Modeling and numerical calculation of piston-like oil displacement for doubly-periodic systems of field development

Prediction of the oil-water boundary movement has great importance to the design problems of oilfield development by waterflooding: knowledge of the nature of the coupled motion between the displaced oil and the displacing water in the reservoir allows us to optimize the system of oil field development. The simplest model of joint oil/water filtering is the model of “multicolored” liquids, which assumes that oil and water have the same or similar physical properties (density and viscosity). In this paper, we consider a more complex model “piston-like” oil-water displacement, which takes into account differences in the viscosity and density of two fluids. An oil reservoir is assumed to be homogeneous and infinite, of fixed thickness, and with constant values of porosity and permeability coefficients. The reservoir is developed by a group of a finite number of production and injection wells recurrent in two directions (doubly-periodic cluster). Filtration of liquids is described by Darcy's law. It is assumed that liquids are weakly compressible, and the pressure in the reservoir satisfies the quasi-stationary diffusion equation. The model of piston-like displacement leads to the discontinuity in the tangential component of the velocity vector at the boundary of oil-water contact. Use of the theory of elliptic functions in conjunction with the generalized Cauchy integrals reduces the problem of finding the current boundaries of oil-water contact to the system of singular integral equations for the tangential and normal components of the velocity vector and the Cauchy problem for the integration of the differential equations of motion of the oil-water contact boundary. An algorithm for the numerical solution of this problem is developed. The monitoring of oil-water boundary motion for different schemes of waterflooding (linear row, four-point, five-point, seven-point, nine-point, etc.) is carried out.