Application of the Discontinuous Galerkin Method to the Study of the Dynamics of Temperature and Pressure Changes in a Formation with an Injection Well and a Hydraulic Fracture

Abstract

Introduction. In this article, the problem of temperature distribution in an oil-bearing formation with a hydraulic fracture and a vertical injection well is numerically modeled. Materials and Methods. To describe the process of temperature distribution in the formation under the action of the fluid injected into the formation, the Fourier-Kirchhoff equation of convective heat transfer is used. To solve this equation, the discontinuous Galerkin method on staggered unstructured grids is used. To describe the process of pressure change in the formation under the action of the injection well, an equation is used that is obtained based on the continuity equation and Darcy’s law. To solve it, the discontinuous Galerkin method on an unstructured triangular grid is used. To parallelize the numerical algorithm, the MPI library is used. Results. The article presents a numerical algorithm and the results of modeling the dynamics of the temperature fields in an oil reservoir with a hydraulic fracture and a vertical injection well. Discussion and Conclusion. A numerical algorithm based on the discontinuous Galerkin method for math modeling of the temperature and pressure fields in a oil-bearing formation with a hydraulic fracture and injection well was developed and implemented. The results obtained for the distribution of temperature and pressure in the fracture are adequate and in good agreement with the specified initial-boundary conditions. Further work in this direction involves modeling on tetrahedral unstructured meshes for a more accurate study of the ongoing processes.

Keywords

• discontinuous galerkin method
• vertical injection well
• hydraulic fracturing
• convective heat transfer equation
• continuity equation
• darcy’s law
• unstructured grids
• spaced grids
• mpi