Analytical study on a moving boundary problem of semispherical centripetal seepage flow of Bingham fluid with threshold pressure gradient

Abstract

It is well known that the Non-Newtonian Bingham fluid flow in porous media does not obey the conventional linear Darcy’s law due to the yield stress for the Bingham fluid: There exists a threshold pressure gradient, which means that the seepage flow only happens when the threshold pressure gradient is overcome. The principle of non-Darcy seepage flow with the threshold pressure gradient is also applicable into the situation of the fluid flow in the low-permeable porous media. Here, a nonlinear moving-boundary mathematical model is built for the semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient in a three-dimensional infinite heavy oil reservoir with the type of Bingham fluid; wherein the moving boundary conditions are incorporated for describing the effect of the threshold pressure gradient. In consideration of the strong nonlinearity of the model, the similarity transformation method is applied into obtaining the exact analytical solution of the model. In order to keep full self-similarity for the model, the inner boundary condition is set as variable flow rate that increases linearly with the time. As a result, an exact analytical solution for the nonlinear moving-boundary mathematical model of semispherical centripetal non-Darcy seepage flow with the threshold pressure gradient is obtained. The existence and the uniqueness of the exact analytical solution are also strictly proved. It is also theoretically proved that as the threshold pressure gradient tends to zero, the exact analytical solution can be reduced to that of a mathematical model of semispherical centripetal Darcy’s seepage flow. The presented exact analytical solution can be used for strictly verifying the validity of the numerical methods for solving the three-dimensional moving boundary models of non-Darcy seepage flow with the threshold pressure gradient in the actual engineering problems. From the exact analytical solution, it is also revealed that when the threshold pressure gradient exists, the spatial pressure distribution exhibits an instructive feature of compact support; as the threshold pressure gradient tends to zero, the sensitivity of its effect on the transient distance of the moving boundary and the transient pressure will grow, which reveals the difficulty in accurately determining the position of the moving boundary by the numerical methods and the serious uncertainty problem in the interpretation of the threshold pressure gradient by the pressure transient analysis method in engineering as the threshold pressure gradient is rather small. Through the comparison of the two different exact analytical solutions that corresponds to the two different models with and without incorporating the moving boundary conditions for describing the effect of the threshold pressure gradient, it is demonstrated that when the moving boundary conditions are not incorporated in the modeling, the effect of the threshold pressure gradient on the spatial pressure distribution, the transient pressure and the productivity index can be overestimated largely. Therefore, it is very necessary to incorporate the moving boundary conditions in the modeling of non-Darcy seepage flow with the threshold pressure gradient. The study in the paper definitely provides solid theoretical basis of fluid mechanics for the relevant engineering applications in the development of heavy oil reservoirs and low-permeable reservoirs in petroleum engineering and in the development of water resources in low-permeable formations in hydraulic engineering.