A projection method for porous media flow

TL;DR

This paper introduces a novel projection-based numerical method for simulating flow and elastic deformation in porous media, demonstrating second-order accuracy and applicability to complex biological gels.

Contribution

A new projection method for coupled porous media flow and elastic deformation that achieves second-order convergence and handles non-linear saddle-point problems.

Findings

Method achieves second-order convergence in space and time.

Successfully applied to phase separating neo-Hookean gels.

Provides an efficient alternative to implicit solvers for porous media flow.

Abstract

Flow through porous, elastically deforming media is present in a variety of natural contexts ranging from large-scale geophysics to cellular biology. In the case of incompressible constituents, the porefluid pressure acts as a Lagrange multiplier to satisfy the resulting constraint on fluid divergence. The resulting system of equations is a possibly non-linear saddle-point problem and difficult to solve numerically, requiring nonlinear implicit solvers or flux-splitting methods. Here, we present a method for the simulation of flow through porous media and its coupled elastic deformation. The pore pressure field is calculated at each time step by correcting trial velocities in a manner similar to Chorin projection methods. We demonstrate the method's second-order convergence in space and time and show its application to phase separating neo-Hookean gels.