Numerical modeling and parallel computations of heat and mass transfer during polymer flooding of non-uniform oil reservoir developing by system of producing and injecting wells

The mathematical and numerical models are developed for computation of interrelated thermal and hydrodynamic processes in the unified oil-producing complex during the polymer flooding of the heterogeneous oil reservoir exploited with a system of arbitrarily located injecting wells and producing wells equipped with submersible multistage electric centrifugal pumps with regulation of their working modes by the surface control stations. The complete differential model includes equations governing non-stationary two-phase three-component filtration in the reservoir and quasi-stationary heat and mass transfer in the wells and working channels of pumps. Special non-linear boundary conditions and dependences simulate the influence of the drossel diameter and the frequency electric current, respectively, on the flow rate and pressure at the wellhead of each producing well and the performance characteristics of all submersible units. The oil field development is also regulated by changes in bottom-hole pressure of each injection well, the concentration of polymer in water solution, its total volume and duration of injection. The problem is numerically solved with the use of the finite difference method and the iterative algorithms with application of technologies of parallel computing. It is shown that parallelization can improve the performance of calculations at several times in comparison with sequential computing.


Introduction
The subject of the study is the interrelated heat and mass transfer in the unified oil-producing complex "the oil reservoira system of arbitrarily located injecting wells and producing wells equipped with multistage electric submersible pumps (ESP)the surface control stations (SCS)" during polymer flooding of the heterogeneous reservoir. Oil field development induces very intricate non-steady filtration motion of oil and water phases in the non-uniform porous medium [1][2][3][4][5][6][7][8][9][10][11][12]. Quasi-stationary thermal and hydrodynamic processes in the multiphase flows in the wells and working channels of ESP are also very complicated, they are accompanied by such factors as the phase transitions at degassing of oil in the pipes and gas dissolution in the channels of pump, compressibility of phases, friction, gravity force, restructuring of gas-liquid flow, inversion of phases, drift motion of disperse components, heat exchange between the flow and rock formation around the well, etc., see [13][14][15][16][17][18][19][20][21][22]. Performance characteristics of submersible unit significantly depend on properties of the pumped heterogeneous mixtures [10,11,16,19,22]. In addition, now the control for the current working modes of ESP are often realized with SCS on the base of analysis of the telemetering data (a feed- forward) and generation of the needed actions (a feedback) to improve the work conditions of submersible equipment, up to its disable in emergency situations [11]. Therefore, computation of these processes and optimization of the oil recovery can be effectively done on the basis of mathematical and numerical modeling.
Previously, we proposed in [9][10][11] a computer model to study the features of thermal and hydrodynamic processes in such oil-producing object in the case of water-oil displacement. In this article, we continue our research and generalize mathematical, numerical and algorithmical models [9 -11] to more complex practical situations when a water polymer solution of the desired concentration is injected into the porous medium with the purpose to create high-viscous moving fields and redirect the filtration flows. Chemical reagent (polymer) is soluble only in water and can enter into the reservoir both from all or some injection wells with different concentration, its total volume, an initial time and the end moment of injection. The sorption of thickener is irreversible and affects the permeability of porous medium, so the resistance factor is a function of the polymer concentration.

Mathematical model
Using [1,[4][5][6]8] and generalizing the mathematical model [9][10][11], the system of conjugated nonlinear two-dimensional differential equations governing simultaneous filtration of the polymer-water solution and the oil phase in the porous reservoir averaged over its thickness, one-dimensional quasistationary momentum (force-balance) and energy equations for the disperse water-oil-gas flow with discrete phases (gas bubbles, water drops or oil drops) in the m -th producing well and the m -th pumping unit, located in this well, can be written in the following form: a) two-phase three-component filtration in the reservoir where p M is the number of the producing wells;  is the time; the sub-indexes " ,, w o g " denote characteristics of water, oil and gas, respectively; x L is the length of the filtration region Equations (1), (2) govern the two-phase isothermal filtration under the Darcy's law without regard to the capillary effects, gravity, compressibility of the phases and the porous medium [1,[4][5][6]8]. Here, x and y are spatial coordinates; P is the pressure; S is the water saturation; C is the mass concentration of the thickener; a is mass amount of thickener that adsorbed in pores;  is the viscosity of water without the thickener at 0 C  ; A , B and p  are empirical coefficients that must be determined throughout the range of values of absolute permeability of the oil reservoir. In this paper we assume that the adsorption   , a a S C  is an equilibrium and irreversible, so that this dependence is determined by the Henry sorption isotherm (4) are written in a framework of the quasi-stationary approximation [14][15][16][17][18][19] for the dispersed water-oil-gas flow in the producing well. In such three-phase mixture the discrete components move as the gas bubbles and drops of water (or oil) inside the continuous (oil or water) phase. In these equations Oz is the vertical coordinate axis directed upwards the well from its beginning on the reservoir roof; P and T are the pressure and the temperature, identical for all phases;  is the mean multiphase mixture density;  is the overall mixture flow velocity; i  , i  , i  , i  and i G are the density, the actual phase velocity, the volumetric concentration, the consumption content and the mass flow rate (debit) of i -th phase, averaged over the well pipe cross- G is the overall mixture debit; ( , ) F P T is the relative gas factor which is defined as the ratio of the mass of gas released from the oil phase at certain Equations (5) govern the thermal and hydrodynamic processes in the channels of the multi-stage pump. They were developed in [16,19] under the assumption that all the phases become highly dispersed in the pump stage and move without slippage, i.e. It should be noted also that this article provides only some basic relationships which define the operating parameters of the pump stages and characteristics of multiphase flows in the pipes and in the porous medium. A complete set of special constitutive relations to close the equations is too much large and it can be found in our publications [4-11, 16, 18-20]. The boundary, initial and conjugation conditions for system of differential equations (1)-(5) are analogous to that, which has been formulated in [9,11]. In addition to these conditions, the initial distribution of the polymer concentration in the region r D and the boundary conditions for the second transport equation (2) for the function C at the bottom-hole of each injection well can be written by analogy with [4][5][6]8] in the following form: ( , ,0) 0, , ,, An important feature of the model is the ability to simulate the control actions from the earth surface on development of the oil reservoir [9][10][11]. First of all, this can be done by using nonlinear boundary conditions

Numerical model
An approximate solution of the system of nonlinear differential equations (1)- (8) is founded by the finite difference method and iterative algorithms. In detail the conservative finite difference scheme, approximating equations (1)-(5) in the grid-points of the discrete domains h em D is discussed in [11] at absence of the second transport equation (2) for concentration C for the case of only water-oil displacement in the reservoir. The following is a brief summary of the principal results of [11] and their extension to the case of polymer flooding. The pressure P is calculated from the quasi-linear elliptic equation , that can be obtained from equations (1), (2). This equation is approximated by the five-point difference scheme of the second order. The appropriate system of algebraic equations in respect to P is solved by the iterative method [4,5,7,11] of a high rate convergence (3-5 iterations) at every time step.
It is need also to note that the water saturation S and concentration C are multivalued function at the boundaries of the producing and injection wells. This specific feature of the problem solution can be taken into consideration by means of the special method [4], which allows us to determine the values of water saturation in eight directions to the well (in the four directions along the coordinate axes and along the four diagonal directions). In this method the average integral values of the water saturation are determined with the use of the unit cells of the special geometry in the vicinity of the borehole of wells which look as the sectors. For example, the difference equations for the saturation S along axis Ox and along the diagonal direction of the first quadrant are written in the following form: The formulas for water saturation and concentration along other directions can be written by analogy with equations (13). To provide the stability of solution equations (9), (11), (13) the time step h  of the scheme is determined in according to the Courant-Friedrichs-Lewy criterion. Systems of first order differential equations (3), (4) and (5) are solved with implicit Euler's schemas. However, their realization is significantly complicated because of a large number of special, usually non-linear constitutive relations. For example, performance characteristics (head H , power N , efficiency  ) of pump stages during pumping of the non-uniform mixtures are computed with the modified procedure [9,16,19] based on nonlinear semi-empirical dependencies [23]. Other similar features of equations also require the use of iterative procedures in calculations. Moreover, the number of the producing wells equipped with ESP-units on the certain oil field can reach several hundred, so the general number of grid points in all the wells may be much more than the number of grid points in the reservoir. This results to that the calculating times of filtration and heat mass transfer in the all wells become comparable. In realization of the complete numerical model the linear pressure behind the drossel at the wellhead of each well is determined by its given value lin m P , which can be much less than an unknown pressure wh m P before him, so as the values wh m  , wh m w , dr m  of density, flow velocity and local resistance coefficient of the regulating drossel in the relationship (7) are unknown and should be determined in solving system (1)- (8). Therefore, this system could be iteratively solved in such a way that the nonlinear boundary conditions (7)  The developed numerical model is implemented in the program package "PolymerFlood" with using of technologies of parallel computing [24,25] and simultaneous visualization of the results of computation. This package interacts with the special program "SCS" which simulates an operation of the surface control stations for regulation of work of equipment in the producing wells. PolymerFlood sends telemetry data and values of the current operating parameters of the submersible pump unit to SCS (a direct relation). In turn, this program analyses incoming data and generates the desired control parameters ( and etc.) for the pump unit. These parameters are sent to PolymerFlood (a feedback). Additionally, there is the interactive interface for changing the parameters * Parallelizing of calculations is effective in the following cases: 1) computation of two-phase threecomponent filtration in heterogeneous reservoir using grids which can contain tens and hundreds thousand mesh points; 2) calculation of thermal and hydrodynamic processes in the producing wells with electric pumps at the grids with a large number of nodes; 3) realization of the direct relation and feedback between PolymerFlood and SCS packages for all the wells and control stations; 4) visualization of the two-dimensional computed and initial characteristics in the filtration area on display when the number of pixels reaches several millions.

The results of numerical experiments
Computational experiments were carried out and analyzed for the concrete oil-extracting complex. Their results shown that parallelization can improve the performance of the calculations at several times in comparison with sequential computing. In particular, it is shown that in depending on the type of CPU and GPU the time of parallel computations at the central processor in solving of filtration problem reduces by 2-3 times, calculation of processes in the system of producing wells accelerates proportional to the number of processors and visualization with the use of graphic coprocessorsat about 4-7 times.

Conclusion
The complete mathematical model to study interrelated heat and mass transfer in the unified oilproducing complex during the polymer flooding of the heterogeneous oil reservoir exploited with a system of injection wells and producing wells equipped with submersible electric centrifugal pumps is proposed. An important feature of the model is the ability to simulate the control actions from the earth surface on the oil reservoir development. Numerical methods are developed for solving the system of corresponding non-linear differential equations. The numerical and algorithmical models are implemented in the program package. The analysis of results of computational experiments has also shown that on-line creation of the moving polymer fields due to injection of thickener in the desired wells significantly increases the uniformity of oil displacement in the heterogeneous reservoir and improves their basic exploitation parameters by redirecting the filtration flows.