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

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.

A poro-viscoelastic substitute model of fine-scale poroelasticity obtained from homogenization and numerical model reduction

“…In addition, the constraint equation (27f) becomes obsolete. Altogether, these model choices give rise to the single-phase viscoelastic macro-scale model proposed in [13,14]. The procedure may, therefore, be called "selective" homogenization.…”

“…This "selective" homogenization approach allows to reduce the macro-scale problem to that of a single-phase viscoelastic material, whereby the "'mode coefficients" play the role of classical internal variables of the macro-scale model. More recently, this selective homogenization towards a viscoelastic macro-scale material was extended to account for Darcy-type seepage in fluid-filled fracture networks [14].…”

“…Generalized Maxwell rheology for the poro-viscoelastic NMR model according to the spectral evolution equation (59).Ê stat represents the stiffness of the stationary problem {χ =R −1 ξ } by solving the generalized eigenvalue problem forŜ andM as presented in[14]. The matrixR contains the eigenvectors of the generalized eigenvalue problem.…”

Viscoelastic substitute models for seismic attenuation caused by squirt flow and fracture leak off

“…Based on this framework, Jänicke et al (2016) establish a numerical model reduction (NMR) procedure that can be used to derive viscoelastic substitute models for poroelastic media in a systematic and numerically highly efficient manner. This computational framework was recently extended to poroelastic media with embedded fluid-saturated fractures (Jänicke et al, 2019).…”

“…In the present contribution, our aim is, first, to use this novel NMR procedure proposed and described by Jänicke et al (2019) to derive viscoelastic substitute models for a range of scenarios considering fluid pressure diffusion associated with the mesoscopic fracture leak-off and the mesoscopic or microscopic squirt flow. Second, we validate the results against reference computations and show that the proposed procedure is able to predict the material properties of the viscoelastic substitute model with high accuracy.…”