Exact analytical solutions of non-Darcy seepage flow problems of one-dimensional Bingham fluid flow in finite long porous media with threshold pressure gradient

Abstract

In the paper, the study on the exact analytical solution of the moving boundary problem of one-dimensional Bingham fluid seepage flow is extended from the infinite long porous media (Liu et al., 2012) to the finite long porous media. Two exact analytical solutions are presented by appropriately relying on some methods of mathematical physics and mathematical techniques. One is for the model with finite closed outer boundary condition; the other is for the model with finite constant pressure outer boundary condition. The existence and the uniqueness of the exact analytical solutions are also strictly proved theoretically. In addition, the numerical solutions of the two models by the finite difference method are also provided. Through the comparison, it is found that these exact analytical solutions have very excellent agreement with the numerical solutions although few terms of the infinite function series existent in the exact analytical solutions have to be retained for the calculation. Furthermore, for the two models, the effect of the threshold pressure gradient on the transient pressure and the transient pressure derivative at the inner boundary for the whole flow process is analyzed through the analytical solutions. Finally, through the comparison of the relevant model solutions, it is concluded that it is very necessary to incorporate the process of moving boundary for the modeling of non-Darcy Bingham fluid flow in finite long porous media with threshold pressure gradient; otherwise, large errors can be introduced in predicting the transient pressure and the transient pressure derivative in the porous media. The presented work can support solid theoretical foundations for the experiment design of measuring the threshold pressure gradient and the pressure transient analysis in the field of inverse problems in the petroleum engineering, which have been widely involved in the development of low-permeable oil reservoirs and heavy oil reservoirs.