The equations governing the flows of a weakly-compressible fluid through a weakly-deformable porous medium accounting for the presence of initial (limit) pressure gradient are discussed. The influence of a quadratic term, which is usually dropped in deriving the linear equations of elastic regime, on the qualitative behaviour of the solutions is studied. The taking account of the additional term is shown to qualiatively change the asymptotics of the Darcy velocity for one-dimensional travelling wave solutions. The change in the wave front motion law for large times is demonstrated by an example of the solution to the axisymmetric problem of fluid injection into a porous bed.
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