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The Effect of Quadratic Gradient Terms on the Borehole Solution in Poroelastic Media
Author(s) -
Wang Yarlong,
Dusseault Maurice B.
Publication year - 1991
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/91wr01552
Subject(s) - poromechanics , compressibility , biot number , mechanics , pressure gradient , porous medium , nonlinear system , quadratic equation , laplace transform , compressible flow , pore water pressure , mathematics , mathematical analysis , physics , geotechnical engineering , geology , geometry , porosity , quantum mechanics
When compressible fluid continuity in a fluid‐saturated compressible porous medium under transient conditions is considered, we cannot obtain a standard linear diffusion equation in terms of pressure unless we ignore the quadratic terms in the pressure gradient expression, for example [∂ p /∂ r ] 2 for cylindrical plane strain coordinates. They are assumed to be so small that their contribution can be ignored in pressure analysis. Thereby, a nonlinear equation can be avoided. During hydraulic fracturing, rapid drawdown, or slug testing, the pressure difference can reach a high value “instantly.” An extremely steep pressure gradient is generated, and it may not be appropriate to neglect quadratic terms. In this paper, an analytical solution for pore pressure coupling with the deformation in a porous medium is developed by taking the quadratic term into account. By Laplace transformation, we obtain a solution for a nonlinear diffusion equation by setting up a fluid continuity equation according to the mass conservation law rather than from energy principles in terms of volume. Deviations from existing solutions are identified in cases of high pressure gradients, and these deviations are related to the compressibility of the pore and injected fluids. It would seem that the new solution gives a more correct early time response in these cases. To calculate the effective stresses and pore pressure, we need to carefully define hydraulic diflfusivity. We have related the coefficients of Biot and Geertsma to those of mass conservation, which are commonly used in hydrology.