An exact analytical solution of moving boundary problem of radial fluid flow in an infinite low-permeability reservoir with threshold pressure gradient

Abstract

Many engineering technologies involve the moving boundary problems for the radial seepage flow with a threshold pressure gradient, such as the well testing in the development of low-permeability reservoirs, heavy oil reservoirs and groundwater resources. However, as a result of the strong nonlinearity, an exact analytical solution of the moving boundary problems for the radial seepage flow with a threshold pressure gradient has not been obtained yet. Here, a dimensionless moving boundary mathematical model for the radial fluid flow in an infinite low-permeability reservoir with a threshold pressure gradient is developed first. The setting of a variable well production rate for an inner boundary condition can make the mathematical model exhibit a full self-similarity property. Second, by introducing some similarity transformations, the nonlinear system of PDEs of the model can be equivalently transformed into a closed pseudo-linear system of ODEs, whose exact analytical solution can be easily obtained. What's more, the existence and the uniqueness of the exact analytical solution to the moving boundary model are also proved strictly through the mathematical analysis. Third, it is also strictly proved that the exact analytical solution can be degenerated to that of the model of the Darcy's radial fluid flow as the threshold pressure gradient approaches to zero. Finally, by a comparison of model analytical solutions, it is demonstrated that the moving boundary conditions must be incorporated in the modeling of the radial seepage flow with a threshold pressure gradient; otherwise, the effect of the threshold pressure gradient on the radial seepage flow can be overestimated largely.