A semi-analytical solution under a variable inner boundary condition for fractured horizontal wells in a rectangular closed reservoir with discrete fracture networks

Abstract

The paper focuses on establishing a semi-analytical model under a variable inner boundary condition for fractured horizontal wells in a rectangular closed reservoir with discrete fracture networks. In this paper, the inner boundary is a piecewise boundary condition, which varies from Newman boundary condition to Dirichlet boundary condition. Rectangular reservoir model and complex fracture model are established respectively, and hydraulic fractures and natural fractures can cross or not. The governing equation is solved under the first-stage inner boundary condition by using the Laplace transform and Fourier transform methods, and the coupling solution is derived by the condition that the pressure and flux at the discrete fracture networks wall are equal. The control equation under the second-stage inner boundary condition is a completely non-homogeneous equation group, which is solved by variable replacement method. The final pressure and flow solutions under the variable inner boundary conditions are obtained in real space through the Stehfest numerical inversion method. This semi-analytical solution is verified accurately for simple fracture case under constant rate condition and variable inner boundary conditions through numerical simulation. The results show rate decline rate of complex fractures is also very fast, and that of simple fractures is relatively slow through the comparison between complex fracture networks and simple fractures under variable inner boundary conditions. The influence of some important parameters on flow rate and pressure change is analyzed in detail under variable inner boundary conditions, mainly including initial rate, limited pressure, fracture networks permeability and initial reservoir pressure.