Construction of a numerical model and algorithm for solving two-dimensional problems of filtration of multicomponent liquids, taking into account the moving "oil-water" interface

. The paper considers a two-dimensional mathematical model of the filtration of a viscous, incompressible fluid in a deformable porous medium. The article describes a mathematical model of the problem at the “oil - water” interface with a system of parabolic differential equations. The problem posed is solved numerically using the differential-difference method based on the longitudinal-transverse scheme and the differential sweep method to determine the unknown boundary. To obtain a differential-difference problem, an algorithmic representation of a hidden scheme of alternating directions (longitudinal-transverse scheme) is used. The resulting system of differential-difference equations with initial conditions is known for each time layer along straight lines along the x axis, and then along each y axis it is solved by a differential sweep along a straight line, where the values just found correspond to time, i.e., the layer is taken as initial conditions. By approximating the differential equations in time, the position of the interface is determined for each time layer.


Introduction
Mathematical and software tools have been developed to solve the problem of filtration with a moving "oil-water" interface and conduct computational experiments that allow visualizing the results of numerical calculations, the main indicators of oil field development, in graphical and animated form.
At the present stage of the development of the oil industry, mathematical modeling is widely used in the design and development of oil fields.It is used to solve the problems of forecasting, monitoring, and managing the process of developing layers.An important area of application of mathematical modeling is the solution of research problems in the theory of filtration, for example, the creation of models for the filtration processes of two-phase fluids in inhomogeneous and porous media, the study of the effect of mechanisms on the reservoir, and the modeling of new technologies.At the same time, in order to increase the efficiency of multi-year deposits, mathematical and numerical modeling of the processes of Numerical modeling can be carried out with the help of through-counting methods, which require the development of new highly efficient computational algorithms and programs.
The process of joint filtration of two or more immiscible liquids in a porous medium is very complex and is characterized by the following features: -the coefficients of the equation depend on time and spatial coordinates.
-the pressure values at the interface between two phases are not known in advance.
-the position of the interface between two phases is determined in the process of solution.

Problem statement
Oil production takes place in the most difficult conditions, and the efficiency of field operations depends on the degree of adequacy of the design and management decisions made.The adequacy of the decisions made depends on the degree of compliance of mathematical models, computational algorithms, software, and tools for analyzing and predicting technological indicators of oil and gas field development with modern requirements.
During the development of oil fields in a water-driven regime, the advancement of contour or bottom waters is observed.Mathematically, such processes are formulated as problems with a moving oil-water interface.
-the presented mathematical model is based on the following assumptions: -the fluids under consideration are non-displaceable.
-the movement of liquids in a porous medium is rectilinear, and in each area of filtration, it obeys the linear Darcy law.
-reservoir conductivity coefficients in the vertical direction are identical; -the properties of liquids in both phases remain unchanged over time.
Based on these assumptions, we will consider a problem with a moving oil-water interface.Here, it is assumed that the reservoir is characterized by a constant thickness h, length L, porosity m, and initial formation pressure н.
Under these assumptions, the mathematical model of the problem with a moving oilwater interface is described by a system of differential equations of the parabolic type: , y , 1 x, y .
The system of equations ( 1) is integrated under the following initial, boundary and internal conditions: , 0, , , , , , , q Hi q -flow rate of the q i th oil well; q Bi q -flow rate of the q i th water well; , q q N M -the number of wells, respectively, in the field of oil and water content; l -velocity vector directed along the internal normal; a -coefficient of oil saturation; We can make the variables of the above equation (1-8) dimensionless.To do this, we introduce the notation as follows.

 
Here are some characteristic pressure values; x P -some characteristic values of reservoir permeability; x k -some characteristic values of reservoir permeability; L - characteristic length.
After passing to dimensionless variables in system (1) with the corresponding boundary conditions (2)-(8), the problem is solved numerically using the longitudinal-transverse scheme for the differential-difference problem and the differential sweep method to determine the unknown boundary.
The filtration area 12 GG  is covered by the grid area formed by a regular grid of coordinate lines: h -grid step corresponding to the axes x and y.

The method of solving the problem
To obtain a differential-difference problem, we use the algorithmic idea of an implicit scheme of alternating directions (longitudinal-transverse scheme).To obtain a differentialdifference problem, we use the algorithmic idea of an implicit scheme of alternating directions (longitudinal-transverse scheme).The transition from layer r to layer 1 r  is made in two stages with step 0.5   .Then solution (1) is found by successive solution of the system of equations: .
where the left and right sweep coefficients ,, j j j u x v x w x are found as solutions to the following Cauchy problems: , 0 0; To solve the Cauchy problems (14), the initial conditions are determined from the boundary conditions (4).Solutions of the third and fourth differential-difference equations (9) on the r+0.5 t and r+1 time with boundary conditions (2)-( 7) are defined similarly.At the same time, taking into account the Cauchy problems (14) for the third and fourth equations are also solved.
The resulting system of differential-difference equations is solved by differential sweep along each of the straight lines Approximating differential equations (7) with respect to  , we obtain a formula for refining the position of the interface at each time layer: where is the velocity vector , l ij directed along the internal normal at the interface; , ˆij l -velocity vector directed along the internal normal at the interface in the previous time layer;  is the angle between the normal and the ox axis;  is the angle between the normal and the axis oy: When specifying the position of the interface, the values , ˆij l are taken from the initial condition (8).
On the border of the filtration area, one of the following conditions can be satisfied: the first kind; second kind and mixed condition.
If the pressure values are known on the boundary of the filtration area, i.e., the first boundary condition is given, then the initial conditions of the Cauchy problem take the following form, respectively, on the left and right parts of the outer boundary  : is given on the outer boundary of the filtration area, i.e., the second boundary condition, then the initial conditions of the Cauchy problem take the following form, respectively, on the left and right parts of the boundary: , then the boundary is impenetrable.
In the case when a condition of the 1st kind is specified on one part of the boundaries of the filtration area, and a condition of the 2nd kind is specified on the other part, i.e. the pressure is given on one part, and the flow on the other part, the initial conditions of the Cauchy problem are determined similarly.
On the internal boundaries of a multiply connected region, the conditions of impermeability and continuity of pressure are specified.These conditions are met automatically at the transition of the interface between two phases when applying the differential sweep method.In the process of successively finding the values ui(x), vi(x), wi(x) during the transition from one phase to another, the previous values of these functions are used as initial conditions.The numerical integration of the Cauchy problem is carried out by the Runge-Kutta method using the procedure for normalizing the sweep coefficients and coefficients of this method.At each iterative step, when calculating the vector ) is substituted in the right side of the system of equations.The normalization procedure can be omitted if the chosen method consistently solves the Cauchy problem.
It should be noted that in the numerical simulation of such processes in a multiply connected domain, the use of the differential-difference method has the following advantages: -there is no need to satisfy special ratios during the transition from one phase to another since the conditions of conjugation are performed automatically; -the Cauchy problems to be solved with respect to the sweep coefficients can be integrated with the desired accuracy using the well-known Runge-Kutta or Kutta-Meyerson methods, for which there is appropriate software; -in the numerical integration of Cauchy problems by the Runge-Kutta method, the procedures for normalizing the fitting coefficients and coefficients of the Runge-Kutta method are used.At the same time, in each iterative step, when calculating the vector ) is substituted in the right side i U of the system of equations.This ensures the stability of the solution of the Cauchy problem; -allows for a thorough account in areas with internal features;; -allows for an absolutely stable computational scheme for the system as a whole.
-the developed computational algorithm is easily implemented on a computer.

Algorithm for solving the problem
The numerical implementation of a discrete model on a computer is built according to the following algorithm (Fig. 1).To solve the problem of filtration with a moving oil-water interface and conduct computational experiments, mathematical and software tools have been developed that allow the visualization of the numerical results of the calculation, the main indicators of the development of oil fields in graphical and animated form.

Results of numerical solution
Consider non-stationary oil-water filtration in a porous medium.The reservoir is developed in a water-driven mode by a system of two well banks with predetermined constant flow rates and the initial reservoir pressure is maintained using four injection wells in the aquifer. Calculations

 
In Tables 1 and 2 show the calculated values of the dimensionless pressure at the interface and in the supply circuit, as well as the positions of the interface at different times.The results obtained show that the difference in oil viscosities significantly affects both the dynamics of the pressure distribution in the reservoir and the nature of the progression of the water-bearing contour.According to these figures, the initial reservoir pressure is almost constant in a certain interval, then drops sharply and reaches its minimum value at the point where the production well is located.With an increase in the reservoir operation time (i.e. with constant oil production from production wells), the pressure drop in the right part of the oil content increases (this part is shown in black on the graph).In the left part of the formation, where three injection wells are located, the pressure increases with time.
Further in Fig. 2-4 shows the options when the viscosity coefficient of oil took the values  Н = 4, 6  8 сПз.Due to the symmetrical arrangement of wells and the distribution of pressure in the reservoir, the results of calculations in the section 0.5 y  are analyzed.
It can be seen from these figures that with a higher viscosity of oil in an oil well, the pressure gradually drops, and with a lower viscosity, the pressure drop stops over time, i.e. the process stabilizes.It depends on the operation of the injection well.
It can be seen from these figures that with a higher viscosity of oil in an oil well, the pressure gradually drops, and with a lower viscosity, the pressure drop stops over time, i.e. the process stabilizes.It depends on the operation of the injection well.
Wells: well 1 -operational; well 2 -injection   All graphs show that the programs give the expected patterns of pressure distribution, which are achieved in the third year of oil field development.Here, also in the center near the injection well, the pressure gradually increases.

Conclusion
According to computational experiments, it was found that the increase in oil pressure caused by displacement and acceleration of production processes in the area of operation of injection wells for a long time (10-20 years) occurs at high values of the viscosity coefficient of the liquid and at low values of oil viscosity.Along the length of the filtration field, the pressure in the layers begins to be provided continuously over time, while the permeability coefficients of the layers are equal to each other and vary within 0.02 d.We can observe in the above-calculated studies that the parallel increase in pressure in the water layer of the liquid depends on the permeability coefficients and the viscosity of the liquid on the wall.All graphs show that the programs give the expected pressure distribution patterns that are achieved in the third year of oil field development.Here, also in the center near the injection well, the pressure is gradually increasing.
Computer modeling and computational experiments have made it possible to determine the main indicators of oil field development in reservoirs at various reservoir parameter settings, and the numerical results obtained are useful for analyzing the development of oil fields in layers of porous media.Thus, the developed model and algorithm, as well as software for calculating the main indicators of oil field development in layers of porous media, can be used in the calculation and design as well as in the development of oil and gas fields.
Web of Conferences 402, 14040 (2023) https://doi.org/10.1051/e3sconf/number of nodes on the line j y and i x respectively; The resulting system of differential-difference equations (9) is solved by differential sweep along each of the straight lines i x with initial conditions known at k   , and then along each of the straight lines j y , where the values just found corresponding to the 0.5 r  th layer are taken as the initial conditions.According to the differential sweep method, the solutions of differential-difference equations (9) on the 0.5 r  th and 1 r  th time layers with boundary conditions (2) - (7) are determined by the formulas.E3S Web of Conferences 402, 14040 (2023) https://doi.org/10.1051/e3sconf/202340214040TransSiberia 2023 as the initial conditions.

Fig. 1 .
Fig. 1.Algorithm for solving the boundary value problem of filtration with a moving oil-water interface in a porous medium.

Fig. 2 .
Fig. 2. Dynamics of pressure changes in the reservoir and wells at.

Fig. 3 .
Fig. 3. Dynamics of pressure changes in the reservoir and wells at.

Table 1 .
Dimensionless values of the pressure at the injection well and the position of the interface at 4

Table 2 .
Dimensionless values of pressure at the injection well and the position of the interface at 8