An analytical model for matrix–fracture gas flow under variable fracture pressure

This paper presents an analytical model for modeling of matrix–fracture fluid transfer in gas systems subjected to variable fracture pressure. In this work, pseudopressure and pseudotime functions are used to linearize the matrix governing flow equation so that the constant fracture pressure solution is obtained in terms of pseudotime function. The superposition theorem is further applied on the obtained solution with respect to pseudotime to account for variable fracture pressure conditions. Knowing that using pseudotime function is an approximate technique to linearize the governing flow equation, it is observed that superposition theorem can be successfully applied with respect to this time function to predict accurate values of matrix–fracture flow rate. To avoid the use of iterative procedures for calculation of pseudotime at any point of time, a pseudotime–time relationship is developed with the aid of a material balance equation and several specific assumptions. Also, specific computational treatments are proposed to reduce the number of time steps for calculation of convolution integral and also to eliminate the error associated with calculations. A fine grid single-porosity numerical solution is prepared to verify the content of this work.