The multilateral well model with complete vertical opening of naturally fractured reservoir

Abstract

A new analytical solution of the diffusivity equation for a mathematical model of a multilateral which penetrates a double-porosity reservoir vertically along the entire thickness has been obtained. The solution of the diffusivity equations is given in the Laplace space and was derived with the assumption of a well constant flow rate and the absence of friction pressure losses in the channels. The analytical solution of the diffusivity equation, written taking into account the presence of a fracture system and pore matrices, contains a modified Bessel function of the first and second kind of zero order and is presented in the form of a matrix equation. The matrix equation is solved using the LU decomposition, and the transfer of the dimensionless pressure from the Laplace space to the Cartesian coordinate system is performed using the Stehfest algorithm. On the basis of the developed numerical algorithm, a parametric analysis of the multilateral well model in a formation with double porosity was carried out. The reservoir flow properties, the parameters of the multilateral well branches, the double porosity model parameters, such as the storativity ratio and the transmissivity ratio, vary. The difference between the calculated parameters of the multilateral well model in a homogeneous formation and in a formation with double porosity is shown. The influence of the storativity ratio and the transmissivity ratio on the pressure drop and the derivative of the pressure drop in the well has been established. It is shown that when the transmissivity ratio decreases by a factor of 10, the start time of the transient regime also increases by a factor of 10. A decrease in the storativity ratio value from 0.01 to 0.005 leads to an increase in the pressure drop at the beginning of the well operation by 14.3% and the pressure drop derivative minimum value of the transient regime reduces by 1.92 times. When the storativity ratio decreases to 0.001, the value of the pressure drop increases by 48.2% and the pressure drop derivative minimum value of the transient regime decreases by 7.5 times.