# Identification of viscoelastic properties from numerical model reduction of pressure diffusion in fluid-saturated porous rock with fractures

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## Abstract

This paper deals with the computational homogenization and numerical model reduction of deformation driven pressure diffusion in fractured porous rock. Exposed to seismic waves, the heterogeneity of the material leads to local fluid pressure gradients which are equilibrated via pressure diffusion. However, a macroscopic observer is not able to measure the diffusion process directly but senses the intrinsic attenuation of an apparently monophasic viscoelastic solid. The aim of this paper is to establish a reliable, yet numerically efficient, computational homogenization method to identify the viscoelastic properties of the macroscopic substitute model. Inspired by the Nonuniform Transformation Field Analysis, we incorporate a Numerical Model Reduction procedure. The proposed method is validated for several scenarios ranging from pressure diffusion in an unfractured poroelastic matrix, via localized pressure diffusion in interconnected fractures embedded in an impermeable matrix, to the fully coupled pressure diffusion both in fractures and the embedding poroelastic matrix.

## Keywords

Poroelasticity with fractures Computational homogenization Numerical model reduction## 1 Introduction

### 1.1 Pressure diffusion in fluid-saturated fractured rock

Squirt flow in interconnected micro-cracks causes dissipation of acoustic waves due to viscous friction [19]. This dissipation mechanism is governed by fluid pressure diffusion. An identical phenomenon occurs at a larger spatial scale in hydraulically interconnected mesoscopic fractures [26], here referred to as squirt-type flow. Two interconnected fractures are differently deformed by the propagating wave mainly as a consequence of their different orientation. Hydro-mechanical coupling induces different fluid pressures in the fractures and, thus, fluid pressure diffusion occurs from one fracture into the other one, causing dissipation. The resulting dissipation is frequency dependent and associated with dispersion of the elastic moduli. These effects of fracture interconnectivity on seismic waves represent a great potential for inferring hydraulic characteristics of subsurface formations through the interpretation of seismic data.

A few numerical upscaling approaches have been used in the recent years to compute, at the level of a Representative Volume Element (RVE), frequency-dependent attenuation and the corresponding dispersion of the elastic moduli caused by squirt-type flow in interconnected fractures [22, 23, 26, 30]. They describe the squirt-type flow within the fractures either by using Biot’s quasi-static equations of poroelasticity or quasi-static, linearized Navier–Stokes equations, whilst the embedding rock matrix is described either as a poroelastic medium or as a monophasic linear-elastic medium based on Hooke’s law.

Numerical simulations of wave propagation can be used to predict the effect of fracture interconnectivity on surface seismic data, when the subsurface geometry and properties are known. However, in this case, model domains much larger than the RVE considered for the previously mentioned numerical upscaling approaches are necessary which, if feasible at all, requires tremendous computational efforts. A solution for this problem is to replace the heterogeneous medium at the macro-scale by an equivalent homogeneous viscoelastic medium [4] exhibiting identical frequency-dependent attenuation and stiffness modulus dispersion. This presumes that the fluid pressure diffusion observed in the heterogeneous medium at the mesoscopic scale is not observed at the macroscopic scale [13]. In [15], the attenuation-dispersion behavior of a heterogeneous poroelastic medium, consisting of a two-dimensional periodic array of circular inclusions, was roughly approximated with a Maxwell-Zener model containing one single Maxwell chain. This procedure matched the maxima of the corresponding frequency-dependent P- and S-wave attenuation curves but failed to predict the high-frequency limits of the stiffness moduli. Hence, it is an open challenge to find a more fundamental approach for the identification of a suitable viscoelastic model that is able to provide a precise match of attenuation-dispersion behavior. Additionally, such an approach should be able to handle more realistic description of heterogeneities and stochastic fracture networks on the meso-level.

### 1.2 Computational homogenization and numerical model reduction

In this paper we develop a Numerical Model Reduction (NMR) technique based on the concepts of computational homogenization. We, therefore, refer to the length scales involved in the problem as shown in Fig. 1. We distinguish between, firstly, the macro-scale with the characteristic length *L* where propagation and attenuation of seismic waves is observed, secondly, the meso-scale with the characteristic length \(l\ll L\) where the squirt-type flow in fracture networks as well as pressure diffusion in the embedding porous rock takes place and, thirdly, the micro-scale with the characteristic length \(\lambda \ll l\) as the scale of discrete grains and pores. In order to connect the micro- and the meso-scale we employ the Theory of Porous Media (TPM) [6, 7] to model the fluid-saturated porous rock as a biphasic poroelastic medium with possibly heterogeneous material properties. In fact, the resulting mesoscopic description of the rock matrix corresponds to Biot’s equations of quasi-static linear consolidation [3]. In addition, the mesoscopic model is enriched by a fracture network, where we treat the fractures as fluid-filled mechanically and hydraulically open sharp interfaces embedded in the poroelastic matrix. Hence, fluid pressure diffusion occurs in the poroelastic matrix as well as along the fractures. We denote the resulting model that couples fluid pressure diffusion on those two topology levels as “hybrid-dimensional interface model” [30].

It is important to remark that quasi-static considerations are admissible and reasonable on the SVE-level due to separation of scales. Hence, the macroscopic wavelength at seismic frequencies is much larger than the SVE size. Moreover, the inertia forces at those low frequencies remain, at the SVE-level, much smaller than the viscous drag forces in the SVE, see [24].

However, the FE\(^2\) approach requires the nested solution of one macroscopic and numerous mesoscopic initial boundary value problems, which leads to tremendous computational costs thus restricting practical applicability to rather coarse meso-scale problems. In this paper, we therefore propose a NMR strategy that significantly reduces the numerical efforts in solving the SVE problems. Our approach is inspired by the Nonuniform Transformation Field Analysis (NTFA), which was initially established for elasto-viscoplastic materials [18, 25] and recently extended towards solids with cohesive interfaces [10], generalized standard media [9, 27], poroelastic composites [12] and transient heat flow [1], to name only a few application fields. The advancement presented in this paper is that we significantly extend the NMR-methodology that was initially proposed for undamaged poroelastic media in [12] towards pressure diffusion in a hybrid-dimensional formulation.

The key ingredient of our scheme is that we approximate the fluid pressure in the hybrid-dimensional domain as a linear combination of so-called pressure modes which we identify via a Proper Orthogonal Decomposition (POD) from training computations on the SVE level. The pressure modes span a reduced basis for the spatial approximation of the mesoscopic pressure diffusion problem and, as a consequence, allow for the identification of the viscoelastic evolution equations of the macroscopic substitute model. Hence, the reduced FE\(^2\) scheme is split into two stages: The numerically expensive training computations on SVE level define the first stage. They need to be executed only once in advance to derive the homogenized substitute model. Therefore, the first stage is often referred to as “offline” phase. The macroscopic substitute model is used in the second stage, called “online” phase, to solve the macroscopic initial boundary value problems. This allows us to access the full sub-scale information during the “online” phase without explicitly solving the underlying SVE problems and, thus, we succeed to reduce the computational costs tremendously.

The paper is organized as follows: In Sect. 2 we describe the hybrid-dimensional modeling approach on the SVE-level. We, therefore, employ a sharp interface model for pressure diffusion along fractures embedded in a poroelastic rock matrix. Section 3 is devoted to the computational homogenization framework and the definition of the mesoscopic initial boundary value problem for the hybrid-dimensional description in a FE\(^2\) sense. In Sect. 4 we develop the novel NMR technique which allows us to derive viscoelastic substitute models for arbitrary diffusion problems. Finally, we validate our methodology in several numerical experiments presented in Sect. 5.

Throughout the manuscript, vector and tensor quantities are written as bold types. Taking into account Einstein’s sum convention we write out simple and double contractions as \(\mathbf{a}\cdot \mathbf{b}=a_i\,b_i\) and \(\mathbf{A}:\mathbf{B}=A_{ij}\,B_{ij}\). In Voigt notation, vector and matrix quantities are written as underlined italic types, for example \(\underline{a}\) and \(\underline{\underline{A}}\).

## 2 Hybrid-dimensional interface model for fractured poroelastic media

We introduce a cubic SVE in the mesoscopic domain \(\varOmega _\Box \) with the volume \(V_\Box :=|\varOmega _\Box |=l^3\), where *l* is the edge length *l* of the SVE. The SVE contains *n* fluid-filled thin fractures in the domain \(\varOmega _F\subset \varOmega _\Box \) with \(V_F:=|\varOmega _F|\). They are considered to be mechanically and hydraulically open. Thus, we assume that the fracture surfaces are not in contact and fluid transport along the fractures is possible. Moreover, we assume that the fractures are penny-shaped and that each fracture can be represented by a symmetry plane.^{1} The fractures are surrounded by a poroelastic matrix occupying \(\varOmega _M\subset \varOmega _\Box \) with \(V_M:=|\varOmega _M|\) as shown in Fig. 2a. Hence, we can state that \(\varOmega _\Box =\varOmega _M\cup \varOmega _F\) whilst \(\varOmega _M\cap \varOmega _F=\emptyset \). The internal surface separating the matrix material from the fracture space is denoted \(\partial \varOmega _M^i\). We define the surface normal vector \(\mathbf{n}\) such that it points from \(\partial \varOmega _M^i\) into \(\varOmega _F\).

For seismic attenuation and pressure diffusion in fracture networks the case of very thin fractures is of major interest. In fact this means that we consider the case that \(V_F\ll V_M\). Since it would be numerically extremely costly to discretize the volume of such thin fractures we aim to simplify the geometrical description of the fracture space. We, therefore, condense the fracture volume towards a set of *n* possibly intersecting planar interfaces \(\partial F_k\), \(k=1,2,\dots ,n\), see Fig. 2b, whilst \(\partial F:=\cup _{k=1}^n\partial F_k\). In the fracture space, we assume all quantities to be homogeneous in thickness direction. In other words, we neglect any gradients perpendicular to \(\partial F_k\). This hybrid-dimensional interface model is discussed in detail in [29].

Hence, we can state that a volume element \(\hbox {d}v\) in the fracture \(\varOmega _{F,k}\) computes as \(\hbox {d}v=\tau \,\hbox {d}a\). Here, we use the interface element \(\hbox {d}a\) of \(\partial F_k\) and introduce the (current) fracture opening \(\tau (\mathbf{x},\,t)\) of \(\partial F_k\). The now one-dimensional intersection of \(\partial F_k\) with the external surface \(\partial \varOmega _\Box \) is denoted \(\partial \partial F_{\Box ,k}\).

In order to distinguish between positive and negative fracture surfaces \(\partial \varOmega _M^{i,\pm }\) we assign the fracture normal vector \(\mathbf{n}_{F}\) perpendicular to the interface \(\partial F_k\). By convention, \(\mathbf{n}_{F}\) shows in the direction of the positive fracture surface \(\partial \varOmega _{M,k}^{i,+}\) and, therefore, \(\mathbf{n}^+:=-\mathbf{n}_{F}\) and \(\mathbf{n}^-:=\mathbf{n}_{F}\).

### 2.1 Poroelastic rock matrix

*s*or

*f*. We adopt the concept of effective stress and use Biot’s quasi-static theory of linear consolidation [3] in a displacement–pressure formulation. This leads us to the strong form of the coupled equation system

*G*and

*K*of the try solid rock skeleton, the Biot–Willis parameter \(\alpha \) and the pore pressure \(p^M\). We assume small deformations and define the linear solid strain tensor \({\varvec{\varepsilon }}= (\mathbf{u}\otimes {\varvec{\nabla }})^\mathrm{sym}\). We execute the volumetric-deviatoric decomposition and define the volumetric strain tensor \({\varvec{\varepsilon }}^\mathrm{vol}=e\,\mathbf{I}/3\) with \(e=\hbox {tr}\,{\varvec{\varepsilon }}\) and the deviatoric strain tensor \({\varvec{\varepsilon }}^\mathrm{dev}={\varvec{\varepsilon }}-{\varvec{\varepsilon }}^\mathrm{vol}\).

Poroelastic material parameters

\(k^s\) | Intrinsic permeability of the solid skeleton |

\(\phi \) | Porosity (\(\phi :=\phi (t>0)=\phi (t=0)\)) |

| Elastic moduli of the dry skeleton |

\(K^s\), \(K^f\) | Bulk moduli of solid grains and pore fluid |

\(\eta ^{fR}\) | Effective dynamic viscosity |

\(\alpha \) | Biot–Willis parameter |

| Storativity |

\(\alpha \) | \(=1-K/K^s\) |

1 / | \(=\phi /K^f + (\alpha -\phi )/K^s\) |

### 2.2 Pressure diffusion along fractures

*l*to fracture

*k*along their intersection \(\partial F_k\cap \partial F_l\). Here, we use the Dirac-function

^{2}\(\delta _{kl}\) at the intersection of fractures

*k*and

*l*. The volumetric fracture strain is defined as

### 2.3 Hydro-mechanical coupling between matrix and fracture space

*p*instead of \(p^M\) and \(p^{F}\).

## 3 Computational homogenization

### 3.1 Scale transition

In the subsequent section, we aim to embed pressure diffusion in fluid-saturated fractured rock in the multiscale framework of computational homogenization and derive a macroscopic substitute model for the hydro-mechanically coupled problem on the meso-level. To this end, we define a SVE for the mesoscopic pressure diffusion problem and impose separation of scales in space and time. This means that macroscopic gradients have a wavelength that is much larger than the SVE size (\(L\gg l\)). Similarly, due to the scaling of spatial gradients, processes within the SVE will occur at much higher rates than those occurring on the macroscopic level. We conclude that, under these conditions, transient pressure diffusion, brought on by macroscopic waves, occurs on the meso-scale only. In standard fashion, we assume the SVE to be periodic and apply periodic boundary conditions. Thereby, we a priori ensure that the amount of fluid stored within the SVE is conserved. It remains to discuss pressure diffusion through the SVE: Firstly, due to separation of time scales and negligible macroscopic time rates, pressure diffusion through the SVE will be stationary. Secondly, due to the linearity of the problem, it does not interfere with the transient processes and is, therefore, ignored. See [13] for a more thorough discussion.

Altogether, the macroscopic substitute medium can be described as a monophasic material. In fact, the attenuation properties of the macro-model represent viscoelastic damping of the Maxwell-Zener type.

### 3.2 Weak form and macro-homogeneity condition

^{3}that are sufficiently regular in \(\varOmega _M\) and \(\partial F\). Furthermore, we introduce the corresponding trial spaces of self-equilibrated fluxes \(\mathbb {T}_M\), \(\mathbb {Q}_M\) and \(\mathbb {Q}_{F}\) that are sufficiently regular on \(\partial \varOmega _M^+\) and \(\partial \partial F^+\). We write the equations for finding \(\mathbf{u},\,p,\,\mathbf{t},\,q_M,\,q_{F}\,\in \,\mathbb {U}_M\times \mathbb {P}_\Box \times \mathbb {T}_M\times \mathbb {Q}_M\times \mathbb {Q}_{F}\) as

## 4 Numerical model reduction for the RVE problem

In this section we aim to enhance the computational homogenization concept in a way that enables us to explicitly derive the macroscopic viscoelastic material properties of the pressure diffusion problem on the sub-scale. We, therefore, extend the NMR technique introduced in [12] towards the deformation-driven hybrid-dimensional pressure diffusion problem.

### 4.1 Decomposition of the sub-scale field quantities

*N*result in very good approximations of the pressure field and the depending quantities. The pressure modes satisfy the condition \(\sum _{a=1}^N\bar{\xi }_a\,p_a=0\) only by the trivial choice \(\bar{\xi }_a=0\), \(a=1,2,\dots ,N\). The quantities \(\bar{\xi }_a\) are called mode activity coefficients. They represent internal variables of the macroscopic substitute medium. Altogether, the current state of the hybrid-dimensional medium on the meso-level at time

*t*is defined by the overall strain \(\bar{{\varvec{\varepsilon }}}(t)\) and by the internal variables \(\bar{\xi }_a(t)\), \(a=1,2,\dots ,N\).

*S*snapshots \(\hat{p}_s\), \(s=1,2,\dots ,S\), of the pressure distribution obtained from training computations. As described in [11, 25], training computations take place on the SVE level and are driven by a macroscopic loading

*N*pressure modes as

Finally, it remains to identify a relation for the temporal evolution of the activity parameters \(\bar{\xi }_a\) which we interpret in the following as internal variables of the macroscopic substitute medium.

### 4.2 Evolution of the internal variables

*N*-dimensional vector \(\underline{\bar{\xi }}^T=[\bar{\xi }_1,\bar{\xi }_2,\dots ,\bar{\xi }_N]^T\) and the vector representation for the macroscopic strain \(\underline{\bar{\varepsilon }}^T=[\bar{\varepsilon }_{11}\), \(\bar{\varepsilon }_{22}\), \(\bar{\varepsilon }_{33}\), \(2\,\bar{\varepsilon }_{12}\), \(2\,\bar{\varepsilon }_{13}\), \(2\,\bar{\varepsilon }_{23}]^T\). We write

*N*Maxwell chains. It is important to remark that the structure of the evolution equation solely results from the unified SVE continuity equation (45) under the assumption of conservation of fluid mass (21)– (22) and employing the additive decomposition (53).

## 5 Numerical experiments

We now aim to demonstrate the capability of our approach to predict the viscoelastic properties of the macroscopic substitute model in a reliable way. We have, therefore, implemented the hybrid-dimensional SVE problem in the Finite Element Software COMSOL Multiphysics and investigate three different cases. In example 1, we study pressure diffusion in the case of a so-called patchy saturation. In this case, pressure diffusion occurs only in the poroelastic matrix material in the absence of fractures. In example 2, we choose a SVE consisting of a simple fracture network embedded in an impermeable matrix. In this case, on the other hand, pressure diffusion only occurs in the fractures. Finally, we investigate the full hybrid-dimensional diffusion problem including pressure diffusion in both the matrix and the fracture network.

### 5.1 Example 1: patchy saturation

Example 1—poroelastic material parameters for water and gas saturation (1 mD\(\,\approx \,\)1e−15 m\(^2\)) in the 2D patchy saturated SVE, see Fig. 3

Rock matrix | Water-saturated | Gas-saturated |
---|---|---|

\(k^s\) (mD) | 600 | 600 |

\(\phi \) (–) | 0.2 | 0.2 |

| 4.2 | 4.2 |

| 7.0 | 7.0 |

\(K^s\) (GPa) | 36.0 | 36.0 |

\(K^f\) (GPa) | 2.3 | 0.02 |

\(\eta ^{fR}\) (mPa s) | 1 | 0.01 |

| 10 | – |

| – | 4.5 |

Example 2—material parameters of the meso-scale problem

Rock | Fractures | |
---|---|---|

| 16.0 | – |

| 16.0 | – |

\(K^f\) (GPa) | – | 2.4 |

\(\eta ^{fR}\) (mPa s) | – | 3.0 |

| 10 | – |

\(a_1\) (m) | – | 8 |

\(\tau _0\) (m) | – | 1e\(\texttt {-}\) 5 |

### 5.2 Example 2: fractures in impermeable matrix

Example 3—material parameters of the meso-scale problem (1 mD \(\approx \) 1e\(\texttt {-}\) 15 m\(^2\))

Rock | Fractures | |
---|---|---|

| 31 | – |

| 37 | – |

\(K^s\) (GPa) | 40 | – |

\(k^s\) (mD) | 1 | – |

\(\phi \) (–) | 0.05 | – |

\(K^f\) (GPa) | 2.4 | 2.4 |

\(\eta ^{fR}\) (mPa s) | 1 | 1 |

| 10 | – |

\(a_1\) (m) | – | 8 |

\(\tau _0\) (m) | – | 6e\(\texttt {-}\) 4 |

*N*of pressure modes that span the reduced basis until we reach an excellent agreement with the reference computation for \(N=6\).

### 5.3 Example 3: fractures in permeable matrix

In Fig. 13 we show example pressure modes. Whilst the modes \(a=4,6\) are related to processes in the fracture space, modes \(a=13,15\) describe processes in the matrix.

Finally, we validate the reduced viscoelastic basis against a reference computation of a uniaxial stress relaxation test on the basis of a fully resolved SVE. The resulting macroscopic total stresses are shown in Figs. 14 and 15, where we observe a very high accuracy of the reduced viscoelastic model.

## 6 Discussion

We establish a NMR procedure that enables us to identify macroscopic viscoelastic substitute models for strain-driven pressure diffusion problems in fractured porous rock in a reliable and numerically highly efficient way. The overall viscoelastic response is caused by squirt-type fluid flow which causes fluid pressure diffusion in interconnected fractures and in the embedding porous rock within a mesoscopic SVE. The fluid mass stored in the SVE is conserved throughout the entire process. This allows us to treat fluid pressure diffusion as a local process which is, on the macro-scale, only observable indirectly via overall frequency-dependent properties such as dispersion of the stiffness moduli and attenuation. Hence, the macroscopic substitute medium represents a monophasic viscoelastic solid. For the mesoscopic model we employ a hybrid-dimensional interface model treating fractures as fluid-filled open conduits embedded in a poroelastic matrix material. Pressure diffusion occurs along the fractures as well as in the poroelastic matrix. Moreover, exchange of fluid mass between fractures and matrix is incorporated.

We employ volume averaging techniques to derive a computational homogenization scheme that links the heterogeneous meso-scale to the viscoelastic macro-model. We propose a NMR technique that is based on the approximation of the space-time dependent fluid pressure in the SVE by a linear combination of pressure modes. Hereby, the pressure modes span a finite-dimensional reduced basis. They are identified via a POD from a set of training computations. Evaluation of the mesoscopic continuity equation together with the pressure decomposition and accordingly decomposed further quantities allows us to derive the evolution equation of the desired viscoelastic substitute model which turns out to be of the Maxwell-Zener type. Finally, we investigate the performance of the proposed method in three different numerical examples. Throughout the examples, we observe a convincing agreement between NMR and reference solutions.

The main achievement of the present contribution is that we establish a general, yet reliable, method that enables us to compute seismic wave propagation on the macro-level with full access to the mesoscopic pressure diffusion effects but without the numerical drawbacks of the FE\(^2\) concept. The numerically expensive solution of mesoscopic boundary value problems is restricted to 6 transient training computations, to 6 computations of the linear-elastic response of the dry solid skeleton and to the solution of *N* linear-elastic eigenstress problems related to the identified pressure modes. All mesoscopic computations are “off-line” precomputations and, therefore, do not burden the “online” computation of the macroscopic wave propagation.

In our ongoing research we will use the proposed method to execute forward simulations of propagation of seismic waves in a cross-hole tomography setting. The goal is to gain a better understanding for the correlation between seismic attenuation and the interconnectivity of stochastic fracture networks.

## Footnotes

- 1.
Note that this assumption is invoked here for the sake of simplicity. Existence of a symmetry plane is not ultimately necessary to set up the interface model.

- 2.
We define the Dirac-function as the distribution satisfying \(\int \limits _{\partial F_k} f(\mathbf{x})\delta _{kl}(\mathbf{x})\,\hbox {d}a = \int \limits _{\partial F_k\cap \partial F_l}f(\mathbf{x})\,\hbox {d}s\).

- 3.
\(\mathbb {P}_\Box \) contains functions that satisfy the continuity constraint (16).

## Notes

### Acknowledgements

This work was supported by the Swedish Research Council (VR) under the Grant Number 2015-05422. The financial support is gratefully acknowledged.

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