Isogeometric Analysis of Coupled Problems in Porous Media

Abstract

Computational modeling of coupled problems in porous media is essential in various disciplines of science and engineering. The porous media of interest here are soils subjected to different physical processes. In particular, hydraulic (H), coupled hydro-mechanical (HM) and thermo-hydro-mechanical (THM) processes are addressed. Development of a fully-coupled THM numerical model targets ground freezing phenomena as the main application problem. The governing equations for the various processes are derived based on porous media theory. The fundamentals of this theory are presented in a general form and discussed by focusing on soils and the application problems. Isogeometric analysis (IGA) is adopted for implementing the governing equations, leading to the development of a code for numerical simulations. One of the main reasons for choosing IGA as a computational framework is the smoothness of the basis functions, which is attractive for better continuity of field variables. The main concepts behind IGA, including its advantages compared to traditional finite element analysis (FEA), are briefly discussed. The first problem studied is steady-state groundwater flow governed by Darcy’s law. Numerical challenges occur in the simulation of groundwater flow problems due to complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain or a combination of these. Adaptive IGA using locally refined (LR) B-Splines is applied to address some of these problems. A posteriori error estimates are calculated to identify which B-Splines should be locally refined. The error estimates are calculated based on recovery of the L2-projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two-dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Classical problems in poroelasticity are next addressed using mixed IGA, i.e. using different polynomial degrees for displacement and pore pressure. The finite element method has been widely applied to such problems and the numerical behavior of the governing equations has been discussed by several researchers. Equal order IGA has recently been applied to poroelasticity. Pressure oscillations at small time steps have been known to be an issue in the simulation of poroelasticity problems. The performance of mixed IGA for smaller time steps is investigated by revisiting Terzaghi’s classical consolidation problem. A numerical study is also performed on the consolidation of a layered soil where a very low permeability layer is known to cause pressure oscillations. It is observed from the numerical studies that mixed IGA improves the accuracy of the pore pressure results compared to equal order simulations, as is known from traditional FEA. The pressure oscillations, however, are not completely removed but were observed to decrease with increasing polynomial degrees. Mixed simulations with a graded mesh refinement were observed to reduce the pore pressure oscillations, revealing the potential of adaptive refinement for such problems. Fully coupled THM processes in ground freezing are then studied using mixed IGA. The governing linear momentum, mass and energy balance equations are formulated by assuming saturated conditions. Strain due to phase change is incorporated into the linear momentum balance equation. This is attained through a simple thermoelastic constitutive equation with temperature dependent strength parameters. The supplementary equations that complete the model include the soil-water characteristic curve and a hydraulic conductivity model. After spatial and temporal discretization, the governing and supplementary equations result in a strongly coupled and highly nonlinear system of equations, which are solved using Newton-Raphson iteration. Numerical studies are performed on one-dimensional freezing and a frost heave problem where experimental data is available. Good agreements were observed between the mixed IGA based simulation of a THM coupled problem in frost heave and the corresponding experimental data found from literature. The continuity of the basis functions in mixed IGA of THM coupled problems implies that prediction of derived quantities, such as fluxes, across knot spans (analogous to elements in traditional FEA) can be controlled and improved. In general, the numerical implementation work resulted in H, HM and THM frameworks for simulation of poro/geomechanics problems using IGA. The frameworks are developed based on IFEM - an object-oriented isogeometric toolbox for the solution of partial differential equations. The developed numerical codes may be used and extended further. In addition to the various application problems studied, the numerical work mainly initiates application of IGA to THM coupled problems in porous media. The features of IGA that are computationally attractive in this context, such as the ability to perform higher-order simulations with ease, can thus be utilized.