Boundary element simulation of petroleum reservoirs with hydraulically fractured wells.

摘要

The boundary element method is applied to solve the linear pressure-diffusion equation of fluid-flow in porous media. The governing parabolic partial differential equation is transformed into the Laplace space to obtain the elliptic modified-Helmholtz equation including the homogeneous initial condition. The free-space Green's functions, satisfying this equation for anisotropic media in two and three dimensions, are combined with the generalized form of the Green's second identity. The resulting boundary integral equation is solved by following the collocation technique and applying the given time-dependent boundary conditions of the Dirichlet or Neumann type. The boundary integrals are approximated by the Gaussian quadrature along each element of the discretized domain boundary. Heterogeneous regions are represented by the sectionally-homogeneous zones of different rock and fluid properties. The final values of the interior pressure and velocity fields and of their time-derivatives are found by numerically inverting the solutions from the Laplace space by using the Stehfest's algorithm.; The main extension of the mostly standard BEM-procedure is achieved in the modelling of the production and injection wells represented by internal sources and sinks. They are treated as part of the boundary by means of special single-node and both-sided elements, corresponding to the line and plane sources respectively. The wellbore skin and storage effects are considered for the line and cylindrical sources. Hydraulically fractured wells of infinite conductivity are handled directly according to the specified constraint type, out of the four alternatives. Fractures of finite conductivity are simulated by coupling the finite element model of their 1D-interior with the boundary element model of their 2D-exterior. Variable fracture width, fractures crossing zone boundaries, “networking” of fractures, fracture-tip singularity handling, or the 3D-description are additional advanced formulations of the proposed model of the hydraulically fractured wells.; Another strong emphasis is put on the realization of the numerical model on a computer using the object-oriented programming. In addition to the graphical editor of input data, a higher-level language is designed to facilitate a universal data interface to the numerical simulator. The final version of the simulator is supplied on a CD-ROM together with the 35 solved example problems.