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** N. N. Smirnova ^{1*} and E. A. Izotov^{2}**

^{1}Department of General and Technical Physics, Faculty of Fundamental and Humanitarian Disciplines, Saint- Petersburg Mining University, St. Petersburg, Russia

^{2}Department of Physical Education, Faculty of Fundamental and Humanitarian Disciplines, Saint-Petersburg Mining University, St. Petersburg, Russia

**Received date: ** 06 May, 2017; **Accepted date:** 08 May, 2017

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Journal of Industrial Pollution Control

The problem is now becoming ever more relevant due to a wide variety of practical applications of filtration heat transfer to the solution of the problems of forecasting and regulating the thermal regime in geotechnological systems. In the development of high-viscosity oil fields, thermodynamic effects on the oil reservoir are particularly promising, one of which is the injection of hot heat carriers into the oil stratum. The article deals with the analysis of the main components of the reducing temperature potential process of the coolant under thermal action on a multi-layered oil reservoir. It is noted that not all heat entering the reservoir is useful. The heat spent on warming up the solid phase of all the structural elements of the inhomogeneous deposit should be attributed to losses. Therefore, the underground part of the geotechnological complex with the oil deposit was chosen for the study. The model problem of heat transfer in an inhomogeneous oil reservoir is considered with the aim of obtaining a simple analytical solution for determining the time for heating the reservoir with residual oil to a predetermined temperature allowing it to begin to be displaced. Solving the conjugate problem of non-stationary heat transfer under the conditions of a multilayer filtration scheme, the equivalent heat equation was used, which has a wide application in solving a number of problems of geothermal thermal physics. The implementation of the method made it possible to obtain analytical dependencies for analysing the temperature fields of oil deposits with a multilayered heterogeneous structure. The results presented in the form of graphical dependencies in dimensionless variables make it possible to use them for various initial parameters of the problem. Jr. of Industrial Pollution Control 33(1)(2017) pp 950-958 www.icontrolpollution.com Research *

Heat losses, Geotechnological systems, Oil reservoir, Heat transfer, Heterogeneous structure

Among the topical scientific areas are downhole methods of fossil minerals mining. These include the methods of thermodynamic impact on the oil reservoir.

This method of influencing oil reservoirs allows to significantly increase their oil recovery. Oil reserves of high-viscosity, paraffinic and resinous oil deposits, not extracted by industrial development methods, reach on average 50% to 75% of the initial geological reserves of oil in the bowels. In Russia, approximately 22% of oil production is carried out due to thermal methods of increasing oil recovery. To resume development of oil deposits, where oil no longer has fluidity, thermal methods for increasing oil recovery are particularly promising.

One of the types of thermodynamic effects on the oil reservoir is the injection of hot heat carriers into the productive strata. Steam, hot water, polyacrylamide solutions (PAA), etc. are used as heat carriers. Stimulating thermal methods of intensifying oil production were used, for example, at the Yuzhno- Sakhalinsk heavy oil field (Sakhalin.info, 2013), at the Usinskoye field by pumping hot polyacrylamide solutions into fractured reservoirs with high viscosity oil, at the Yaregsky deposit using the steam injection method of extraction (Information Agency "Devon", 2017).

In particular, the favorable geothermal conditions of the Achi-Su area (Northern Dagestan) served as a pretext for the proposal of the Mining University (former St. Petersburg State Mining Institute SPGGI) (Dyadkin, 1987) to create an experimental geothermal circulation system with the possibility usage of geothermal subsoil warmth to resume the development of a multi-layer oil reservoir with highviscosity oil from the previously exploited oil field in the same area.

Evaluation of the introducing innovative feasibility or improving the efficiency of existing energy technology solutions requires the development of methods for calculating processes related to the transfer of energy in terrestrial and underground conditions, characterized by a variety of technical performance and natural structures.

The efficiency of the heat process depends on the value of the heat used in the formation, that is, the heat used directly to reduce the viscosity of the oil. It is known that the share of all types of heat loss inherent in this geotechnological method can on average be 40% to 60% of the heat content in the energy source used.

It is necessary to know the heat loss in terrestrial communications and wells in order to determine the productivity of used heat, as well as the losses associated with non-stationary heat exchange processes in the underground part of the technological complex. Therefore, an integral part of the system approach to solving practical problems of optimization and calculation of energy and process parameters by economic criteria are the analysis of the temperature regime of the entire geotechnological complex and accounting for all types of heat loss.

Various aspects of the thermal regime of wells and oil layers have been studied in connection with the development and implementation of methods for intensifying oil production by a wide range of specialists (Lauwerier, 1955; Ramey, 1962; Sheinman, *et al*., 1969; Rubinshtein, 1972; Nigmatulin, 1978; Jamalov, 1978; Palacio, 1990; Zheltov, *et al*., 1997; Dyadkin, Gendler and Smirnova, 1993; Mirzadjanzadeh, Hasanov and Bakhtizin, 2004; Fedorov and Shevelev, 2004; 2005; Filippov and Akhmetova, 2011; Ruzin, *et al*., 2013; Ahmedov, Akhmedova and Akhmedova, 2016 and others).

The main problem in the study of the thermal regime of thermodynamic technological systems is the influence on the oil reservoir lies in the multifactor and extremely complex functional connection between the conditions and the results of the operation of geotechnological systems. And as a consequence, the existence of numerous physical models, calculation schemes and methods for solving such problems in a variety of settings. There is a need to choose the optimal complexity of setting the problem, depending on the purpose of the research. The criterion of choice can be the achievement of the required accuracy of establishing its main technological parameters.

In most cases, the description of the temperature field of complex oil deposits is based on the concept of interpenetrating media exchanging mass and energy with each other (Rakhmatulin, 1956; Nigmatulin, 1978; Egorov and Salamatin, 1984). A model of a natural permeable reservoir imitating a homogeneous medium in which the real characteristics of the medium are replaced by equivalent ones with effective properties were introduced into the practice of calculating thermal processes during filtration by Rubinshtein, 1972. Thermal resistance of solid particles of a permeable reservoir is considered negligible here. Another approach, with possible consideration of the thermal resistance in models of a heterogeneous medium, further complicates the formulation of the problem.

The application of numerical methods in these cases makes it possible to perform calculations of heat transfer processes taking into account many real operating conditions and parameters. With such "exact" statements using the great possibilities of computer technology, mathematical modeling is certainly relevant.

However, the excessive complexity of the model, as well as the uncertainty of the initial information about real environments, limits the value of the computational experiment and calls into question the results obtained.

On the other hand, a significant simplification of the initial system of equations and approximate methods of solutions lead to an inevitable deviation from reality and require a serious physical justification for all the conventions for setting the problem and for the experimental verification of the obtained results.

Despite this, the use of asymptotic approximations, approximate methods of solving, purposeful experiments and the creation of physically grounded models, ensure the obtaining of simple analytical results. With their help, you can make operational calculations for the early stages of design.

Therefore, with the minimal amount and degree of reliability of the initial information, the use of analytical methods in engineering calculations remains relevant.

The amount of heat directly entering the oil reservoir is one of the most important parameters of the heat treatment process. Therefore, for its definition in the assessment methodology, thermal losses in terrestrial communications, in wells and losses through the sole and roof to surrounding rocks.

The efficiency of the process depends on the ratio of the amount of heat that is useful in the formation to the amount originally spent on its production. In determining the coefficient of thermal efficiency of the impact on the reservoir under useful heat, the literature usually refers to the heat accumulated in the reservoir volume.

In fact, it is more correct to include heat losses spent on warming up the solid phase (particles) of the permeable reservoir and impermeable layers. A productivity should be considered only the heat expended on the heating of the oil contained in it.

The relative part of the energy received in the reservoir is determined as follows:

Types of heat loss in fractions relative to the received amount of heat (for example, from the steam generator) are indicated in **Fig. 1**.

In **Fig. 1**, the fraction of the thermal power supplied to the reservoir, necessary for its warming up, etc. structural elements (particles) of the permeable collector ηp and for heating impermeable layers η_{L}.

Typically, the heat loss in the steam generator (or other energy source) is constant. Losses in the pipeline are also almost constant. Estimation of their magnitude is carried out according to known dependences for stationary heat transfer. Kutateladze, 1990 and is determined mainly by geometric and thermophysical parameters of heat exchange surfaces, thermal insulation and the length of the pipeline.

The heat losses in the wells and oil reservoirs depend on the time and they are non-stationary.

In the wellbore, non-stationary heat transfer occurs between the injection agent and the rock massif. The heat losses are directly dependent on the mechanism of heat transfer (convective or conductive) to the surrounding well, an almost unlimited rock mass. And the temperature difference between the coolant and the surrounding rocks.

If in the rocks the heat transfer mechanism is conductive, then the intensity of the conductive transfer and the amplitude of the temperature disturbance gradually decrease. Over time, the layer of rocks affected by the process in the radial direction grows. In this case, along the vertical, in the direction of motion of the coolant, its value decreases in comparison with the value at the wellhead of the injection well. This layer plays the role of thermal resistance. Over time, its increase and leads to a decrease in heat loss.

A large number of issues related to the forecast of heat losses in the wells can be solved on the basis of the coefficient of non-stationary heat transfer (Dyadkin, *et al*., 1993), using numerical methods (Shulyupin and Chermoshentseva, 2007), based on an "average exact" solution (Filippov, *et al*., 2010) and many other methods of solution. Practically in all works it is confirmed that a significant nonstationary of the process is observed in the initial period of the coolant injection. For example, at the well of the Mutnovsky deposit, non-stationary heat exchange exerts a noticeable influence on the operating parameters only in the first two days from the injection (Shulyupin and Chermoshentseva, 2007). With the passage of time, the problem acquires a quasistationary character.

Numerous studies carried out not only by the authors listed here made possible to develop many valid methods for calculating the dynamics of temperature fields in a mountain massif surrounding wells of various purposes. The estimates made in them allow us to assume that the heat loss in the well can be increased to 2% to 3% for every 100 m of the barrel length from the heat content of the energy input using heat insulation measures.

With the transition to the development of deep-seated oil deposits, the loss of the temperature potential of the energy carrier is significantly increased.

The greatest complexity and practical interest, from the point of view of the heat conduction theory are the processes in the oil reservoir itself.

Losses in the top and the soil of the reservoir also depend on the time. For the simplest case of low-power strata with temperature averaging over the section of the formation, these losses can be determined using the analytical dependencies proposed in the works (Lauwerier, 1955; Gringarten, *et al*., 1975; Smirnova and Soloviev, 1982). The solutions were obtained on the basis of the heat exchange model for the fluid flow in a channel surrounded by a semi-infinite solid mass, using the methods used in the classical theory of heat conduction (Carslow and Eger, 1964).

The role of heat loss in the roof and sole in the overall balance decrease, as time passes, as well as with high thickness of the formation.

Special difficulties in the theoretical analysis of heat exchange processes are associated with the high heterogeneity of oil deposits. Experimental work in this direction is known when implementing the technology of selective thermoinjection of stratified heterogeneous layers (Jamalov, 1978), theoretical studies based on numerical modeling of the mechanism of oil recovery of heterogeneous layers (Ruzin, *et al*., 2013), layered influence -uniform structure of the formation on the development indices (Ahmedov, *et al*., 2016).

The low temperature potential of the energy carrier in traditional methods of displacement of oil leads to the formation of crystalline hydrates and paraffin in the pores of the oil layer. Further development of the deposit and additional recovery of residual oil is possible only with the use of thermal methods.

On the previously used oil field, as an object of application of thermodynamic influence, different variants of heating a multi-layer oil reservoir can be used to resume the development of reservoirs with residual, fixed high-viscosity oil.

In the case of high filtration resistance at the contact of sandy and clay layers in oil strata, hydraulic fractures are produced. The heat carrier is pumped into the cracks, from which the oil layers and the interlayers of clay are conductively heated. The corresponding physical model of the process is a system of parallel cracks. In such a system, an analytical solution of the conjugate problem of non-stationary heat.

While goal setting and solution of the conjugate problem of non-stationary heat transfer in conditions of a multilayer filtration scheme, the equivalent heat equation is used (Smirnova, 1978). The method became widespread in solving problems of geothermal thermal physics (Dyadkin, *et al*., 1993; Alekseenko, *et al*., 2016) and in the theory of heat and mass transfer (Nustrov and Saifulaev, 1981).

The principle of method is in replacing the integral describing the non-stationary heat flux from the structural elements of the medium by differential approximation.

The implementation of the method makes it possible to obtain an analytical dependence for the analysis of the temperature fields of oil deposits with an inhomogeneous structure.

A complex manifold will be modeled by a system of alternating permeable and impermeable layers (**Fig. 2**).

A hot coolant with a temperature of t0 is injected into the permeable layer (model-filtration in the layer of balls) and raises the temperature of adjacent impermeable interlayers (model-filtration in a system of parallel plates). The initial temperature in the system is T0. The minimum required temperature at the bottom of the production well and in the center of impenetrable interlayers is determined by the melting temperature of paraffin-tar-asphaltene substances and T_{min}=50°C is adopted. When this temperature is reached, it is possible to start the coolant injection directly into previously impermeable layers.

Within the framework of reservoir models, as a heterogeneous medium with a periodic isotropic structure of elements of the classical form (plates, spheres, cylinders), taking into account their thermal resistance, analytical solutions of a complex of individual problems of nonstationary heat transfer in one-dimensional filtration have already been obtained in works (Romm, 1972; Smirnova, 1978; 1990; Dyadkin, *et al*., 1993).

The mathematical formulation of the problem includes a system of heat balance equations for a liquid phase (heat carrier) with internal heat sources, differential heat equations for solids of the simplest form (ball, plate), and conditions for singlevaluedness.

Differential heat balance equation for the average permeable reservoir of the dimensionless temperature of a unit volume of liquid

(1)

Initial and Boundary Conditions

θ = 1 when x = 0; θ = 0 when

(2)

By introducing into the heat balance equation, the liquid phase of internal heat sources q1 and q2. Interphase heat transfer is taken into account. The specific heat flux to the surface of the particles of the porous layer q_{1} and the specific heat flux to the impermeable layers q_{2} are determined from the solution of the differential equations of thermal conductivity for the solid phase corresponding to the form of the elements of the heterogeneous medium (sphere, plate) (Carslow and Eger, 1964; Lykov, 1967, 1978). In this case, the time variable of the ambient temperature and the finite rate of phase-to-phase heat transfer in the formulation of the problem with boundary conditions of the third kind are taken into account.

Substituting these solutions into equation (1) and taking into account the notation for dimensionless variables, we obtain a cumbersome, rather complex, integro-differential equation.

(3)

Using the method of the equivalent heat equation, equation (3) can be reduced to a differential equation with a second-time derivative

(4)

Since equation (4) is of the second order, it is necessary to add one more time to the initial conditions, in time. A physically justified condition will be the fact that the value of the heat content in the heat exchange zone in question is finite, and at large times the phase temperature does not change.

In the new variables, the initial and boundary conditions are:

θ = 1 when X=0, θ = 0 when

F_{0}* ≤ 0, θ = 1 when F_{0}* → ∞ (5)

The use of this method is justified by the length of the processes being studied, and hence the possibility of investigating the asymptotic behavior of the unknown function, as well as by the existence of an asymptotic expansion for the integrals equation (3) describing the nonstationary heat transfer from the structural elements of a heterogeneous medium. (Lavrentyev and Shabat, 1965).

The solution of the system (4)-(5) was obtained using the operational Laplace method (Lavrentyev and Shabat, 1965).

As a result of the solution of the system (4)-(5), the dependences for the temperature of the coolant and the temperature of the heated impermeable layer were obtained.

Coolant temperature

This solution is formally analogous to the solutions in the work (Smirnova, 1978; 1990), however, the coefficients A and B in this formulation have a more complicated form.

Coefficients:

Here: *μ _{n}* and

It is practically sufficient to use the six first roots of these equations.

The temperature of the heated impermeable layer is obtained as a function of time and two coordinates (X-along the direction of the coolant flow and Y-along the thickness of the reservoir) and, with the known expression for the coolant temperature θ, is determined by the formula:

For boundary conditions of the third kind, the coefficients A_{p}, B_{p}, ÃÂÃÂ_{L}, ÃÂÃâ_{L} and ÃÂÃÅ¡_{1}, ÃÂÃÅ¡_{2} are calculated in accordance with the shape of the chosen model of the structural elements of the heterogeneous medium (sphere, plate) and can be determined using the tables of the work (Smirnova,1990).

The graphs of the dependences obtained are presented in **Fig. 3-6**.

At known temperatures of the heat carrier and collector, the amount of heat lost to the heating of all structural elements of the formation is determined by the well-known heat balance formulas.

Representing the result in dimensionless variables when solving a problem and constructing graphs makes it suitable to use with any initial parameters of the problem. The presented range of dimensionless variables does not cover all possible parameters of the problem encountered in practice.

In the **Fig. 3** and **4**, the profile of the dimensionless temperature of an impermeable layer is shown for different process times and a fixed coordinate value. From the values of the dimensionless time, the time for their heating τ is determined, i.e., the time at which the specified temperature is reached in the center of these strata at a certain distance from the injection well. It is possible to determine the length of the formation heated to the technologically necessary temperature for a specific time interval. With a large reservoir length, the temperature in the center of the formation at the bottom of the injection well will naturally be higher than at the bottom of the production well.

Determination of the well of the coolant temperature (**Fig. 5** and **6**) in different ranges of the coordinate and time makes it possible to estimate the heat losses associated with the heating of the oil deposit at a known flow rate of the coolant and the distance between the injection and production wells.

For a comparative evaluation of the options for warming the seams due to the filtration of the coolant along the fractures of the fracturing and along the permeable layer. **Fig. 7** shows the dependence of the temperature of the impermeable layer on the length of hydraulic fracturing cracks at different times of the heating process.

1. Review of the state of research, three thermal effects on the oil reservoir have shown the possibility of a wide choice of justified methods for calculating the processes of stationary and non-stationary heat and mass transfer in pipelines and wells for use in the methodology for estimating heat losses.

2. Analysis of all the reasons for the reduction in the potential of the energy carrier under the thermal impact on multilayer oil deposits showed that the heat used for heating the solid phase (particles) of the permeable reservoir and impermeable layers will be an essential component in the method for estimating heat losses. It is useful to use only the heat spent for heating the oil contained in it.

3. The complication of the model for the analysis of heat transfer in the underground part of the geotechnological complex, with the minimal amount and degree of reliability of the initial information about real environments, limits the value of mathematical modeling using numerical methods, leaving the priority behind simple valid models and analytical solutions suitable for operational engineering calculations and More complex physical statements.

4. For essentially heterogeneous media, such as multi-layer inhomogeneous deposits, the most correct is the formulation of the conjugate heat transfer problem taking into account the thermal resistance of the reservoir structural elements.

5. Application of the method of the equivalent heat equation allows one to obtain analytical dependencies for analyzing the temperature of the coolant and the temperature field of the oil deposit with a multilayered heterogeneous structure. The obtained dependences are used in determining the time and amount of heat for warming up the oilcontaining impermeable layers to the temperature providing the increase in fluidity of the oil.

6. The proposed approach allows to take into account a significant part of the heat losses in the method of their evaluation with thermal impact on the heterogeneous oil layer.

7. Further improvement of the methodology can be carried out in the direction of a reasonable choice of transport coefficients and interphase heat transfer, included in the calculation formulas and including all the conventions of this approach.

*ÃâÃâ¦, ÃâÃÆ*-longitudinal and transverse coordinates; τ-time; *t* is the temperature of the liquid (coolant); t0- the temperature of the liquid at the inlet; T is the temperature of the layer (layer); T_{0}-initial temperature of the formation; *u* is the real velocity; b-halfthickness of the impermeable layer; l half-thickness of the permeable layer; R_{p}-radius of particles of the permeable layer; λ_{L}, *ÃÂÃÂ° _{L}*-effective thermal conductivity and thermal diffusivity of layers; λ

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