Mathematical model as a means of predicting effectiveness of well intervention for near-wellbore space

The conclusion about the appropriateness of using this or that type of well intervention equipment should be based not only on the technological and economic efficiency but also on a mathematical model. The existing geological and field restrictions for the selection of impact objects for each type of well intervention should not also be overlooked due to the fact that in this case the use of a mathematical model is more appropriate. A fundamentally new method of multifunctional regressive mathematical modeling for evaluating the technological efficiency of processing the bottom-hole zone with an increase in the oil recovery coefficient is described. A new algorithm for determining the dependence of the technology efficiency is proposed, the obtained multivariate regression model is checked for statistical reliability and significance, the forecast for the increase in oil recovery coefficient is calculated


Introduction
The last stage of an oil and gas field development is characterized by the importance of searching for the most effective technologies aimed to increase oil recovery and well injection capacity with simultaneous energy savings in the pressure maintenance system. The effective application of one technology in a particular area does not always guarantee its success in another one due to the heterogeneity of reservoir layers. Therefore, it is necessary to treat each site of the field individually, taking into account the analysis of the results concerning well intervention on the basis of the methods of correlation and statistical analysis of the geological and physical as well as geological-field characteristics of the development targets, taking into account the level of economic profitability of the well stock. In this connection, there arose the need to develop new criteria for the efficiency of technologies aimed to increase the oil recovery coefficient on the basis of a mathematical apparatus and stochastic approach. In the works [1,2], mathematical methods for defining oil recovery coefficient were first proposed. A formula denoting a simple oil recovery coefficient by factor multipliers has been presented: (1) where η1 is displacement efficiency, η2 is volumetric displacement efficiency. It is known that the main criterion for changing the oil recovery coefficient occurs under the influence of three main geological and physical factors being macro-and microinhomogeneities of a layer, viscous forces, and surface tension forces.  [3] in assessing a deposits productivity factor at the stage of the initial design documents development and modeling oil recovery processes in the fields that have been in operation for a long time.
A methodology for assessing the potential effectiveness of new technologies in the development of the oil zone is proposed in [4]. In [5,6], the role of a multi-parameter analysis of the technologies effectiveness was considered. In studies [6], the use of the Nernst -Planck -Poisson equation is proposed. It is an applied equation in terms of its applicability to the description of phenomena in various media. Algorithms for data mining and machine data processing that provide the effectiveness of a modeling alternative under the conditions when the basic physical relationships between system variables are very complex and non-linear have been proposed [7].
Researchers [8] propose a mathematical model that enables to accurately assess the effectiveness of hydraulic fracturing. Linear regression analysis is suggested for unifying the obtained research results [9][10][11][12]].

The purpose and objectives of the study
The aim of the study was to determine a unified regressional dependence of well intervention effectiveness aimed to increase the oil recovery coefficient on the parameters and indicators of the development status. To achieve this goal, the following tasks were set: 1. To analyse the experience of using statistical methods for determining the effectiveness of the recovery factor and the technologies used. 2. To derive the regression equation. 3. To study the effectiveness of the derived regression equation for small samples.

Materials and methods
A sufficiently large sample of measurements is required for obtaining reliable results, which is not always possible in real production conditions. The path to constructing a regressive equation in the form of a polynomial of the second degree was chosen as the main method for solving this problem.
For this purpose, the geological and production results of the use of thermal oil and gas treatment (thermogas barometric treatment) at 13 production wells of the Pashensky field were systematized. The results are presented in the form of a summary table 1.
It is known that the most common way of processing experimental data is the method of regression analysis, which enables to obtain a mathematical description of the technological process on the basis of experimental data in the form of an algebraic power polynomial. It is known that with an increase in the number of its members, the reliability of the mathematical description of the technological process increases.
Processing experimental data has shown that in most cases the results of an experiment in the form of a tabular function are reflected with a sufficient approximation by a full cubic polynomial, so that the number of the polynomial members can be reduced without significant loss of calculation accuracy.
Of course, in this case, there arises the question of the sample presentability and the results reliability, which in turn leads to the formal use of statistical methods of analysis in solving the problem under study. Despite the low reliability of the formally applied probabilistic-statistical methods, it can be stated that they are widely used to predict the oil recovery coefficient in the absence of multidimensional filtration models of liquid hydrocarbon deposits. In table 1, the following notation is used: y is an increase in flow rate (t per day), x1 is a reservoir pressure (MPa), x2 is an oil saturation coefficient (unit), x3 is a permeability coefficient (μm2), x4 is oilbearing thickness (m), x5 is a porosity coefficient (d.ed), x6 is proppant mass in the layer (t), x7 is a final pressure (MPa), x8 is a water cut coefficient (%), x9 is an initial oil recovery coefficient (d.ed), x10 is a current oil recovery coefficient (unit fraction), x11 is proppant concentration (kg/m 3 ). For a table function  (table 1), we compose a polynomial of degree 2 having the following form: Let us study the correlation between y and x1, x2, x3, ..., x6. We obtain the following results (table 2). It follows from table 2 that a close relationship between these components was not detected, with the exception of x1, x3, x9, x10.    Using the Data Analysis function of the MS Excel program we perform a correlation analysis of the data in Table 3 for the presence of indirect relationships.
Since there is no close relationship between the components, all of them can be included in the equation. We carry out a correlation analysis for the presence of indirect connections between x2 • x3, ..., x2 • x11 based on the obtained results, shown in table 4  Observing the tight connection between the variables, excluding those with a weak connection, we come to the equation in the following primitive form: ... ...  According to the graph (figure), the difference between the actual and calculated results is minimal. If we turn to the graph, we will determine that the deviations from the actual and the forecast results are very small.

Conclusion
The authors have proposed a methodology for assessing the forecast of technological efficiency of measures aimed to increase oil recovery from productive reservoirs of oil fields. This methodology is based on the analysis of the results of experimental and industrial testing of a specific EOR technique (increasing oil recovery of productive reservoirs). According to the results obtained, a correlation analysis was carried out in order to assess the magnitude and range of diagnostic criteria that ensure the expected technological effect. The authors have been working for many years on the use of methods of