Finite Element Models for Multiscale Theory of Porous Media

Abstract

Theories to describe porous media aim to represent a combination of a solid matrix and a connected pore space, which may be occupied by fluids. Porous media applications are found in structural materials, geomechanics, biological tissues, and chemical filtration processes, amongst others. Early developments are attributed to Biot who, based on experimental data, derived a set of governing equations for consolidation problems. Concomitantly, other theories were based on mixture theory and volume averaging concept, establishing a clear connection between micro and macro scales. The work presented by de la Cruz and Spanos (dCS) is an example of a volume-averaging theory. The Biot (BT) theory can be seen as a simplified version of the dCS formulation, since the former assumes a unique energy potential, restrains solid-fluid deformations to reciprocal interactions, and does not account for fluid viscous dissipation terms on the fluid stress definition. Due to the complexity of the dCS theory governing equations, correspondent numerical models have been scarce in literature. To fill this gap, this research focuses on the development of novel finite element (FE) models for the dCS theory. The main applications are consolidation and wave propagation problems in water-saturated rock formations. Results are compared to the BT formulation, and the circumstances which can lead to discrepancies between these theories are studied. First, a novel FE method is developed for quasi-static solid deformations and transient pore fluid flow, where dynamic effects are neglected. The governing equations are written in terms of solid displacement, fluid pressure, and porosity. Fully implicit time integration and a mixed-element formulation are employed to ensure stability. The convergence rate of the dCS FE model is shown to be optimal in a one-dimensional consolidation problem, considering that rates are dependent on how significant the solid-fluid coupling terms are compared to the uncoupled terms. Two-dimensional examples further attest the robustness of the implementation and are shown to reproduce BT model results as a special case. Non-reciprocal solid-fluid interactions, inherent to the dCS theory, lead to significant differences depending on the properties of the porous media (e.g., permeability) and problem specific constraints. Extending the transient formulation to include inertia (acceleration) terms, a three-field dCS FE model for dynamic porous media is presented, now formulated in terms of solid displacement, fluid pressure, and fluid displacement. Due to fluid viscous dissipation terms, wave propagation in the dCS theory yields an additional rotational wave compared to BT theory. Besides the introduction of non-reciprocal solid fluid interactions, the dCS model further accounts for a dimensionless parameter related to the macroscopic shear modulus. Space and time convergence rates are demonstrated in a one-dimensional case. A dimensionless analysis performed in the dCS framework led to negligible differences between BT and dCS models except when assuming high fluid viscosity. Domains with small characteristic lengths resulted in BT and dCS damping terms in the same order of magnitude. One- and two-dimensional examples showed that the dCS non-reciprocal interactions and the macroscopic shear modulus parameter are responsible for modifying wave patterns. A two-dimensional injection well simulation with water and slickwater showed higher wave attenuation for the latter. Following the derivation of the dCS dynamic formulation, wave propagation phenomena in porous media is analyzed. The dCS slow S wave is studied, essential in the representation of fluid vorticity and shear motion at high frequencies. Differences between dCS and BT results appear at high frequency range. Solid-fluid non-reciprocal interactions and changes in the macroscopic solid shear modulus lead to distinct P wave patterns. The influence of permeability, porosity, and dynamic viscosity is evaluated, showing that wave patterns are generally most affected at the ultrasonic frequency range. At low frequencies, BT and dCS yield the same results for saturated rocks, except when the slow P wave is non-dissipative. While BT theory incorporates a correction factor to reproduce fluid behavior at high frequency, the dCS model naturally accounts for this effect due to the complete fluid stress tensor. High-frequency results from both theories, nonetheless, are discrepant. Since the dCS equations can be written in terms of different main variables, they can be chosen according to the problem setup. In this sense, a three-field dCS formulation is written in terms of solid displacement, fluid pressure, and relative fluid velocity. The verification study is in agreement with BT results, highlighting how the addition of fluid viscous dissipation terms does not influence load bearing. However, it is essential in the representation of fluid vorticity motion at high frequencies and in wave reflection/transmission problems. Optimal convergence rates for a one-dimensional example are obtained for same-order linear elements. Two-dimensional examples with sandstone and shale layers show how waves are transmitted and reflected at the domain interface at the low- and high-frequency regimes. The research development and findings reported in this thesis consist in novel FE models to represent the dCS porous media theory. The results show how the dCS formulation is able not only to recover BT results but further circumvent gaps in the BT theory. The dCS FE framework herein presented is also a foundation for future studies in the area. Examples are the expansion of wave propagation studies with a reduced number of assumptions, the combination of a inertial fracture flow model with a porous media representation of fractured rocks, the simulation of nonlinear solid matrix behavior, and the development of a dCS FE model for inertia-driven flow in porous media.