Body-Force Effects in Homogenised Models of Anisotropic Double Poroelastic Materials

Abstract

We develop a rigorous multiscale homogenisation framework for determining the effective behaviour of materials composed of a poroelastic matrix containing embedded poroelastic subphases, each governed by fully anisotropic and heterogeneous Biot equations and subject to body forces. Starting from local balance laws for momentum, Darcy flow, and mass conservation, and assuming scale separation, periodic microstructure, and macroscopic uniformity, we derive a closed system of Biot-type equations at the macroscale. The resulting effective coefficients explicitly capture the influence of microstructural geometry, elastic and hydraulic heterogeneity, and mesoscale forcing. The framework yields well-posed cell problems for the effective elasticity tensor, hydraulic conductivity tensor, and Biot-type coupling coefficients, establishing a unified multiscale theory that extends and subsumes existing single-phase, composite, and double-poroelastic models. As an illustrative application, we specialise the theory to magnetic Lorentz forcing and show that body forces with gradient structure induce additional pressure-like terms in the effective Darcy law, rather than appearing solely as averaged source terms. The proposed framework sets the agenda for modelling complex poroelastic materials with interacting subphases, with applications in biomechanics, engineered composites, and geomechanics.

Introduction

Purely elastic or purely hydraulic models fail to describe many saturated materials, as they neglect the feedback between deformation of the solid matrix and transport of interstitial fluid. In such settings, stresses, pore pressure, and fluid flux evolve in a coupled manner. This coupled behaviour is captured by the Theory of Poroelasticity, [1–4], which links momentum balance, Darcy flow, and mass conservation within a unified continuum framework. This framework applies to a wide range of physical scenarios where interactions occur between a deformable solid and the pore-scale fluid. Such models are particularly relevant in biological and geomechanical systems, where microscale heterogeneity strongly influences macroscopic response (see [5–12]).

The systems considered here are inherently multiscale, with processes occurring at distinct spatial and temporal scales that interact non-trivially. Multiscale modelling provides a systematic framework for capturing such interactions. It has been employed primarily in materials science to link atomistic or microscale structure to macroscopic mechanical behaviour [13], in geomechanics to couple pore-scale flow with reservoir-scale deformation [14, 15], and in biomechanics to connect cellular or tissue microstructure with organ-level function [16, 17]. Similar multiscale frameworks underpin modern simulations of battery systems [18], climate and weather prediction [19], and turbulent flows [20]. These diverse applications illustrate the broad utility of multiscale approaches and that they are particularly valuable when microscale heterogeneity strongly influences effective material response and cannot be represented accurately within a single-scale description.

In this work, we focus on multiscale deformable porous media with three characteristic length scales. At the smallest scale, fluid–solid interactions occur at the porescale, which is much smaller than the macroscale of the material. Intermediate (meso) scales also arise, such as local heterogeneities in poroelastic properties (see Fig. 1). Our analysis concentrates on the meso- and macroscales, rather than a full three-scale framework ([21] for elastic composites).

Fig. 1.

A 2D sketch representing a cross-section of the 3D domain $\mathrm{\Omega }$. The matrix ${\mathrm{\Omega }}_m$ is shown in teal and the subphase domain ${\mathrm{\Omega }}_s$ is in orange. There is an interface $\mathrm{\Gamma }$ between the two domains. We also can see that when zooming in on each of the two domains we have two different porous media with a fluid flowing in the pores

We refer to the porescale microstructure because the governing equations employed here arise from upscaling fluid–structure interactions between the solid and fluid phases [22–24]. Specifically, we adopt the most general Biot’s anisotropic, heterogeneous, compressible equations for poroelasticity, rather than embracing simplifications such as the isotropic or incompressible limits. Importantly, in our formulation we also assume that a body force acts on each domain, thereby entering the balance equations at both the mesoscale and macroscale.

The governing equations at the macroscale are obtained by upscaling the local balance equations, incorporating both pore-scale interactions and the imposed body forces. This upscaling can be achieved through homogenisation techniques (see [25] and [26]). Here, we employ the asymptotic homogenisation method [27–30], which exploits the separation between mesoscale and macroscale to decouple spatial variations. In this approach, the relevant fields are expanded as power series in the ratio of the two scales. The resulting balance equations describe the poromechanics of the material at the macroscale, with coefficients encoding microstructural information and the effects of body forces. These coefficients are computed by solving differential problems on the mesoscale.

Asymptotic homogenisation has become a standard tool for deriving effective poroelastic models, beginning with early analyses of wave propagation and effective Biot-type behaviour [22, 31]. Subsequent developments have broadened the framework to incorporate additional physical mechanisms, including appositional growth between fluid and solid phases [32], vascularised poroelastic materials [33], and heterogeneous poroelastic composites [34]. More recently, a substantial body of work has advanced the multiscale theory of poroelastic and multiphase materials through rigorous upscaling from microstructural principles. This includes the derivation of double-poroelastic models [35], homogenised balance laws for nonlinear and electrostrictive composites [36, 37], and micromechanical analyses of effective stiffness in poroelastic media [38]. Further contributions have addressed porosity and diffusion in biological tissues [23], the impact of microstructural alterations such as those induced by myocardial infarction [39], and coupled electro-mechanical homogenisation in cardiac tissue [40]. Collectively, these works demonstrate the versatility of multiscale modelling in capturing the macroscale behaviour of complex poroelastic systems while retaining detailed information about microscale geometry, anisotropy, and multiphysics interactions. However, existing homogenisation frameworks for poroelastic and double-poroelastic media typically neglect phase-dependent body forces or restrict attention to isotropic or incompressible regimes. As a result, the influence of heterogeneous forcing on effective coupling coefficients remains poorly understood.

In this work, we determine the effective behaviour of a material comprising a poroelastic matrix and embedded poroelastic subphases (fibres, inclusions, strata), all interacting under the influence of body forces. Each phase is assumed anisotropic and heterogeneous. Our motivation is to study materials with multiple poroelastic phases interacting at the mesoscale. This structure has been considered previously, e.g. [41], where poroelastic cells were embedded in an extracellular matrix, and equations were derived to study tissue consolidation. Biological tissues, with poroelastic subphases such as cells and collagen fibres [42], provide natural examples.

Here, we adopt the classical two-scale asymptotic homogenisation framework under assumptions of scale separation and periodicity, as is standard in multiscale poromechanics. We generalise [35] and upscale the full interaction between matrix and subphases, each governed by Biot’s anisotropic, heterogeneous, compressible poroelasticity, and each subject to body forces. The mesoscale, where subphases are resolved, is assumed much smaller than the global domain. Upscaling is carried out while enforcing continuity of stresses, displacements, pressures, fluxes, and body force contributions across interfaces. The resulting global model is of Biot-type, with coefficients encoding microstructural properties and body force effects, computed from local differential problems on a finite subset of the domain.

This general formulation is useful since different classes of body force may be transmitted to the macroscale in different ways. Some forcing contributions enter the homogenised equations through averaged source terms, whereas others have additional structure that can modify the effective constitutive relations themselves. To illustrate this distinction, we later specialise the theory to magnetic Lorentz forcing. This is motivated by magnetic-resonance-type settings in biological tissues, where electrically conducting interstitial fluids may experience electromagnetic body forces. In this case, the force may depend on gradients of electromagnetic potentials, and therefore is not captured solely by a simple averaged source term. The Lorentz-force example consequently demonstrates that body-force structure can alter the effective Darcy law itself, producing additional pressure-like contributions at the macroscale.

The paper is organised as follows. In Sect. 2 we consider the quasi-static, multiphase problem consisting of the governing equations for both the poroelastic matrix and the poroelastic subphases with all phases subjected to body forces, and the appropriate interface conditions. The governing equations for the matrix and the subphases are the equations of Biot’s anisotropic, heterogeneous, compressible poroelasticity. In Sect. 3 we introduce the two-scale asymptotic homogenisation method and exploit the scale separation between the inter-subphase distance (the mesoscale) and the overall size of the domain (the macroscale) to derive the macroscale system of PDEs. In Sect. 4 we specialise our force to the magnetic Lorentz force and then in Sect. 5 compare the similarities and differences in the derivations. Section 6 concludes our work by discussing limitations of the model and by providing further perspectives.

Problem Formulation

We will now introduce our problem. We build on the double-poroelastic framework introduced in [35], extending it to anisotropic, heterogeneous phases subject to phase-dependent body forces.

We consider a set $\mathrm{\Omega }\in {\mathbb{R}}^3$ where $\mathrm{\Omega }$ is the union of the poroelastic matrix ${\mathrm{\Omega }}_m$, and a collection of disjoint poroelastic subphases where we can write $\overline{\mathrm{\Omega }}={\overline{\mathrm{\Omega }}}_m\cup {\overline{\mathrm{\Omega }}}_s$. Physically, we regard ${\mathrm{\Omega }}_m$ as a connected poroelastic matrix phase and ${\mathrm{\Omega }}_s$ as a collection of embedded poroelastic subphases, which may represent fibres, inclusions, cellular regions, or layered structures depending on the application, see [37] for further geometrical discussion. Both phases are themselves saturated porous materials containing a deformable solid skeleton and an interstitial fluid. The purpose of the present mesoscale model is therefore to describe how two distinct poroelastic materials interact through their common interface and collectively determine the apparent macroscale behaviour of the composite material when subjected to an applied body force. We provide a sketch of a cross-section of the domain $\mathrm{\Omega }$ in Fig. 1.

To create our model we begin by writing the equations that govern each domain as well as the appropriate interface conditions that will close the problem.

The two phases need balance equations that will govern their mechanical behaviour. These are given by

$$\begin{array}{cc}\hfill \mathrm{\nabla }·{\mathsf{T}}_m={\mathbf{f}}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 1a

$$\begin{array}{cc}\hfill \mathrm{\nabla }·{\mathsf{T}}_s={\mathbf{f}}_s& \text{in}{\mathrm{\Omega }}_s.\hfill \end{array}$$ 1b

The stress tensors ${\mathsf{T}}_m$ and ${\mathsf{T}}_s$ are the stress tensors in each compartment and are those of poroelastic materials. The body forces ${\mathbf{f}}_m$ and ${\mathbf{f}}_s$ here are kept general for the derivation and will be specified later depending on specific applications.

We model each solid phase using Biot’s theory of anisotropic, heterogeneous, compressible poroelasticity [22–24]. Accordingly, the stress tensors ${\mathsf{T}}_m$ and ${\mathsf{T}}_s$ appearing in (1a) and (1b) are prescribed by

$$\begin{array}{cc}\hfill {\mathsf{T}}_m={\mathbb{C}}_m:\zeta ({\mathbf{u}}_m)-{\mathit{\alpha }}_m{p}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 2a

$$\begin{array}{cc}\hfill {\mathsf{T}}_s={\mathbb{C}}_s:\zeta ({\mathbf{u}}_s)-{\mathit{\alpha }}_s{p}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 2b

where

$$\begin{array}{c}\hfill \zeta (\bullet )=\frac{\mathrm{\nabla }(\bullet )+{(\mathrm{\nabla }(\bullet ))}^{\text{T}}}{2}\end{array}$$ 3

denotes the symmetric part of the gradient operator. We have that ${\mathbf{u}}_m$ and ${\mathbf{u}}_s$ are the elastic displacements in the matrix and the subphase respectively and $p_m$ and $p_s$ are the pressures in the matrix and subphase respectively. The ${\mathbb{C}}_m$ and ${\mathbb{C}}_s$ are the effective elasticity tensors that describe the matrix and the subphase. These tensors can be obtained by carrying out a homogenisation process at the finer hierarchical level i.e., at the porescale shown in Fig. 1. These tensors ${\mathbb{C}}_m$ and ${\mathbb{C}}_s$ are the effective elasticity tensors obtained in [22, 23] for a standard poroelastic material. These effective elasticity tensors possess major and minor symmetries as proved in [29]. This means we can write the fourth rank effective elasticity tensors ${\mathbb{C}}_m$ and ${\mathbb{C}}_s$ in components as ${\mathcal{C}}_{\mathit{pqrs}}^m$ and ${\mathcal{C}}_{\mathit{pqrs}}^s$, for $p,q,r,s=1,2,3$. Therefore we have

$$\begin{array}{c}\hfill {\mathcal{C}}_{\mathit{pqrs}}^m={\mathcal{C}}_{\mathit{qprs}}^m={\mathcal{C}}_{\mathit{pqsr}}^m={\mathcal{C}}_{\mathit{rspq}}^m,\end{array}$$ 4a

$$\begin{array}{c}\hfill {\mathcal{C}}_{\mathit{pqrs}}^s={\mathcal{C}}_{\mathit{qprs}}^s={\mathcal{C}}_{\mathit{pqsr}}^s={\mathcal{C}}_{\mathit{rspq}}^s.\end{array}$$ 4b

The second rank tensors ${\mathit{\alpha }}_m$ and ${\mathit{\alpha }}_s$ appearing in (2a) and (2b) are the effective Biot tensors of coefficients in the matrix and subphase respectively. These tensors are obtained from homogenisation at a finer hierarchical scale and quantify the coupling between pore pressure and mechanical deformation. Physically, they describe how changes in fluid pressure contribute to the total stress and how deformation of the solid skeleton influences fluid content. In the classical isotropic Biot theory, these tensors reduce to scalar Biot coefficients multiplied by the identity tensor. The tensorial form adopted here allows this coupling to depend on direction and therefore accommodates anisotropic microstructures and material behaviour.

Remark 1

For comparison, in the classical isotropic linear poroelastic model the elasticity tensor reduces to the Lamé elasticity tensor and the Biot tensor reduces to ${\alpha }_{\beta }\mathsf{I}$, where $\beta =m,s$. The constitutive law then takes the familiar form

$${\mathsf{T}}_{\beta }=2{\mu }_{\beta }\zeta ({\mathbf{u}}_{\beta })+{\lambda }_{\beta }(\mathrm{\nabla }·{\mathbf{u}}_{\beta })\mathsf{I}-{\alpha }_{\beta }{p}_{\beta }\mathsf{I}.$$

The present formulation therefore generalises the standard isotropic Biot model to anisotropic and heterogeneous poroelastic phases.

In the subphases and the matrix we also have Darcy’s law governing the fluid flow. That is,

$$\begin{array}{cc}\hfill {\mathbf{w}}_m=-{\mathsf{K}}_m(\mathrm{\nabla }{p}_m+{\mathbf{f}}_{\mathit{mf}})& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 5a

$$\begin{array}{cc}\hfill {\mathbf{w}}_s=-{\mathsf{K}}_s(\mathrm{\nabla }{p}_s+{\mathbf{f}}_{\mathit{sf}})& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 5b

where ${\mathsf{K}}_m$ and ${\mathsf{K}}_s$ are the hydraulic conductivities in the matrix and subphase respectively, and ${\mathbf{w}}_m$ and ${\mathbf{w}}_s$ are the relative fluid-solid velocities in the matrix and subphase respectively. These are subject to the body forces ${\mathbf{f}}_{\mathit{mf}}$ and ${\mathbf{f}}_{\mathit{sf}}$.

The last governing equation of each compartment is the conservation of mass equation given by

$$\begin{array}{cc}\hfill \frac{{\dot{p}}_m}{M_m}=-{\mathit{\alpha }}_m:\zeta ({\dot{\mathbf{u}}}_m)-\mathrm{\nabla }·{\mathbf{w}}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 6a

$$\begin{array}{cc}\hfill \frac{{\dot{p}}_s}{M_s}=-{\mathit{\alpha }}_s:\zeta ({\dot{\mathbf{u}}}_s)-\mathrm{\nabla }·{\mathbf{w}}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 6b

in the matrix and subphase. The quantities $M_m$ and $M_s$ are the Biot moduli in the matrix and subphase respectively. These coefficients are obtained from homogenisation at finer hierarchical scales and depend on the local pore geometry, porosity, fluid compressibility, and elastic properties of the solid skeleton. Physically, the Biot modulus characterises the storage capacity of the poroelastic material. Its inverse measures the variation in fluid content induced by a variation in pore pressure under prescribed deformation. Consequently, larger values of $M_m$ and $M_s$ correspond to systems with lower fluid-storage capacity, while smaller values indicate that fluid content changes more readily in response to pressure variations. These have been proved to be positive definite in [29].

The conservation of mass equations describe the balance between fluid storage, deformation-induced changes in pore volume, and fluid transport. The terms involving the Biot tensors account for the change in fluid content caused by local expansion or contraction of the porous skeleton, while the divergence of the Darcy velocity represents the net fluid flux leaving the phase. These equations therefore provide a continuum description of swelling, consolidation, and local reconfiguration of the pore space resulting from coupled fluid-solid interactions.

Finally we need to close the problem by prescribing conditions on the interface $\mathrm{\Gamma }$. These are continuity of stress, continuity of elastic displacements and the continuity of fluxes and pressures between the subphases and the matrix

$$\begin{array}{cc}\hfill {\mathsf{T}}_m\mathbf{n}={\mathsf{T}}_s\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 7a

$$\begin{array}{cc}\hfill {\mathbf{u}}_m={\mathbf{u}}_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 7b

$$\begin{array}{cc}\hfill {\mathbf{w}}_s·\mathbf{n}={\mathbf{w}}_m·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 7c

$$\begin{array}{cc}\hfill p_m=p_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 7d

where $\mathbf{n}$ is the normal to the interface $\mathrm{\Gamma }$ pointing into the subphase.

The problem is also to be closed by appropriate boundary conditions on the external boundary $\partial \mathrm{\Omega }$. The latter could be, for example, of Dirichlet-Neumann type, as noted in [43]. The conditions on the external boundary typically do not play a role in the derivation of results carried out by formal asymptotic homogenisation.

Now that we have set up the problem we are able to perform a multiscale analysis in order to derive the new model. To do this we will (a) non-dimensionalise the problem we have set up in this section, (b) we will introduce two well-separated length scales, (c) apply the asymptotic homogenisation technique to the non-dimensionalised problem, and (d) derive the novel effective governing equations for the double poroelastic material with applied body force.

Multiscale Analysis

We begin the multiscale analysis by collecting the governing equations introduced in the previous section. We have

$$\begin{array}{cc}\hfill \mathrm{\nabla }·{\mathsf{T}}_m={\mathbf{f}}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 8a

$$\begin{array}{cc}\hfill \mathrm{\nabla }·{\mathsf{T}}_s={\mathbf{f}}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 8b

$$\begin{array}{cc}\hfill {\mathsf{T}}_m={\mathbb{C}}_m:\zeta ({\mathbf{u}}_m)-{\mathit{\alpha }}_m{p}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 8c

$$\begin{array}{cc}\hfill {\mathsf{T}}_s={\mathbb{C}}_s:\zeta ({\mathbf{u}}_s)-{\mathit{\alpha }}_s{p}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 8d

$$\begin{array}{cc}\hfill {\mathbf{w}}_m=-{\mathsf{K}}_m(\mathrm{\nabla }{p}_m+{\mathbf{f}}_{\mathit{mf}})& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 8e

$$\begin{array}{cc}\hfill {\mathbf{w}}_s=-{\mathsf{K}}_s(\mathrm{\nabla }{p}_s+{\mathbf{f}}_{\mathit{sf}})& \text{in}{\mathrm{\Omega }}_s\hfill \end{array}$$ 8f

$$\begin{array}{cc}\hfill \frac{{\dot{p}}_m}{M_m}=-{\mathit{\alpha }}_m:\zeta ({\dot{\mathbf{u}}}_m)-\mathrm{\nabla }·{\mathbf{w}}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 8g

$$\begin{array}{cc}\hfill \frac{{\dot{p}}_s}{M_s}=-{\mathit{\alpha }}_s:\zeta ({\dot{\mathbf{u}}}_s)-\mathrm{\nabla }·{\mathbf{w}}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 8h

$$\begin{array}{cc}\hfill {\mathsf{T}}_m\mathbf{n}={\mathsf{T}}_s\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 8i

$$\begin{array}{cc}\hfill {\mathbf{u}}_m={\mathbf{u}}_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 8j

$$\begin{array}{cc}\hfill {\mathbf{w}}_s·\mathbf{n}={\mathbf{w}}_m·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 8k

$$\begin{array}{cc}\hfill p_m=p_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 8l

The problem (8a)–(8l) is closed by prescribing suitable external boundary conditions on $\partial \mathrm{\Omega }$. The latter could be, for example, of Dirichlet-Neumann type, as noted in [43]. For the purposes of the formal asymptotic homogenisation carried out here, the precise form of the external boundary conditions does not influence the derivation of the effective equations. The analysis relies on the presence of two well-separated characteristic length scales. We denote the average size of the whole domain by L and this is the macroscale. The scale at which we clearly see the subphases resolved from the surrounding matrix is denoted by d, the mesoscale.

The Asymptotic Homogenisation Technique

The homogenisation procedure follows the standard two-scale asymptotic framework developed in [30, 40], here extended to multiphase poroelastic media subject to phase-dependent body forces. We assume that the characteristic length scale of the mesoscale structure, denoted by d, is much smaller than the characteristic length scale of the whole domain, denoted by L. Thus,

$$\begin{array}{c}\hfill ϵ=\frac{d}{L}\ll 1.\end{array}$$ 9

This separation of scales motivates the introduction of a fast spatial variable

$$\begin{array}{c}\hfill \mathbf{y}=\frac{\mathbf{x}}{ϵ},\end{array}$$ 10

where $\mathbf{x}$ and $\mathbf{y}$ are treated as independent variables representing the macro- and mesoscale coordinates, respectively. The gradient operator is therefore transformed according to

$$\begin{array}{c}\hfill \mathrm{\nabla }\to {\mathrm{\nabla }}_x+\frac{1}{ϵ}{\mathrm{\nabla }}_y.\end{array}$$ 11

Each dependent variable is assumed to admit an asymptotic expansion in integer powers of $ϵ$, namely

$$\begin{array}{c}\hfill {\varphi }^{ϵ}(\mathbf{x},\mathbf{y},t)=\sum _{l=0}^{\infty }{ϵ}^l{\varphi }^{(l)}(\mathbf{x},\mathbf{y},t),\end{array}$$ 12

where $\varphi$ denotes any of the fields appearing in the governing equations.

Throughout the derivation, the fields are assumed to be periodic in the mesoscale variable $\mathbf{y}$ over a representative cell. This assumption allows the mesoscale problems generated by the asymptotic expansion to be posed on a single periodic cell rather than on the full material microstructure. We also assume macroscopic uniformity of the cell geometry, so that the cell does not vary explicitly with $\mathbf{x}$. This permits differentiation under the integral sign, for example

$$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}_x·(\bullet )d\mathbf{y}={\mathrm{\nabla }}_x·{\int }_{\mathrm{\Omega }}(\bullet )d\mathbf{y},\end{array}$$ 13

where $(\bullet )$ denotes a vector or tensor quantity.

Multiple Scales Expansion

We now apply the two-scale expansion introduced in Sect. 3.1 to the dimensionless system (8a)–(8l). The full expanded system, obtained after applying $\mathrm{\nabla }\mapsto {\mathrm{\nabla }}_x+{ϵ}^{-1}{\mathrm{\nabla }}_y$, is given in Appendix A.1. Here, we record the consequences of the first two orders in $ϵ$, since these are the terms required to obtain the effective macroscale model.

At leading order, the displacement and pressure fields are independent of the mesoscale variable and are common to both phases. In particular, the ${ϵ}^0$ equations imply

$${\mathbf{u}}_m^{(0)}={\mathbf{u}}_s^{(0)}={\mathbf{u}}^{(0)}(\mathbf{x},t),p_m^{(0)}=p_s^{(0)}=p^{(0)}(\mathbf{x},t).$$

The ${ϵ}^1$ equations determine the first-order corrector fields, which account for mesoscale heterogeneity and phase contrast. These correctors enter the effective constitutive laws and therefore determine the macroscale poroelastic response. We now derive the corresponding cell problems in three steps: first for the elastic displacement correctors and effective stress, then for the pressure correctors and effective Darcy law, and finally for the averaged conservation of mass equation.

Poroelastic Problem

We begin with the mechanical part of the coupled problem. Combining the leading-order momentum balance with the first-order constitutive relations yields a cell problem for the first-order displacement correctors ${\mathbf{u}}_m^{(1)}$ and ${\mathbf{u}}_s^{(1)}$. These correctors describe how mesoscale elastic heterogeneity and pressure coupling modify the effective stress at the macroscale. Therefore by taking (A4a), (A4b) with (A8c) and (A8d) along with the interface conditions (A4i) and (A8j) we can write

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathbf{u}}_m^{(1)}))=-{\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_x({\mathbf{u}}^{(0)}))+{\mathrm{\nabla }}_y·(p^{(0)}{\mathit{\alpha }}_m)& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 14a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathbf{u}}_s^{(1)}))=-{\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_x({\mathbf{u}}^{(0)}))+{\mathrm{\nabla }}_y·(p^{(0)}{\mathit{\alpha }}_s)& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 14b

$$\begin{array}{cc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathbf{u}}_m^{(1)})-{\mathbb{C}}_s{\zeta }_y({\mathbf{u}}_s^{(1)}))\mathbf{n}=(({\mathbb{C}}_s-{\mathbb{C}}_m){\zeta }_x({\mathbf{u}}^{(0)})-({\mathit{\alpha }}_s-{\mathit{\alpha }}_m)p^{(0)})\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 14c

$$\begin{array}{cc}\hfill {\mathbf{u}}_s^{(1)}={\mathbf{u}}_m^{(1)}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 14d

The problem (14a–14d) has a unique solution up to a $\mathbf{y}$-constant function. Exploiting the linear structure of the problem, we introduce the following ansatz

$$\begin{array}{cc}& {\mathbf{u}}_m^{(1)}={\mathcal{B}}_m{\zeta }_x({\mathbf{u}}^{(0)})+{\mathbf{b}}_m{p}^{(0)}+c_1(\mathbf{x}),\hfill \end{array}$$ 15a

$$\begin{array}{cc}& {\mathbf{u}}_s^{(1)}={\mathcal{B}}_s{\zeta }_x({\mathbf{u}}^{(0)})+{\mathbf{b}}_s{p}^{(0)}+c_2(\mathbf{x}),\hfill \end{array}$$ 15b

where we have that $c_1(\mathbf{x})$ and $c_2(\mathbf{x})$ are $\mathbf{y}$-constant functions and we have the third order tensors ${\mathcal{B}}_m$ and ${\mathcal{B}}_s$ which satisfy the cell problems given by

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathcal{B}}_m))=-{\mathrm{\nabla }}_y·{\mathbb{C}}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 16a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathcal{B}}_s))=-{\mathrm{\nabla }}_y·{\mathbb{C}}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 16b

$$\begin{array}{cc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathcal{B}}_m)-{\mathbb{C}}_s{\zeta }_y({\mathcal{B}}_s))\mathbf{n}=({\mathbb{C}}_s-{\mathbb{C}}_m)\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 16c

$$\begin{array}{cc}\hfill {\mathcal{B}}_m={\mathcal{B}}_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 16d

as well as the vectors ${\mathbf{b}}_m$ and ${\mathbf{b}}_s$ which satisfy the problem

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathbf{b}}_m))={\mathrm{\nabla }}_y·{\mathit{\alpha }}_m& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 17a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathbf{b}}_s))={\mathrm{\nabla }}_y·{\mathit{\alpha }}_s& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 17b

$$\begin{array}{cc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathbf{b}}_m)-{\mathbb{C}}_s{\zeta }_y({\mathbf{b}}_s))\mathbf{n}=-({\mathit{\alpha }}_m-{\mathit{\alpha }}_s)\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 17c

$$\begin{array}{cc}\hfill {\mathbf{b}}_m={\mathbf{b}}_s& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 17d

The problems (16a–16d) and (17a–17d) are further supplemented with periodic conditions on $\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }$. Uniqueness is enforced by prescribing zero average of the sum of auxiliary fields ${\mathcal{B}}_m$, ${\mathcal{B}}_s$, ${\mathbf{b}}_m$ and ${\mathbf{b}}_s$, over their respective subdomains. That is

$$\begin{array}{c}\hfill {⟨{\mathcal{B}}_m⟩}_m+{⟨{\mathcal{B}}_s⟩}_s=0,{⟨{\mathbf{b}}_m⟩}_m+{⟨{\mathbf{b}}_s⟩}_s=0.\end{array}$$ 18

Now that we have expressions for the leading order elastic displacements we can write the leading order effective stress tensors in the matrix and subphase respectively as

$$\begin{array}{cc}\hfill {\mathsf{T}}_m^{(0)}& ={\mathbb{C}}_m{\zeta }_y({\mathcal{B}}_m{\zeta }_x({\mathbf{u}}^{(0)})+{\mathbf{b}}_m{p}^{(0)})+{\mathbb{C}}_m{\zeta }_x({\mathbf{u}}^{(0)})-{\mathit{\alpha }}_m{p}^{(0)}\hfill \\ \hfill & =({\mathbb{C}}_m{\mathbb{L}}_m+{\mathbb{C}}_m){\zeta }_x({\mathbf{u}}^{(0)})+({\mathbb{C}}_m{\mathit{\tau }}_m-{\mathit{\alpha }}_m)p^{(0)},\hfill \end{array}$$ 19

where we have the auxiliary tensors

$$\begin{array}{c}\hfill {\mathbb{L}}_m={\zeta }_y({\mathcal{B}}_m)\text{and}{\mathit{\tau }}_m={\zeta }_y({\mathbf{b}}_m),\end{array}$$ 20

and

$$\begin{array}{cc}\hfill {\mathsf{T}}_s^{(0)}& ={\mathbb{C}}_s{\zeta }_y({\mathcal{B}}_s{\zeta }_x({\mathbf{u}}^{(0)})+{\mathbf{b}}_s{p}^{(0)})+{\mathbb{C}}_s{\zeta }_x({\mathbf{u}}^{(0)})-{\mathit{\alpha }}_s{p}^{(0)}\hfill \\ \hfill & =({\mathbb{C}}_s{\mathbb{L}}_s+{\mathbb{C}}_s){\zeta }_x({\mathbf{u}}^{(0)})+({\mathbb{C}}_s{\mathit{\tau }}_s-{\mathit{\alpha }}_s)p^{(0)},\hfill \end{array}$$ 21

where we have used the notation

$$\begin{array}{c}\hfill {\mathbb{L}}_s={\zeta }_y({\mathcal{B}}_s)\text{and}{\mathit{\tau }}_s={\zeta }_y({\mathbf{b}}_s).\end{array}$$ 22

The fourth-rank tensors ${\mathbb{L}}_m$ and ${\mathbb{L}}_s$ represent mesoscale strain localisation induced by elastic heterogeneity and phase contrast, analogous to strain concentration tensors in classical composite theory. The second rank tensors ${\mathit{\tau }}_m$ and ${\mathit{\tau }}_s$ quantify the additional mesoscale deformation induced by pressure loading, beyond that predicted by classical Biot coupling.

We require a balance equation for the effective stresses that takes into consideration both the subphases and the matrix. Integrating the ${ϵ}^1$ momentum balance (A8a) and (A8b) over the matrix and subphase, using periodicity on the external cell boundary and traction continuity on $\mathrm{\Gamma }$, gives

$${\mathrm{\nabla }}_{\mathbf{x}}·\left({⟨{\mathsf{T}}_m^{(0)}⟩}_m,+,{⟨{\mathsf{T}}_s^{(0)}⟩}_s\right)={⟨{\mathbf{f}}_m^{(0)}⟩}_m+{⟨{\mathbf{f}}_s^{(0)}⟩}_s.$$

The detailed cancellation of the cell-boundary and interface terms is reported in Appendix A.1. Thus,

$${\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_{\text{eff}}={⟨{\mathbf{f}}_m^{(0)}⟩}_m+{⟨{\mathbf{f}}_s^{(0)}⟩}_s,$$

where

$$\begin{array}{cc}\hfill {\mathsf{T}}_{\text{eff}}& ={⟨{\mathsf{T}}_m^{(0)}⟩}_m+{⟨{\mathsf{T}}_s^{(0)}⟩}_s\hfill \\ \hfill & =({⟨{\mathbb{C}}_m{\mathbb{L}}_m+{\mathbb{C}}_m⟩}_m+{⟨{\mathbb{C}}_s{\mathbb{L}}_s+{\mathbb{C}}_s⟩}_s){\zeta }_x({\mathbf{u}}^{(0)})+({⟨{\mathbb{C}}_m{\mathit{\tau }}_m-{\mathit{\alpha }}_m⟩}_m\hfill \\ \hfill & +{⟨{\mathbb{C}}_s{\mathit{\tau }}_s-{\mathit{\alpha }}_s⟩}_s)p^{(0)}\hfill \end{array}$$ 23

At this stage, we have derived the constitutive law and the balance equation for the solid portion of our material. We have determined the leading order displacement correctors in both the matrix and subphases (${\mathbb{L}}_m$, ${\mathbb{L}}_s$, ${\mathit{\tau }}_m$ and ${\mathit{\tau }}_s$) and identified the associated cell problems defining the elastic and pressure coupling tensors. These correctors encode the influence of mesoscale heterogeneity and phase contrast on the effective stress–strain and stress–pressure relations.

The Darcy Flow Problem

We can use the balance equations (A4g) and (A4h) with the interface conditions (A4k) and (A8l) to write the following problem for the first order pressures in each phase $p_m^{(1)}$ and $p_s^{(1)}$

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·{\mathbf{w}}_m^{(0)}=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 24a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·{\mathbf{w}}_s^{(0)}=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 24b

$$\begin{array}{cc}\hfill p_m^{(1)}=p_s^{(1)}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 24c

$$\begin{array}{cc}\hfill {\mathbf{w}}_m^{(0)}·\mathbf{n}={\mathbf{w}}_s^{(0)}·\mathbf{n}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 24d

Using the expressions (A8e) and (A8f) that we have for ${\mathbf{w}}_m^{(0)}$ and ${\mathbf{w}}_s^{(0)}$ we rewrite the problem (24a–24d) in terms of the pressures

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(1)})=-{\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_x{p}^{(0)})+{\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)})& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 25a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(1)})=-{\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_x{p}^{(0)})+{\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)})& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 25b

$$\begin{array}{cc}\hfill p_m^{(1)}=p_s^{(1)}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 25c

$$\begin{array}{cc}\hfill ({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(1)}-{\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(1)})·\mathbf{n}=(({\mathsf{K}}_s-{\mathsf{K}}_m){\mathrm{\nabla }}_x{p}^{(0)}+({\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}-{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}))·\mathbf{n}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 25d

This problem then admits a unique solution up to a $\mathbf{y}$-constant function (see [44, 45]). By exploiting the linearity we are able to propose the following ansatz

$$\begin{array}{cc}& p_m^{(1)}={\mathbf{P}}_m·{\mathrm{\nabla }}_x{p}^{(0)}+s_m(x,t)+c_3(\mathbf{x}),\hfill \end{array}$$ 26a

$$\begin{array}{cc}& p_s^{(1)}={\mathbf{P}}_s·{\mathrm{\nabla }}_x{p}^{(0)}+s_s(x,t)+c_4(\mathbf{x}),\hfill \end{array}$$ 26b

where we have that $c_3(\mathbf{x})$ and $c_4(\mathbf{x})$ are $\mathbf{y}$-constant functions and the vectors ${\mathbf{P}}_m$ and ${\mathbf{P}}_s$ and the scalar functions $s_m(x,t)$ and $s_s(x,t)$ satisfy the cell problems given by

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·(({\mathrm{\nabla }}_y{\mathbf{P}}_m){\mathsf{K}}_m^{\text{T}})=-{\mathrm{\nabla }}_y·{\mathsf{K}}_m^{\text{T}}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 27a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·(({\mathrm{\nabla }}_y{\mathbf{P}}_s){\mathsf{K}}_s^{\text{T}})=-{\mathrm{\nabla }}_y·{\mathsf{K}}_s^{\text{T}}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 27b

$$\begin{array}{cc}\hfill {\mathbf{P}}_m={\mathbf{P}}_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 27c

$$\begin{array}{cc}\hfill (({\mathrm{\nabla }}_y{\mathbf{P}}_m){\mathsf{K}}_m^{\text{T}}-({\mathrm{\nabla }}_y{\mathbf{P}}_s){\mathsf{K}}_s^{\text{T}})\mathbf{n}={({\mathsf{K}}_s-{\mathsf{K}}_m)}^{\text{T}}\mathbf{n}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 27d

and

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathrm{\nabla }}_y{s}_m))={\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)})& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 28a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathrm{\nabla }}_y{s}_s))={\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)})& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 28b

$$\begin{array}{cc}\hfill s_m=s_s& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 28c

$$\begin{array}{cc}\hfill ({\mathsf{K}}_m({\mathrm{\nabla }}_y{s}_m)-{\mathsf{K}}_s({\mathrm{\nabla }}_y{s}_s))·\mathbf{n}=({\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}-{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)})·\mathbf{n}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 28d

The cell problem is supplemented by periodic conditions on the boundary $\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }$ and for the sake of uniqueness of solution we place a further condition on ${\mathbf{P}}_m$ and ${\mathbf{P}}_s$. We impose a zero-average condition on the sum of the auxiliary fields over the two subdomains, that is

$$\begin{array}{c}\hfill {⟨{\mathbf{P}}_m⟩}_m+{⟨{\mathbf{P}}_s⟩}_s=0.\end{array}$$ 29

We now wish to find the expressions for the leading order Darcy’s law in the subphases and matrix respectively. We can use the expressions (26a) and (26b) for the first order pressures $p_m^{(1)}$ and $p_s^{(1)}$ in (A8e) and (A8f) and by applying the integral average we obtain

$$\begin{array}{cc}\hfill {⟨{\mathbf{w}}_m^{(0)}⟩}_m& =-{⟨{\mathsf{K}}_m{({\mathrm{\nabla }}_y{\mathbf{P}}_m)}^{\text{T}}⟩}_m{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m⟩}_m{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathrm{\nabla }}_y{s}_m⟩}_m+{⟨{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}⟩}_m\hfill \\ \hfill & =-{⟨{\mathsf{K}}_m{\mathsf{R}}_m+{\mathsf{K}}_m⟩}_m{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathbf{s}}_m⟩}_m+{⟨{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}⟩}_m,\hfill \end{array}$$ 30

in the matrix where we have used the notation

$$\begin{array}{c}\hfill {\mathsf{R}}_m={({\mathrm{\nabla }}_y{\mathbf{P}}_m)}^{\text{T}}\text{and}{\mathbf{s}}_m={\mathrm{\nabla }}_y{s}_m\end{array}$$ 31

and we obtain

$$\begin{array}{cc}\hfill {⟨{\mathbf{w}}_s^{(0)}⟩}_s& =-{⟨{\mathsf{K}}_s{({\mathrm{\nabla }}_y{\mathbf{P}}_s)}^{\text{T}}⟩}_s{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_s⟩}_s{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_s{\mathrm{\nabla }}_y{s}_s⟩}_s+{⟨{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}⟩}_s\hfill \\ \hfill & =-{⟨{\mathsf{K}}_s{\mathsf{R}}_s+{\mathsf{K}}_s⟩}_s{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_s{\mathbf{s}}_s⟩}_s+{⟨{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}⟩}_s,\hfill \end{array}$$ 32

in the subphase where we have used the notation

$$\begin{array}{c}\hfill {\mathsf{R}}_s={({\mathrm{\nabla }}_y{\mathbf{P}}_s)}^{\text{T}}\text{and}{\mathbf{s}}_s={\mathrm{\nabla }}_y{s}_s\end{array}$$ 33

These expressions are now combined to derive the effective Darcy law governing the macroscopic flow. We have

$$\begin{array}{cc}\hfill {\mathbf{w}}_{\text{eff}}:& ={⟨{\mathbf{w}}_m^{(0)}⟩}_m+{⟨{\mathbf{w}}_s^{(0)}⟩}_s\hfill \\ \hfill & =-({⟨{\mathsf{K}}_m{\mathsf{R}}_m+{\mathsf{K}}_m⟩}_m+{⟨{\mathsf{K}}_s{\mathsf{R}}_s+{\mathsf{K}}_s⟩}_s){\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathbf{s}}_m⟩}_m+{⟨{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}⟩}_m\hfill \\ \hfill & -{⟨{\mathsf{K}}_s{\mathbf{s}}_s⟩}_s+{⟨{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}⟩}_s,\hfill \end{array}$$ 34

where we can define an effective hydraulic conductivity tensor as

$$\begin{array}{c}\hfill \mathsf{W}={⟨{\mathsf{K}}_m{\mathsf{R}}_m+{\mathsf{K}}_m⟩}_m+{⟨{\mathsf{K}}_s{\mathsf{R}}_s+{\mathsf{K}}_s⟩}_s\end{array}$$ 35

and this means that Darcy’s law can be rewritten as

$$\begin{array}{c}\hfill {\mathbf{w}}_{\text{eff}}=-\mathsf{W}{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathbf{s}}_m⟩}_m+{⟨{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}⟩}_m-{⟨{\mathsf{K}}_s{\mathbf{s}}_s⟩}_s+{⟨{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}⟩}_s.\end{array}$$ 36

The resulting expressions define an effective Darcy law in which the hydraulic conductivity tensor incorporates both phase-wise conductivities and mesoscale correctors accounting for heterogeneity and body-force effects.

Conservation of Mass

The final equation required to close the macroscale system is the averaged conservation of mass equation. Integrating the ${ϵ}^1$ mass balance equations over the matrix and subphase, and using flux continuity across $\mathrm{\Gamma }$, gives an effective mass balance involving the macroscale pressure, the effective relative fluid–solid velocity, and the first-order displacement correctors. The intermediate averaging steps are given in Appendix A.2.

Substituting the first-order displacement correctors (15a)–(15b) into this averaged balance yields

$$\begin{array}{cc}\hfill {\dot{p}}^{(0)}& =-\overline{\mathcal{M}}({\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}+\overline{\mathit{\alpha }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})),\hfill \end{array}$$ 37

where

$$\begin{array}{c}\hfill \overline{\mathcal{M}}:=\frac{{⟨M_m⟩}_m{⟨M_s⟩}_s}{{⟨M_m⟩}_m+{⟨M_s⟩}_s+{⟨M_m⟩}_m{⟨M_s⟩}_s\left({⟨{\mathit{\alpha }}_m:{\mathit{\tau }}_m⟩}_m,+,{⟨{\mathit{\alpha }}_s:{\mathit{\tau }}_s⟩}_s\right)},\end{array}$$ 38

and

$$\begin{array}{c}\hfill \overline{\mathit{\alpha }}:={〈{\mathit{\alpha }}_m,+,{\mathit{\alpha }}_s,+,{\mathbb{L}}_m^{\text{T}},:,{\mathit{\alpha }}_m,+,{\mathbb{L}}_s^{\text{T}},:,{\mathit{\alpha }}_s〉}_{\mathrm{\Omega }}.\end{array}$$ 39

The scalar coefficient $\overline{\mathcal{M}}$ plays the role of an effective Biot modulus for the double poroelastic system, while $\overline{\mathit{\alpha }}$ is the corresponding effective Biot tensor. The latter contains the phase-wise Biot tensors together with corrector terms arising from mesoscale compressibility contrasts and displacement–pressure coupling across the matrix–subphase interface.

We have now derived all the equations required to be able to state our macroscale model.

Macroscale Model

By assembling the results derived above, we obtain a closed system of macroscale equations describing the effective poroelastic response of this double poroelastic material subject to body forces. The system consists of a balance of linear momentum, a modified Darcy law, and a conservation of mass equation, all expressed in terms of the leading order elastic displacement ${\mathbf{u}}^{(0)}$, the leading order body forces, the relative fluid-solid velocity ${\mathbf{w}}_{\text{eff}}$ and the pressure $p^{(0)}$. The model is given by

$$\begin{array}{cc}& {\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_{\text{eff}}={⟨{\mathbf{f}}_m^{(0)}⟩}_m+{⟨{\mathbf{f}}_s^{(0)}⟩}_s,\hfill \end{array}$$ 40a

$$\begin{array}{cc}& {\mathsf{T}}_{\text{eff}}=({⟨{\mathbb{C}}_m{\mathbb{L}}_m+{\mathbb{C}}_m⟩}_m+{⟨{\mathbb{C}}_s{\mathbb{L}}_s+{\mathbb{C}}_s⟩}_s){\zeta }_x({\mathbf{u}}^{(0)})+({⟨{\mathbb{C}}_m{\mathit{\tau }}_m-{\mathit{\alpha }}_m⟩}_m\hfill \\ \hfill & +{⟨{\mathbb{C}}_s{\mathit{\tau }}_s-{\mathit{\alpha }}_s⟩}_s)p^{(0)}\hfill \end{array}$$ 40b

$$\begin{array}{cc}& {\dot{p}}^{(0)}=-\overline{\mathcal{M}}({\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}+{⟨{\mathit{\alpha }}_m+{\mathit{\alpha }}_s+{\mathbb{L}}_m^{\text{T}}:{\mathit{\alpha }}_m+{\mathbb{L}}_s^{\text{T}}:{\mathit{\alpha }}_s⟩}_{\mathrm{\Omega }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})),\hfill \end{array}$$ 40c

$$\begin{array}{cc}& {\mathbf{w}}_{\text{eff}}=-\mathsf{W}{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathbf{s}}_m⟩}_m+{⟨{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}⟩}_m-{⟨{\mathsf{K}}_s{\mathbf{s}}_s⟩}_s+{⟨{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}⟩}_s\hfill \end{array}$$ 40d

We have the macroscale balance equation (40a) for the effective stress tensor (40b) subject to the leading order body forces. The effective stress comprises the coefficient tensors ${\mathbb{C}}_m+{\mathbb{C}}_m{\mathbb{L}}_m$ and ${\mathbb{C}}_s+{\mathbb{C}}_s{\mathbb{L}}_s$ which are found by solving (16a)–(16d) and account for differences in elastic properties at different points in the microstructure. The terms ${\mathbb{L}}_m$ and ${\mathbb{L}}_s$ can be thought of as corrector terms to the classical averaging of the elastic properties. The effective stress also contains a term relating to the fluid pressure where the coefficient of this term is to be found by solving (17a)–(17d). This coefficient encodes differences in the compressibility at different points in the microstructure. The problems to be solved for the coefficients in the effective stress are similar to those found for elastic composites in [46, 47], poroelastic composites in [34, 38, 39] and double poroelastic materials [35].

We then have (40c) which is the macroscale conservation of mass equation. This equation comprises the divergence of the relative fluid-solid velocity ${\mathbf{w}}_{\text{eff}}$ and the Biot’s tensor of coefficients $\overline{\mathit{\alpha }}$ applied to the leading order strains. The Biot’s tensor that we obtain here (39) comprises the Biot’s tensors from the subphase and the matrix as well as the two additional corrector contributions arising due to the fact that we are accounting for changing compressibility at different points in the microstructure.

And finally we have Darcy’s law (40d) with the modified hydraulic conductivity tensor $\mathsf{W}$ (35). This tensor consists of the hydraulic conductivities ${\mathsf{K}}_m$ and ${\mathsf{K}}_s$ from the subphase and matrix as well as two additional terms. The extra terms ${\mathsf{K}}_m{\mathsf{R}}_m$ and ${\mathsf{K}}_s{\mathsf{R}}_s$ account for the differences in the hydraulic conductivities at different points in the microstructure. The hydraulic conductivity tensor $\mathsf{W}$ can be found by solving cell problem (27a)–(27d). Darcy’s law also contains the additional leading order forcing contributions and phase-averaged sources to be computed by solving cell problem (28a)–(28d).

Although the derivation above is written in the most general anisotropic form, the effective tensors simplify considerably when the periodic cell possesses additional material or geometrical symmetries. For example, isotropic, transversely isotropic, orthotropic, or layered microstructures reduce the number of independent entries of the effective elasticity tensor, hydraulic conductivity tensor, and Biot-type coupling tensors. In such cases fewer independent cell problems are required. Thus, the fully anisotropic formulation presented here should be viewed as the general framework from which practically simpler models can be obtained by imposing the relevant symmetry class.

Specialising to the Application of Magnetic Lorentz Force

In this section, we specialise the general body-force framework derived above to the case of magnetic Lorentz forcing. This example is chosen because it represents a structured body force: rather than entering the governing equations as a prescribed source alone, it depends on gradients of electromagnetic potentials and is therefore coupled to the transport structure of the problem. Such forces are relevant in magnetic-resonance-type settings involving saturated biological tissues or porous materials with electrically conducting interstitial fluids.

The purpose of this section is not to develop a full electromagnetic model, but to demonstrate how the mathematical structure of a particular body force is transmitted through the homogenisation procedure. In particular, the Lorentz-force example shows that body forces with gradient structure can generate additional pressure-like contributions in the effective Darcy law, rather than appearing only as averaged forcing terms in the macroscale equations.

We define the magnetic Lorentz forcing in the matrix and subphase by

$$\begin{array}{c}\hfill {\mathbf{f}}_{\mathit{mf}}={\mathbf{j}}_m\times \mathbf{B},{\mathbf{j}}_m=-{\mathsf{G}}_m\mathrm{\nabla }{\varphi }_m,\end{array}$$ 41

and

$$\begin{array}{c}\hfill {\mathbf{f}}_{\mathit{sf}}={\mathbf{j}}_s\times \mathbf{B},{\mathbf{j}}_s=-{\mathsf{G}}_s\mathrm{\nabla }{\varphi }_s.\end{array}$$ 42

Here, ${\mathbf{j}}_{\alpha }$ denotes the current density, ${\mathsf{G}}_{\alpha }$ is the electrical conductance tensor, ${\varphi }_{\alpha }$ is the electric potential, $\mathbf{B}$ is the imposed magnetic field, and $\alpha =m,s$ denotes the matrix and subphase respectively.

We present below the key changes induced by Lorentz forcing. The full order-by-order expansion, the longer auxiliary cell problems, and the intermediate averaging steps are provided in Appendix B.

Lorentz-Modified Darcy’s Law

We first examine the phase-wise Darcy laws, since this is where the Lorentz force enters the coupled system most directly. Substituting (41) and (42) into the Darcy laws gives

$$\begin{array}{cc}\hfill {\mathbf{w}}_m& =-{\mathsf{K}}_m\mathrm{\nabla }{p}_m+{\mathsf{K}}_m({\mathsf{G}}_m\mathrm{\nabla }{\varphi }_m\times \mathbf{B}),\hfill \end{array}$$ 43

$$\begin{array}{cc}\hfill {\mathbf{w}}_s& =-{\mathsf{K}}_s\mathrm{\nabla }{p}_s+{\mathsf{K}}_s({\mathsf{G}}_s\mathrm{\nabla }{\varphi }_s\times \mathbf{B}).\hfill \end{array}$$ 44

The important distinction from the generic body-force case is that the Lorentz terms contain gradients of electromagnetic potentials. As a result, the leading-order pressure is no longer, in general, represented by a single phase-independent quantity. Instead, the order ${ϵ}^0$ problem gives

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(0)})-{\mathrm{\nabla }}_y·\left({\mathsf{K}}_m,({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})\right)=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 45a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(0)})-{\mathrm{\nabla }}_y·\left({\mathsf{K}}_s,({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})\right)=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 45b

$$\begin{array}{cc}\hfill \left({\mathsf{K}}_m,{\mathrm{\nabla }}_y,p_m^{(0)},-,{\mathsf{K}}_m,({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})\right)·\mathbf{n}=\left({\mathsf{K}}_s,{\mathrm{\nabla }}_y,p_s^{(0)},-,{\mathsf{K}}_s,({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})\right)·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 45c

$$\begin{array}{cc}\hfill p_m^{(0)}=p_s^{(0)}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 45d

Thus, by linearity, we write

$$\begin{array}{cc}\hfill p_m^{(0)}& ={\widehat{p}}_m+\overline{p}(\mathbf{x},t),\hfill \end{array}$$ 46

$$\begin{array}{cc}\hfill p_s^{(0)}& ={\widehat{p}}_s+\overline{p}(\mathbf{x},t),\hfill \end{array}$$ 47

where ${\widehat{p}}_m$ and ${\widehat{p}}_s$ are Lorentz-induced mesoscale pressure correctors and $\overline{p}(\mathbf{x},t)$ is independent of $\mathbf{y}$. The pressure correctors satisfy

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_y{\widehat{p}}_m)-{\mathrm{\nabla }}_y·\left({\mathsf{K}}_m,({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})\right)=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ 48a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_y{\widehat{p}}_s)-{\mathrm{\nabla }}_y·\left({\mathsf{K}}_s,({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})\right)=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ 48b

$$\begin{array}{cc}\hfill \left({\mathsf{K}}_m,{\mathrm{\nabla }}_y,{\widehat{p}}_m,-,{\mathsf{K}}_m,({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})\right)·\mathbf{n}=\left({\mathsf{K}}_s,{\mathrm{\nabla }}_y,{\widehat{p}}_s,-,{\mathsf{K}}_s,({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})\right)·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ 48c

$$\begin{array}{cc}\hfill {\widehat{p}}_m={\widehat{p}}_s& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ 48d

These equations are supplemented with periodic boundary conditions on $\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }$ and a zero-average condition to ensure uniqueness.

Proceeding to the order ${ϵ}^1$ problem, we use the ansatz

$$\begin{array}{cc}\hfill p_m^{(1)}& ={\mathbf{d}}_m·{\mathrm{\nabla }}_x\overline{p}+t_m+c(\mathbf{x}),\hfill \end{array}$$ 49

$$\begin{array}{cc}\hfill p_s^{(1)}& ={\mathbf{d}}_s·{\mathrm{\nabla }}_x\overline{p}+t_s+c(\mathbf{x}),\hfill \end{array}$$ 50

where ${\mathbf{d}}_m,{\mathbf{d}}_s$ are hydraulic correctors and $t_m,t_s$ collect the remaining Lorentz-induced contributions. Their cell problems are stated in Appendix B.1. Defining

$$\begin{array}{c}\hfill {\mathsf{D}}_m={({\mathrm{\nabla }}_y{\mathbf{d}}_m)}^{\mathrm{T}},{\mathsf{D}}_s={({\mathrm{\nabla }}_y{\mathbf{d}}_s)}^{\mathrm{T}},{\mathbf{t}}_m={\mathrm{\nabla }}_y{t}_m,{\mathbf{t}}_s={\mathrm{\nabla }}_y{t}_s,\end{array}$$ 51

the effective Darcy law is

$$\begin{array}{cc}\hfill {\mathbf{w}}_{\text{eff}}& =-\mathsf{W}{\mathrm{\nabla }}_x\overline{p}-{⟨{\mathsf{K}}_m{\mathrm{\nabla }}_x{\widehat{p}}_m⟩}_m-{⟨{\mathsf{K}}_s{\mathrm{\nabla }}_x{\widehat{p}}_s⟩}_s-{⟨{\mathsf{K}}_m{\mathbf{t}}_m⟩}_m-{⟨{\mathsf{K}}_s{\mathbf{t}}_s⟩}_s+{\mathbf{F}}_L,\hfill \end{array}$$ 52

where

$$\begin{array}{c}\hfill \mathsf{W}={⟨{\mathsf{K}}_m⟩}_m+{⟨{\mathsf{K}}_s⟩}_s+{⟨{\mathsf{K}}_m{\mathsf{D}}_m⟩}_m+{⟨{\mathsf{K}}_s{\mathsf{D}}_s⟩}_s,\end{array}$$ 53

and

$$\begin{array}{cc}\hfill {\mathbf{F}}_L& ={⟨{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})⟩}_m+{⟨{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})⟩}_s\hfill \\ \hfill & +{⟨{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)})⟩}_m+{⟨{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)})⟩}_m\hfill \\ \hfill & +{⟨{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)})⟩}_s+{⟨{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)})⟩}_s.\hfill \end{array}$$ 54

Equation (52) shows that the Lorentz force does not simply enter as an averaged source term. It generates mesoscale pressure correctors and additional driving terms in the effective pressure–flow relation.

Lorentz-Modified Poroelastic Problem

We next consider the effect of Lorentz forcing on the mechanical part of the coupled problem. For simplicity, we assume that the Lorentz force acting on the solid components is the same in the matrix and subphase:

$$\begin{array}{c}\hfill {\mathbf{f}}_m={\mathbf{f}}_s=-\mathsf{G}\mathrm{\nabla }\varphi \times \mathbf{B}.\end{array}$$ 55

Here, $\mathsf{G}$ is the conductance tensor, $\varphi$ is the electric potential and $\mathbf{B}$ is the magnetic field. These quantities are, in general, distinct from ${\mathsf{G}}_m$, ${\mathsf{G}}_s$, ${\varphi }_m$ and ${\varphi }_s$ appearing in (41)–(42).

The Lorentz-modified poroelastic cell problem differs from Sect. 3.3 because the leading-order pressures in the two phases are retained separately. The first-order displacement correctors are therefore written as

$$\begin{array}{cc}\hfill {\mathbf{u}}_m^{(1)}& ={\mathcal{B}}_m{\zeta }_x({\mathbf{u}}^{(0)})+{\mathbf{b}}_m{p}_m^{(0)}+{\tilde{\mathbf{b}}}_m{p}_s^{(0)}+{\mathbf{e}}_m+c_1(\mathbf{x}),\hfill \end{array}$$ 56a

$$\begin{array}{cc}\hfill {\mathbf{u}}_s^{(1)}& ={\mathcal{B}}_s{\zeta }_x({\mathbf{u}}^{(0)})+{\mathbf{b}}_s{p}_s^{(0)}+{\tilde{\mathbf{b}}}_s{p}_m^{(0)}+{\mathbf{e}}_s+c_2(\mathbf{x}).\hfill \end{array}$$ 56b

The correctors ${\mathcal{B}}_m,{\mathcal{B}}_s,{\mathbf{b}}_m,{\mathbf{b}}_s,{\tilde{\mathbf{b}}}_m$ and ${\tilde{\mathbf{b}}}_s$ are defined in Appendix B.2. The genuinely new Lorentz-induced mechanical correctors ${\mathbf{e}}_m$ and ${\mathbf{e}}_s$ solve

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathbf{e}}_m))& =-\mathsf{G}{\mathrm{\nabla }}_y{\varphi }^{(0)}\times \mathbf{B}\hfill & \hfill \text{in}{\mathrm{\Omega }}_m,\end{array}$$ 57a

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathbf{e}}_s))& =-\mathsf{G}{\mathrm{\nabla }}_y{\varphi }^{(0)}\times \mathbf{B}\hfill & \hfill \text{in}{\mathrm{\Omega }}_s,\end{array}$$ 57b

$$\begin{array}{ccc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathbf{e}}_m))\mathbf{n}& =({\mathbb{C}}_s{\zeta }_y({\mathbf{e}}_s))\mathbf{n}\hfill & \hfill \text{on}\mathrm{\Gamma },\end{array}$$ 57c

$$\begin{array}{ccc}\hfill {\mathbf{e}}_m& ={\mathbf{e}}_s\hfill & \hfill \text{on}\mathrm{\Gamma }.\end{array}$$ 57d

These problems are supplemented with periodic boundary conditions on $\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }$ and zero-average conditions for uniqueness.

Define

$$\begin{array}{c}\hfill {\mathbb{L}}_m={\zeta }_y({\mathcal{B}}_m),{\mathit{\tau }}_m={\zeta }_y({\mathbf{b}}_m),{\tilde{\mathit{\tau }}}_m={\zeta }_y({\tilde{\mathbf{b}}}_m),{\mathit{\theta }}_m={\zeta }_y({\mathbf{e}}_m),\end{array}$$ 58

and

$$\begin{array}{c}\hfill {\mathbb{L}}_s={\zeta }_y({\mathcal{B}}_s),{\mathit{\tau }}_s={\zeta }_y({\mathbf{b}}_s),{\tilde{\mathit{\tau }}}_s={\zeta }_y({\tilde{\mathbf{b}}}_s),{\mathit{\theta }}_s={\zeta }_y({\mathbf{e}}_s).\end{array}$$ 59

The leading-order stresses are then

$$\begin{array}{cc}\hfill {\mathsf{T}}_m^{(0)}& =({\mathbb{C}}_m{\mathbb{L}}_m+{\mathbb{C}}_m){\zeta }_x({\mathbf{u}}^{(0)})+({\mathbb{C}}_m{\mathit{\tau }}_m-{\mathit{\alpha }}_m)p_m^{(0)}+{\mathbb{C}}_m{\tilde{\mathit{\tau }}}_m{p}_s^{(0)}+{\mathbb{C}}_m{\mathit{\theta }}_m,\hfill \end{array}$$ 60

$$\begin{array}{cc}\hfill {\mathsf{T}}_s^{(0)}& =({\mathbb{C}}_s{\mathbb{L}}_s+{\mathbb{C}}_s){\zeta }_x({\mathbf{u}}^{(0)})+({\mathbb{C}}_s{\mathit{\tau }}_s-{\mathit{\alpha }}_s)p_s^{(0)}+{\mathbb{C}}_s{\tilde{\mathit{\tau }}}_s{p}_m^{(0)}+{\mathbb{C}}_s{\mathit{\theta }}_s.\hfill \end{array}$$ 61

For the solid Lorentz force, the leading-order body force entering the effective momentum balance is

$$\begin{array}{c}\hfill {\mathbf{f}}^{(0)}=-\mathsf{G}{\mathrm{\nabla }}_y{\varphi }^{(1)}\times {\mathbf{B}}^{(0)}-\mathsf{G}{\mathrm{\nabla }}_y{\varphi }^{(0)}\times {\mathbf{B}}^{(1)}-\mathsf{G}{\mathrm{\nabla }}_x{\varphi }^{(0)}\times {\mathbf{B}}^{(0)}.\end{array}$$ 62

Hence the Lorentz-modified effective momentum balance is

$$\begin{array}{c}\hfill {\mathrm{\nabla }}_x·{\mathsf{T}}_{\text{eff}}={⟨{\mathbf{f}}^{(0)}⟩}_m+{⟨{\mathbf{f}}^{(0)}⟩}_s,\end{array}$$ 63

where

$$\begin{array}{cc}\hfill {\mathsf{T}}_{\text{eff}}& ={⟨{\mathsf{T}}_m^{(0)}⟩}_m+{⟨{\mathsf{T}}_s^{(0)}⟩}_s\hfill \\ \hfill & =\left({⟨{\mathbb{C}}_m{\mathbb{L}}_m+{\mathbb{C}}_m⟩}_m,+,{⟨{\mathbb{C}}_s{\mathbb{L}}_s+{\mathbb{C}}_s⟩}_s\right){\zeta }_x({\mathbf{u}}^{(0)})+{〈{\mathbb{C}}_m,{\mathit{\tau }}_m,-,{\mathit{\alpha }}_m〉}_m{p}_m^{(0)}\hfill \\ \hfill & +{〈{\mathbb{C}}_s,{\mathit{\tau }}_s,-,{\mathit{\alpha }}_s〉}_s{p}_s^{(0)}+{〈{\mathbb{C}}_m,{\tilde{\mathit{\tau }}}_m〉}_m{p}_s^{(0)}+{〈{\mathbb{C}}_s,{\tilde{\mathit{\tau }}}_s〉}_s{p}_m^{(0)}\hfill \\ \hfill & +{〈{\mathbb{C}}_m,{\mathit{\theta }}_m〉}_m+{〈{\mathbb{C}}_s,{\mathit{\theta }}_s〉}_s.\hfill \end{array}$$ 64

Remark 2

We have assumed that the external loads are the same in the solid phase, while they may differ in the fluid phase. In the fluid phase, the role of the external body force is encoded in Darcy’s law, and local conservation of mass enforces continuity of the normal component of the velocity at the interface, as in [48]. For elastic-type problems with vector or scalar potential body forces, the boundary conditions involve continuity of tractions and displacements. In this setting, compatibility of the cell problems requires care when external loads are discontinuous across the interface; alternative formulations, such as decompositions involving inelastic stress contributions [37], can accommodate discontinuous loads by incorporating them into the traction condition.

Lorentz-Modified Conservation of Mass

We now derive the macroscale conservation of mass equation in the presence of Lorentz forcing, without assuming the existence of a phase-independent leading-order pressure. Accordingly, the leading-order pressures in the matrix and subphase are denoted by $p_m^{(0)}$ and $p_s^{(0)}$, respectively.

Integrating the phase-wise mass balances over the cell, using flux continuity across $\mathrm{\Gamma }$, and substituting the displacement correctors (56a)–(56b) gives

$$\begin{array}{cc}& {\mathcal{A}}_m{\dot{p}}_m^{(0)}+{\mathcal{A}}_s{\dot{p}}_s^{(0)}=-{\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}-\overline{\mathit{\alpha }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)}),\hfill \end{array}$$ 65

where

$$\begin{array}{cc}\hfill {\mathcal{A}}_m& :={⟨M_m⟩}_m^{-1}+{⟨{\mathit{\alpha }}_m:{\mathit{\tau }}_m⟩}_m+{⟨{\mathit{\alpha }}_s:{\tilde{\mathit{\tau }}}_s⟩}_s,\hfill \end{array}$$ 66

$$\begin{array}{cc}\hfill {\mathcal{A}}_s& :={⟨M_s⟩}_s^{-1}+{⟨{\mathit{\alpha }}_s:{\mathit{\tau }}_s⟩}_s+{⟨{\mathit{\alpha }}_m:{\tilde{\mathit{\tau }}}_m⟩}_m,\hfill \end{array}$$ 67

and

$$\begin{array}{c}\hfill \overline{\mathit{\alpha }}:={〈{\mathit{\alpha }}_m,+,{\mathbb{L}}_m^{\mathrm{T}},:,{\mathit{\alpha }}_m〉}_m+{〈{\mathit{\alpha }}_s,+,{\mathbb{L}}_s^{\mathrm{T}},:,{\mathit{\alpha }}_s〉}_s.\end{array}$$ 68

The intermediate averaging steps are given in Appendix B.3. The coefficients ${\mathcal{A}}_m$ and ${\mathcal{A}}_s$ are effective storage coefficients. Equivalently, one may define phase-wise effective Biot-type moduli by ${\overline{M}}_m={\mathcal{A}}_m^{-1}$ and ${\overline{M}}_s={\mathcal{A}}_s^{-1}$.

Equation (65) shows that Lorentz effects enter the conservation of mass through the modified effective flux, while the separation of the leading-order pressures produces additional storage and coupling terms.

The Macroscale Model for a Double Poroelastic Material Subject to the Magnetic Lorentz Force

Collecting the previous results, the Lorentz-modified macroscale model is

$$\begin{array}{cc}& {\mathrm{\nabla }}_x·{\mathsf{T}}_{\text{eff}}={⟨{\mathbf{f}}^{(0)}⟩}_m+{⟨{\mathbf{f}}^{(0)}⟩}_s,\hfill \end{array}$$ 69a

$$\begin{array}{cc}& {\mathsf{T}}_{\text{eff}}=\left({⟨{\mathbb{C}}_m{\mathbb{L}}_m+{\mathbb{C}}_m⟩}_m,+,{⟨{\mathbb{C}}_s{\mathbb{L}}_s+{\mathbb{C}}_s⟩}_s\right){\zeta }_x({\mathbf{u}}^{(0)})+{〈{\mathbb{C}}_m,{\mathit{\tau }}_m,-,{\mathit{\alpha }}_m〉}_m{p}_m^{(0)}\hfill \\ \hfill & +{〈{\mathbb{C}}_s,{\mathit{\tau }}_s,-,{\mathit{\alpha }}_s〉}_s{p}_s^{(0)}+{〈{\mathbb{C}}_m,{\tilde{\mathit{\tau }}}_m〉}_m{p}_s^{(0)}+{〈{\mathbb{C}}_s,{\tilde{\mathit{\tau }}}_s〉}_s{p}_m^{(0)}+{〈{\mathbb{C}}_m,{\mathit{\theta }}_m〉}_m\hfill \\ \hfill & +{〈{\mathbb{C}}_s,{\mathit{\theta }}_s〉}_s,\hfill \end{array}$$ 69b

$$\begin{array}{cc}& {\mathcal{A}}_m{\dot{p}}_m^{(0)}+{\mathcal{A}}_s{\dot{p}}_s^{(0)}=-{\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}-\overline{\mathit{\alpha }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)}),\hfill \end{array}$$ 69c

$$\begin{array}{cc}& {\mathbf{w}}_{\text{eff}}=-\mathsf{W}{\mathrm{\nabla }}_x\overline{p}-{⟨{\mathsf{K}}_m{\mathrm{\nabla }}_x{\widehat{p}}_m⟩}_m-{⟨{\mathsf{K}}_s{\mathrm{\nabla }}_x{\widehat{p}}_s⟩}_s-{⟨{\mathsf{K}}_m{\mathbf{t}}_m⟩}_m-{⟨{\mathsf{K}}_s{\mathbf{t}}_s⟩}_s+{\mathbf{F}}_L.\hfill \end{array}$$ 69d

Here $\mathsf{W}$ is defined in (53), ${\mathbf{F}}_L$ in (54), and ${\mathbf{f}}^{(0)}$ in (62).

This specialisation highlights a key distinction between generic body forces and forces possessing gradient or curl structure. In the Lorentz case, the forcing does not simply enter the homogenised equations as an averaged source term. It also modifies the effective pressure and flow relations through additional corrector contributions, thereby altering the effective constitutive behaviour of the double-poroelastic medium.

Comparison Between the General Force Model and the Magnetic Lorentz Force Model

We now want to compare the expressions obtained in the general macroscale model derived in Sect. 3.6 with the Lorentz force model derived in Sect. 4.4.

We begin by considering Darcy’s law, and compare (69d) with the general expression (40d). For convenience we restate (40d) as

$$\begin{array}{c}\hfill {\mathbf{w}}_{\text{eff}}=-\mathsf{W}{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathbf{s}}_m⟩}_m+{⟨{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}⟩}_m-{⟨{\mathsf{K}}_s{\mathbf{s}}_s⟩}_s+{⟨{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}⟩}_s\end{array}$$ 70

where

$$\begin{array}{c}\hfill {\mathbf{f}}_{\mathit{mf}}=-{\mathsf{G}}_m\mathrm{\nabla }{\varphi }_m\times \mathbf{B}\text{and}{\mathbf{f}}_{\mathit{sf}}=-{\mathsf{G}}_s\mathrm{\nabla }{\varphi }_s\times \mathbf{B}\end{array}$$ 71

Applying the multiple scales expansion we obtain

$$\begin{array}{c}\hfill ϵ{\mathbf{f}}_{\mathit{mf}}^{ϵ}=-{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ}-ϵ{\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ}\end{array}$$ 72

$$\begin{array}{c}\hfill ϵ{\mathbf{f}}_{\mathit{sf}}^{ϵ}=-{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ}-ϵ{\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ}\end{array}$$ 73

We can equate the powers of $ϵ$ in these two forces. For ${ϵ}^0$ we obtain

$$\begin{array}{c}\hfill -{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)}=0\text{and}-{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)}=0\end{array}$$ 74

and the coefficients of ${ϵ}^1$ give

$$\begin{array}{c}\hfill {\mathbf{f}}_{\mathit{mf}}^{(0)}=-{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)}-{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)}-{\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)}\end{array}$$ 75

$$\begin{array}{c}\hfill {\mathbf{f}}_{\mathit{sf}}^{(0)}=-{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)}-{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)}-{\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)}\end{array}$$ 76

We can substitute these expressions into (70) to obtain

$$\begin{array}{cc}\hfill {\mathbf{w}}_{\text{eff}}& =-\mathsf{W}{\mathrm{\nabla }}_x{p}^{(0)}-{⟨{\mathsf{K}}_m{\mathbf{s}}_m⟩}_m-{⟨{\mathsf{K}}_s{\mathbf{s}}_s⟩}_s+{⟨{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})⟩}_m\hfill \\ \hfill & +{⟨{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})⟩}_s+{⟨{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)})⟩}_m\hfill \\ \hfill & +{⟨{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)})⟩}_s+{⟨{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)})⟩}_m\hfill \\ \hfill & +{⟨{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)})⟩}_s\hfill \end{array}$$ 77

Comparing (77) and (69d), we see that (69d) differs from the generic Darcy law (77) because, in the Lorentz case, the body force is generated by electrical currents of the form ${\mathbf{j}}_{\alpha }=-{\mathbb{G}}_{\alpha }\mathrm{\nabla }{\varphi }_{\alpha }$, $\alpha =m,s$ and therefore depends explicitly on the gradient of an electromagnetic potential. When this structure is carried through the homogenisation procedure, the curl-free component of the Lorentz force can be rewritten as the gradient of an additional magnetic pressure. This produces the extra pressure-type contribution that appears in (69d), while the remaining non–curl-free component enters as a standard forcing term, in direct analogy with the generic body-force contributions already present in (77).

By contrast, if the chosen body force does not contain a gradient, divergence, or any structure that can be recast as a potential, then no such decomposition occurs. Most classical body forces in poroelasticity such as gravity, buoyancy, or drag enter the homogenised Darcy law purely as source terms and do not modify the pressure-gradient structure. The magnetic Lorentz force is exceptional because its dependence on $\mathrm{\nabla }{\varphi }_{\alpha }$, $\alpha =m,s$ allows part of its homogenised contribution to behave exactly like a pressure gradient at the macroscale.

The stress balance equations (40a) and (69a) will coincide when the leading order magnetic Lorentz force is used in (40a). However, the influence of magnetic forcing appears in the effective stress tensor (69b) where the leading order pressure is no longer unified.

This lack of unified leading order pressure also influences the conservation of mass equation (69c). Comparing with (40c) we can see that Lorentz effects enter via the effective flux and via the Biot’s moduli.

Conclusion

In this work, we have developed a rigorous multiscale framework for determining the effective behaviour of materials composed of a poroelastic matrix containing embedded poroelastic subphases, each governed by the fully anisotropic and heterogeneous Biot equations and each subject to body forces. By applying the asymptotic homogenisation technique [29, 30, 44], under assumptions of scale separation, periodic microstructure, and macroscopic uniformity, we derived a closed system of Biot-type equations that captures the coupled mechanical and fluid-transport response of the composite at the macroscale.

Unlike existing double-poroelastic and poroelastic composite models, the present framework explicitly incorporates body forces acting at the mesoscale and tracks their influence through to the effective macroscale equations. The resulting effective coefficients capture not only elastic and hydraulic heterogeneity, but also force-induced couplings that are absent from the classical Biot theory.

A central outcome of the analysis is the explicit form of the effective stress tensor, hydraulic conductivity tensor, and Biot-type coupling coefficients, each obtained from well-posed cell problems on the representative microstructural domain. These quantities reveal how mesoscale interactions between matrix and subphases shape the macroscale poromechanical response, and they generalise existing models for single-phase, composite, and double-poroelastic materials [34, 35, 38].

By specialising the framework to the case of magnetic Lorentz forcing, we demonstrated that body forces possessing a gradient structure lead to qualitatively different effective behaviour. In particular, such forces generate additional pressure-like contributions in the homogenised Darcy law, rather than appearing solely as averaged source terms. This highlights the importance of correctly accounting for the microscale structure of applied forces when modelling coupled poroelastic systems.

The framework presented here provides a foundation for analysing a broad class of multiphase poroelastic materials, including biological tissues, engineered composites, and geomechanical systems with embedded inclusions or fibres. The present analysis relies on several simplifying assumptions, including mesoscale periodicity, macroscopic uniformity of the microstructure, and quasi-static, linear poroelastic behaviour. While these assumptions enable a tractable and rigorous derivation, relaxing them would allow the framework to capture a wider range of physically relevant phenomena.

A natural extension of the present work is the relaxation of the macroscopic uniformity assumption [23], allowing the mesoscale geometry and material coefficients to vary slowly in space. In such locally periodic settings, additional transport and drift terms arise in the homogenised equations, reflecting the evolution of the underlying microstructure.

Extending the framework to nonlinear poroelastic constitutive laws or evolving microstructures [36, 40] and modelling microstructural alterations such as those induced by disease or remodelling [39] would require the solution of parameter-dependent cell problems and the development of consistent update strategies for the effective coefficients, posing both analytical and computational challenges. These challenges make the resulting cell problems computationally expensive to solve. Emerging techniques in the literature aim to overcome this issue, for example those in [49, 50], making extensions of this kind more computationally feasible.

This work could be extended to account for discontinuous volume loads governing the balance of linear momentum of the poroelastic phases by adapting the formulation for electrostrictive composites [37], in which forces can be written as the divergence of suitable inelastic stress components, such as the Maxwell stress tensor.

From a computational perspective, the explicit characterisation of the effective coefficients through cell problems facilitates numerical implementation using finite element or reduced-order methods. Developing efficient solution strategies, particularly in three dimensions and for highly anisotropic materials, remains an important direction for future research. Extending these computational approaches to realistic geometries, including biological imaging, would further enable quantitative predictions for application-specific scenarios.

From a biological perspective, Lorentz forces generated during MRI arise from the interaction between induced ionic currents and the applied magnetic field and are known to produce measurable tissue displacements (e.g., [51, 52]). In a double-poroelastic tissue, where intracellular (subphase) and extracellular (matrix) compartments may differ in electrical conductivity and hydraulic properties, such forces would not be distributed uniformly at the mesoscale. This suggests that MRI-induced electromagnetic forcing could drive compartment-specific pressure variations and relative fluid motion, potentially modulating perfusion, interstitial transport, or mechanotransductive signalling. Consequently, the multiscale coupling identified in our analysis may provide a mechanistic link between electromagnetic contrasts and measurable poromechanical responses in heterogeneous biological tissues.

Appendix A Technical Details of the Homogenisation Derivation

This appendix contains the technical details supporting the homogenisation procedure presented in Sect. 3. The main text focuses on the key cell problems, effective coefficients, and resulting macroscale equations, while the order-by-order expansions and intermediate averaging steps are collected here for completeness. This organisation preserves the reproducibility of the derivation while keeping the main presentation focused on the structure and interpretation of the effective model.

A.1 Order-by-Order Expansion of the Two-Scale Problem

In this appendix we provide the order-by-order details of the two-scale expansion used in Sect. 3. Applying the transformations (11) and (12) to the dimensionless system (8a)–(8l), and using the periodicity of the representative cell, gives

$$\begin{array}{cc}\hfill ϵ{\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_m^{ϵ}+{\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_m^{ϵ}=ϵ{\mathbf{f}}_m^{ϵ}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A1a

$$\begin{array}{cc}\hfill ϵ{\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_s^{ϵ}+{\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_s^{ϵ}=ϵ{\mathbf{f}}_s^{ϵ}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A1b

$$\begin{array}{cc}\hfill ϵ{\mathsf{T}}_m^{ϵ}={\mathbb{C}}_m:{\zeta }_y({\mathbf{u}}_m^{ϵ})+ϵ{\mathbb{C}}_m:{\zeta }_x({\mathbf{u}}_m^{ϵ})-ϵ{\mathit{\alpha }}_m{p}_m^{ϵ}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A1c

$$\begin{array}{cc}\hfill ϵ{\mathsf{T}}_s^{ϵ}={\mathbb{C}}_s:{\zeta }_y({\mathbf{u}}_s^{ϵ})+ϵ{\mathbb{C}}_s:{\zeta }_x({\mathbf{u}}_s^{ϵ})-ϵ{\mathit{\alpha }}_s{p}_s^{ϵ}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A1d

$$\begin{array}{cc}\hfill ϵ{\mathbf{w}}_m^{ϵ}=-{\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{ϵ}-ϵ{\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{ϵ}+ϵ{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A1e

$$\begin{array}{cc}\hfill ϵ{\mathbf{w}}_s^{ϵ}=-{\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{ϵ}-ϵ{\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{ϵ}+ϵ{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A1f

$$\begin{array}{cc}\hfill ϵ\frac{{\dot{p}}_m^{ϵ}}{M_m}=-{\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{ϵ})-ϵ{\mathit{\alpha }}_m:{\zeta }_x({\dot{\mathbf{u}}}_m^{ϵ})-{\mathrm{\nabla }}_y·{\mathbf{w}}_m^{ϵ}-ϵ{\mathrm{\nabla }}_x·{\mathbf{w}}_m^{ϵ}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A1g

$$\begin{array}{cc}\hfill ϵ\frac{{\dot{p}}_s^{ϵ}}{M_s}=-{\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{ϵ})-ϵ{\mathit{\alpha }}_s:{\zeta }_x({\dot{\mathbf{u}}}_s^{ϵ})-{\mathrm{\nabla }}_y·{\mathbf{w}}_s^{ϵ}-ϵ{\mathrm{\nabla }}_x·{\mathbf{w}}_s^{ϵ}& \text{in}{\mathrm{\Omega }}_{\text{s}},\hfill \end{array}$$ A1h

$$\begin{array}{cc}\hfill {\mathsf{T}}_m^{ϵ}\mathbf{n}={\mathsf{T}}_s^{ϵ}\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A1i

$$\begin{array}{cc}\hfill {\mathbf{u}}_m^{ϵ}={\mathbf{u}}_s^{ϵ}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A1j

$$\begin{array}{cc}\hfill {\mathbf{w}}_m^{ϵ}·\mathbf{n}={\mathbf{w}}_s^{ϵ}·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A1k

$$\begin{array}{cc}\hfill p_m^{ϵ}=p_s^{ϵ}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ A1l

Substituting expansions of the form (12) into (A1a)–(A1l) and equating terms with equal powers of $ϵ$ gives the hierarchy of problems used in Sect. 3 to derive the effective model.

For fields that retain a dependence on the mesoscale variable, we use the phase average

$$\begin{array}{c}\hfill {⟨\phi ⟩}_{\alpha }=\frac{1}{|\mathrm{\Omega }|}{\int }_{{\mathrm{\Omega }}_{\alpha }}\phi (\mathbf{x},\mathbf{y},t)d\mathbf{y},\alpha =m,s,\end{array}$$ A2

where the integral is taken over one representative periodic cell and $|\mathrm{\Omega }|=|{\mathrm{\Omega }}_m|+|{\mathrm{\Omega }}_s|$. We also define the average over the full cell by

$$\begin{array}{c}\hfill {⟨{\phi }_m+{\phi }_s⟩}_{\mathrm{\Omega }}=\frac{1}{|\mathrm{\Omega }|}\left({\int }_{{\mathrm{\Omega }}_m},{\phi }_m,(\mathbf{x},\mathbf{y},t),,d,\mathbf{y},+,{\int }_{{\mathrm{\Omega }}_s},{\phi }_s,(\mathbf{x},\mathbf{y},t),,d,\mathbf{y}\right).\end{array}$$ A3

Equating terms of order ${ϵ}^0$ gives

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_m^{(0)}=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A4a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_s^{(0)}=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A4b

$$\begin{array}{cc}\hfill {\mathbb{C}}_m:{\zeta }_y({\mathbf{u}}_m^{(0)})=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A4c

$$\begin{array}{cc}\hfill {\mathbb{C}}_s:{\zeta }_y({\mathbf{u}}_s^{(0)})=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A4d

$$\begin{array}{cc}\hfill {\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(0)}=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A4e

$$\begin{array}{cc}\hfill {\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(0)}=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A4f

$$\begin{array}{cc}\hfill {\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{(0)})+{\mathrm{\nabla }}_y·{\mathbf{w}}_m^{(0)}=0& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A4g

$$\begin{array}{cc}\hfill {\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{(0)})+{\mathrm{\nabla }}_y·{\mathbf{w}}_s^{(0)}=0& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A4h

$$\begin{array}{cc}\hfill {\mathsf{T}}_m^{(0)}\mathbf{n}={\mathsf{T}}_s^{(0)}\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A4i

$$\begin{array}{cc}\hfill {\mathbf{u}}_m^{(0)}={\mathbf{u}}_s^{(0)}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A4j

$$\begin{array}{cc}\hfill {\mathbf{w}}_m^{(0)}·\mathbf{n}={\mathbf{w}}_s^{(0)}·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A4k

$$\begin{array}{cc}\hfill p_m^{(0)}=p_s^{(0)}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ A4l

Equations (A4c) and (A4d), together with periodicity, imply that the leading-order displacements are independent of the mesoscale variable. Hence

$$\begin{array}{cc}\hfill {\mathbf{u}}_m^{(0)}& ={\mathbf{u}}_m^{(0)}(\mathbf{x},t),\hfill \end{array}$$ A5a

$$\begin{array}{cc}\hfill {\mathbf{u}}_s^{(0)}& ={\mathbf{u}}_s^{(0)}(\mathbf{x},t).\hfill \end{array}$$ A5b

The interface condition (A4j) then gives

$$\begin{array}{c}\hfill {\mathbf{u}}^{(0)}={\mathbf{u}}_m^{(0)}={\mathbf{u}}_s^{(0)}.\end{array}$$ A6

Similarly, (A4e) and (A4f), together with (A4l), imply that the leading-order pressure is independent of $\mathbf{y}$ and common to both phases:

$$\begin{array}{c}\hfill p^{(0)}=p_m^{(0)}(\mathbf{x},t)=p_s^{(0)}(\mathbf{x},t).\end{array}$$ A7

Equating terms of order ${ϵ}^1$ gives

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_m^{(0)}+{\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_m^{(1)}={\mathsf{f}}_m^{(0)}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A8a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_s^{(0)}+{\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_s^{(1)}={\mathsf{f}}_s^{(0)}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A8b

$$\begin{array}{cc}\hfill {\mathsf{T}}_m^{(0)}={\mathbb{C}}_m:{\zeta }_y({\mathbf{u}}_m^{(1)})+{\mathbb{C}}_m:{\zeta }_x({\mathbf{u}}_m^{(0)})-{\mathit{\alpha }}_m{p}_m^{(0)}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A8c

$$\begin{array}{cc}\hfill {\mathsf{T}}_s^{(0)}={\mathbb{C}}_s:{\zeta }_y({\mathbf{u}}_s^{(1)})+{\mathbb{C}}_s:{\zeta }_x({\mathbf{u}}_s^{(0)})-{\mathit{\alpha }}_s{p}_s^{(0)}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A8d

$$\begin{array}{cc}\hfill {\mathbf{w}}_m^{(0)}=-{\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(1)}-{\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{(0)}+{\mathsf{K}}_m{\mathbf{f}}_{\mathit{mf}}^{(0)}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A8e

$$\begin{array}{cc}\hfill {\mathbf{w}}_s^{(0)}=-{\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(1)}-{\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{(0)}+{\mathsf{K}}_s{\mathbf{f}}_{\mathit{sf}}^{(0)}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A8f

$$\begin{array}{cc}\hfill \frac{{\dot{p}}_m^{(0)}}{M_m}=-{\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{(1)})-{\mathit{\alpha }}_m:{\zeta }_x({\dot{\mathbf{u}}}_m^{(0)})-{\mathrm{\nabla }}_y·{\mathbf{w}}_m^{(1)}-{\mathrm{\nabla }}_x·{\mathbf{w}}_m^{(0)}& \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ A8g

$$\begin{array}{cc}\hfill \frac{{\dot{p}}_s^{(0)}}{M_s}=-{\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{(1)})-{\mathit{\alpha }}_s:{\zeta }_x({\dot{\mathbf{u}}}_s^{(0)})-{\mathrm{\nabla }}_y·{\mathbf{w}}_s^{(1)}-{\mathrm{\nabla }}_x·{\mathbf{w}}_s^{(0)}& \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ A8h

$$\begin{array}{cc}\hfill {\mathsf{T}}_m^{(1)}\mathbf{n}={\mathsf{T}}_s^{(1)}\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A8i

$$\begin{array}{cc}\hfill {\mathbf{u}}_m^{(1)}={\mathbf{u}}_s^{(1)}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A8j

$$\begin{array}{cc}\hfill {\mathbf{w}}_m^{(1)}·\mathbf{n}={\mathbf{w}}_s^{(1)}·\mathbf{n}& \text{on}\mathrm{\Gamma },\hfill \end{array}$$ A8k

$$\begin{array}{cc}\hfill p_m^{(1)}=p_s^{(1)}& \text{on}\mathrm{\Gamma }.\hfill \end{array}$$ A8l

These ${ϵ}^0$ and ${ϵ}^1$ problems are the starting point for the cell problems for the displacement correctors, pressure correctors, and effective fluxes derived in Sect. 3.

Averaging of the Momentum Balance

For completeness, we provide the averaging step used to obtain the effective momentum balance in Sect. 3.3. Integrating the ${ϵ}^1$ momentum Eqs.  (A8a) and (A8b) over the matrix and subphase domains gives

$$\begin{array}{cc}& {\int }_{{\mathrm{\Omega }}_m}{\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_m^{(0)}\text{d}\mathbf{y}+{\int }_{{\mathrm{\Omega }}_m}{\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_m^{(1)}\text{d}\mathbf{y}+{\int }_{{\mathrm{\Omega }}_s}{\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_s^{(0)}\text{d}\mathbf{y}+{\int }_{{\mathrm{\Omega }}_s}{\mathrm{\nabla }}_{\mathbf{y}}·{\mathsf{T}}_s^{(1)}\text{d}\mathbf{y}\hfill \\ \hfill & -{\int }_{{\mathrm{\Omega }}_m}{\mathbf{f}}_m^{(0)}\text{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_s}{\mathbf{f}}_s^{(0)}\text{d}\mathbf{y}=0.\hfill \end{array}$$ A9

Using macroscopic uniformity, the ${\mathrm{\nabla }}_{\mathbf{x}}$ derivatives may be taken outside the cell integrals. Applying the divergence theorem to the ${\mathrm{\nabla }}_{\mathbf{y}}$ terms yields

$$\begin{array}{cc}& {\mathrm{\nabla }}_{\mathbf{x}}·{⟨{\mathsf{T}}_m^{(0)}⟩}_m+{\mathrm{\nabla }}_{\mathbf{x}}·{⟨{\mathsf{T}}_s^{(0)}⟩}_s+{\int }_{\partial {\mathrm{\Omega }}_m\backslash \mathrm{\Gamma }}{\mathsf{T}}_m^{(1)}{\mathbf{n}}_{\partial {\mathrm{\Omega }}_m}\text{d}S-{\int }_{\mathrm{\Gamma }}{\mathsf{T}}_m^{(1)}\mathbf{n}\text{d}S\hfill \\ \hfill & +{\int }_{\partial {\mathrm{\Omega }}_s\backslash \mathrm{\Gamma }}{\mathsf{T}}_s^{(1)}{\mathbf{n}}_{\partial {\mathrm{\Omega }}_s}\text{d}S+{\int }_{\mathrm{\Gamma }}{\mathsf{T}}_s^{(1)}\mathbf{n}\text{d}S-{⟨{\mathbf{f}}_m^{(0)}⟩}_m-{⟨{\mathbf{f}}_s^{(0)}⟩}_s=0.\hfill \end{array}$$ A10

The boundary contributions on $\partial {\mathrm{\Omega }}_m\backslash \mathrm{\Gamma }$ and $\partial {\mathrm{\Omega }}_s\backslash \mathrm{\Gamma }$ cancel by periodicity. The interface terms on $\mathrm{\Gamma }$ cancel by the traction-continuity condition (A8i). Hence,

$$\begin{array}{c}\hfill {\mathrm{\nabla }}_{\mathbf{x}}·{⟨{\mathsf{T}}_m^{(0)}⟩}_m+{\mathrm{\nabla }}_{\mathbf{x}}·{⟨{\mathsf{T}}_s^{(0)}⟩}_s={⟨{\mathbf{f}}_m^{(0)}⟩}_m+{⟨{\mathbf{f}}_s^{(0)}⟩}_s.\end{array}$$ A11

Defining

$$\begin{array}{c}\hfill {\mathsf{T}}_{\text{eff}}={⟨{\mathsf{T}}_m^{(0)}⟩}_m+{⟨{\mathsf{T}}_s^{(0)}⟩}_s,\end{array}$$ A12

we obtain the effective momentum balance

$$\begin{array}{c}\hfill {\mathrm{\nabla }}_{\mathbf{x}}·{\mathsf{T}}_{\text{eff}}={⟨{\mathbf{f}}_m^{(0)}⟩}_m+{⟨{\mathbf{f}}_s^{(0)}⟩}_s.\end{array}$$ A13

A.2 Averaging of the Conservation of Mass Equation

For completeness, we provide the averaging step used to obtain the effective conservation of mass equation in Sect. 3.5. Integrating the ${ϵ}^1$ mass balance equations (A8g) and (A8h) over ${\mathrm{\Omega }}_m$ and ${\mathrm{\Omega }}_s$ gives

$$\begin{array}{cc}& {\int }_{{\mathrm{\Omega }}_m}\frac{{\dot{p}}^{(0)}}{M_m}\text{d}\mathbf{y}+{\int }_{{\mathrm{\Omega }}_s}\frac{{\dot{p}}^{(0)}}{M_s}\text{d}\mathbf{y}=-{\int }_{{\mathrm{\Omega }}_m}{\mathit{\alpha }}_m:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})\text{d}\mathbf{y}-{\mathrm{\nabla }}_x·{\int }_{{\mathrm{\Omega }}_m}{\mathbf{w}}_m^{(0)}\text{d}\mathbf{y}\hfill \\ \hfill & -{\int }_{{\mathrm{\Omega }}_s}{\mathit{\alpha }}_s:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})\text{d}\mathbf{y}-{\mathrm{\nabla }}_x·{\int }_{{\mathrm{\Omega }}_s}{\mathbf{w}}_s^{(0)}\text{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_m}{\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{(1)})\text{d}\mathbf{y}\hfill \\ \hfill & -{\int }_{{\mathrm{\Omega }}_s}{\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{(1)})\text{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_m}{\mathrm{\nabla }}_y·{\mathbf{w}}_m^{(1)}\text{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_s}{\mathrm{\nabla }}_y·{\mathbf{w}}_s^{(1)}\text{d}\mathbf{y}.\hfill \end{array}$$ A14

Applying the divergence theorem to the final two terms, using periodicity on the external cell boundary and the flux-continuity condition (A8k) on $\mathrm{\Gamma }$, these boundary contributions cancel. Hence,

$$\begin{array}{cc}\hfill (\frac{{⟨M_m+M_s⟩}_{\mathrm{\Omega }}}{{⟨M_m⟩}_m{⟨M_s⟩}_s}){\dot{p}}^{(0)}& =-{⟨{\mathit{\alpha }}_m+{\mathit{\alpha }}_s⟩}_{\mathrm{\Omega }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})-{\mathrm{\nabla }}_x·{⟨{\mathbf{w}}_m^{(0)}+{\mathbf{w}}_s^{(0)}⟩}_{\mathrm{\Omega }}\hfill \\ \hfill & -{〈{\mathit{\alpha }}_m,:,{\zeta }_y,({\dot{\mathbf{u}}}_m^{(1)}),+,{\mathit{\alpha }}_s,:,{\zeta }_y,({\dot{\mathbf{u}}}_s^{(1)})〉}_{\mathrm{\Omega }}.\hfill \end{array}$$ A15

Using the first-order displacement correctors (15a)–(15b), we obtain

$$\begin{array}{cc}& (\frac{{⟨M_m+M_s⟩}_{\mathrm{\Omega }}}{{⟨M_m⟩}_m{⟨M_s⟩}_s}){\dot{p}}^{(0)}=-({⟨{\mathit{\alpha }}_m+{\mathit{\alpha }}_s⟩}_{\mathrm{\Omega }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})+{\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}\hfill \\ \hfill & +{〈{\mathbb{L}}_m^{\text{T}},:,{\mathit{\alpha }}_m,+,{\mathbb{L}}_s^{\text{T}},:,{\mathit{\alpha }}_s〉}_{\mathrm{\Omega }}:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})+{〈{\mathit{\alpha }}_m,:,{\mathit{\tau }}_m,+,{\mathit{\alpha }}_s,:,{\mathit{\tau }}_s〉}_{\mathrm{\Omega }}{\dot{p}}^{(0)}).\hfill \end{array}$$ A16

Rearranging (A16) gives the effective conservation of mass equation reported in Sect. 3.5.

Appendix B Details of the Lorentz-Force Specialisation

This appendix contains the order-by-order calculations used to obtain the Lorentz-modified effective equations in Sect. 4. The main text retains the key Lorentz-induced cell problems and the final effective equations, while the repeated expansions, longer auxiliary cell problems, and intermediate averaging steps are collected here.

B.1 Lorentz-Modified Darcy Flow

Starting from (43)–(44), the two-scale expansion gives

$$\begin{array}{cc}\hfill ϵ{\mathbf{w}}_m^{ϵ}& =-{\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{ϵ}-ϵ{\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{ϵ}+{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{ϵ}\times {\mathsf{B}}^{ϵ}+ϵ{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{ϵ}\times {\mathsf{B}}^{ϵ},\hfill \end{array}$$ B1

$$\begin{array}{cc}\hfill ϵ{\mathbf{w}}_s^{ϵ}& =-{\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{ϵ}-ϵ{\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{ϵ}+{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{ϵ}\times {\mathsf{B}}^{ϵ}+ϵ{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{ϵ}\times {\mathsf{B}}^{ϵ}.\hfill \end{array}$$ B2

Using these expressions in the phase-wise conservation of mass equations and multiplying through by ${ϵ}^2$ gives

$$\begin{array}{cc}& {ϵ}^2\frac{{\dot{p}}_m^{ϵ}}{M_m}=-ϵ{\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{ϵ})-{ϵ}^2{\mathit{\alpha }}_m:{\zeta }_x({\dot{\mathbf{u}}}_m^{ϵ})+{ϵ}^2{\mathrm{\nabla }}_x·({\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{ϵ})-{ϵ}^2{\mathrm{\nabla }}_x·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ}))\hfill \\ \hfill & +ϵ{\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{ϵ})+ϵ{\mathrm{\nabla }}_x·({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{ϵ})-ϵ{\mathrm{\nabla }}_x·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ}))\hfill \\ \hfill & +{\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{ϵ})-ϵ{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ}))-{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ})),\hfill \end{array}$$ B3

$$\begin{array}{cc}& {ϵ}^2\frac{{\dot{p}}_s^{ϵ}}{M_s}=-ϵ{\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{ϵ})-{ϵ}^2{\mathit{\alpha }}_s:{\zeta }_x({\dot{\mathbf{u}}}_s^{ϵ})+{ϵ}^2{\mathrm{\nabla }}_x·({\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{ϵ})-{ϵ}^2{\mathrm{\nabla }}_x·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ}))\hfill \\ \hfill & +ϵ{\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{ϵ})+ϵ{\mathrm{\nabla }}_x·({\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{ϵ})-ϵ{\mathrm{\nabla }}_x·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ}))\hfill \\ \hfill & +{\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{ϵ})-ϵ{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ}))-{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ})).\hfill \end{array}$$ B4

The corresponding flux and pressure interface conditions are

$$\begin{array}{cc}& \left(ϵ,{\mathsf{K}}_m,{\mathrm{\nabla }}_x,p_m^{ϵ},-,ϵ,{\mathsf{K}}_m,({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ}),+,{\mathsf{K}}_m,{\mathrm{\nabla }}_y,p_m^{ϵ},-,{\mathsf{K}}_m,({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{ϵ}\times {\mathbf{B}}^{ϵ})\right)·\mathbf{n}\hfill \\ \hfill & =\left(ϵ,{\mathsf{K}}_s,{\mathrm{\nabla }}_x,p_s^{ϵ},-,ϵ,{\mathsf{K}}_s,({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ}),+,{\mathsf{K}}_s,{\mathrm{\nabla }}_y,p_s^{ϵ},-,{\mathsf{K}}_s,({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{ϵ}\times {\mathbf{B}}^{ϵ})\right)·\mathbf{n},\hfill \end{array}$$ B5

$$\begin{array}{cc}& p_m^{ϵ}=p_s^{ϵ}\hfill \end{array}$$ B6

Equating terms of order ${ϵ}^0$ gives the pressure problem (45a)–(45d). The decomposition (46)–(47) gives the pressure-corrector problem (48a)–(48d).

Equating terms of order ${ϵ}^1$ in (B3)–(B6) gives

$$\begin{array}{cc}& {\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{(0)})+{\mathrm{\nabla }}_x·({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(0)})-{\mathrm{\nabla }}_x·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)}))\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)}))+{\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(1)})\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)}))-{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)}))=0\hfill \\ \hfill & \text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ B7a

$$\begin{array}{c}\hfill \\ \hfill & {\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{(0)})+{\mathrm{\nabla }}_x·({\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(0)})-{\mathrm{\nabla }}_x·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)}))\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)}))+{\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(1)})\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)}))-{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)}))=0\hfill \\ \hfill & \text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ B7b

$$\begin{array}{c}\hfill \\ \hfill & (({\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{(0)}-{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})+{\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(1)}-{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)})\hfill \\ \hfill & -{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)}))·\mathbf{n}=({\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{(0)}-{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})+{\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(1)}\hfill \\ \hfill & -{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)})-{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)}))·\mathbf{n}\text{on}\mathrm{\Gamma },\hfill \end{array}$$ B7c

$$\begin{array}{cc}& p_m^{(1)}=p_s^{(1)}\text{on}\mathrm{\Gamma }.\hfill \end{array}$$ B7d

The ansatz (49)–(50) leads to the cell problems for ${\mathbf{d}}_m,{\mathbf{d}}_s$:

$$\begin{array}{ccc}& {\mathrm{\nabla }}_y·({\mathrm{\nabla }}_y{\mathbf{d}}_m{\mathsf{K}}_m^T)=-{\mathrm{\nabla }}_y·{\mathsf{K}}_m^T\hfill & \hfill \text{in}{\mathrm{\Omega }}_m,\end{array}$$ B8a

$$\begin{array}{ccc}& {\mathrm{\nabla }}_y·({\mathrm{\nabla }}_y{\mathbf{d}}_s{\mathsf{K}}_s^T)=-{\mathrm{\nabla }}_y·{\mathsf{K}}_s^T\hfill & \hfill \text{in}{\mathrm{\Omega }}_s,\end{array}$$ B8b

$$\begin{array}{ccc}& \left(({\mathrm{\nabla }}_y{\mathbf{d}}_m),{\mathsf{K}}_m^T,-,({\mathrm{\nabla }}_y{\mathbf{d}}_s),{\mathsf{K}}_s^T\right)\mathbf{n}={({\mathsf{K}}_s-{\mathsf{K}}_m)}^T\mathbf{n}\hfill & \hfill \text{on}\mathrm{\Gamma },\end{array}$$ B8c

$$\begin{array}{ccc}& {\mathbf{d}}_m={\mathbf{d}}_s\hfill & \hfill \text{on}\mathrm{\Gamma }.\end{array}$$ B8d

The scalar correctors $t_m,t_s$ solve

$$\begin{array}{cc}& {\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_y{t}_m)+{\mathrm{\nabla }}_y·({\mathsf{K}}_m{\mathrm{\nabla }}_x{\widehat{p}}_m)+{\mathrm{\nabla }}_x·({\mathsf{K}}_m{\mathrm{\nabla }}_y{\widehat{p}}_m)-{\mathrm{\nabla }}_x·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)}))\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)}))-{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)}))\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)}))=0\text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ B9a

$$\begin{array}{cc}& {\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_y{t}_s)+{\mathrm{\nabla }}_y·({\mathsf{K}}_s{\mathrm{\nabla }}_x{\widehat{p}}_s)+{\mathrm{\nabla }}_x·({\mathsf{K}}_s{\mathrm{\nabla }}_y{\widehat{p}}_s)-{\mathrm{\nabla }}_x·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)}))\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)}))-{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)}))\hfill \\ \hfill & -{\mathrm{\nabla }}_y·({\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)}))=0\text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ B9b

$$\begin{array}{cc}& ({\mathsf{K}}_m{\mathrm{\nabla }}_y{t}_m+{\mathsf{K}}_m{\mathrm{\nabla }}_x{\widehat{p}}_m-{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathbf{B}}^{(0)})-{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathbf{B}}^{(1)})\hfill \\ \hfill & -{\mathsf{K}}_m({\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathbf{B}}^{(0)}))·\mathbf{n}=({\mathsf{K}}_s{\mathrm{\nabla }}_y{t}_s+{\mathsf{K}}_s{\mathrm{\nabla }}_x{\widehat{p}}_s-{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathbf{B}}^{(0)})\hfill \\ \hfill & -{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathbf{B}}^{(1)})-{\mathsf{K}}_s({\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathbf{B}}^{(0)}))·\mathbf{n}\text{on}\mathrm{\Gamma },\hfill \end{array}$$ B9c

$$\begin{array}{cc}& t_m=t_s\text{on}\mathrm{\Gamma }.\hfill \end{array}$$ B9d

These problems are supplemented with periodic boundary conditions and zero-average conditions.

The leading order relative fluid-solid velocities are

$$\begin{array}{cc}\hfill {\mathbf{w}}_m^{(0)}& =-{\mathsf{K}}_m{\mathrm{\nabla }}_y{p}_m^{(1)}-{\mathsf{K}}_m{\mathrm{\nabla }}_x{p}_m^{(0)}+{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathsf{B}}^{(0)}+{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathsf{B}}^{(1)}\hfill \\ \hfill & +{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathsf{B}}^{(0)},\hfill \end{array}$$ B10

$$\begin{array}{cc}\hfill {\mathbf{w}}_s^{(0)}& =-{\mathsf{K}}_s{\mathrm{\nabla }}_y{p}_s^{(1)}-{\mathsf{K}}_s{\mathrm{\nabla }}_x{p}_s^{(0)}+{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathsf{B}}^{(0)}+{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathsf{B}}^{(1)}\hfill \\ \hfill & +{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathsf{B}}^{(0)}.\hfill \end{array}$$ B11

Substituting (46)–(50) and averaging gives

$$\begin{array}{cc}\hfill {⟨{\mathbf{w}}_m^{(0)}⟩}_m& =-{⟨{\mathsf{K}}_m{\mathsf{D}}_m+{\mathsf{K}}_m⟩}_m{\mathrm{\nabla }}_x\overline{p}-{⟨{\mathsf{K}}_m{\mathbf{t}}_m⟩}_m-{⟨{\mathsf{K}}_m{\mathrm{\nabla }}_x{\widehat{p}}_m⟩}_m\hfill \\ \hfill & +{⟨{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(1)}\times {\mathsf{B}}^{(0)}⟩}_m+{⟨{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_y{\varphi }_m^{(0)}\times {\mathsf{B}}^{(1)}⟩}_m+{⟨{\mathsf{K}}_m{\mathsf{G}}_m{\mathrm{\nabla }}_x{\varphi }_m^{(0)}\times {\mathsf{B}}^{(0)}⟩}_m,\hfill \end{array}$$ B12

$$\begin{array}{cc}\hfill {⟨{\mathbf{w}}_s^{(0)}⟩}_s& =-{⟨{\mathsf{K}}_s{\mathsf{D}}_s+{\mathsf{K}}_s⟩}_s{\mathrm{\nabla }}_x\overline{p}-{⟨{\mathsf{K}}_s{\mathbf{t}}_s⟩}_s-{⟨{\mathsf{K}}_s{\mathrm{\nabla }}_x{\widehat{p}}_s⟩}_s\hfill \\ \hfill & +{⟨{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(1)}\times {\mathsf{B}}^{(0)}⟩}_s+{⟨{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_y{\varphi }_s^{(0)}\times {\mathsf{B}}^{(1)}⟩}_s+{⟨{\mathsf{K}}_s{\mathsf{G}}_s{\mathrm{\nabla }}_x{\varphi }_s^{(0)}\times {\mathsf{B}}^{(0)}⟩}_s.\hfill \end{array}$$ B13

Adding (B12) and (B13) gives the effective Darcy law (52).

B.2 Lorentz-Modified Poroelastic Cell Problems

The Lorentz-modified first-order poroelastic problem is

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathbf{u}}_m^{(1)}))& =-{\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_x({\mathbf{u}}^{(0)}))+{\mathrm{\nabla }}_y·(p_m^{(0)}{\mathit{\alpha }}_m)-\mathsf{G}{\mathrm{\nabla }}_y{\varphi }^{(0)}\times \mathsf{B}\text{in}{\mathrm{\Omega }}_m,\hfill \end{array}$$ B14a

$$\begin{array}{cc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathbf{u}}_s^{(1)}))& =-{\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_x({\mathbf{u}}^{(0)}))+{\mathrm{\nabla }}_y·(p_s^{(0)}{\mathit{\alpha }}_s)-\mathsf{G}{\mathrm{\nabla }}_y{\varphi }^{(0)}\times \mathsf{B}\text{in}{\mathrm{\Omega }}_s,\hfill \end{array}$$ B14b

$$\begin{array}{cc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathbf{u}}_m^{(1)})-{\mathbb{C}}_s{\zeta }_y({\mathbf{u}}_s^{(1)}))\mathbf{n}& =(({\mathbb{C}}_s-{\mathbb{C}}_m){\zeta }_x({\mathbf{u}}^{(0)})-{\mathit{\alpha }}_s{p}_s^{(0)}+{\mathit{\alpha }}_m{p}_m^{(0)})\mathbf{n}\text{on}\mathrm{\Gamma },\hfill \end{array}$$ B14c

$$\begin{array}{cc}\hfill {\mathbf{u}}_m^{(1)}& ={\mathbf{u}}_s^{(1)}\text{on}\mathrm{\Gamma }.\hfill \end{array}$$ B14d

The ansatz (56a)–(56b) leads to the following corrector problems. The elastic correctors solve

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathcal{B}}_m))& =-{\mathrm{\nabla }}_y·{\mathbb{C}}_m\hfill & \hfill \text{in}{\mathrm{\Omega }}_m,\end{array}$$ B15a

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathcal{B}}_s))& =-{\mathrm{\nabla }}_y·{\mathbb{C}}_s\hfill & \hfill \text{in}{\mathrm{\Omega }}_s,\end{array}$$ B15b

$$\begin{array}{ccc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathcal{B}}_m)-{\mathbb{C}}_s{\zeta }_y({\mathcal{B}}_s))\mathbf{n}& =({\mathbb{C}}_s-{\mathbb{C}}_m)\mathbf{n}\hfill & \hfill \text{on}\mathrm{\Gamma },\end{array}$$ B15c

$$\begin{array}{ccc}\hfill {\mathcal{B}}_m& ={\mathcal{B}}_s\hfill & \hfill \text{on}\mathrm{\Gamma }.\end{array}$$ B15d

The pressure correctors associated with $p_m^{(0)}$ solve

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\mathbf{b}}_m))& ={\mathrm{\nabla }}_y·{\mathit{\alpha }}_m\hfill & \hfill \text{in}{\mathrm{\Omega }}_m,\end{array}$$ B16a

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\tilde{\mathbf{b}}}_s))& =0\hfill & \hfill \text{in}{\mathrm{\Omega }}_s,\end{array}$$ B16b

$$\begin{array}{ccc}\hfill ({\mathbb{C}}_m{\zeta }_y({\mathbf{b}}_m)-{\mathbb{C}}_s{\zeta }_y({\tilde{\mathbf{b}}}_s))\mathbf{n}& ={\mathit{\alpha }}_m\mathbf{n}\hfill & \hfill \text{on}\mathrm{\Gamma },\end{array}$$ B16c

$$\begin{array}{ccc}\hfill {\mathbf{b}}_m& ={\tilde{\mathbf{b}}}_s\hfill & \hfill \text{on}\mathrm{\Gamma }.\end{array}$$ B16d

The pressure correctors associated with $p_s^{(0)}$ solve

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_m{\zeta }_y({\tilde{\mathbf{b}}}_m))& =0\hfill & \hfill \text{in}{\mathrm{\Omega }}_m,\end{array}$$ B17a

$$\begin{array}{ccc}\hfill {\mathrm{\nabla }}_y·({\mathbb{C}}_s{\zeta }_y({\mathbf{b}}_s))& ={\mathrm{\nabla }}_y·{\mathit{\alpha }}_s\hfill & \hfill \text{in}{\mathrm{\Omega }}_s,\end{array}$$ B17b

$$\begin{array}{ccc}\hfill ({\mathbb{C}}_m{\zeta }_y({\tilde{\mathbf{b}}}_m)-{\mathbb{C}}_s{\zeta }_y({\mathbf{b}}_s))\mathbf{n}& =-{\mathit{\alpha }}_s\mathbf{n}\hfill & \hfill \text{on}\mathrm{\Gamma },\end{array}$$ B17c

$$\begin{array}{ccc}\hfill {\tilde{\mathbf{b}}}_m& ={\mathbf{b}}_s\hfill & \hfill \text{on}\mathrm{\Gamma }.\end{array}$$ B17d

The Lorentz-induced mechanical correctors are given in (57a)–(57d) in the main text. All these cell problems are supplemented with periodic boundary conditions on $\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }$. The auxiliary fields are assigned zero average over their respective subdomains for uniqueness.

B.3 Averaging of the Lorentz-Modified Conservation of Mass

Starting from the order ${ϵ}^1$ conservation of mass equations in ${\mathrm{\Omega }}_m$ and ${\mathrm{\Omega }}_s$, and retaining distinct leading-order pressures, we integrate over the two subdomains to obtain

$$\begin{array}{cc}& {\int }_{{\mathrm{\Omega }}_m}\frac{{\dot{p}}_m^{(0)}}{M_m}\mathrm{d}\mathbf{y}+{\int }_{{\mathrm{\Omega }}_s}\frac{{\dot{p}}_s^{(0)}}{M_s}\mathrm{d}\mathbf{y}=-{\int }_{{\mathrm{\Omega }}_m}{\mathit{\alpha }}_m:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})\mathrm{d}\mathbf{y}-{\mathrm{\nabla }}_x·{\int }_{{\mathrm{\Omega }}_m}{\mathbf{w}}_m^{(0)}\mathrm{d}\mathbf{y}\hfill \\ \hfill & -{\int }_{{\mathrm{\Omega }}_s}{\mathit{\alpha }}_s:{\zeta }_x({\dot{\mathbf{u}}}^{(0)})\mathrm{d}\mathbf{y}-{\mathrm{\nabla }}_x·{\int }_{{\mathrm{\Omega }}_s}{\mathbf{w}}_s^{(0)}\mathrm{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_m}{\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{(1)})\mathrm{d}\mathbf{y}\hfill \\ \hfill & -{\int }_{{\mathrm{\Omega }}_s}{\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{(1)})\mathrm{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_m}{\mathrm{\nabla }}_y·{\mathbf{w}}_m^{(1)}\mathrm{d}\mathbf{y}-{\int }_{{\mathrm{\Omega }}_s}{\mathrm{\nabla }}_y·{\mathbf{w}}_s^{(1)}\mathrm{d}\mathbf{y}.\hfill \end{array}$$ B18

Applying the divergence theorem to the final two terms, and using periodicity and the interfacial flux-continuity condition, these terms cancel. Introducing

$$\begin{array}{c}\hfill {\mathbf{w}}_{\text{eff}}:={⟨{\mathbf{w}}_m^{(0)}⟩}_m+{⟨{\mathbf{w}}_s^{(0)}⟩}_s,\end{array}$$ B19

we obtain

$$\begin{array}{cc}\hfill \frac{{⟨{\dot{p}}_m^{(0)}⟩}_m}{{⟨M_m⟩}_m}+\frac{{⟨{\dot{p}}_s^{(0)}⟩}_s}{{⟨M_s⟩}_s}& =-({⟨{\mathit{\alpha }}_m⟩}_m+{⟨{\mathit{\alpha }}_s⟩}_s):{\zeta }_x({\dot{\mathbf{u}}}^{(0)})-{\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}\hfill \\ \hfill & -{⟨{\mathit{\alpha }}_m:{\zeta }_y({\dot{\mathbf{u}}}_m^{(1)})⟩}_m-{⟨{\mathit{\alpha }}_s:{\zeta }_y({\dot{\mathbf{u}}}_s^{(1)})⟩}_s.\hfill \end{array}$$ B20

Using the displacement correctors (56a)–(56b) and differentiating with respect to time gives

$$\begin{array}{cc}\hfill \frac{{⟨{\dot{p}}_m^{(0)}⟩}_m}{{⟨M_m⟩}_m}+\frac{{⟨{\dot{p}}_s^{(0)}⟩}_s}{{⟨M_s⟩}_s}& =-{\mathrm{\nabla }}_x·{\mathbf{w}}_{\text{eff}}-({⟨{\mathit{\alpha }}_m⟩}_m+{⟨{\mathit{\alpha }}_s⟩}_s):{\zeta }_x({\dot{\mathbf{u}}}^{(0)})\hfill \\ \hfill & -({⟨{\mathbb{L}}_m^{\mathrm{T}}:{\mathit{\alpha }}_m⟩}_m+{⟨{\mathbb{L}}_s^{\mathrm{T}}:{\mathit{\alpha }}_s⟩}_s):{\zeta }_x({\dot{\mathbf{u}}}^{(0)})\hfill \\ \hfill & -{⟨{\mathit{\alpha }}_m:{\mathit{\tau }}_m⟩}_m{\dot{p}}_m^{(0)}-{⟨{\mathit{\alpha }}_s:{\mathit{\tau }}_s⟩}_s{\dot{p}}_s^{(0)}\hfill \\ \hfill & -{⟨{\mathit{\alpha }}_s:{\tilde{\mathit{\tau }}}_s⟩}_s{\dot{p}}_m^{(0)}-{⟨{\mathit{\alpha }}_m:{\tilde{\mathit{\tau }}}_m⟩}_m{\dot{p}}_s^{(0)}.\hfill \end{array}$$ B21

Rearranging (B21) gives (65) in the main text.

Author Contributions

Conceptualisation: L.M., R.P.; Investigation: L.M., R.P.; Formal analysis: L.M.; Writing-original draft preparation: L.M.; Writing-review and editing: L.M., R.P. All authors reviewed and agreed to the manuscript.

Funding

R.P. is partially supported by EPSRC, UK Grant EP/T017899/1, as well as EPSRC grants EP/V521917/1 and EP/Z000580//1 - INI. L.M. is the PI of, and R.P. is a key network partner, on the INI-HIMR Network: Biological Tissues – Modelling, Mechanics and Applications (August 2025-July 2027) supported by EPSRC EP/V521917/1 - Heilbronn Institute and EP/Z000580/1 - INI.

Data Availability

No datasets were generated or analysed during the current study.

Declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.