DOI: https://doi.org/10.24028/gzh.0203-3100.v41i2.2019.164465

### Mathematical modeling of nonlinear displacement processes in oil LEF-layer by methods of complex analysis and summary representations

#### Abstract

The approach to the modeling of nonlinear displacement processes (one and two-phase filtration) in heterogeneous oil deformable layers is developed, taking into account the inverse effect of the potential of the velocity field and the flow function on the conductivity of the medium. We constructed a method and computational technology for solving the corresponding boundary value problems for nonlinear-layered triple-connected curvilinear domains, bounded by equipotential lines and flow lines, on the basis of the synthesis of numerical methods of quasiconformal mappings and summary representations for differential equations with discontinuous coefficients in combination with domain decomposition by Schwartz method. Quasi-ideal processes in nonlinearly double-layered horizontal LEF-layers, whose geometry of heterogeneity zones is unknown in advance, is described by the corresponding boundary value problems obtained on the basis of the Darcy law and the continuity equation with the coefficient of layer permeability, which is given by a piecewise-constant function with ruptures along the searched equipotentials and lines of flow. The coefficient of conductivity the medium is given as a piecewise-constant function, that is dependent on the searched quasipotential and function of flow, with unknown line dividing layers (lines gap conductance coefficient) that is along the searched equipotential lines and flow lines and that is finding in the process of solving the problem. The proposed algorithms automatically solve the problem of choice of nodes and building a dynamic grid, finding of the unknown dividing lines of areas constancy coefficient of conductivity the medium, the calculation of the velocity field and calculate other characteristic parameters of the process. The decomposition of the domain on the layers of the constancy of permeability coefficient allows us to solve problems in more «comfortable» subdomain than the original problem the whole domain, and allows make in parallel the computational process, since calculations in the subdomain at each iterative step are independent of each other and can be done in parallel with the use of modern computer technology.

#### Keywords

reservoir of oil; doubly-layered medium; quasiconformal mappings; complex quasipotential; summary representations method; domain decomposition; alternating method by Schwarz; LEF-layer

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