Flow in deformable porous media. Part 2 Numerical analysis – the relationship between shock waves and solitary waves

Abstract

Using numerical schemes, this paper demonstrates how viscous resistance to volume changes modifies the simplest shock wave solutions presented in Part 1. For an initial condition chosen to form a step-function shock, viscous resistance causes the shock to disperse into a rank-ordered wavetrain of solitary waves. Large obstructions in flux produce large-amplitude, slow-moving wavetrains while smaller shocks shed small-amplitude waves. While the viscous resistance term is initially important over a narrow boundary layer, information about obstructions in the flux can propagate over many compaction lengths through the formation of non-zero wavelength porosity waves. For large-amplitude shocks, information can actually propagate backwards relative to the matrix. The physics of dispersion is discussed and a physical argument is presented to parameterize the amplitude of the wavetrain as a function of the amplitude of the predicted shock. This quantitative relationship between the prediction of shocks and the development of solitary waves also holds when mass transfer between solid and liquid is included. Melting causes solitary waves to decrease in amplitude but the process is reversible and freezing can cause small perturbations in the fluid flux to amplify into large-amplitude waves. These model problems show that the equations governing volume changes of the matrix are inherently time dependent. Perturbations to steady-state solutions propagate as nonlinear waves and these problems demonstrate several initial conditions that do not relax to steady state. If these equations describe processes such as magma migration in the Earth, then these processes should be inherently episodic in space and time.