A multiparameter mathematical model for the problem of nonlinear filtration of fluids in two-layer media

In the realm of fluid dynamics and porous media, the nonlinear filtration of fluids in two-layer media presents a complex and intriguing challenge that demands a comprehensive mathematical framework for accurate representation and analysis. This paper introduces a multiparameter mathematical model designed to elucidate the intricacies of fluid filtration in heterogeneous environments, specifically focusing on two-layer media. The authors research the construction of a multiparameter mathematical model and numerical solution of nonlinear problems of the theory of filtration of unstructured and structured fluids in hydrodynamically connected multilayer formations with different characteristics. The developed models and solution algorithms were tested on hypothetical data.


Introduction
Understanding fluid flow in porous media is pivotal in numerous industrial, environmental, and geological applications.However, existing models often fall short in capturing the nonlinearity inherent in the filtration process within stratified media.To address this gap, our proposed mathematical model integrates multiple parameters to describe the nuanced interactions between the fluid and the porous layers.
Mathematical modeling of the problem of fluid filtration in single-layer and multilayer porous media for Newtonian and non-Newtonian fluids is considered in [1][2][3].There are works [4][5] where individual problems of the filtration process of gas and oil in single-layer and multi-layer deformable porous media are studied with an individual interpretation of physical and mechanical processes in a porous medium, as well as approaches to solving them.In [6][7] the process of constructing mathematical models was improved by optimizing the construction of multiparameter mathematical models.
By introducing parameterized functions into the mathematical model as well as the boundary conditions, a multi-parameter model was built that managed to combine all known types of filtration laws and the corresponding mathematical models.The constructed multiparameter model made it possible to develop parameterized multifunctional computational algorithms and the corresponding mathematical software.It made it possible to classify models and solution algorithms, and researchers were able to obtain the boundary value problem they needed from a multiparameter model.
The multi-parameter mathematical model allows the use of all stages of computing technology.In this work, an attempt is made to apply this technology to a two-layer medium.

Methods
Let the filtration region under study (Ω) consist of two ( 1 and  2 ) subregions, where  1 is well permeable (horizontal characteristics of the formation and fluid prevail over vertical ones and fluid movements occur in horizontal directions), and the other  2 is poorly permeable (vertical characteristics prevail over horizontal and fluid movement occurs vertically).
1 is saturated with anomalously structured fluids, from  2 there is a flow of fluid to the lower region  1 .
Problem (1) -(14) is nonlinear and numerical methods are used to solve it: the iteration method, the method of straight lines and the difference version of the flow sweep [11].
Problem (1)-( 14) includes 13 different laws and calculation algorithms are based on the fact that to which group does this or that mathematical model belong, since the rules of computational processes for them are different and have their own specifics, which made it possible to group these laws , in three groups.The first group includes laws: I, VII, VIII, and they are characterized by the existence of disturbed and undisturbed areas.The second group includes laws: II, VI, IX, XII, XIII, where the disturbance covers the entire filtration area, with unknown moving boundaries between different filtration zones.The third group contains laws: III, IV, V, X, XI.For them there are no unknown boundaries of disturbances and each of them contains an individual characteristic nonlinearity.

Results and discussion
To illustrate the capabilities of the developed computational algorithms, the following test data was used.Let in point  0 = 0 there is a well with a flow rate.
Table 2, for Law VI, shows the values of the pressure function for different t and x and the corresponding positions of the disturbance boundary.To establish the speed of propagation of disturbance boundaries in the vicinity of the source, the same problem was solved for  0 = 0, and the well was located at point  1 = 0,5; with the corresponding flow rate  1 = 0,1718, the initial disturbance boundary  1 2 ).Fragments of partial results on the dynamics of advancement of the left and right borders are given in Table 3.  Table 5 shows the results of calculating the value of the function in the left part of the formation when the source is located at point  = 0,5, as well as  = 10 −2 .
The calculation results allow us to conclude that the function change curve for laws III, IV, IX lies between laws V and I, and in this case the minimum pressure will be at the bottom of the source.

Conclusion
By introducing this multiparameter mathematical model, we aim to contribute to the advancement of the field of nonlinear filtration, offering a more nuanced and realistic portrayal of fluid dynamics in twolayer media.This research has the potential to inform decision-making processes in industries ranging from environmental engineering to oil and gas exploration, providing valuable insights for optimizing processes and mitigating potential challenges in fluid filtration applications.
It should be noted that the constructed multi-parameter model of a relatively two-layer reservoir reflects all existing laws and mathematical models of fluid filtration and is a compact and also implementable form of representation of mathematical models.The developed model and corresponding computational algorithms can be used in calculating the technical indicators of fluid development in multilayer porous media.
We can conclude that when carrying out an iteration to determine the position of the left and right boundaries of perturbations of a point source for the problem of the law with an initial gradient (law I), you can use a two-way iteration or the method of "shuttle" iterations which allows rapid convergence when determining the position boundaries, which makes it possible to clarify the values of fluid flow from area  2 to area  1 .

Table 1 .
The value accepted by the parameters when choosing the filtration law of the mathematical model.

Table 2 .
Dynamics of pressure changes and disturbance boundaries under a curvilinear filtration law.

Table 3 .
Changing the boundaries of disturbances over time for the central well according to Law I.

Table 4
shows the average values of the function (, ) for separately selected filtration laws in the integral sense.

Table 4 .
Average pressure values in the vicinity of the disturbance source for two values of .

Table 5 .
Pressure values on the left side of the filtration area, where the source is located in the middle part of the formation for the five filtration laws.

Table 6 .
Dynamics of the integral change in the pressure function in the area  1 as well as the fraction of flowing fluids for various values .

Table 7 .
The value of the flow between layers.