Abstract
The object of research is a numerical simulation of the process of two-dimensional two-phase filtration of viscoplastic oil and water, taking into account the gravitational forces, some properties of liquids, as well as relative phase permeabilities and capillary forces.As is known, the problems of multiphase filtration have specific features. Therefore, there is a need to develop difference schemes in adaptive grids that reduce the artificial viscosity and oscillation of the numerical solution. They also make it possible to obtain acceptable results with a small number of nodes in the computational grid.To take into account the singularities of the solution, a difference-iteration method is used in moving grids. Based on the computational experiment, the influence of the initial pressure gradient and gravity on the displacement process is investigated.Economical difference schemes that combine the advantages of explicit and implicit schemes are constructed and make it possible to reduce the two-dimensional problem to a chain of one-dimensional problems. A difference-iterative method is also proposed in moving grids for solving two-dimensional (axisymmetric) non-stationary filtration problems of anomalous liquids, by means of which an iterative process is constructed to find the distribution of water saturation.The carried out calculations to determine the influence of gravity on the displacement process have shown that at z=0, even at low productive-bed thicknesses, gravitational forces influence the displacement process. And over time this influence increases: if at the time t=0.08 on the circuit the difference of water saturation was 0.0077; at t=0.24–0.0122, then at t=1.04 it becomes equal to 0.0292.It is shown that when modeling the process without taking gravity into account it is expedient to simplify the geometry of the filtration region, i. e., to consider a plane-radial flow in view of the considerable simplicity of the calculations.The developed algorithms can be used for hydro-gas dynamic calculations related to the development and operation of oil fields containing anomalous oil.
Highlights
Mathematical modeling of oil production processes, as a rule, reduces to solving boundary value problems for systems of nonlinear partial differential equations
Analytical solutions can be obtained by sub stantially simplifying the models of real processes, when most of the main parameters are not taken into account, for example, the inhomogeneity of the seams, the nonstationarity of the operating modes, the compressibility of phases, the complexity of the geometry of the filtration region, etc
Calculations show that in the two-dimensional problem the water saturation value on the injection well taking into account G1 is 0.3520, and at the radial filtration it is 0.3528. They differ insignificantly and at the same time, in both cases, at this moment the water advances to the same distance – 50 m. It follows that in modeling the process without taking gravity into account, it is expedient to simplify the geometry of the filtration region, i. e. consider the plane-radial flow, in view of the considerable simplicity of the calculations
Summary
Mathematical modeling of oil production processes, as a rule, reduces to solving boundary value problems for systems of nonlinear partial differential equations. Analytical solutions can be obtained by sub stantially simplifying the models of real processes, when most of the main parameters are not taken into account, for example, the inhomogeneity of the seams, the nonstationarity of the operating modes, the compressibility of phases, the complexity of the geometry of the filtration region, etc. Such solutions are undoubtedly theoretical and methodical, but their practical significance is significantly limited. It is possible to obtain qualitatively and quantitatively acceptable results in the entire region with a small number of nodes in the computational grid, excluding zones where there are singularities of the solution, for example, zones of large gradients
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