Development of mathematical methods for processing tracer studies of wells

. The use of tracer methods for studying oil wells makes it possible to determine the efficiency of oil field development. However, the reliability of the results of such studies largely depends on both the correctly selected indicators and the methods of interpretation. The currently developed theory of multiphase filtration and numerical analysis methods make it possible to use appropriate models to interpret the results of using various types of indicators. Of particular interest are mathematical models of processes occurring in oil-saturated rocks when using separable tracers.


Introduction
Using tracer methods for studying oil wells, they determine the directions of movement of formation fluids and water injected into the formation, the distribution of flows in the formations and between individual wells and the sources of their watering, hydrodynamic connections over the area and section of deposits, residual oil saturation, the efficiency of the oil displacement process, the degree of influence individual wells on it, the mode of their drainage and injection, etc. [1][2][3].
The method using indicators (tracers) is based on the injection of water with an indicator (or an aqueous solution of a reagent) and subsequent monitoring of its progress, which is carried out by periodically taking fluid samples from the mouths of control production wells.Determination of the concentration of indicators in the aqueous phase is carried out in laboratory conditions.
Stable indicators such as ammonium thiocyanate, potassium thiocyanate, uranine, fluorescein, eosin, urea, or others that have the appropriate properties necessary for studies to control the movement of injected water in oil fields can be used as indicators for the preparation of labeled liquids.
The presence of several indicators with the same hydrodynamic properties makes it possible to evaluate the influence of several injection wells (influence coefficient) on one of the surrounding observation production wells.In this case, different indicators are simultaneously pumped into each of these injection wells.Samples of produced products are periodically taken from the wellheads of production wells and a physical and chemical analysis of the formation water is carried out for the presence of each indicator.
Currently, new ultradisperse multicolor fluorescent indicators with a highly sensitive luminescent photometric method have been developed and widely tested for their quantitative determination in any environment, including formation waters.A set of studies in this case is described in [4].
Due to the growing interest in the quantitative assessment of the residual oil saturation parameter, the SWCTT method has recently become widespread [5].Some experts consider the SWCTT method to be the best choice due to its demonstrated accuracy and reasonable cost.
Currently, to determine the residual oil saturation, core analysis and well logging are carried out.But both methods give values that are average over very small volumes of the formation: core sampling does not take into account the anisotropy of the formation and cannot be sufficiently representative, since it cannot be sampled at a sufficient distance from well.Logging, like core testing, provides insight into the formation just a few centimeters from the wellbore.In this regard, the advantage of SWCTT studies is that they make it possible to analyze residual oil saturation in a fairly large volume around the well (the tracer front moves up to 10-50 meters from the wellbore in the radial direction) and in reservoir conditions, and not using models of reservoir fluid in surface laboratory conditions.The use of indicator methods in the development of oil and gas fields makes it possible to obtain objective quantitative information about the direction and speed of fluid movement in the studied formations, to evaluate the main filtration parameters and the presence of interlayer flows, and also to obtain reliable information about the technical condition of wells.And the use of technologies with separable tracers makes it possible to determine the residual oil saturation, the effectiveness of the development and application of oil recovery methods.
Depending on what information we need, several types of tracer tests can be used.The most common tracer studies:  Interwell Tracer Test (IWTT) -Interwell tracer test: To demonstrate the connection between production and injection wells. Single Well Chemical Tracer Test (SWCTT) -Single well chemical tracer test to determine residual oil saturation around the wellbore. Two-Well Tracer Test (TWTT) -Tracer test on two wells: to determine the residual oil saturation along the path of the tracer from one well to another.Initially, in order to monitor the pressure maintenance system, as well as enhanced oil recovery methods, only the IWTT method was used to study the direction of filtration flows between production and injection wells.Later, the use of SWCTT and TWTT also made it possible to determine the residual oil saturation around one or between two wells.
The tracers used in IWTT are water-soluble substances that are injected into the injection chamber.well and then monitored in production wells by collecting samples according to a predetermined schedule.Since there is no ideal tracer for monitoring the flow of fluids through oil reservoirs, tracer selection is always carried out through laboratory studies aimed at selecting the optimal tracer for each specific reservoir.[3] For TWTT, two tracers with different phase equilibrium coefficients are used.These two tracers, one soluble only in the aqueous phase and the other soluble in both the aqueous and oil phases, are injected into the formation through an injection well and withdrawn through a production well.The time difference between the arrival of the two tracers, due to chromatographic separation, is used to calculate the residual oil along the tracer's path between the two wells.
Tracers soluble in oil and water are characterized by phase equilibrium Kd.This coefficient is unique for each tracer and has a certain value only under the conditions of its determination in the laboratory.
In order for the use of a tracer to give good results, it must meet the following conditions:  Move at the speed of pumped water. Not present or present in very low concentrations in formation water. Be stable: formation fluids or bacteria should not decompose it. Do not adsorb on the rock. Detectable even in very small quantities. Be pumped and produced in safe conditions. Be cheap and easily accessible.
The separation tracer for SWCTT is selected depending on two main parameters: formation temperature and formation water salinity.These two parameters control the extent of the hydrolysis reaction that occurs during the shutdown period to produce the secondary tracer.Proper tracer selection using these two parameters is critical to ensure that after the well is shut in, enough primary tracer remains for detection during production.Hydrolysis of 10% of the primary tracer is generally considered a normal lower limit for the total test time: injection time, shutdown time, then tracer extraction time.The sensitivity of SWTT begins to decrease when 50% of the primary tracer is hydrolyzed during the test.
The peculiarity of the SWCTT method is that the study does not use a mixture of tracers, but one tracer (active or primary tracer), which during the study is hydrolyzed to form a secondary tracer (passive).
There are the following options for selecting tracers [5]: Injected tracer (active) -ethyl acetate (ethyl formate):  Concentration: 0.8% -1.2%. Phase equilibrium constant (oil/water): laboratory studies. Rate constant of the hydrolysis reaction: adjusting the GDM to the actual indicators of tracer production.The resulting tracer (passive) is ethanol:  Concentration: regulated by the holding time (the holding period is selected in such a way that no more than 10% -50% of the initial concentration of the injected tracer interacts, otherwise we will get too small an amount of the injected tracer in the extracted product in order to consider the research results representative). Phase equilibrium constant (oil/water): soluble only in water.

Results
In work [4], the results of the use of fluoprescent tracers (Figure 1) are given and a method of processing the determination of the concentration of indicators in formation water is considered.Figure 2 shows the results of microscopic examination of fluorescent tracers.The separable tracer method is based on the chromatographic separation of the mobile components of injected water with the addition of a primary tracer on the stationary phase in the rock, which in this case is residual oil.During the filtration of the solution, the components of the mobile phase (tracers) are distributed between the water and oil phases, while the tracers have different affinities for water, as a result of which the most hydrophilic component is more soluble in the water phase, and the most hydrophobic component is transferred to the oil phase to a greater extent.Since this process is dynamic, at each moment of time an equilibrium is established in each part of the system, which can be described by the phase equilibrium constant of the i-th tracer in the oil-water system.A hydrophilic tracer (does not pass into the oil phase) is called passive, and a hydrophobic tracer is called active (it exists in both oil and water phases).
A feature of the SWCTT method is that not a mixture of tracers is used for the study, but one tracer (active or primary tracer), which is hydrolyzed during the study to form a secondary tracer (passive).There are the following options for selecting tracers: Injected tracer (active) -ethyl acetate (ethyl formate).The resulting tracer (passive) is ethanol.
Step by step, the whole process is reduced to the following steps [6]:  Injection of formation water (to displace mobile oil beyond the expected radius of the study). Pumping the tracer into the well, pushing it with water. Waiting for the reaction for 1-5 days (depending on reservoir conditions) [7], this is necessary for partial hydrolysis of the primary tracer with the formation of a secondary tracer. Well development (liquid samples are taken from the well at a certain frequency).
To avoid difficulties when injecting tracers, it is necessary to analyze the water quality (especially suspended solids) and any precipitation trends [8][9].
As a result of determining the concentration of tracers in liquid samples, the dependences of the concentration of tracers on the time of study are obtained.In this case, oil saturation is a function of the time of registration of maximum tracer concentrations [10].
The distribution of ethyl formate and ethanol concentrations is shown on the example of a model that does not take into account the process of temperature change during the tracer test and is displayed in Figure 3.
The figures show the dynamics of changes in the concentration of the injected tracer (ethyl formate) at different stages of injection and production of tracers.Also, these figures show the view of the constructed radial model itself. The oil phase is stationary. The chemical reaction occurs only in the aqueous phase. Equilibrium is reached by mass transfer.
The so-called Black Oil model is considered, which describes the filtration of a threecomponent fluid in the form of the laws of conservation of the masses of the components: (1) Designations: j = w, o, g -water, and two hydrocarbon fractions, heavy and light, abbreviated as oil and gas; W, L, G -water, liquid and gaseous phases of the hydrocarbon mixture.Oil and gas do not mix with water, so w and W are synonyms.P α , S α -pore pressures and saturations (volume concentrations) of phase α; mporosity; ρ L,G -densities of the liquid and gaseous hydrocarbon phases as mixtures of two hydrocarbon components without taking into account porosity and water saturation (i.e.density per unit volume of the phase, these are the values that appear in Darcy's law when gravity is taken into account); ρ w is the density of water; Cjα are mass concentrations corresponding to the mass concentration of the hydrocarbon component j (j =o,g) in phase α; uα-phase speed α; q jM-functions of mass sources of fluid component j; P cow , P cgocapillary pressures.
Constitutive relationships: Here: is the absolute permeability, which is, generally speaking, a complete symmetric tensor; µα ,k kr -viscosity and relative permeability of phase α.
The system of mass balance equations is supplemented by an equation describing the movement of tracers.The evolution of the mass concentration of tracer C moving along with water is described by the transport equation: (3) Where m is porosity,  w is density, S w is saturation (volume concentration), w is flow, Q w is source, , is mass concentration of ethyl formate and ethanol, respectively.w и Q w are obtained from the solution of the main system and do not depend on C.
In order to obtain a universal method for interpreting tracer tests, it is necessary to abandon the analysis of the results by the analytical method and interpret the results using hydrodynamic modeling, according to the mathematical model.
In this work, the model was adapted to the experimental results.Model adaptation refers to the process of minimizing the difference between the curves obtained in the course of solving the model and the curves constructed from experimental data.Adaptation occurs by varying various parameters in the model, which, in turn, are called adaptation tools.
The specifics of adapting the model to the actual data:  A radial model is built (the radius of the model is calculated from the volumes of tracer injection). If tracers of the material balance are present, then first adaptation is carried out on the dynamics of their concentrations.In this case, the adaptation tools are: grid (cell sizes, number of layers), dispersion and diffusion parameters, selection of inhomogeneity.
In our case, the variable are the residual oil saturation and the rate coefficient of the chemical reaction occurring at the holding stage.
Determination of residual oil saturation carried out by adapting the model to experimental data.By selecting the rate coefficient of the hydrolysis reaction and the value of the residual oil saturation during adaptation, the relative error between the values of tracer concentrations calculated in the model and constructed based on the experimental results is minimized.
Figure 4 displays all the curves that were plotted for different values of the reaction rate coefficient and residual oil saturation, and one of the solutions is optimal (marked in the figure with a bold curve).

Conclusion
The scheme for conducting the SWCTT study, as well as the physical and chemical aspects of this study, is described.Criteria were given to select an active tracer.An analysis of the existing methods for interpreting field test results was also carried out, during which the shortcomings of the analytical method for processing the results were established and it was concluded that it was necessary to create numerical interpretation methods.
Further, the implementation of the SWCTT method in a real field was considered in detail.The main result of this work is the creation of a mathematical model of the SWCTT process, which allows a fairly accurate interpretation of real field studies.
As a result of building hydrodynamic models and selecting the coefficient of residual oil saturation in them to approximate their results to real field ones carried out on wells, the optimal values of residual oil saturation and the rate coefficient of the hydrolysis chemical reaction were obtained.The values of residual oil saturation are the main results of the interpretation, which allow us to evaluate the displacement efficiency in the implementation of the reservoir pressure maintenance system.Thus, a technique for interpreting the separable tracer method for actual results has been developed.

Fig. 3 .
Fig. 3.The concentration of ethyl formate at the initial time and at the time of well shutdown.Assumptions made when constructing the SWCTT mathematical model:  Liquids are incompressible.

Fig. 4 .
Fig.4.Adaptation of the hydrodynamic model by experimental points.