The work is devoted to numerical modeling of the processes of heat and mass transfer of a two-phase fluid in the environment of a production well, which is necessary for monitoring the development of fractured-porous reservoirs. This work proposes an efficient approach to constructing a solution to the problem. To solve the problem, a model of the “double medium” type is used, where the pore part of the reservoir is considered as the first medium, and the system of natural fractures is considered as the second medium. For the resulting mathematical model, the difference schemes with time weights are constructed based on the algorithm of splitting by physical processes, which ensure the correctness and consistency of fluxes in the fracture system and the pore reservoir. In the numerical solution, the approximations of differential operators obtained in the framework of the finite difference method are used. For the parallel implementation of the developed numerical approach, the domain decomposition method and the matrix sweep algorithm are chosen. The program implementation is made using the MPI standard. Computational experiments are carried out, the results of which confirm the effectiveness of the developed numerical algorithm and its parallel implementation. In numerical experiments, the distributions of pressure and temperature near an operating production well are obtained, on the basis of which it is possible to adjust the operation of wells in order to increase production.

Mass transfer in a carbonate reservoir of a fractured-porous type is considered. Such reservoirs have a natural system of destruction in the form of fractures and cavities. In this paper, a mathematical model of the fluid redistribution between a porous-type matrix and a network of natural fractures is proposed and studied. The resulting system of differential equations is quasi-linear and quite complex. A number of difficulties arise in its numerical solution. First, the system contains a large number of unknown functions. Second, the nature of the nonlinearity of the equations is such that the corresponding linearized system no longer has the property of the self-adjointness of the spatial differential operators. To solve this problem, the method of splitting by physical processes and the approximation of differential operators by the finite difference method are used. The resulting split mesh model is equivalent to the discrete original balance equations of the system (conservation of mass components of fluids and total energy of the system) written in the divergent form. This approach is based on a nonlinear approximation of mesh functions in time, which depends on the fraction of the volume occupied by fluids in the pores, and is simple to implement. This paper presents the results of numerical calculations and analyzes the space-time dynamics of pressure change processes.

This work considers the isothermal process of incompressible viscous fluid filtration in an oil-saturated, fractured-porous reservoir. A study of the pressure and water saturation distribution process is carried out for a case in which a production well is put into operation. For this problem, i.e., a mathematical model in a two-dimensional formulation, a numerical method and a parallel algorithm are proposed. The mathematical model of two-phase filtration is written in accordance with the classical laws of continuum mechanics and Darcy’s law and also includes a function of fluid exchange between low-permeability pores and high-permeability natural fractures within the framework of the Warren–Root model. The numerical solution is based on the finite-difference method and a splitting scheme of physical processes and spatial coordinates. For a split system with respect to piezoconductivity, an implicit finite-difference scheme with fixed saturations is constructed, and with respect to saturation transfer, explicit and implicit difference schemes are constructed. For parallel implementation of the developed numerical approach, a method based on geometric parallelism is selected. Testing of the developed method is performed using the example of calculating liquid mass transfer for a wide range of parameters. To verify the model, the obtained calculated pressure curves are compared with field data recorded by a deep-well measuring device. The results allow for estimation of the distribution of reservoir pressure and water saturation depending on the permeability of the fracture set and the pore part. The obtained results allow for monitoring of well operations, reducing unexpected accident risks and optimizing the development system in order to increase oil production in fractured-porous reservoirs. Computational experiments confirm the efficiency of the developed numerical algorithm and its parallel implementation.

The process of two-phase filtration in a carbonate formation of fractured-pore type is considered. A mathematical model in a spatially two-dimensional formulation is proposed, a numerical method for solution and a parallel algorithm for its implementation are developed. The mathematical model is based on the Buckley-Leverett approach. The reservoir takes into account the exchange of fluids between low-permeability pores and natural fracturing, specified within the framework of the dual porosity model. The numerical algorithm is based on the use of the finite difference method and the splitting scheme by physical processes. To speed up calculations, a parallel algorithm based on two-dimensional domain decomposition is used. Numerical experiments were carried out, which showed that the developed algorithm is highly efficient and allows one to calculate the necessary characteristics of the modeled physical process.

Authors consider processes the filtration process in the reservoir. The model of the dual porosity of Warren and Root is usedto study the processes. The pores (matrix) are represented by rectangular parallelepipeds in this model, which have high porosity and low permeability. The low permeable pores are separated by a system of natural fracture that has high permeability and low porosity. Filtration of liquid in the formation is carried out through a system of fracture, and the matrix is a reservoir that continuously feeds the network of fracture. The redistribution of fluid between the matrix and fracture depends on the shape and size of the matrix blocks, the ratio of pore permeability and fracture system. The pressure distribution in the "system of fracture-matrix" system is described by the piezoconductivity equations. An analytical solution for the Warren and Rootmodel cannot be obtained in a general form, approximate solutions exist for special cases. The results of the computer simulation and the numerical solution of this problem is presented in this paper. The problem was approximated using an implicit difference scheme. The matrix sweep method was used for the calculation. Model pressure recovery curves were obtained. Analysis showed that the specific conductivity coefficient depends on the size of the matrix blocks and possible to evaluate the process of manifestation of the effect of dual porosity.

The flowing process of multiphase flows in single-pore reservoirs is considered. The multiphase flow in the well is formed by mixing oil, formation water and petroleum gas. A mathematical model is presented that describes the process of mass transfer of a two-phase fluid in a single-pore reservoir during a hydrodynamic study on a well. Based on the model, a difference scheme is constructed. The scalar sweep method is used as a method for solving a system of linear algebraic equations. The analysis of the influence of various characteristics of productive formations, reflecting their influence on the pressure dynamics, was carried out. The optimal time for conducting a hydrodynamic study on a production well was calculated. Modeling was carried out in the developed compact software.