous model withembedded Darcy–Stokes flow. Table V presents the permeability values for various soil types [33].This table serves as the basis for the selection of the subdomain ΩL⊆ΩG where the

heterogeneous model, that is, Darcy (GM) with embedded Darcy–Stokes (LM) is employed. For

Figure 22. Distorted mesh: (a) coarsest mesh, (b) velocity vx field, (c) velocity vy field, and (d) pressurefield.

Figure 23. Convergence rates for distorted meshes: (a) Darcy region and (b) Stokes region.



the numerical test cases presented, permeability κ⩾ 1 was the criteria for the selection of the zoneΩL⊆ΩG with coexisting GM and LM, and κ< 1 was the criteria for spatial domain ΩG\ΩL

governed by GM alone.Figure 25 shows the (a) 3D pressure profile and the (b) 3D absolute velocity profile for Darcy

model in zones I and II and IV and the heterogeneous model in zone III. A higher resultant velocityis observed in zone III.

7.5. Highly heterogeneous permeability field

As a final problem, we consider a physically motivated example in which the permeability of themedium exhibits significant variation throughout the domain. Let ΩG be a 1200� 2200 rectangulardomain discretized into a grid of 60� 220 linear quadrilateral elements. The value of the isotropic

Figure 24. Modified checkerboard domain.

Table V. Permeability values for various unconsolidated sediments.

Medium Permeability (cm/s)

Clay 10�9–10�6

Silt, sandy silt, clayley sands, and till 10�6–10�4

Silty sands and fine sands 10�5–10�3

Well-sorted sands and glacial outwash 10�3–10�1

Well-sorted gravels 10�2–1

Figure 25. (a) Pressure profile and (b) absolute velocity profile for Darcy flow with κ = 0.00001, τ = 0.5 inzones I, II, and IV and heterogeneous flow, τ ¼ κ

μ in zone III.



permeability coefficient within each element is taken from the top layer of the 85 layers pro-vided in the data set for the 10th Society of Petroleum Engineers (SPE) comparative solutionproject [34], the distribution of which is plotted on a log scale in Figure 26(a); the same perme-ability layer was used in [35]. The data set contains orthotropic permeability coefficients foreach 20� 10� 2 element where the values are equal in the x–y plane (κx= κy) and a distinctvalue for κz; therefore, we elect to use the common value κ = κx= κy for the 2D analysesperformed for this work. The viscosity of the fluid in the porous medium is specified as μ = 0.3.Herein, we compare the results obtained from three different models: Darcy flow in the entire

domain, Darcy–Stokes in the entire domain, and a heterogeneous model with Darcy–Stokes appliedas the LM within select subdomains. The objective is to verify that the Darcy results can becorrected through the heterogeneous modeling method to approach the results obtained from theDarcy–Stokes model, thereby realizing a cost savings by applying the more complex modeljudiciously only within regions of interest. In this manner, we provide upscaling of the physicsthrough the mathematical model of the underlying physics. In all cases, the normal component ofthe boundary conditions are prescribed as shown in Figure 26(b), where v·n¼ 0 on all edges exceptalong strips in the upper right and lower left on which the normal component of the flux (pressurefor Darcy and combined stress for Darcy–Stokes) is prescribed. Also, the value p= 100 isprescribed to the node in the lower-left corner to fix the constant part of the pressure. For the globalDarcy–Stokes model, zero tangential Neumann conditions are specified along the entire boundary.Finally, the boundaries of the local Darcy–Stokes subdomains are depicted in Figure 26(b), whichare selected to enclose regions in which the permeability is relatively higher. Recall from Section 5that homogeneous Dirichlet boundary conditions are assigned to the local problems to allow forsuperposition of the solutions. For simplicity, the same grid size is employed for both the LMand GM so that the meshes coincide within these subdomains. Because the discrete function spacesfor the GM and LM coincide, namely, Co-conforming, a projection is not required within thecalculation of the residual term for the LM in (45). Thus, the discrete global solution can be directlysubstituted into the LM forcing terms. We remark that the method admits the use of a finer grid sizeto resolve the LM, for which a more elaborate implementation may be required.The pressure field obtained from all three simulations is nearly identical; the result for the Darcy

model is shown in Figure 27. The region of low permeability in the bottom portion of the domainhinders the flow and causes the main pressure drop to occur along the vertical flow through this

Figure 26. Heterogeneous permeability field: (a) log permeability and (b) problem description.



region. However, the velocity profiles computed from the three models do exhibit distinctivefeatures, as presented in Figures 28 and 29. For instance, the x-component of velocity in theDarcy simulation possesses sharp features in the zones of high permeability that may be aninaccurate representation of physics; these features are diffused out in the Darcy–Stokes modelin the center. Within the three subdomains where the LM is added via the heterogeneousmodeling method, the Darcy solution is corrected to be in closer agreement with the Darcy–Stokes solution. In particular, notice the region in the lower-right corner of each plot ofFigure 28: While the Darcy solution indicates a red contour zone, the Darcy–Stokes and hetero-geneous model exhibit only yellow contours. Similar behavior is also noted in the contours ofthe vy contour plots, particularly in the upper-right and lower-right regions. Thus, the heteroge-neous modeling method is seen to effectively correct the Darcy solution to approach the Darcy–Stokes solution. The benefit is that the cost of evaluating the second derivatives for the Laplaceoperator and the bubble functions for the stabilization tensor are localized only to the subregionsrather than the whole domain. These features can result in substantial reduction in the cost ofcomputation in 3D calculations over larger domains.

Figure 27. Pressure field from Darcy model.

Figure 28. Velocity vx from each model: Darcy model (left), Darcy–Stokes model (center), and heteroge-neous model (right).



8. CONCLUSIONS

A new method for overlapping and nonoverlapping coupling of hierarchically refined mathematicalmodels is presented that yields a heterogeneous modeling paradigm for porous media flows. In theheterogeneous modeling method, coarse-scale features are resolved via global model (GM), whilethe fine-scale features are resolved via a mathematically refined set of equations termed as the localmodel (LM). The proposed framework accommodates stabilized methods for Darcy, Stokes, andDarcy–Strokes equations. The Darcy and Darcy–Stokes models are coupled via the hierarchicalmultiscale modeling framework proposed in Masud and Scovazzi [12]. For nonoverlapping decom-position of the models, an interface coupling operator is derived that accommodates matching andnonmatching meshes along the interface. In addition, these operators accommodate differentmechanical material properties across the common interface and provide an effective way to matchsolutions from different PDEs that may in fact live in different spaces of functions. This modelingmethod not only couples physics-based mathematical models but also, because of the integral natureof the interface terms, provides a unification of solutions in a consistent fashion. The method is appliedto linear triangle and quadrilateral elements with equal-order pressure–velocity combinations.Numerical tests show stable and convergent behavior for both element types for a variety ofbenchmark problems. Test cases with nonoverlapping decomposition with Darcy and Stokes acrossthe common interface, and overlapping decomposition with GM (Darcy) and LMs (Darcy–Stokes)are presented to show the range of applicability of the method.

APPENDIX A

In this section we design a model problem for the convergence rate study of Darcy–Stokesequations. Consider (25), set fx= fy= 0 and rearrange the problem:

�∇p ¼ μκν� μΔν (A1)

The two components of ∇p are as follows:

�p;x ¼μκvx � μ 2vx;xx þ vx;yy þ vy;xy

� �(A2)

�p;y ¼μκvy � μ 2vy;yy þ vy;xx þ vx;xy

� �(A3)

Figure 29. Velocity vy from each model: Darcy model (left), Darcy–Stokes model (center), and heteroge-neous model (right).



We assume the velocity field to be of the following form:

vx x; yð Þ ¼ � κμ2π cos 2πxð Þ sin 2πyð Þ (A4)

vy x; yð Þ ¼ � κμ2π sin 2πxð Þ cos 2πyð Þ (A5)

Substituting (A6) and (A7) into (A4) and (A5) yield the following components of the pressure field:

�p;x ¼ 2π 1þ 16κπ2� �

cos 2πxð Þ sin 2πyð Þ (A6)

�p;y ¼ 2π 1þ 16κπ2� �

sin 2πxð Þ cos 2πyð Þ (A7)

Integrating (A6) with respect to x yields the following:

p ¼ ∫p;x d x ¼ 1þ 16κπ2� �

sin 2πxð Þ sin 2πyð Þ þ C (A8)

The same expression is obtained via integrating with respect to y. Setting the mean value of pressure tobe zero yields the expression for the exact pressure field:

p ¼ 1þ 16κπ2� �

sin 2πxð Þ sin 2πyð Þ (A9)

From (26), we can evaluate the volumetric flow rate as follows:

φ ¼ vx;x þ vy;y ¼ 8π2κμ

sin 2πxð Þ sin 2πyð Þ (A10)

ACKNOWLEDGEMENTS

G. Hlepas was supported by the Department of Defense SMART scholarship program. T. Truster wassponsored by an NSF Graduate Research Fellowship. These supports are gratefully acknowledged.

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