On the preferential flow patterns induced by transverse

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Comput. Methods Appl. Mech. Engrg. 353 (2019) 570–592www.elsevier.com/locate/cma

On the preferential flow patterns induced by transverse isotropy andnon-Darcy flow in double porosity media

Qi Zhanga, Jinhyun Choob, Ronaldo I. Borjaa,∗

a Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USAb Department of Civil Engineering, The University of Hong Kong, Pokfulam, Hong Kong

Received 30 January 2019; received in revised form 7 April 2019; accepted 25 April 2019Available online 6 June 2019

Abstract

Fluid flow in isotropic porous media with one porosity scale is a well understood process and a common scenario innumerous simulations published in the literature. However, there exists a class of porous materials that exhibit two porosityscales with strong permeability contrast between the two scales. Examples of such materials are aggregated soils and fracturedsedimentary rocks such as shale. In sedimentary rocks, fluid could flow through the micro-fractures at the larger scale as well asthrough the nanometer-size pores of the rock matrix at the smaller scale. In this paper, we shall refer to the larger and smallerpores of sedimentary rocks as the micro-fractures and nanopores, respectively. Due to preferentially oriented micro-fractures inthe rock, fluid could flow predominantly in the direction of the discontinuities, resulting in an anisotropic flow pattern at thelarger scale. We idealize such material as a transversely isotropic medium with respect to fluid flow. In addition, the nanoporesof sedimentary rocks such as shale are so small that Darcy’s law may not hold at this scale. To better understand the impactof non-Darcy flow on the overall flow pattern, we present a hydromechanical model for materials with two porosity scales thataccommodates both transverse isotropy at the larger scale and non-Darcy flow at the smaller scale. Even though this studyis motivated by shale properties, the discussion revolves around a generic material with two porosity scales whose fluid flowcharacteristics are similar to those of shale. The overarching goal of this paper is to better understand the impacts of transverseisotropy and non-Darcy flow on the fluid flow pattern in this material.c⃝ 2019 Elsevier B.V. All rights reserved.

Keywords: Double porosity; Mixed finite element; Non-Darcy flow; Preferential flow; Transverse isotropy

1. Introduction

Fluid flow through porous media is an intriguing research topic in theoretical and computational poromechanics.Generally speaking, there are two classes of methods used for analyzing this problem: discrete methods andcontinuum methods [1]. In this paper, we consider a continuum method based on mixture theory [2–7]. Novel finiteelement methods have been proposed, and meaningful numerical simulation results have been generated based onthis theory [8–12]. A key advantage of mixture theory is its capability to handle complex pore structures withrelatively simple homogenized quantities such as volume fractions and overall stiffness, which could otherwise bevery complicated to quantify [13]. However, this simplification is feasible only if the pore sizes are approximately

∗ Corresponding author.E-mail address: [email protected] (R.I. Borja).

https://doi.org/10.1016/j.cma.2019.04.0370045-7825/ c⃝ 2019 Elsevier B.V. All rights reserved.

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uniform [14], which means that there is only one porosity scale of interest [15,16]. On the other hand, fissuresand micro-fractures could also transmit fluids particularly if they are connected and continuous [17–20]. Somefissures can be seen with the naked eye, so-called macro-fractures which are always incorporated in 3D geologicalmodeling [21], but there are also micro-fractures that can only be seen with the aid of a microscope [22,23]. Due tosignificant difference in the pore sizes, fluid transmission through the fissures and through the matrix pores couldbe very distinctive. In general, fissures serve as the primary conduit for fluid flow, whereas the smaller pores in therock matrix generally play a supporting role in the fluid transmission, and for the most part simply discharge fluidinto the fissures [24–28]. This distinctive flow mechanism poses new challenges to the modeler, because the muchsimpler single-porosity formulation is no longer sufficient to capture such flow mechanism.

In 1960, Barenblatt et al. [17] introduced the concept of double porosity for materials with two distinctive porosityand permeability scales. Subsequently, this concept has been used in petroleum engineering, subsurface hydrology,geotechnical engineering, and many other branches of earth sciences [29–38]. In this conceptual framework, twopore regions are assumed to overlap within the solid matrix. In the context of aggregated soils, they are usuallytermed macropores and micropores [13], whereas in the present context for sedimentary rocks, we shall call themas the micro-fractures and nanopores. It must be noted that the macro-fractures described earlier constitute another(much larger) porosity scale, which is not covered in the present discussion. Also, attempts had been made in thepast to replace a double-porosity structure with an equivalent single-porosity formulation [39–41], but the grosssimplification often produced misleading results [14,42].

One important application of double porosity concept is the solution of coupled solid deformation/fluid flowproblems in poromechanics [43–49]. Efforts along this line have resulted in numerous publications over the last30 years, see [50–55] for a sampling of these publications. A crucial element missing in most formulations is athermodynamic basis for the so-called effective stress. In the paper by Borja and Koliji [56], an expression for theeffective stress has been derived using mixture theory and continuum principles of thermodynamics. Because of thevolume averaging employed in mixture theory, the volume fractions played a central role in the expression for theeffective stress. Following [56], Choo and Borja [13] developed a stabilized mixed finite element for deformablesaturated double porosity media, which they later generalized to the unsaturated flow regime [42]. Subsequently,Borja and Choo [57] extended the formulation to the finite deformation range by incorporating an evolving internalstructure into the constitutive description of the solid mechanical response.

Sedimentary rocks such as shale can exhibit strong anisotropy in both their mechanical and fluid flow behav-iors [58,59]. The existence of bedding planes arising from natural sediment deposition and the preferred fractureorientation developed from previous loading history often result in structure-induced anisotropy [60]. A numberof papers have been published on anisotropic fluid flow, with the fluid flow equations generally uncoupled fromsolid deformation [61–66]. A special case of anisotropy is transverse isotropy that can impact both the mechanicaland fluid flow responses [67–71]. For fluid flow, transverse isotropy is reflected in the hydraulic conductivity(i.e. permeability) through the orientation of the unit normal vector to the plane of isotropy. Experimental methodshave been explored to measure the impact of anisotropy on the hydraulic conductivity [72–74]. However, despitethese efforts a coupled hydromechanical model encapsulating transverse isotropy is yet to emerge.

In poromechanics, the default constitutive theory is Darcy’s law. While this law may hold in a majority ofcases, it is hardly valid when one deals with tight and shale reservoirs where the pore sizes are on the order ofnanometers. Experimental studies have shown that the fluid flux deviates from that calculated using Darcy’s lawdue to the interaction between fluid particles and the solid pore wall. In addition, Wang and Sheng [75] pointed outthat non-Darcy flows for gas and liquid can be quite different. For gas flow, Knudsen diffusion, slippage flow, andgas desorption significantly increase the total amount of gas flux in unconventional reservoir. Numerous theoreticalmodels based on Knudsen number Kn have been developed to calculate the apparent permeability and predict gasproduction [76–87]. However, for liquid flow, the Knudsen number loses its meaning, since there is no desorptionof liquid. Instead, as reported in the literature [88–90], the interaction forces between liquid and rock lead to theformation of a boundary layer on which the liquid exhibits higher viscosity. As a result, the average fluid flowvelocity tends to be lower than the value calculated from Darcy’s law. This phenomenon is known in the literatureas low-velocity non-Darcy flow.

It has been argued that when the pressure gradient is extremely low, say, lower than a threshold pressuregradient (TPG), the boundary layer would prevent the fluid to flow. In addition to the TPG, a pseudo thresholdpressure gradient (PTPG) has also been defined in the literature as the intersection between the extension of

572 Q. Zhang, J. Choo and R.I. Borja / Computer Methods in Applied Mechanics and Engineering 353 (2019) 570–592

the linear flow plot and the horizontal axis, see [91–93]. However, due to limitations in experimental setupand measurement accuracy, the TPG cannot be measured directly, and some authors even argue that suchthreshold does not exist. Despite controversies, mathematical models describing low-velocity non-Darcy flow areprevalent in the literature [75,94–97]. In single-porosity models the non-Darcy flow changes the magnitude of thepermeability, whereas in double-porosity models the interface permeability k also depends on the permeability of thenanopores [17,98,99]. In other words, in double-porosity media the term “non-Darcy flow” pertains not only to theflow of fluid through the nanopores, but also to the discharge of fluid from the nanopores into the micro-fractures.In this work, we will only consider non-Darcy flow for liquid.

The aim of this paper is to develop a thermodynamically consistent framework for hydromechanical modelingof double-porosity media in which fluid flow through the micro-fractures exhibits transverse isotropy while fluidflow through the nanopores follows a low-velocity non-Darcy law. It is the first time, to the authors’ knowledge,that these combined effects are integrated within the framework of computational poromechanics. Essentialingredients for combining these mechanisms are presented under the assumption of infinitesimal deformation andfull saturation. The paper is organized as follows: Conservation laws for double porosity media are first formulatedincorporating a thermodynamically consistent effective stress. Constitutive frameworks are then developed for thesolid matrix, micro-fractures, and nanopores, including the mass transfer between the nanopores and micro-fractures.Subsequently, a three-field mixed finite-element formulation is developed for the numerical solution of fully coupledflow and deformation problems. Numerical examples are presented to demonstrate the impacts of double porosity,transverse isotropy, and non-Darcy flow on the preferential flow patterns in such material.

2. Conservation laws

In this section, we review the basic double porosity formulation developed in [13,42,56], in which the balanceof mass is formulated with respect to the larger and smaller porosity scales separately, while the balance of linearmomentum is formulated with respect to the total mixture. We also present an expression for the rate of changeof internal energy that will prove to be useful for incorporating transverse isotropy and non-Darcy flow into thetheory.

2.1. Balance of mass

Consider a double-porosity sedimentary rock in which the fluid fully saturates both the micro-fractures andnanopores. We use the concept of volume fractions [100] to smear the masses of solid and fluid over the entiredifferential volume dV of the mixture as follows,

φs=

dVs

dV, φM

=dVM

dV, φm

=dVm

dV, (1)

where index s refers to the solid, index M to the fluid in the micro-fractures, and index m to the fluid in thenanopores. Throughout this paper, we use superscripts to denote partial quantities, and subscripts to denote intrinsicquantities of a constituent. Volume fractions satisfy the closure relation

φs+

∑i=m,M

φi= 1 . (2)

The partial mass densities can be defined using the volume fractions and intrinsic mass densities according tothe equations

ρs= φsρs , ρM

= φMρ f , ρm= φmρ f , (3)

where index f denotes the pore fluid itself such as water, and ρ with upper and lower indices denote partialand intrinsic mass densities of each constituent, respectively. Hereafter, we assume that the fluid is incompressibleand the processes are isothermal. Pore fractions define the portions of the total pore volume occupied by the twodominant pore regions, which is given as:

ψM=

φM

1 − φs, ψm

=φm

1 − φs,

∑i=M,m

ψ i= 1 . (4)


Adopting the material time derivative following the solid phase motion, the balance of mass for water in themicro-fractures and nanopores is given by the pair of equations [4,6]:

dρM

dt+ ρM div v + div wM = cM , (5)

dρm

dt+ ρm div v + div wm = cm . (6)

Here, v is the velocity of the solid matrix, cM and cm are the mass transfer terms between the two pore scales,which satisfy the closure condition cM

+ cm= 0, and wM and wm are the Eulerian relative flow vectors defined as

wM = ρM (vM − v) , (7)

wm = ρm (vm − v) , (8)

where vM and vm are intrinsic fluid velocities in the micro-fractures and nanopores, respectively. If ρ f = constant,then we can simplify Eqs. (5) and (6) to the form

dφM

dt+ φM div v + div q M =

cM

ρ f, (9)

dφm

dt+ φm div v + div qm =

cm

ρ f, (10)

where q M = wM/ρ f and qm = wm/ρ f are the superficial (Darcy) velocity vectors (average relative velocity ofseepage per unit total area), as defined in [6,9,101].

Assuming barotropic flow for the solid grains [102], we can expand the first terms in Eqs. (9) and (10) as follows,

dφM

dt=

(1 − φs) dψM

dt+ ψM

(φs

Ks

dps

dt+ φs div v

), (11)

dφm

dt=

(1 − φs) dψm

dt+ ψm

(φs

Ks

dps

dt+ φs div v

), (12)

where Ks and ps are the bulk modulus of and intrinsic pressure in the solid grain. Rewriting Eqs. (9) and (10)using Eqs. (11) and (12) yields(

1 − φs) dψM

dt+ ψM div v +

ψMφs

Ks

dps

dt+ div q M =

cM

ρ f, (13)

(1 − φs) dψm

dt+ ψm div v +

ψmφs

Ks

dps

dt+ div qm =

cm

ρ f. (14)

The third terms in the foregoing equations contain the intrinsic solid pressure ps , which we can eliminate by makingthe substitution [4]

φs dps

dt= −K div v , (15)

where K is the bulk modulus of the solid matrix. Inserting Eq. (15) into Eqs. (13) and (14), we obtain(1 − φs) dψM

dt+ ψM B div v + div q M =

cM

ρ f, (16)

(1 − φs) dψm

dt+ ψm B div v + div qm =

cm

ρ f, (17)

where B = 1 − K/Ks is the so-called Biot coefficient. Eqs. (16) and (17) are the mass balance equations for thefluids in the micro-fractures and nanopores, respectively.

2.2. Balance of linear momentum

The balance of linear momentum for the entire mixture under quasi-static condition takes the form

div σ + ρg = c , (18)


where σ is the total Cauchy stress, ρ = ρs+ ρM

+ ρm is the total mass density of the mixture, g is the gravityacceleration vector, and c is the momentum exerted by the mass transfer between the micro-fractures and nanopores.The expression for c is given as follows:

c =

∑i=m,M

ci (vi − v) . (19)

2.3. Balance of internal energy

To solve Eqs. (16)–(18), we need to impose some constitutive relationships, which can be established with theaid of the first law of thermodynamics. Following the developments for unsaturated flow presented in [4], andconsidering only the mechanical terms, the rate of change of internal energy per unit volume of the mixture can bewritten as:

ρe = σ s: ϵs + σ M

: ϵM + σ m: ϵm +

12

cMvM · vM +12

cmvm · vm , (20)

where e is the internal energy per unit total mass of the mixture, σ s, σ M and σ m are partial Cauchy stress tensor,ϵs , ϵM and ϵm are infinitesimal strain tensors. From Eq. (20), we know that σ s is work-conjugate to ϵs ; however,there are other possible constitutive stresses that are also work-conjugate to ϵs and we need to extract them fromthe remaining terms in Eq. (20).

By adopting the definition of Ks , K , B, and assuming an isotropic fluid pressure, the expression for the rate ofchange of internal energy can be re-arranged following the lines presented in [42,56] to yield

ρe = (σ + B p1) : ϵs +

∑i=m,M

(vi − v) ·(gradφi) pi

+(1 − φs) (pM − pm)

dψM

dt−

∑i=m,M

ci[

pi

ρ f−

12vi · vi

].

(21)

In the above equation, pM is the intrinsic fluid pressure in the micro-fractures, pm is the intrinsic fluid pressure inthe nanopores, 1 is the second-order identity tensor,

σ ′= σ + B p1 (22)

is the effective Cauchy stress tensor, and p is the mean pore fluid pressure given by

p = ψM pM + ψm pm . (23)

Note the similarity of Eq. (23) with the expressions presented in [42,56,103].Using the symbol

⟨, . . . ,

⟩to denote energy-conjugate pairing, we can rewrite Eq. (21) in a more abstract form

as follows,

ρe =⟨σ ′, ϵs

⟩+

∑i=m,M

⟨vr

i , φi , pi

⟩+

⟨1 − φs,Π , ˙ψM

⟩−

∑i=m,M

⟨ ci

ρ f, pi , vi

⟩,

(24)

where vri = vi − v is the relative velocity of fluid with respect to solid at pore scale i (i = M,m), and

Π = pM − pm is the fluid pressure difference between the two pore scales. Eq. (24) yields four energy-conjugaterelations corresponding to four constitutive relations that must be established to relate the relevant field variables.Two of these constitutive relations (the terms with the summation sign) are defined separately at each pore scale.Specific constitutive models for these energy-conjugate pairs are presented in the following section.

2.4. Entropy inequality

Consider a free energy function per unit volume of the form

Ψ = Ψ (ϵs, urM , ur

m) , (25)


where uri is defined such that ur

i = vri for i = m,M . Implied in the above equation is that ϵs is fully elastic (for

the inelastic response, see Ref. [104]). The Clausius–Duhem inequality for isothermal process reads

D = ρe − Ψ ≥ 0 , (26)

where

Ψ =∂Ψ

∂ϵs: ϵs +

∂Ψ

∂urm

· vrm +

∂Ψ

∂urM

· vrM . (27)

Substituting Eqs. (21) and (27) into (26) and using the standard Coleman argument yields the constitutive relations

σ ′=∂Ψ

∂ϵs, (gradφm)pm =

∂Ψ

∂urm

(gradφM )pM =∂Ψ

∂urM, (28)

along with the reduced dissipation inequality

D = (1 − φs)Π ψM−

∑i=m,M

ci[ pi

ρ f−

12vi · vi

]≥ 0 . (29)

In the infinitesimal deformation regime ψM= 0, and so the reduced dissipation inequality becomes

D =

∑i=m,M

ci[1

2vi · vi −

pi

ρ f

]≥ 0 . (30)

The above inequality imposes constraints on the constitutive laws, as explained in the next section.

3. Constitutive laws

To close the boundary value problem, we assume the following constitutive relations. The first energy-conjugatepair on the right hand side of Eq. (24) suggests that we must establish a constitutive law between the deformationof the solid matrix ϵs and the effective stress σ ′. Here we assume isotropic linear elasticity, which is given by theconstitutive relation

σ ′= Ce

: ϵs , (31)

where

Ce=

∂2Ψ

∂ϵs ⊗ ∂ϵs= K (1 ⊗ 1)+

3K (1 − 2ν)1 + ν

(I −

13

1 ⊗ 1), (32)

K is the elastic bulk modulus, ν is Poisson’s ratio, and I is the fourth-order identity tensor. In one-dimensionalanalysis where the lateral strain is zero, the relevant elastic modulus is the constrained modulus D, also known asthe longitudinal modulus or P-wave modulus, which is calculated from K and ν as

D =3K (1 − ν)

1 + ν. (33)

The second energy-conjugate pair on the right hand side of Eq. (24) suggests that we must establish constitutivelaws linking the relative velocity, volume fraction, and pore fluid pressures for the two pore scales. Here, we combinethe effects of relative velocity and volume fraction through the superficial (Darcy) velocity qi = φivr

i , and thenestablish relations between qi and pi . For the micro-fractures, we assume that Darcy’s law is still valid, which isgiven by the linear relation

q M = −kM

µ f·(grad pM − ρ f g

), (34)

where µ f is the dynamic viscosity of fluid, and kM is the intrinsic hydraulic conductivity (or permeability, forshort) tensor. For micro-fractures preferentially oriented on a plane whose unit normal vector is n, the intrinsicpermeability tensor kM takes the form

kM = kM⊥n ⊗ n + kM∥ (1 − n ⊗ n) , (35)


where kM⊥ is the intrinsic permeability in the direction perpendicular to the micro-fractures, and kM∥ is the intrinsicpermeability in the direction parallel to the micro-fractures. The above mathematical expression for kM characterizesa transversely isotropic material with respect to fluid flow.

For the nanopores, we assume a low-velocity non-Darcy flow following the work of Hansbo [95], who establishedthe non-Darcy constitutive relation for this condition under 1D flow. The following constitutive relations are anextension of the work by Hansbo to general 3D flow (in component j = 1, . . . , ndim):

(qm

)j =

⎧⎪⎪⎪⎨⎪⎪⎪⎩−ρ f g km

(im) j

ξµ f ξ i ξ−1

1

sgn[(im) j

] (im) j

< i1 ,

−ρ f g km

µ f

[(im) j

− i1ξ − 1ξ

]sgn

[(im) j

] (im) j

≥ i1 ,

(36)

where km is the maximum intrinsic permeability of the nanopores, ξ ≥ 1 and i1 > 0 are non-Darcy flow modelparameters, g is the scalar gravity acceleration constant, and im is the gradient of fluid potential hm , which is definedas

hm = z +pm

ρ f g, (37)

in which z denotes the elevation head. The value of i1 in Eq. (36) may be interpreted as the transition gradientrequired to overcome the maximum binding energy of mobile pore fluid [95]. From Eq. (36), we see that Darcy’slaw is recovered automatically by setting ξ = 1. Note that the above constitutive relation does not contain a TPGparameter. To summarize, this non-Darcy flow model is a combination of a linear flow part and a nonlinear flowpart.

The third energy-conjugate pair on the right hand side of Eq. (24) suggests that we must establish a constitutiverelation between the pore pressure difference (pM − pm) and the pore fraction ψM . This constitutive relation isrelevant under the condition of finite deformation where the pore fractions change with time. However, underinfinitesimal deformation the volume of the pore spaces is not expected to change significantly, and so it isreasonable to not consider this relation.

The last energy-conjugate pair on the right hand side of Eq. (24) suggests that we must establish constitutiverelations relating the mass transfer term, pore fluid pressure, and fluid velocity. From the second pair, wehave already considered the constitutive relationship between qi and pore pressure pi , and under infinitesimaldeformation, we can readily calculate vr

i from qi . Furthermore, the solid skeleton velocity v can be calculatedfrom the displacement field, and so we can readily add v and vr

i to obtain vi . Thus, we can theoretically expressvi in terms of pi , and so we only need to consider the constitutive relationship between the diffusive mass transferterm ci and pore fluid pressures. A number of expressions have been proposed in the literature accounting for suchmass transfer (e.g., [18,98,99,105]). Here, we adopt semi-empirical equations similar to that proposed in [98] asfollows (with minor modification for the unit of cM and cm)

cM

ρ f= α

kµ f

(pm − pM) ,cm

ρ f= α

kµ f

(pM − pm) , α =β

a2 γ , (38)

where k is an interface permeability whose value is elaborated subsequently, α is a mass transfer coefficient (MTC),a is the characteristic length of the micro-fractures spacing, β is a dimensionless coefficient that accounts for thesolid geometry, and γ is a dimensionless scaling coefficient suggested to be 0.4 to fit experimental results [98].Note that this relation automatically satisfies the closure condition cM

+ cm= 0. Furthermore, substituting this

relation into the reduced dissipation inequality (30) yields

D = αkµ f

(pm − pM )2+

cM

2(vM · vM − vm · vm) ≥ 0 . (39)

The first term on the right-hand side is always greater than zero. Now, for the second term to be greater than zero,cM > 0 if ∥vr

M∥ > ∥vrm∥ (fluid in the nanopores drains into the micro-fractures) and cM < 0 if ∥vr

M∥ < ∥vrm∥

(fluid in the micro-fractures enters into the nanopores). The latter scenario is possible if pm < pM and the point issufficiently far away from the drainage boundary that the fluid in the micro-fractures is moving very slowly.

While the flux of fluid through the nanopores may be lower than that estimated from Darcy’s law due to boundarylayer effects [75,97], the interface permeability k could also decrease as the pressure gradient decreases. This


statement is supported in [106], which revealed small-scale multiple and simultaneous rates of mass transfer due tovariations of aquifer properties in space. In [106], the multi-rate series mass transfer coefficients α j = j2π2 Da/a2

control transfer between zones of mobile and immobile contaminant for spherical diffusion model, where Da/a2

characterizes the diffusion property of immobile zone. Motivated by this particular multi-rate mass transfer model,we make an analogy between the immobile zone and the nanopores and propose a variable interface permeabilityof the form

k =

⎧⎪⎨⎪⎩kmax + kmin

2+

kmin − kmax

2cos

[π

(imax

i1

)ξ]imax < i1 ,

kmax imax ≥ i1 ,

(40)

where kmax and kmin are the maximum and minimum interface permeabilities, respectively, and

imax = maxj∈1,...,ndim

(im) j

. (41)

From the foregoing equations, we can define a dimensionless interface permeability ratio (IPR)

r =kmax

kmin. (42)

When r = 1, the interface permeability k becomes a trivial constant, whereas when r > 1, the decrease in kdue to boundary layer effect is analogous to that which occurs when a small value of Da/a2 is specified for claylayers or clay pods [106], which is also interpreted as non-Darcy flow since k controls the fluid flux between themicro-fractures and nanopores. Later we shall show that this variation in k also affects the entire fluid flow patterns.In addition, since in general k will not be a constant, this composite non-linear mass transfer term can be regardedas a supplement for the coupling terms proposed in [99,105].

4. Mixed finite element formulation

This section formulates a three-field u/pM/pm mixed finite element solution to an initial boundary value problemof coupled solid deformation–fluid diffusion with double porosity following the lines of [13,42,57,107]. Here, weonly highlight the salient features of the formulation with an emphasis on how to handle transverse isotropy withrespect to fluid flow and a non-Darcy constitutive law. The relevant three-field conservation laws include the balanceof linear momentum,

div σ + ρg = c , (43)

and the balance of mass for the two pore scales, with both dψM/dt and dψm/dt set to zero:

ψM B div v + div q M =cM

ρ f, (44)

ψm B div v + div qm =cm

ρ f. (45)

These conservation laws must be combined with relevant initial and boundary conditions (both essential and natural)for the unknown fields.

For balance of linear momentum over the problem domain Ω , the corresponding variational equation takes theform ∫

Ω

symm(∇η) :(σ ′

− B p1)

dΩ =

∫Ω

η · ρg dΩ −

∫Ω

η · c dΩ +

∫Γt

η · t dΓ , (46)

where η is the weighting function for displacement, Γ is the boundary of Ω , and Γt is the portion of Γ on whichthe traction vector t is specified. For balance of mass for fluid in the micro-fractures, the variational equation is∫

Ω

ωMψM B div v dΩ −

∫Ω

gradωM · q M dΩ =

∫Ω

ωMcM

ρ fdΩ +

∫ΓhM

ωM hM dΓ , (47)


where ωM is the weighting function for fluid pressure in the micro-fractures, and hM is the normal fluid flux onthe portion ΓhM of the total boundary Γ . For the nanopores, the corresponding variational equation is∫

Ω

ωmψm B div v dΩ −

∫Ω

gradωm · qm dΩ =

∫Ω

ωmcm

ρ fdΩ +

∫Γhm

ωm hm dΓ , (48)

with similar definitions for ωm , hm , and Γhm .Rate-dependence is embedded in Eqs. (47) and (48) through the solid velocity vector v. These two equations

must be time-integrated either in their variational form or after they have been converted into their matrix form.Here, we perform time-integration of the variational equation using the first-order accurate, unconditionally stablebackward difference scheme [108]. For the micro-fractures, the time-integrated equation takes the form∫

Ω

ωMψM B div (u − un) dΩ − ∆t

∫Ω

gradωM · q M dΩ = ∆t∫Ω

ωMcM

ρ fdΩ + ∆t

∫ΓhM

ωM hM dΓ , (49)

where un is the displacement field u evaluated at time tn , ∆t = tn+1 − tn is the time increment, and all the statevariables inside the integral sign are evaluated at time tn+1. For the nanopores, we have∫

Ω

ωmψm B div (u − un) dΩ − ∆t

∫Ω

gradωm · qm dΩ = ∆t∫Ω

ωmcm

ρ fdΩ + ∆t

∫Γhm

ωm hm dΓ . (50)

Next, we formulate the matrix equations in residual form. For the balance of linear momentum, the residualmatrix equation is given by

Ru =

∫Ω

BTσ ′

− B p1

dΩ +

∫Ω

NT c dΩ − FEXT , (51)

where N is the displacement shape function matrix, B is the strain–displacement transformation matrix, and FEXT

is the external nodal force vector induced by the body force and surface tractions. Here, we remark thatσ ′

− B p1

are stresses written in Voigt form. For the balance of fluid mass, the corresponding finite element matrix equationsare

RM =

∫Ω

NTψM B div (u − un) dΩ − ∆t∫Ω

ETq M dΩ − ∆t∫Ω

NT cM

ρ fdΩ − GEXT (52)

for the micro-fractures, and

Rm =

∫Ω

NTψm B div (u − un) dΩ − ∆t∫Ω

ETqm dΩ − ∆t∫Ω

NT cm

ρ fdΩ − HEXT (53)

for the nanopores. In the foregoing equations, N is the shape function matrix for the fluid pressures, E = ∇ N , andGEXT and HEXT are external fluid flux vectors.

Due to non-Darcy flow, the residuals defined above are nonlinear in the nodal solid displacement and fluidpressure vectors in the micro-fractures and nanopores, respectively. To solve for these primary variables in anonlinear setting, we employ the Newton–Raphson iteration scheme. The iteration requires that we construct theoverall residual vector

R(X) =

⎧⎨⎩ Ru

RM

R m

⎫⎬⎭ , X =

⎧⎨⎩ Xu

X M

Xm

⎫⎬⎭ , (54)

and evaluate the tangent operator

R′ (X) =

⎡⎣ A B1 C1B2 D E1C2 E2 F

⎤⎦ , (55)

where Xu , X M , and Xm are the nodal values of u, pM , and pm , respectively. The individual matrices comprisingthe tangent operator R′ (X) are as follows:

A =

∫Ω

BTCe B dΩ , (56)


B1 =

∫Ω

NTχ1 dΩ − B∫Ω

ψM bT N dΩ , (57)

C1 =

∫Ω

NTχ2 dΩ − B∫Ω

ψm bT N dΩ , (58)

B2 = B∫Ω

ψM NTb dΩ , (59)

D =∆tµ f

∫Ω

ETkM E dΩ − ∆t∫Ω

NTχ3 dΩ , (60)

E1 = −∆t∫Ω

NTχ4 dΩ , (61)

C2 = B∫Ω

ψm NTb dΩ , (62)

E2 = −∆t∫Ω

NTχ5 dΩ , (63)

F = −∆t∫Ω

ETχ6 E dΩ − ∆t∫Ω

NTχ7 dΩ , (64)

where b = B1, Ce is the elasticity tensor Ce written in matrix form, and χ1, . . . ,χ7 are complex vector or

matrix expressions emanating from non-Darcy flow (please see Appendix for a full representation of these terms).In general, the tangent operator R′ (X) is non-symmetric.

Stabilization is needed when equal-order interpolation is used for displacement and pressure degrees of freedom,particularly when the permeabilities are very low and/or the time increment is very small [109]. We refer the readersto Eq. (66)–(71) in Section 3 of Choo and Borja [13] for relevant materials on this topic.

5. Numerical results and discussion

The foregoing formulation has been implemented in Geocentric, a massively parallel finite-element code forgeomechanics that is built on the deal.II finite-element library [110,111], p4est mesh handling library [112], andthe Trilinos project [113]. Due to the large size of the linear system stemming from a three-field mixed formulation,we have utilized the 3 × 3 block preconditioned Newton–Krylov solver built into the code, as proposed by Whiteand Borja [114] and extended by Choo and Borja [13]. The implementation of this code has been verified bychecking to see that under isotropic condition and Darcy flow (setting ξ = r = 1), the results of double-porositysimulations coincide with those presented in [13], and that when either of the two pore fractions is reduced to zero(i.e. single porosity condition) and under isotropic Darcy flow, the results coincide with those presented in [101].With reference to the mixed finite elements employed in this paper, Q2P1P1 pertains to mixed elements employingsecond-order (biquadratic Lagrangian) interpolation for displacement and first-order (linear) interpolation for thetwo pressures, while Q1P1P1s refers to equal-order linear interpolation for displacement and two pressure fieldswith stabilization.

5.1. Isotropic versus transversely isotropic flows

We consider a 2D rectangular double porosity medium 10 m wide and 5 m deep, and discretized into 3200Q2P1P1 mixed quadrilateral elements (26,082 DOFs for displacement and 3321 DOFs for each pressure type). Thetop surface is subjected to plane strain strip loading. The geometric profile and boundary conditions are shown inFig. 1. We assume no water exfiltration under the strip loading, and for the remaining portion of the top surfacewe assume a homogeneous zero fluid pressure for the micro-fractures and no-flow boundary condition for thenanopores. Thus, fluid can only discharge to the ground surface through the micro-fractures, whereas fluid in thenanopores can only discharge into the micro-fractures, see [13,57]. Note here that we use a full mesh instead of ahalf mesh to define the problem domain, see [13,101].

The following material parameters are assumed in the simulations: volume fractions φM= 0.1 and φm

= 0.15;fluid viscosity µ f = 10−6 kPa s; mass transfer parameter α = 480 m−2; fluid density ρ f = 1000 kg/m3; solid


Fig. 1. Geometry and boundary conditions for plane strain strip loading on a double-porosity medium.

bulk modulus K = 3000 kPa; solid Poisson’s ratio ν = 0.15; micro-fracture permeability on the plane of isotropykM∥ = 10−10 m2; nanopore permeability km = 5 × 10−16 m2; interface permeability kmax = 5 × 10−16 m2; andBiot coefficient B = 1. Darcy’s law is assumed on both pore scales. We then investigate the impact of transverseisotropy with respect to fluid flow by varying the permeability ratio r = kM⊥/kM∥, with r = 1 for isotropic flowand r < 1 for transversely isotropic flow.

Setting r = 1 (isotropic flow), Fig. 2 shows the pore pressure accumulation generated by a surface strip loadof p0 = 150 kPa applied instantaneously. Values reported in this and subsequent figures are so-called excesspore pressures, i.e. pressures in addition to the hydrostatic steady-state values that dissipate with time. Significantdifference can be observed between the two pore pressure fields pM and pm , both quantitatively and qualitatively.The nanopore pressure pm is much higher than the micro-fracture pressure pM , which is a basic non-equilibriumflow pattern in double porosity formulation due to permeability difference between the two pore scales [24–27,42].Non-equilibrium flow promotes fluid transfer from the smaller pore scale to the larger pore scale; however, flowwithin the smaller pore scale itself is relatively insignificant due to lower permeability and the fact that fluid withinthis pore scale is prevented from draining to the top surface. The last statement can be verified by increasing themagnitude of km and plotting the pressure evolution with time, as shown in Fig. 3. By comparing with Fig. 2, wesee that the pressure distributions are nearly the same, suggesting that in a double porosity medium the permeabilityin the smaller pore scale does not determine the resulting flow pattern [42].

To fully understand the impacts of transverse isotropy on the flow pattern, we specify r = kM⊥/kM∥ = 0.05and keep the value of kM∥ at 10−10 m2. Figs. 4 and 5 show instantaneous pore pressure accumulation at micro-fracture orientations θ = 10 and θ = 0, respectively. The impacts of transverse isotropy can be summarized asfollows. First, transverse isotropy in permeability changes the distribution of pore pressure in the micro-fracturesimmediately after the application of surface load, which is affirmed by Figs. 4(a) and 5(a). More importantly, theresulting contours are skewed towards the direction of anisotropy as long as r is sufficiently small. This is becausewhen kM∥ is at least one order of magnitude higher than kM⊥, the fluid will always choose to flow along the micro-fractures to decrease the pressure gradient; as a result, we have nearly the same pM values along the orientation ofthe micro-fractures. Similar results have been reported in single porosity uncoupled formulations of [63–65].

Second, a lower value of kM⊥ hampers the overall drainage capability of the porous medium. Thus, higher porepressures build up in the micro-fractures, as seen in Figs. 4(a) and 5(a), compared with Fig. 2(a) when kM⊥ is higher.The orientation θ , together with the imposed drainage boundary conditions, also affects the drainage paths, which inturn influences the pressure accumulation in the micro-fractures. In Fig. 4(a), since the micro-fractures are inclined(θ = 10), fluid could flow along the micro-fractures and drain to the top surface. However, with the horizontallyoriented micro-fractures considered in Fig. 5(a), fluid could only flow perpendicular to the micro-fractures, resultingin higher pressure accumulation.


Fig. 2. Pressure evolution for isotropic Darcy flow in a double porosity medium with km = 5 × 10−16 m2. Color bars are excess porepressures in kPa. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Pressure evolution for isotropic Darcy flow in a double porosity medium with km = 1 × 10−15 m2. Color bars are excess porepressures in kPa. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Comparing Figs. 4(b) and 5(b) with Fig. 2(b), we see that the spatial distribution of the nanopore pressure pm is

hardly affected by the transversely isotropic fluid flow behavior in the micro-fractures due to the fact that the excess


Fig. 4. Instantaneous pore pressure accumulation at t = 0+ for a transversely isotropic double porosity medium with r = 0.05 and θ = 10.Color bars are excess pore pressures in kPa. (For interpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

Fig. 5. Instantaneous pore pressure accumulation at t = 0+ for a transversely isotropic double porosity medium with r = 0.05 and θ = 0.Color bars are excess pore pressures in kPa. (For interpretation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

Fig. 6. Contour of nanopore pressure pm at t = 800 s: (a) isotropic flow through the micro-fractures; (b) transversely isotropic flow throughthe micro-fractures, with r = 0.05 and θ = 10. Color bars are excess pore pressures in kPa. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

fluid pressure in the nanopores does not have enough time to drain to the micro-fractures. Furthermore, the nearlysymmetric spatial variation of pm shown in Figs. 4(b) and 5(b) arises from the isotropy in the mechanical responseof the solid skeleton. Work is underway to investigate how this initial pore pressure response in the nanopores maybe influenced by anisotropy in the mechanical response of the solid skeleton arising, for example, from the existenceof bedding planes.

Finally, as time goes on, fluid pressures in both the micro-fractures and nanopores dissipate, and the pressuredifference between the two pore scales gradually becomes smaller and smaller. After a certain period of time, ananisotropic unsymmetric pattern of nanopore pressure distribution appears, as shown in Fig. 6(b) (compare thiswith the symmetric pattern shown in Fig. 6(a), for example). Due to strong coupling between fluid flow and solid


Fig. 7. Ground surface displacement u y at time t = 400 s. Note: transversely isotropic has θ = 10.

deformation, this anisotropic fluid flow pattern leads to unsymmetric ground surface displacement, which is depictedin Fig. 7. The upshot from these simulations is that in double-porosity media, it is the permeability of the larger porescale that determines the overall flow pattern. This is essentially the same conclusion reached by Choo et al. [42]for the case of unsaturated flow.

5.2. Darcy versus non-Darcy flows

We consider the same boundary-value problem solved in the last section, but now compare the flow patternsgenerated by Darcy and non-Darcy flows in an isotropic double-porosity medium. Recall that the non-Darcyconstitutive law considered in this paper degenerates to Darcy’s law depending on the material parameters ξ andr . A value of ξ = 1 yields Darcy’s law for flow through the nanopores, whereas a value of r = 1 results in aconstant value of interface permeability k. In the present simulations, we thus assign values of ξ = 1.3 and r = 10for parameters characterizing a non-Darcy flow, as well as assume a value of i1 = 2 for the transition gradient.

Fig. 8 compares the mean excess pore pressures p that develop immediately after the application of thestrip surface load assuming Darcy and non-Darcy flows through the nanopores. Because the load is appliedinstantaneously, there is not much difference between the two pore pressure contours, as expected. However, aftera certain period of time the difference between the two contours becomes more apparent. Fig. 9 portrays thedissipation patterns at time t = 400 s suggesting significant difference in the mean pore pressure distributions. Thesimulation that employs Darcy’s law shows faster pore pressure dissipation, as expected, whereas the simulation thatemploys a non-Darcy constitutive law tends to slow down the dissipation process and leads to smaller ground surfacesettlements in central region shown in Fig. 10. Note, however, that in the absence of anisotropy, the displacementpattern remains symmetric for both Darcy and non-Darcy flow simulations due to the symmetric loading condition.

The distribution of mean pore pressures generated by non-Darcy flow deserves further elaboration. At thebeginning of pressure dissipation, imax > i1 along the central region, thus maximizing the interface permeabilityk (i.e. k = kmax). Therefore, no significant difference is observed between Darcy and non-Darcy flows. As timegoes on, however, the nanopore pressure dissipation causes imax to decrease, which in turn causes k to decrease,thus slowing down the mass transfer from the nanopores to the micro-fractures. We thus see that the low-velocitynon-Darcy constitutive law adopted in this work mainly impacts the flow pattern in the form of mass transfer, andthat the flow through the nanopores itself is not so much impacted by the fact that it does not obey Darcy’s law.The contour of pore pressures for the non-Darcy flow case shown in Fig. 9 is a unique result of the simulation andcannot be explained from using one’s intuition alone.


Fig. 8. Instantaneous mean pore pressure p immediately after the application of strip load: (a) Darcy flow through the nanopores, and(b) non-Darcy flow through the nanopores, with ξ = 1.3, r = 10, and i1 = 2. Color bars are excess pore pressures in kPa. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Mean pore pressure p at time t = 400 s: (a) Darcy flow through the nanopores, and (b) non-Darcy flow through the nanopores,with ξ = 1.3, r = 10, and i1 = 2. Color bars are excess pore pressures in kPa. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Fig. 10. Ground surface displacement u y at time t = 400 s.


Fig. 11. Geometry of the 1/8 sphere porous object in the positive octant of the Cartesian space. Triangles are planes of isotropy; pressurep0 = 1 kPa is applied on the surface of the 1/8 sphere (rainbow color), which also serves as a drainage boundary. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this article.)

5.3. 3D consolidation of 1/8 of a sphere

In this third and final example, we consider the consolidation of a portion of a sphere located within thepositive octant of the Cartesian coordinate system. Except for double porosity and an imposed anisotropy describedsubsequently, this example is similar to Cryer’s problem for a single-porosity medium, which describes theconsolidation of a single-porosity full sphere [115,116]. Here, we recast the problem into a double-porosityformulation and combine it with the low-velocity non-Darcy constitutive law within the nanopores, along withtransverse isotropy in the micro-fractures. The configuration of the problem, shown in Fig. 11, corresponds to 1/8of a unit sphere similar to that employed in the simulations of [13,107]. The 3D domain is discretized into 27,648mixed hexahedral elements. Because of the large system of equations generated by the increased dimension ofthe problem, we use equal-order trilinear interpolation (i.e. low-order) for displacement and pressures, along withstabilization [13] for the mixed elements, e.g. Q1P1P1s, resulting in 150,165 DOFs (90,099 for displacement and30,033 for each pressure type).

The material parameters are as follows: volume fraction for the micro-fractures φM= 0.25; volume fraction for

the nanopores φm= 0.4; fluid viscosity µ f = 10−6 kPa s; mass transfer coefficient α = 1760 m−2; fluid mass

density ρ f = 1000 kg/m3; solid bulk modulus K = 1000 kPa; solid Poisson’s ratio ν = 0.25; and Biot coefficientB = 1. For the micro-fractures, we assume an intrinsic permeability of kM∥ = 10−11 m2 on the plane of isotropy. Wealso set r = 1 for the simulation that assumes isotropy in the micro-fractures, and r = 0.02 for another simulationthat assumes transverse isotropy in the micro-fractures, along with a unit vector n =

(√3/3,

√3/3,

√3/3

)defining

the axis of cross anisotropy. Note that the direction of cross anisotropy is radial only along the diagonal of thepositive octant. Thus, the 1/8 of a sphere volume used in this example cannot be interpreted as a prototype sectorof a full sphere. As for the nanopores, we consider both Darcy and non-Darcy flows. For Darcy flow we assumea nanopore intrinsic permeability of km = 10−14 m2 and an interface permeability of kmax = 10−14 m2, while fornon-Darcy flow we take ξ = 1.3, i1 = 2, and r = 10. The normal surface load is p0 = 1 kPa, while the stabilizationparameter is assumed to have a value τ = 1 (see [13,101] for details).

Fig. 12 shows the time evolutions of the mean excess pore pressure p at the origin of the 1/8 of a spherevolume. Four curves are presented in this figure representing permutations of isotropic and transversely isotropicflows through the micro-fractures (note that flow through the nanopores is always isotropic), and Darcy and non-Darcy flows through the nanopores (note that flow through the micro-fractures is always Darcy). All curves emanate


Fig. 12. Time-variation of mean excess pore pressure p at origin. Iso = isotropic; Aniso = anisotropic.

from the value p0 = 1 kPa at time t = 0+, as expected. For a single-porosity problem with Darcy flow, the so-calledMandel–Cryer effect [115,116], in which the time evolution of excess pore pressure shows a momentary increasefollowed by a monotonic dissipation to zero, is a characteristic feature of the solution to the consolidation of asphere problem. Within the context of double porosity consolidation, however, this pattern is reproduced only bythe two simulations that assumed transverse isotropy of flow through the micro-fractures, although the non-Darcysimulation also tends to dampen this effect even with the enhancing effect of transverse isotropy. None of the twoisotropic flow simulations reproduced the Mandel–Cryer effect.

Figs. 13 and 14 depict the time evolutions of the fluid pressures in the micro-fractures and nanopores, respectively.Note that their initial values at t = 0+ differ appreciably depending on the fluid flow properties, even as theirweighted sums are all equal to the applied surface load of 1 kPa (see Fig. 12). This could have significantimplications for the ensuing fluid flow, since the mass transfer depends on the prevailing pressure differencebetween the two pore scales, and flow through the micro-fractures depends on the prevailing gradient at this porescale. Because of such complex interplay between flows through the two pore scales, along with mass exchangestranspiring between them, it is not surprising to see that the Mandel–Cryer effect may be prominent for one flowpattern but not for the other. It is worthy of note, however, that the Mandel–Cryer effect is prominent in theanisotropic/Darcy simulations at both the larger and smaller pore scales.

Finally, Fig. 15 shows a comparison of excess nanopore pressures within the 1/8-sphere domain, with and withoutstabilization. Both Fig. 15(a) and 15(b) show the effect of transverse isotropy in which the excess pore pressures areapproximately constant on the planes of isotropy, rather than on surfaces at constant radial distances away from theorigin, which would be the case if the material was isotropic. However, the unstabilized solution also exhibits porepressure oscillation not only in the radial direction but also on the plane of isotropy, with a checkerboard patternwithin a finite-thickness zone that is parallel to the plane of isotropy. As seen from this figure, the oscillation iscompletely eliminated by the proposed stabilization.

6. Closure

We have presented a hydromechanical framework for porous materials exhibiting two dominant porosity scalesthat accommodates transverse isotropy induced by distributed micro-fractures and non-Darcy flow through thenanometer-scale pore spaces. This study has been motivated by the fluid flow properties of common sedimentaryrocks such as shale, where fluid could flow through the micro-fractures of the rock matrix as well as through thenanopores of the organics embedded in the rock matrix. It is the first time, to the knowledge of the authors, thatthese two important features have been integrated into the framework of double-porosity poromechanics. One of


Fig. 13. Time-variation of excess micro-fracture pore pressure pM at origin. Iso = isotropic; Aniso = anisotropic.

Fig. 14. Time-variation of excess nanopore pressure pm at origin. Iso = isotropic; Aniso = anisotropic.

the important findings in this work is the fact that the instantaneous fluid pressures that develop within the largerand smaller scale pores due to deformation of the solid matrix could be significantly impacted by the fluid flowproperties of the porous material even if the solid is deforming under undrained condition. Since the pore pressuredifference between the two pore scales determines the rate of mass transfer between them, the fluid flow propertiesof a double-porosity porous medium could significantly impact the ensuing flow patterns. This was demonstratedin the 1/8 of a sphere problem that resembles Cryer’s problem: the so-called Mandel–Cryer effect may or may notbe reproduced by the theory depending on the isotropy or lack of it in the fluid flow, as well as on whether ornot the flow obeys Darcy’s law. Work is now underway to integrate the presented fluid flow formulation within theframework of computational plasticity to accommodate inelastic deformation [104,117–119] as well as anisotropyof the solid matrix arising, for example, from the existence of bedding planes [70,71,120,121].


Fig. 15. Contour of excess nanopore pressure pm for transversely isotropic-Darcy flow at t = 1.5 s: (a) stabilized formulation with Q1P1P1s;and (b) unstabilized formulation with Q1P1P1. Note that km and k have been reduced from 10−14 to 10−16 m2 to highlight the pore pressureoscillation. Color bar is excess pore pressure in kPa. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Acknowledgments

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office ofBasic Energy Sciences, Geosciences Research Program, under Award Number DE-FG02-03ER15454. Support formaterials and additional student hours were provided by the National Science Foundation under Award NumberCMMI-1462231. The second author acknowledges the support by the Research Grants Council of Hong Kongunder Grant Number: ECS 27205918.

Appendix. Non-Darcy terms in submatrices of consistent tangent operator

We need to calculate the expression for ∂qm/∂ im and ∂ k/∂ im . From basic linear algebra, we know that ∂qm/im

is a ndim × ndim diagonal matrix, and the diagonal element is given as follows ( j = 1, . . . , ndim):

(∂qm

∂ im

)j j

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩−ρ f g km

(im) j

ξ−1

µ f i ξ−11

(im) j

< i1 ,

−ρ f g km

µ f

(im) j

≥ i1 .

(A.1)

The term ∂ k/im is a vector whose size is ndim, and in our work we write it in a row vector form. The expressionis given below

(∂ k∂ im

)jmax

=

⎧⎪⎨⎪⎩kmax − kmin

2sin

[π

(imax

i1

)ξ]πξ i ξ−1

max

i ξ1sgn (imax) imax < i1 ,

0 imax ≥ i1 ,(∂ k∂ im

)j = jmax

= 0 ,

(A.2)

where

jmax = argmaxj∈1,...,ndim

(im) j

. (A.3)

Now we can give the detailed expressions for χ1, . . .χ7 under the assumption of infinitesimal deformation,i.e., φM and φm are approximately constant. The expressions are given below:

χ1 =∂ c∂ pM

=αkρ f

φmµ fqm N −

αkρ f

φMµ fq M N −

cM

φMµ fkM E , (A.4)


χ2 =∂ c∂ pm

=αρ f

φmµ f

[(pM − pm)

ρ f gqm

∂ k∂ im

E +k(pM − pm)

ρ f g∂qm

∂ imE − kqm N

]+

αρ f

φMµ f

[(pm − pM)

ρ f gq M

∂ k∂ im

E + kq M N],

(A.5)

χ3 =∂

(cM/ρ f

)∂ pM

= −αkµ f

N , (A.6)

χ4 =∂

(cM/ρ f

)∂ pm

=αkµ f

N +α(pm − pM)

µ f ρ f g∂ k∂ im

E , (A.7)

χ5 =∂

(cm/ρ f

)∂ pM

=αkµ f

N , (A.8)

χ6 =1ρ f g

∂qm

∂ im, (A.9)

χ7 =∂

(cm/ρ f

)∂ pm

= −αkµ f

N +α(pM − pm)

µ f ρ f g∂ k∂ im

E . (A.10)

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