Flow in deformable porous media. Part 1 Simple analysis

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Abstract
Many processes in the Earth, such as magma migration, can be described by the flow of a low-viscosity fluid in a viscously deformable, permeable matrix. The purpose of this and a companion paper is to develop a better physical understanding of the equations governing these two-phase flows. This paper presents a series of analytic approximate solutions to the governing equations to show that the equations describe two different modes of matrix deformation. Shear deformation of the matrix is governed by Stokes equation and can lead to porosity-driven convection. Volume changes of the matrix are governed by a nonlinear dispersive wave equation for porosity. Porosity waves exist because the fluid flux is an increasing function of porosity and the matrix can expand or compact in response to variations in the fluid flux. The speed and behaviour of the waves depend on the functional relationship between permeability and porosity. If the partial derivative of the permeability with respect to porosity, ∂kϕ/∂ϕ, is also an increasing function of porosity, then the waves travel faster than the fluid in the pores and can steepen into porosity shocks. The propagation of porosity waves, however, is resisted by the viscous resistance of the matrix to volume changes. Linear analysis shows that viscous stresses cause plane waves to disperse and provide additional pressure gradients that deflect the flow of fluid around obstacles. When viscous resistance is neglected in the nonlinear equations, porosity shock waves form from obstructions in the fluid flux. Using the method of characteristics, we quantify the specific criteria for shocks to develop in one and two dimensions. A companion paper uses numerical schemes to show that in the full equations, viscous resistance to volume changes causes simple shocks to disperse into trains of nonlinear solitary waves.
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