Mathematical Models of Consolidation and Compaction

Various models for the flow of a slightly compressible fluid through a saturated deformable porous medium are described. The classical porous medium equation can model the evolution of porosity of the medium and the pressure of the fluid in one dimension. This primitive model has been extended to reflect detailed mechanics of the medium by the Biot system that consists of a diffusion equation for the fluid flow coupled to a momentum equation for the deformable porous solid consolidation. The small-strain constitutive laws may include nonlinear or degenerate relations of elasto-visco-plastic type or variational inequalities of contact mechaincs. We shall describe the non-linear evolution equations in Hilbert space that characterize these problems for which the existence or uniqueness of a global weak or strong solution is proved by means of monotonicity methods. In order to exploit the available data for the variation of solid porosity with fluid pressure, the simple porous medium model in $\Re$ can be extended to a two-phase system in $\Re^3$ that contains general visco-elastic media with hysteresis arising from irreversible damage. This leads to a mathematical theory in $L^1$ for compaction in sedimentary basins.

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