# Foundations and Their Practical Implications for the Constitutive Coefficients of Poromechanical Dual-Continuum Models

- 229 Downloads

## Abstract

A dual-continuum model can offer a practical approach to understanding first-order behaviours of poromechanically coupled multiscale systems. To close the governing equations, constitutive equations with models to calculate effective constitutive coefficients are required. Several coefficient models have been proposed within the literature. However, a holistic overview of the different modelling concepts is still missing. To address this we first compare and contrast the dominant models existing within the literature. In terms of the constitutive relations themselves, early relations were indirectly postulated that implicitly neglected the effect of the mechanical interaction arising between continuum pressures. Further, recent users of complete constitutive systems that include inter-continuum pressure coupling have explicitly neglected these couplings as a means of providing direct relations between composite and constituent properties, and to simplify coefficient models. Within the framework of micromechanics, we show heuristically that these explicit decouplings are in fact coincident with bounds on the effective parameters themselves. Depending on the formulation, these bounds correspond to end-member states of isostress or isostrain. We show the impacts of using constitutive coefficient models, decoupling assumptions and parameter bounds on poromechanical behaviours using analytical solutions for a 2D model problem. Based on the findings herein, we offer recommendations for how and when to use different coefficient modelling concepts.

## Keywords

Dual-continuum Poromechanics Constitutive models Micromechanics Multiscale porous media## 1 Introduction

Many natural and manufactured geomaterials exhibit strong heterogeneities in their material properties, owing to the existence of porous constituents at various length scales. Examples of multiscale systems that are commonly encountered in subsurface operations include fissured or fractured rock and soil aggregates (Warren et al. 1963; Kazemi et al. 1976; Nelson 2001; Gerke 2006; Koliji 2008; Romero et al. 2011). Modelling of such materials is invaluable in understanding how these systems behave in response to extraneous activities. In general, modelling can be done using either explicit (e.g. discrete fracture matrix models) or implicit methods (e.g. continuum approaches) (Berre et al. 2018).

With respect to fractured systems, using explicit methods can be computationally prohibitive at large scales (Karimi-Fard et al. 2006; Gong 2007; Garipov et al. 2016). Additionally, explicit methods may require data (e.g. spatial data) that are not obtainable without direct access (Berkowitz 2002; Blessent et al. 2014). In cases where field-scale modelling of multiscale systems is required, implicit representations are then often preferred. The most common type of implicit model is the dual-continuum (or double-porosity) model, originally attributed to Barenblatt et al. (1960). In this, the dual-material is considered as the superposition of two overlapping continua, which communicate through a mass transfer term. Continua are defined on the basis of their material properties. For example, fractures (or inter-aggregate pores) generally have high permeabilities and poor storage capabilities, vice versa the matrix. Although less detailed than their explicit counterparts, dual-continuum models can provide practical and valuable insight into the first-order behaviours of multiscale systems. Further, use of these models is desirable due to the low number of fitting parameters that allow for efficient calibration to historical data.

Multiscale systems can also exhibit strong coupling between deformation and fluid flow, and vice versa. This phenomenon is known as poromechanical coupling (Rutqvist and Stephansson 2003) and is described by the well-established poromechanical theory (see, for example, Biot 1941, 1977; Detournay and Cheng 1995; Coussy 1995, 2004; Wang 2000; De Boer 2012; Cheng 2016).

Aifantis (1977, 1979) and Wilson and Aifantis (1982) were the first to introduce the generalised notion of deformation within the dual-continuum setting. Further offerings then came from Elsworth and Bai (1992), Lewis and Ghafouri (1997), and Bai et al. (1999). However, all of these models implicitly neglected the effects of coupling between pressures of different pore domains due to their postulation of the form of the constitutive equations. The absence of these pressure couplings was shown to give unphysical responses by Khalili (2003). Specifically, the author’s results showed discontinuous pressure jumps in the matrix and fracture continua that were incompatible with the prescribed boundary conditions. The cause of the observations made by Khalili (2003) still remains an open question.

More recent users of these later constitutive/coefficient models have explicitly decoupled pore domain pressures when expressing the constitutive relations in terms of stress and continuum pressures (pure stiffness setting) (see, for example, Nguyen and Abousleiman 2010; Kim et al. 2012; Mehrabian and Abousleiman 2014; Mehrabian 2018). This has been done as a form of non-algebraic closure and to provide explicit relations between composite and constituent properties, resulting in simplified coefficient models. However, such decoupling assumptions have been made without discussing the origin and sensitivities that may arise as a result.

The aim of this paper is to formulate a set of recommendations for how and when to use different constitutive modelling concepts. In doing we show the impacts of making implicit and explicit decoupling assumptions. In the case of the latter, we use heuristic arguments from micromechanics to show that these assumptions are coincident with bounds on the effective parameters themselves.

We structure the paper as follows: In Sect. 2 we introduce the governing and constitutive equations pertinent to double-porosity materials. For the latter set of equations, we support their form using arguments from the energy approach to poromechanics (Coussy 2004). Section 3 presents the most prevalent modelling approaches for calculating the effective poromechanical coefficients. Section 4 details the origins of explicit assumptions made on constitutive/coefficient models within the framework of micromechanics. From here we offer upscaling recommendations for constituent moduli when composite moduli may not be available. In Sect. 5 we use analytical solutions to the double-porosity Mandel problem to explore the physical implications, and relevance, of different coefficient models and decoupling assumptions. We conclude by way of offering recommendations for how and when to use coefficient models in light of (a) intrinsic fracture stiffness effects and (b) pressure decoupling assumptions between pore domains. Throughout this paper, our reference multiscale material is that of a naturally fractured system. Such systems can be considered as void space inclusion composites or stiff inclusion composites (Fig. 1).

We note that work has been done on the determination of effective properties of multiple-porosity materials via homogenisation methods (e.g. Berryman 2006 and Levin et al. 2012). However, equivalent continuum models can fail to provide insight into processes occurring at the different porosity scales due to use of an averaged flow field (Berre et al. 2018). In contrast, this work is concerned with double-porosity materials for which two distinct flow fields exist. Upscaling of such flow fields for inelastic materials has been addressed by periodic homogenisation (Arbogast et al. 1990), but such a treatment for deformable materials is, to the best of the authors knowledge, still missing. Given this context, the introduction of the phenomenological approaches described herein for the determination of constitutive coefficients is desirable due to their ease of use, and resulting explicit relations to underling properties.

## 2 Double-Porosity Mathematical Model

*m*and

*f*refer to matrix and fracture continua, respectively. It is assumed that the poroelastic double-porosity material is isotropic and is saturated by a slightly compressible fluid which can undergo isothermal flow. Under the assumptions of quasi-static deformations and infinitesimal transformations, the momentum balance for the dual medium recovered as

*G*are the drained bulk and shear moduli of the dual medium, respectively (Coussy 2004). Coefficients \(b_\alpha \) can be thought of as effective Biot coefficients and relate changes in effective Lagrangian porosity to skeletal straining under drained conditions. Coefficients \(\frac{1}{N_{\alpha }}\) relate changes in the Lagrangian porosity of continuum \(\alpha \) to changes in fluid pressure of the same medium, whilst the skeleton remains constrained and fluid pressure in continuum \(\beta \) remains constant. Finally, \(\frac{1}{Q}\) is a coupling coefficient that relates changes in the Lagrangian porosity of continuum \(\alpha \) and pressure changes in continuum \(\beta \).

*b*is the single-porosity Biot coefficient (Berryman and Wang 1995).

*d*denotes the average spacing between the fractures.

## 3 Models of Constitutive Coefficients

We require substantiation of the constitutive coefficients in Eqs. (17) and (23) to (24). One option is direct measurement of these effective parameters. However, this approach is predicted to be non-trivial for dual-continua. For example, isolating matrix and fracture contributions would be challenging.

- (i)
Khalili and Valliappan (1996)—Constituent mechanical properties, assuming the high permeability, low storage continuum is all void space (no intrinsic fracture properties),

- (ii)
Borja and Koliji (2009)—Constituent pore fractions, assuming the high permeability, low storage continuum is all void space,

- (iii)
Berryman (2002)—Constituent mechanical properties, including intrinsic fracture properties.

### 3.1 Model Approaches and Assumptions

#### 3.1.1 Khalili and Valliappan (1996)

The authors take a top-down approach, postulating macroscopic balance laws and an effective stress expression. Closure for the model equations therein is then sought through thought experiments that isolate volumetric changes of the constituents. Superposition due to linearity, and Betti’s reciprocal work theorem finally allow for recovery of the macroscopic behaviour in terms of constituent responses. In doing, expressions for the constitutive coefficients are identified.

Poromechanical material coefficients by author

\(b_f\) | \(b_m\) | \(\frac{1}{M_f}\) | \(\frac{1}{Q}\) | \(\frac{1}{M_m}\) | |
---|---|---|---|---|---|

Khalili and Valliappan (1996) | \(1-\frac{K_{dr}}{K_m}\) | \(\frac{K_{dr}}{K_m}-\frac{K_{dr}}{K_s}\) | \(\frac{\phi ^0_f}{K_l} + \frac{b_f - \phi ^0_f}{K_m}\) | \(\frac{b_f - \phi ^0_f}{K_s} - \frac{b_f - \phi ^0_f}{K_m}\) | \(\frac{\phi ^0_m}{K_l} + \frac{b_m - \phi ^0_m}{K_s} - \frac{1}{Q}\) |

Borja and Koliji (2009)\(\text {}^+\) | \(\psi ^0_f b\) | \(\psi ^0_m b\) | \(\frac{\phi ^0_f}{K_l}\) | 0 | \(\frac{\phi ^0_m}{K_l}\) |

Berryman (2002)\(\text {}^{++}\) | \(b_f^*\frac{K_{dr}-K_m}{K_{f}-K_m}\) | \(b^*_m\frac{K_{dr}-K_{f}}{K_m-K_{f}}\) | \(A_{33} - \frac{(b_f)^2}{K_{dr}}\) | \(A_{23} - \frac{b_mb_f}{K_{dr}}\) | \(A_{22} - \frac{(b_m)^2}{K_{dr}}\) |

#### 3.1.2 Borja and Koliji (2009)

Comparison of Eqs. (31) with (17) leads to the following relations for effective Biot coefficients, \(b_m = \psi ^0_m b\) and \(b_f = \psi ^0_f b\).

Borja and Koliji (2009) identify the requirement for a constitutive expression for \(\psi _\alpha \) based on energy conjugacy with \(p_\alpha \) (their Eq. (76)). However, explicit constitutive equations for \(\psi _\alpha \) remain, to the best of the current authors’ knowledge, an open question. We do note, however, that Borja and Choo (2016) develop a framework that allows for the tracking of pore fraction evolutions numerically.

A summary of the coefficient models from Borja and Koliji (2009), under the explicit assumption \(\frac{\partial \psi _\alpha }{\partial t}\approx 0\), mapped to the constitutive model of Eqs. (17) and (23) to (24) is shown in Table 1.

#### 3.1.3 Berryman (2002)

The motivation behind the approach by Berryman (2002) is to formulate coefficient models using intrinsic fracture properties. Therefore, contrary to the previous models, no assumption is made on the values of \(\phi ^*_f\) and \(K_f\). Inclusion of intrinsic fracture properties is concurrent with the fracture continuum having an associated stiffness.

Expressions for \(A_{11}{-}A_{33}\) are finally recovered (Table 2), using the uniform expansion/contraction thought experiments described above. With the relations in Eq. (38), we get material coefficient formulations pertaining to the conventional mixed compliance setting (\(\epsilon \) and \(p_\alpha \) as primary variables) (Table 1).

Berryman (2002) material coefficient formulations for the pure stiffness constitutive model

Coefficient | Berryman (2002) formulation |
---|---|

\(A_{11}\) | \(\frac{1}{K_{dr}}\) |

\(A_{12}\) | \(\frac{b^*_m}{K_m}\frac{1-K_{f}/K_{dr}}{1-K_{f}/K_m}\) |

\(A_{13}\) | \(\frac{b^*_{f}}{K_{f}}\frac{1-K_m/K_{dr}}{1-K_m/K_{f}}\) |

\(A_{22}\) | \(\frac{v_mb^*_m}{B^*_mK_m}-\left( \frac{b^*_m}{1-K_m/K_{f}}\right) ^2\left\{ \frac{v_m}{K_m}+\frac{v_{f}}{K_{f}}-\frac{1}{K_{dr}}\right\} \) |

\(A_{23}\) | \(\frac{K_mK_{f}b^*_mb^*_{f}}{(K_{f}-K_m)^2}\left\{ \frac{v_m}{K_m}+\frac{v_{f}}{K_{f}}-\frac{1}{K_{dr}}\right\} \) |

\(A_{33}\) | \(\frac{v_{f}b^*_{f}}{B^*_{f}K_{f}}-\left( \frac{b^*_{f}}{1-K_{f}/K_m}\right) ^2\left\{ \frac{v_m}{K_m}+\frac{v_{f}}{K_{f}}-\frac{1}{K_{dr}}\right\} \) |

#### 3.1.4 In Sum

Coefficient models from Khalili and Valliappan (1996) and Borja and Koliji (2009) both make an underlying void space assumption for the high permeability, low storage continuum. Models from Borja and Koliji (2009) make use of continuum pore fractions, but still require a final closure relationship for the evolution of each pore fraction. Finally, models from Berryman (2002) make no underlying assumption on the intrinsic porosity, and thus the stiffness of the high permeability, low storage continuum.

In terms of pressure decoupling assumptions, we have two types. The first are implicit assumptions for which the constitutive relations are postulated without inter-continuum pressure coupling. The second are explicit assumptions for which the full constitutive model is the starting point (or the requirement for constitutive expressions is at least identified in the case of Borja and Koliji (2009)). The explicit decoupling assumptions are then made so as to provide relations between material properties and to simplify coefficient models (e.g. due to non-algebraic closure). However, the physical justifications and/or implications of these assumptions still remain an open question.

It is of interest to investigate how differences in coefficient models, in addition to decoupling assumptions, may impact poromechanical behaviour. This is pursued in the remaining sections.

## 4 Micromechanics of Dual-Continua

To establish the physical implications of explicit decoupling assumptions such as taking \(A_{23}=A_{32}=0\), we make use of the theoretical framework of micromechanics. Micromechanics is used as a tool to relate the macroscopic behaviour and properties of a composite to those of its underlying constituents (microscale) (Nemat-Nasser and Hori 1993). In the following we consider the dual medium as a composite of two (poro-) elastic materials with each material having its own intrinsic constitutive model.

### 4.1 Effective Elastic Properties

*any*REV within a large macroscopic body.

With the averaging operation defined, we proceed to formulate the effective elastic property problem. In doing we make use of the works of Hill (1963) and Hashin (1972). Additionally, we assume that the composite is drained and thus make no distinction between effective and total stress fields.

For given underlying constituent properties, \(K_{dr}\) (resp. \(S_{dr}\)) can take on a range of values, existing between certain well-defined lower and upper bounds, due to the geometrical variability of the problem. As a result, the effective coefficients also exhibit a bounded range of values due to their dependence on \(K_{dr}\) (Table 1). This dependence is further explored in the following section.

### 4.2 Physical Implications of Explicit Decoupling

In the following we show that explicit decoupling assumptions are coincident with effective coefficient bounds. Bounds are attractive as they provide useful estimates of effective properties of interest, as well as a means to verify the values of these properties (Torquato 1991).

We use arguments from micromechanics to show heuristically that the inverse to explicit decoupling is to assume that Eqs. (52) or (53) can be calculated directly by considering certain limiting behaviours and thus bounds on \(K_{dr}\). Bounds, from which explicit decouplings naturally arise, on effective properties then follow from the bounds on \(K_{dr}\).

#### 4.2.1 Isostrain: \(\frac{1}{Q}=0\)

*isostrain*

From the first equality in Eq. (54), we expect \(\frac{1}{Q} \le 0\) since matrix porosity must reduce in order to accommodate the pressure-driven fracture expansion (see also similar arguments in Berryman and Wang (1995)). Thus, based on the arguments described in this section, we can infer that the explicit decoupling assumption \(\frac{1}{Q} = 0\) is concurrent with an *upper bound* on \(\frac{1}{Q}\).

#### 4.2.2 Incompressible Grain Isostrain: \(\frac{1}{N_\alpha }=\frac{1}{Q}=0\)

We now consider the coefficient models from Borja and Koliji (2009) under the assumption \(\frac{\partial \psi _\alpha }{\partial t}\approx 0\) made by Choo and Borja (2015) and Choo et al. (2016).

In Sect. 3.1.2, we identified that \(\frac{\partial \psi _\alpha }{\partial t}\approx 0\) amounts to \(\frac{1}{N_\alpha }=\frac{1}{Q}=0\) when mapping to the constitutive model shown in Eqs. (17) to (19). It is of interest to see under what conditions the result \(\frac{1}{N_\alpha }=\frac{1}{Q}=0\) arises when starting from the void space coefficient models built from constituent mechanical properties (i.e. those from Khalili and Valliappan (1996)).

The set of explicit assumptions: \(\frac{1}{N_\alpha }=\frac{1}{Q}=0\), can easily be derived from the microscale by first considering isostrain (and thus \(\frac{1}{Q}=0\)). With the resulting bounds arising from isostrain, Eqs. (60) to (61), along with the assumption \(K_s=\infty \), we obtain \(\frac{1}{N_\alpha }=0\) using the coefficient models of Khalili and Valliappan (1996) (Table 1, with Eq. (26) to decompose \(\frac{1}{M_\alpha }\)). We therefore refer to conditions resulting in \(\frac{1}{N_\alpha }=\frac{1}{Q}=0\) as *incompressible grain isostrain*

As far as parameters in the balance of mass are concerned, Eqs. (33) and (34) are identical when assuming \(\frac{\partial \psi _\alpha }{\partial t}\approx 0\) in the former and incompressible grain isostrain using the constituent mechanical property void space coefficient models in the latter. However, differences in mass balance behaviour may be introduced through the way in which \(b_\alpha \) is modelled. It is therefore of interest to see how the effective Biot coefficients calculated using the respective void space coefficient models under incompressible grain isostrain compare to the bounds established in Eqs. (60) to (61).

Under the incompressible grain assumption, the upper bound for \(b_m\) now reads \(b_m=v_m\). When using \(K^{V}_{dr}=v_mK_m\) in the coefficient models of Khalili and Valliappan (1996), we can see that the bounds in Eqs. (60) to (61) are naturally recovered (see Table 1). In contrast, from Table 1 the effective Biot coefficients calculated using the models of Borja and Koliji (2009) are equivalent to pore fractions, since \(b=1\) for incompressible grains. Due to the differences in effective Biot coefficients (and thus other constitutive parameters), we expect disparity in poroelastic behaviour when using the two sets of void space coefficient models.

#### 4.2.3 Isostress: \(A_{23}(=A_{32})=0\)

*isostress*

*b*for the single-porosity model (see Dormieux et al. 2006).

### 4.3 On Moduli Upscaling

In Nguyen and Abousleiman (2010), Kim et al. (2012), Kim and Moridis (2013), Mehrabian and Abousleiman (2014, 2015) and more recently Mehrabian (2018), isostress is implicitly assumed. Upscaling of constituent moduli is then admitted through the Reuss average. This raises the question as to whether this is a reasonable approach to upscaling or not?

When \(G_{f}=0\), such as in a fluid suspension geometry, the HS lower bound and the Reuss bound coincide. It follows that in this geometry, with the stiffness of a void space fracture phase (\(K_f=0\)), both the Reuss and HS lower bound result in \(K^{{R}}_{dr}=K^{{HS}+}_{dr}=0\). However, in situations when the fracture phase has an intrinsic stiffness, the Reuss and HS lower bounds may be significantly different (Watt et al. 1976).

Bounds can be used as a first approach to upscaling under certain geometries. If a fracture network completely percolates a matrix, then it will have the maximum effect of weakening the rock (Watt et al. 1976). The effective bulk modulus of the composite will then coincide with the HS lower bound (Boucher 1974; Watt et al. 1976). In the general case fractures are likely to have an associated stiffness (Bandis et al. 1983). We thus recommend using the HS lower bound over the Reuss average as a first approach to upscaling for such geometries. This procedure is also in line with the assumptions built into the continuum approach: The continuum assumption is linked to one’s ability to define an REV over which properties can be averaged. For a fractured system, such an REV cannot be justified if the system is poorly connected (Berkowitz 2002). Use of the HS lower bound, as a first approach to moduli averaging, thus supports the notion of a well-connected isotropic dense fracture network, over which an REV could be defined.

When the underlying composite geometry precludes the use of bounds as methods for upscaling, one must use other methods of averaging. Comprehensive summaries of such approaches can be found in the works of Aboudi (1992), Nemat-Nasser and Hori (1993) and Torquato (2002), for example.

## 5 Qualitative Analysis Using the Mandel Problem

We use solutions to the double-porosity Mandel problem (Nguyen and Abousleiman 2010), to investigate the physical impacts of different coefficient modelling concepts and assumptions, on the poromechanical response of a dual-continuum material. We consider implicit assumptions, where the constitutive model starts with no pressure coupling, and explicit assumptions, where the full constitutive model, Eqs. (17) and (23) to (24), is the starting point, but pressure coupling is neglected, leading to bounds being used for the calculation of \(K_{dr}\) (and thus the calculation of the effective constitutive coefficients).

We first investigate the effects of considering the fractured dual medium as a void space inclusion composite or stiff inclusion composite as assumed by the coefficient models of Khalili and Valliappan (1996) and Berryman (2002), respectively. Second, we study the effects of decoupling assumptions.

### 5.1 Double-Porosity Mandel Problem

The problem geometry is described as an infinitely long (rectangular) cuboid domain such that the plane-strain condition holds (i.e. \(u_y=0\) and \(q_{m,y}=q_{f,y}=0\)) (Fig. 4). The domain is sandwiched between two impermeable, rigid plates, and is free to displace both laterally and vertically. A constant compressive force, \(\int ^a_{-a} \sigma _{zz} \mathrm{d}x= -2Fa\), is applied at the rigid plate boundaries, \(\varGamma _\mathrm{N}\) and \(\varGamma _\mathrm{S}\) (north and south boundaries, respectively). The east and west boundaries, \(\varGamma _\mathrm{E}\) and \(\varGamma _\mathrm{W}\), respectively, are then free to drain such that \(p_m=p_f=0\) at these boundaries.

Due to the symmetry of the problem, only a quarter of the domain need be considered.

*v*is Poisson’s ratio. Substitution of Eqs. (69) and (2) in (1) yields

*f*(

*t*) is an integration function. Use of Eq. (72) in (34) leads to a set of diffusion equations written entirely in terms of continuum pressures. Solutions to the resulting system of equations are presented in Nguyen and Abousleiman (2010), and Mehrabian and Abousleiman (2014). Further, these solutions allow for calculation of vertical stress and strain (see Nguyen and Abousleiman 2010 or Mehrabian and Abousleiman 2014).

### 5.2 Data for Analysis

For the qualitative analysis we use a quarter of a \(2\text { m}\times 2\text { m}\) deformable porous domain. The studied domain is subjected to a constant top boundary force, \(\int ^1_0 -2\times 10^6\,\mathrm{Pa\,d}x = -2\) MPa m. Where possible we use values for material properties that are typically encountered in naturally fractured carbonate reservoirs. The mechanical properties are then \(K_{m}=20\) GPa, \(K_{s}=70\) GPa, and \(v=0.2\) (Wang 2000). Values for \(K_{dr}\) and \(K_f\) are problem dependent. In cases where no rigorous justification is used, we choose values for \(K_{dr}\) and \(K_{f}\) arbitrarily (denoted by a superscript \(\dagger \)). Fluid properties are for that of water: \(\rho ^0_l = 1000 \text { kg m}^{{-3}}\), \(\mu _l = 1 \text { cp}\) and \(K_l = 2.3 \text { GPa}\). Rock properties related to fluid storage and flow are \(\phi ^0_m=0.05\), \(k_m=0.01 \text { md}\) and \(\phi ^0_f=0.01\) (Nelson 2001). Effective permeability of the fracture is assumed to be \(k_f\approx 1000 \text { md}\). With the cubic law (Witherspoon et al. 1980), this corresponds to an aperture, \(a_f=7\times 10^{-5}\text { m}\), and fracture spacing, \(d=2.8\times 10^{-2} \text { m}\).

### 5.3 Test Cases

We consider one test case to investigate the differences between void space and stiff inclusion composite coefficient models, and three test cases to investigate implicit and explicit decoupling assumptions. In each case, analytical solutions to the double-porosity Mandel problem are compared. Differences in solutions for each case then arise due to the parameter permutations described in the case descriptions that follow.

#### 5.3.1 Case 1: Intrinsic Fracture Properties

In Case 1 we are interested in comparing the differences that arise when considering intrinsic fracture properties. In particular, it is of interest to investigate if coefficient models from Khalili and Valliappan (1996) could still be used even when a fracture has an associated phase stiffness.

From Tables 1 and 2, we hypothesise that provided that \(\phi ^*_f\approx 1\) and the fracture phase stiffness is orders of magnitude lower than the grain stiffness (\(K_f\ll K^f_s\)), effects arising due to deviations of intrinsic fracture coefficients under the void space assumption (i.e. deviations from: \(b^*_f=1\), \(\frac{1}{M^*_\alpha } = \frac{1}{K_l}\) and \(B^*_\alpha = 1\)) are negligible. Void space coefficient models could then be used in place of coefficient models that include intrinsic fracture properties. When \(\phi ^*_f=1\), we use coefficient models from Khalili and Valliappan (1996), and when \(\phi ^*_f<1\), we use coefficient models from Berryman (2002) with \(K^f_s=K_s\).

Strictly speaking, use of void space coefficient models implies \(K_f=0\). However, the aim of this test case is to highlight the effect of missing physics by not including intrinsic poromechanical parameters within coefficient model formulations. To do this, and to ensure nonzero values of \(K_{dr}\), we use the same upscaled bulk modulus for both sets of coefficient models, which is calculated with a nonzero \(K_f\). We consider a composite medium with a network of fractures that completely dissociates the matrix. Upscaling is then done through the HS lower bound. To test our hypothesis, we use combinations of various values of \(\phi ^*_f\) and \(K_f\).

We compare results from coefficient models calculated using a fracture modulus several orders of magnitude lower than the solid grain modulus, (\(K^{\dagger }_f=\frac{K^f_s}{1750}\)), versus ones that use a fracture modulus only an order of magnitude lower than the solid modulus, (\(K^{\dagger }_f=\frac{K^f_s}{35}\)).

#### 5.3.2 Case 2: Implicit Decoupling Assumptions

For Case 2 we investigate the impact of implicit decoupling assumptions. This is particularly poignant considering the use of such decoupled constitutive models in the recent works of Alberto et al. (2019) and Hajiabadi and Khoei (2019). To mimic implicit assumptions, we consider \(\frac{1}{Q}=0\) and \(A_{23}=0\) and make no acknowledgement of these assumptions with respect to relations between mechanical properties. When considering \(\frac{1}{Q}=0\) and \(A_{23}=0\), we use coefficient models from Khalili and Valliappan (1996) and Berryman (2002), with \(K_f=0\), respectively. We reference these results against ones for which no decoupling is made with coefficient models from Khalili and Valliappan (1996).

Use of the void space models implies \(K_f=0\). Since we do not enforce relations on \(K_{dr}\) (a fundamental difference between the implicit assumptions considered here and explicit assumptions), we take an arbitrary value of \(K^{\dagger }_{dr}=10 \text { GPa}\). As a result, bounds on the effective constitutive coefficients are not enforced.

#### 5.3.3 Case 3: Explicit Decoupling Assumption—Isostrain

We investigate the effect of assuming isostrain at the microscale whilst making use of coefficient models from Khalili and Valliappan (1996). Under isostrain the composite and constituent bulk moduli are linked by the Voigt average (\(K^{V}_{dr}= 19.8 \text { GPa}\)). This leads naturally to bounds on the effective constitutive coefficients, with \(\frac{1}{Q}=0\) representing an upper bound. We compare the isostrain results to those computed when using coefficient models with a composite bulk modulus coinciding with the HS upper bound and an arbitrary value (\(K^{{HS}+}_{dr}=19.5 \text { GPa}\) and \(K^{\dagger }_{dr}=10 \text { GPa}\), respectively).

Further, we investigate the disparity between results when using the void space coefficient models of Borja and Koliji (2009) under the assumption of \(\frac{\partial \psi _\alpha }{\partial t}=0\), and Khalili and Valliappan (1996) under the assumption of incompressible grain isostrain. We use \(K^{V}_{dr}= 19.8 \text { GPa}\) and \(K_s=\infty \) for both sets of coefficient models in this latter isostrain investigation.

#### 5.3.4 Case 4: Explicit Decoupling Assumption—Isostress

For Case 4 we study the effect of assuming isostress at the microscale. In previous works the coefficient models of Berryman (2002) have been used with an explicit decoupling assumption (\(A_{23}=0\)) that implies isostress (Kim et al. 2012; Mehrabian and Abousleiman 2014; Mehrabian 2018).

To avoid cases where \(K_{dr}=0\), we consider the fracture phase to have the following properties: \(\phi ^*_f=0.7\) with \(K^f_s=K_s\) and \(K^{\dagger }_f=\frac{K_m}{500}\). Coefficient models from Berryman (2002) are then used.

We compare results arising when calculating the composite bulk modulus with the Reuss average (\(K^{R}_{dr}=3.3 \text { GPa}\)), the HS lower bound (\(K^{{HS}-}_{dr}=5.7 \text { GPa}\)), and an arithmetic average of the HS bounds (\(K^{{AHS}}_{dr}=12.7 \text { GPa}\)). The latter modulus is tested in analogy to a dual system with inclusions that do not have the maximum weakening effect on the host material. One example would be a fracture system composed of a network of open and closed fractures. Another would be aggregate material.

## 6 Results and Discussions

In the following, we show results of the test cases described above for the double-porosity Mandel problem. Results are given in terms in evolutions of matrix and fracture pressures, and vertical strain with time.

To aid in our analysis for pressure and vertical strain, we introduce the following notions of the *instantaneous* problem and the *time-dependent* problem. In both cases, mechanical equilibrium is governed, for this system, by Eq. (70). In the instantaneous problem, fluid pressure for continuum \(\alpha \) can be shown to be a state function of total stress and fluid pressure \(\beta \). In the time-dependent problem, continuum pressures are governed by way of the diffusion equation, Eq. (34).

*t*(0), to a loaded state, \(t(0^+)\), upon application of instantaneous loading. Under such conditions the rate of loading is infinitely faster than the rate of inter-continuum fluid transfer (Coussy 2004). Consequently, each continuum is undrained, \(m^{t(0^+)}_{l,\alpha }=m^{t(0)}_{l,\alpha }\). From Eqs. (23) and (24), together with the undrained condition (\(\text {d}m_{l,\alpha }=0\)) we can recover

### 6.1 Case 1: Intrinsic Fracture Properties

If the lower bound in Eq. (76) for a given \(K_f\) is close to one, then changes in intrinsic fracture porosity are negligible due to the proximity of the lower and upper bounds. This is the case when \(K_f\ll K^f_s\). If the fracture phase stiffness is *not* orders of magnitude lower than the solid stiffness, then the lower bound of \(B^*_f\) may be significantly less than one. In this case, changes in \(B^*_f\) cannot be captured when using void space coefficient models, and thus, early time fracture pressure is underestimated as \(\phi ^*_f\) decreases (Fig. 5c).

### 6.2 Case 2: Implicit Decoupling

Pressure and vertical strain results for the implicit decoupling assumptions test case are presented in Fig. 6. When assuming \(A_{23}=0\), the matrix and fracture pressure evolutions are almost identical to the reference case (Fig. 6a), with vertical strains also being correspondingly very similar (Fig. 6b). When assuming \(\frac{1}{Q}=0\), the early time matrix and fracture pressures are measurably lower than the reference case (Fig. 6a). The early time vertical strain is greater when \(\frac{1}{Q}=0\) than the strain in the reference case (Fig. 6b).

The results in Fig. 6a suggest that the assumption \(\frac{1}{Q}=0\) has the most noticeable effect on the poromechanical behaviour of the dual medium. As discussed in Sect. 4.2.1 we would expect \(\frac{1}{Q}<0\). From Eq. (74), setting \(\frac{1}{Q}=0\) thus has the effect of removing a pressure source from continuum \(\alpha \). Removal of this poromechanical coupling explains the lower than expected induced matrix and fracture pressures. From Eq. (73) we can see \(\text {d}p_\alpha \propto \epsilon ^{-1}\). Underestimated pressures therefore explain the overestimated strain when taking \(\frac{1}{Q}=0\).

The current results suggest that assuming \(A_{23}=0\) is a reasonable implicit assumption to make. However, we advise caution when interpreting this result. Based on the results in this section it would be easy, and incorrect, to use them as a justification for assumptions made at the microscale. In Sect. 4.2 it was shown that explicit assumptions affect all of the constitutive coefficients due to bounds on bulk moduli. The remainder of this results section aims at qualitatively supporting these findings.

### 6.3 Case 3: Explicit Decoupling—Isostrain

An alternative, heuristic approach to explaining the pressure distributions in Fig. 7a can be achieved by considering the required geometry that would be necessary to give a \(K_{dr}\) that is lower than the Voigt average. Departure from this upper bound occurs when inclusions are arranged so as to weaken the composite. They then take on a greater portion of the distributed stress. We therefore observe a greater induced fracture pressure when using a lower value of \(K_{dr}\) compared to the Voigt bound, due to the proportionality between stress and pressure.

The early time vertical strains in Fig. 7b can be explained by way of Eq. (73) or through the heuristic argument. With the latter, towards the upper bounds of \(K_{dr}\) the matrix supports the majority of deformation. Since the matrix is stiff, deformation is low. In contrast, when fractures are arranged such that they have a greater weakening effect on the solid, deformation is high.

At later times vertical strain between both the upper bound and arbitrary \(K_{dr}\) cases diverges. Towards the upper bounds for \(K_{dr}\), \(b_f < b_m\). In the case of isostrain, this fact is easily seen from the relations in Eqs. (60) to (61). The magnitude of \(b_m\) means that deformation is more strongly coupled to differences in matrix pressure relative to fracture pressure by way of momentum balance, Eq. (70). This explains the growth in vertical strain separation at later times shown in Fig. 7b.

Of further interest, considering these both represent upper bounds on the composite bulk modulus, is the difference in early time fracture pressures between the isostrain and HS upper bound cases. The early time fracture pressure associated with the HS upper bound is over double that of the isostrain case. This highlights the need for caution before making assumptions on the distribution of strain between constituents.

Under isostrain, we would expect the distribution of stress required to maintain strain uniformity between matrix and fractures to lead to disparate matrix and fracture pressures. The result \(\text {d}p_m=\text {d}p_f\) therefore suggests that the closure condition \(\frac{\partial \psi _\alpha }{\partial t}\approx 0\) may be an even stronger assumption than incompressible grain isostrain alone.

Figure 8b shows vertical strain is lower at early times when using coefficient models from Khalili and Valliappan (1996) under incompressible grain isostrain. This can be explained by Eq. (73), which is affected by differences in \(b_\alpha \) arising from each set of coefficient models.

In light of the discussions in Sect. 4.2.2 and the results presented herein, assuming \(\frac{\partial \psi _\alpha }{\partial t}\approx 0\) appears to be a strong closure assumption to make. Thus, we suggest further development of a constitutive model for \(\psi _\alpha \), along with its relationship to the constitutive model shown in Eqs. (17) to (19).

### 6.4 Case 4: Explicit Decoupling—Isostress

Figure 9a shows further impacts of the upscaling method on matrix and fracture pressure evolutions. A stiffer composite bulk modulus leads to a lower induced fracture pressure and earlier onset of pressure diffusion in the same continuum. This is the case when using the arithmetic mean of the HS bounds. Non-monotonic rises in matrix and fracture pressures are more pronounced for more compliant composites. This is the case when assuming isostress (Reuss average) and the HS lower bound. Additionally, more compliant composites exhibit a faster decrease in matrix pressure at later times when compared to the stiffer modulus case.

Figure 9b shows pronounced distinctions in vertical strain for the three different upscaling cases. As expected, the stiffer arithmetic mean of the HS bounds shows the lowest deformation. Of more interest, considering that they both correspond to lower bounds, are the difference in strains between the cases of isostress and HS lower bound. Vertical strain at late times is approximately 75% larger when assuming isostress compared to when using the HS lower bound.

The cause of the difference in vertical strains between the isostress and HS lower bound cases can be explained using similar discussions as to those used in Sects. 5.3.2 and 5.3.3. First, using the heuristic argument the HS lower bound is higher than the Reuss bound, suggesting that the matrix is capable of supporting a greater distribution of strain. This explains the difference in vertical strain at early times. Late time differences can be explained by the differences in fracture pressure. In contrast to Sect. 5.3.3, towards the lower bounds for \(K_{dr}\), \(b_f > b_m\). The magnitude of \(b_f\) means that deformation is more strongly coupled to differences in fracture pressure relative to matrix pressure by way of momentum balance, Eq. (70).

With a view towards multi-continuum generalisations, based on the results in Sect. 4.2.3 and the qualitative results herein, we recommend care before assuming isostress. This stress distribution has strong geometrical implications that without experimental substantiation to prove otherwise, would seem unlikely to hold within a multi-continuum material.

## 7 Conclusion

- (i)
Constituent mechanical properties; assuming the high permeability, low storage continuum is all void space (no intrinsic fracture properties),

- (ii)
Constituent pore fractions; assuming the high permeability, low storage continuum is all void space,

- (iii)
Constituent mechanical properties, including intrinsic fracture properties.

The second set of recommendations is formed on the basis of our investigations into implicit and explicit decoupling assumptions. In both cases mechanical coupling between continuum pressures is neglected. In the former we showed that implicit assumptions can lead to the removal of pressure sources, leading to physically unreliable results. We therefore recommend the use of a full constitutive system where possible.

Even with a full constitutive system, explicit assumptions have been made as a passage to simplifying relations between composite and constituent moduli without considering the physical implications of their use. In this case we showed that explicit decoupling assumptions are coincident with bounds on composite moduli that arise naturally under end-member states of isostrain and isostress. However, for isotropic composite materials, it is well known that the bounds obtained under isostrain and isostress can be loose, and that tighter bounds using similar quantities are readily available within the literature. Our qualitative investigations showed clear differences in poromechanical behaviour when using these different bounds.

To conclude, bounds arising from isostrain and isostress states, which are concurrent with explicit decoupling assumptions, can provide a useful means for guiding our intuition into multiscale poromechanical behaviour, given their ease of computation. However, for practical subsurface applications, we recommend against the use of explicit decoupling assumptions, as they have physical and geometrical implications that are unlikely to be justified within isotropic multiscale materials.

## Notes

### Acknowledgements

The authors are very grateful for the funding provided to them by the National Environmental Research Council to carry out this work, as well as to the reviewers who helped to improve this work.

## References

- Aboudi, J.: Mechanics of Composite Materials: A Unified Micromechanical Approach. Elsevier, Amsterdam (1992)Google Scholar
- Aifantis, E.C.: Introducing a multi-porous medium. Dev. Mech.
**8**(3), 209–211 (1977)Google Scholar - Aifantis, E.C.: On the response of fissured rocks. Dev. Mech.
**10**, 249–253 (1979)Google Scholar - Alberto, J., Cordero, R., Mejia, E.C., Roehl, D.: Dual permeability models for fluid flow in deformable fractured media. J. Pet. Sci. Eng.
**175**(2018), 644–653 (2019)Google Scholar - Arbogast, T., Douglas Jr., J., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal.
**21**(4), 823–836 (1990)Google Scholar - Bai, M., Meng, F., Elsworth, D., Abousleiman, Y., Roegiers, J.: Numerical modelling of coupled flow and deformation in fractured rock specimens. Int. J. Numer. Anal. Methods Geomech.
**23**(2), 141–160 (1999)Google Scholar - Bandis, S., Lumsden, A., Barton, N.: Fundamentals of rock joint deformation. In: International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 20, pp. 249–268. Elsevier (1983)Google Scholar
- Barenblatt, G., Zheltov, I., Kochina, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech.
**24**(5), 1286–1303 (1960)Google Scholar - Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media, vol. 4. Springer, Berlin (2012)Google Scholar
- Berkowitz, B.: Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour.
**25**(8–12), 861–884 (2002)Google Scholar - Berre, I., Doster, F., Keilegavlen, E.: Flow in fractured porous media: a review of conceptual models and discretization approaches. Transp. Porous Media (2018). https://doi.org/10.1007/s11242-018-1171-6 CrossRefGoogle Scholar
- Berryman, J.G.: Extension of poroelastic analysis to double-porosity materials: new technique in microgeomechanics. J. Eng. Mech.
**128**(8), 840–847 (2002)Google Scholar - Berryman, J.G.: Effective medium theories for multicomponent poroelastic composites. J. Eng. Mech.
**132**(5), 519–531 (2006)Google Scholar - Berryman, J.G., Pride, S.R.: Models for computing geomechanical constants of double-porosity materials from the constituents properties. J. Geophys. Res. Solid Earth
**107**(B3), ECV 2-1–ECV 2-14 (2002)Google Scholar - Berryman, J.G., Wang, H.F.: The elastic coefficients of double-porosity models for fluid transport in jointed rock. J. Geophys. Res.: Solid Earth
**100**(B12), 24611–24627 (1995)Google Scholar - Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys.
**12**(2), 155–164 (1941)Google Scholar - Biot, M.A.: Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion. Int. J. Solids Struct.
**13**(6), 579–597 (1977)Google Scholar - Blessent, D., Jørgensen, P.R., Therrien, R.: Comparing discrete fracture and continuum models to predict contaminant transport in fractured porous media. Groundwater
**52**(1), 84–95 (2014)Google Scholar - Borja, R.I., Choo, J.: Cam-clay plasticity, part VIII: a constitutive framework for porous materials with evolving internal structure. Comput. Methods Appl. Mech. Eng.
**309**, 653–679 (2016)Google Scholar - Borja, R.I., Koliji, A.: On the effective stress in unsaturated porous continua with double porosity. J. Mech. Phys. Solids
**57**(8), 1182–1193 (2009)Google Scholar - Boucher, S.: On the effective moduli of isotropic two-phase elastic composites. J. Compos. Mater.
**8**(1), 82–89 (1974)Google Scholar - Callari, C., Federico, F.: FEM validation of a double porosity elastic model for consolidation of structurally complex clayey soils. Int. J. Numer. Anal. Methods Geomech.
**24**(4), 367–402 (2000)Google Scholar - Cheng, A.H.-D.: Poroelasticity, vol. 27. Springer, Berlin (2016)Google Scholar
- Choo, J., Borja, R.I.: Stabilized mixed finite elements for deformable porous media with double porosity. Comput. Methods Appl. Mech. Eng.
**293**, 131–154 (2015)Google Scholar - Choo, J., White, J.A., Borja, R.I.: Hydromechanical modeling of unsaturated flow in double porosity media. Int. J. Geomech.
**16**(6), D4016002 (2016)Google Scholar - Coussy, O.: Mechanics of Porous Continua. Wiley, Hoboken (1995)Google Scholar
- Coussy, O.: Poromechanics. Wiley, Hoboken (2004)Google Scholar
- De Boer, R.: Theory of Porous Media: Highlights in Historical Development and Current State. Springer, Berlin (2012)Google Scholar
- Detournay, E., Cheng, A.H.D.: Fundamentals of poroelasticity. In: Analysis and Design Methods, pp. 113–171. Pergamon (1995)Google Scholar
- Dormieux, L., Kondo, D., Ulm, F.-J.: Microporomechanics. Wiley, Hoboken (2006)Google Scholar
- Elsworth, D., Bai, M.: Flow-deformation response of dual-porosity media. J. Geotech. Eng.
**118**(1), 107–124 (1992)Google Scholar - Fornells, P., García-Aznar, J.M., Doblaré, M.: A finite element dual porosity approach to model deformation-induced fluid flow in cortical bone. Ann. Biomed. Eng.
**35**(10), 1687–1698 (2007)Google Scholar - Garipov, T.T., Karimi-Fard, M., Tchelepi, H.A.: Discrete fracture model for coupled flow and geomechanics. Comput. Geosci.
**20**(1), 149–160 (2016)Google Scholar - Gerke, H.H.: Preferential flow descriptions for structured soils. J. Plant Nutr. Soil Sci.
**169**(3), 382–400 (2006)Google Scholar - Gong, B.: Effective models of fractured systems. Ph.D. thesis, Stanford University (2007)Google Scholar
- Hajiabadi, M.R., Khoei, A.R.: A bridge between dual porosity and multiscale models of heterogeneous deformable porous media. Int. J. Numer. Anal. Methods Geomech.
**43**(1), 212–238 (2019)Google Scholar - Hashin, Z.: Theory of fiber reinforced materials. NASA CR-1974 (1972)Google Scholar
- Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids
**11**(2), 127–140 (1963)Google Scholar - Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids
**11**(5), 357–372 (1963)Google Scholar - Karimi-Fard, M., Gong, B., Durlofsky, L.J.: Generation of coarse-scale continuum flow models from detailed fracture characterizations. Water Resour. Res.
**42**(10), 1–13 (2006)Google Scholar - Kazemi, H., Merrill Jr., L.S., Porterfield, K.L., Zeman, P.R.: Numerical simulation of water-oil flow in naturally fractured reservoirs. Soc. Pet. Eng. J.
**16**(06), 317–326 (1976)Google Scholar - Khalili, N.: Coupling effects in double porosity media with deformable matrix. Geophys. Res. Lett.
**30**(22), 2153 (2003)Google Scholar - Khalili, N.: Two-phase fluid flow through fractured porous media with deformable matrix. Water Resour. Res.
**44**(5), 1–12 (2008)Google Scholar - Khalili, N., Khabbaz, M.H., Valliappan, S.: An effective stress based numerical model for hydro-mechanical analysis in unsaturated porous media. Comput. Mech.
**26**(2), 174–184 (2000)Google Scholar - Khalili, N., Selvadurai, A.P.S.: A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity. Geophys. Res. Lett.
**30**(24), 1–5 (2003)Google Scholar - Khalili, N., Valliappan, S.: Unified theory of flow and deformation in double porous media. Eur. J. Mech. A Solids
**15**(2), 321–336 (1996)Google Scholar - Kim, J., Moridis, G.J.: Development of the T+M coupled flow–geomechanical simulator to describe fracture propagation and coupled flow–thermal–geomechanical processes in tight/shale gas systems. Comput. Geosci.
**60**, 184–198 (2013)Google Scholar - Kim, J., Sonnenthal, E.L., Rutqvist, J.: Formulation and sequential numerical algorithms of coupled fluid/heat flow and geomechanics for multiple porosity materials. Int. J. Numer. Methods Eng.
**92**(5), 425–456 (2012)Google Scholar - Koliji, A.: Mechanical behaviour of unsaturated aggregated soils. Ph.D. thesis, EPFL (2008)Google Scholar
- Levin, V., Kanaun, S., Markov, M.: Generalized Maxwell’s scheme for homogenization of poroelastic composites. Int. J. Eng. Sci.
**61**, 75–86 (2012)Google Scholar - Lewis, R.W., Ghafouri, H.R.: A novel finite element double porosity model for multiphase flow through deformable fractured porous media. Int. J. Numer. Anal. Methods Geomech.
**21**(11), 789–816 (1997)Google Scholar - Lim, K.T., Aziz, K.: Matrix-fracture transfer shape factors for dual-porosity simulators. J. Pet. Sci. Eng.
**13**(3–4), 169–178 (1995)Google Scholar - Loret, B., Rizzi, E.: Strain localization in fluid-saturated anisotropic elastic-plastic porous media with double porosity. J. Mech. Phys. Solids
**47**(3), 503–530 (1999)Google Scholar - Mehrabian, A.: The poroelastic constants of multiple-porosity solids. Int. J. Eng. Sci.
**132**, 97–104 (2018)Google Scholar - Mehrabian, A., Abousleiman, Y.N.: Generalized biot’s theory and mandel’s problem of multiple-porosity and multiple-permeability poroelasticity. J. Geophys. Res. Solid Earth
**119**(4), 2745–2763 (2014)Google Scholar - Mehrabian, A., Abousleiman, Y.N.: Gassmann equations and the constitutive relations for multiple-porosity and multiple-permeability poroelasticity with applications to oil and gas shale. Int. J. Numer. Anal. Methods Geomech.
**39**(14), 1547–1569 (2015)Google Scholar - Nelson, R.: Geologic Analysis of Naturally Fractured Reservoirs. Gulf Professional Publishing, Houston (2001)Google Scholar
- Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier, Amsterdam (1993)Google Scholar
- Nguyen, V.X., Abousleiman, Y.N.: Poromechanics solutions to plane strain and axisymmetric mandel-type problems in dual-porosity and dual-permeability medium. J. Appl. Mech.
**77**(1), 011002 (2010)Google Scholar - Pao, W.K., Lewis, R.W.: Three-dimensional finite element simulation of three-phase flow in a deforming fissured reservoir. Comput. Methods Appl. Mech. Eng.
**191**(23–24), 2631–2659 (2002)Google Scholar - Reuss, A.: Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle. ZAMM J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik
**9**(1), 49–58 (1929)Google Scholar - Romero, E., Della Vecchia, G., Jommi, C.: An insight into the water retention properties of compacted clayey soils. Géotechnique
**61**(4), 313 (2011)Google Scholar - Rutqvist, J., Stephansson, O.: The role of hydromechanical coupling in fractured rock engineering. Hydrol. J.
**11**(1), 7–40 (2003)Google Scholar - Taron, J., Elsworth, D., Min, K.B.: Numerical simulation of thermal–hydrologic–mechanical–chemical processes in deformable, fractured porous media. Int. J. Rock Mech. Min. Sci.
**46**(5), 842–854 (2009)Google Scholar - Torquato, S.: Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev.
**44**(2), 37–76 (1991)Google Scholar - Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, Berlin (2002)Google Scholar
- Tuncay, K., Corapcioglu, M.Y.: Effective stress principle for saturated fractured porous media. Water Resour. Res.
**31**(12), 3103–3106 (1995)Google Scholar - Tuncay, K., Corapcioglu, M.Y.: Wave propagation in fractured porous media. Transp. Porous Media
**23**(3), 237–258 (1996)Google Scholar - Voigt, W.: Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Springer, Berlin (1928)Google Scholar
- Wang, H.F.: Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, Princeton (2000)Google Scholar
- Wang, K., Sun, W.: A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning. Comput. Methods Appl. Mech. Eng.
**334**, 337–380 (2018)Google Scholar - Warren, J., Root, P.J., et al.: The behavior of naturally fractured reservoirs. Soc. Petrol. Eng. J.
**3**(03), 245–255 (1963)Google Scholar - Watt, J.P., Davies, G.F., O’Connell, R.J.: The elastic properties of composite materials. Rev. Geophys.
**14**(4), 541–563 (1976)Google Scholar - Wilson, R.K., Aifantis, E.C.: On the theory of consolidation with double porosity. Int. J. Eng. Sci.
**20**(9), 1009–1035 (1982)Google Scholar - Witherspoon, P.A., Wang, J.S., Iwai, K., Gale, J.E.: Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour. Res.
**16**(6), 1016–1024 (1980)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.