Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient

The dimensionless mathematical models of one-dimensional flow in the semi-infinite long porous media with threshold pressure gradient are built for the two cases of constant flow rate and constant production pressure on the inner boundaries. Through formula deduction, it is found that the velocity of the moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on the moving boundary, which is very different from the classical heat-conduction Stefan problems. However, by introducing some similarity transformation from Stefan problems, the exact analytical solutions of the dimensionless mathematical models are obtained, which can be used for strict validation of approximate analytical solutions, numerical solutions and pore-scale network modeling for the flow in porous media with threshold pressure gradient. Comparison curves of the dimensionless pressure distributions and the transient dimensionless production pressure under different values of dimensionless threshold pressure gradient are plotted from the exact analytical solutions of problems of the flow in semi-infinite long porous media with and without threshold pressure gradient. It is shown that for the case of constant flow rate the effect of the dimensionless threshold pressure gradient on the dimensionless pressure distributions and the transient dimensionless production pressure is not very obvious; in contrast, for the case of constant production pressure the effect on the dimensionless pressure distributions is more obvious especially at the larger dimensionless distance near the moving boundary; and for the case of constant production pressure, the smaller the dimensionless threshold pressure gradient is, the larger the dimensionless pressure is, and the further the pressure disturbance area reaches.