Numerical Simulation of Phase Transitions in Porous Media with Three-Phase Flows Considering Steam Injection into the Oil Reservoir

Abstract

This study focuses on the analysis of an approach to the simulation of the phase transition in porous media when hot steam is injected into the oil reservoir. The reservoir is assumed to consist of a porous medium with homogeneous thermal properties. Its porous space is filled with a three-phase mixture of steam, water, and oil. The problem is considered in a non-stationary and non-isothermal formulation. Each phase is considered to be incompressible, with constant thermal properties, except for the dynamic viscosity of oil, which depends on the temperature. The 1D mathematical model of filtration, taking into account the phase transition, consists of continuity, Darcy, and energy equations. Steam injection and oil production in the model are conducted via vertical or horizontal wells. In the case of horizontal wells, the influence of gravity is also taken into account. The Lee model is used to simulate the phase transition between steam and water. The convective terms in the balance equations are calculated without accounting for artificial diffusion. Spatial discretization of the 1D domain is carried out using the finite volume method, and time discretization is implemented using the inverse (implicit) Euler scheme. The proposed model is analyzed in terms of the accuracy of the phase transition simulation for various sets of independent phases and combinations of continuity equations. In addition, we study the sensitivity of the model to the selected independent phases, to the time step and spatial mesh parameters, and to the intensity of the phase transition. The obtained results allow us to formulate recommendations for simulations of the phase transition using the Lee model.

1. Introduction

The need to study the behavior of multiphase flows in porous media often arises in practice in the development of oil reservoirs. In particular, multiphase flows arise during the implementation of enhanced oil recovery technologies, such as gas injection, thermal injection, and chemical injection [1,2]. The most common method at present is the injection of hot steam into the oil reservoir. The technology itself can be classified based on the features of the technical implementation (cyclic steam stimulation, steam flooding, in-situ combustion, steam-assisted gravity drainage, etc.) [3,4,5,6], but the general idea is as follows: high-temperature steam is injected into the reservoir, the movement of which leads to heating of the rock, which contains oil, and the displacement of oil to production wells (Figure 1). Due to the increase in the mobility of oil during the heating process, the displacement of the oil increases, and, consequently, the total production increases. In this case, during the steam injection process, three characteristic zones are formed in the reservoir [7,8,9]: (1) a zone of non-isothermal oil displacement by hot steam with a temperature above the phase transition temperature; (2) a zone of non-isothermal displacement of oil by condensed water at a temperature above the initial temperature in the reservoir; and (3) a zone of isothermal displacement of oil by water at the initial reservoir temperature.

Many questions arise during the development of deposits by steam injection into the reservoir. How can steam be used efficiently to maximize reservoir heating? How long will it take for steam to break through into a production well? What physical processes take place in the reservoir at a particular stage of reservoir development? All these questions require scientific research to be conducted. An important part of such research is laboratory experiments [5,10]. Interpretation of the results obtained during the experiments, their generalization, and the identification of regularities is impossible without the use of mathematical modeling.

First of all, the simulation of steam injection into the reservoir requires consideration of the filtration of a complex three-phase steam–water–oil mixture in the reservoir. Under these conditions, it is important to correctly account for the phase transition between steam and water. A great deal of work has been dedicated to simulating the phase transition between steam and water. In [11,12], the Stefan problem and the boiling process were studied using a phase transition model based on the explicit calculation of the heat flux jump near the phase transition boundary. Khaware et al. [13] studied the formation of bubbles during boiling using the Thermal Phase Change Model in the ANSYS Fluent analytical software. He et al. [14] considered the phase transition in a porous medium using an enthalpy approach. However, the most widely used phase transition model in the literature is the Lee model. This model has been used in a wide range of problems: Bahreini et al. [15] considered simulations of vapor–liquid phase change processes for the Stefan problem and the two-dimensional film boiling problem; De Schepper et al. [16] studied the evaporation of hydrocarbon feedstock in the process of steam cracking; Wu et al. [17] analyzed the boiling of a refrigerant during its flow in a serpentine pipe; Lee et al. [18] studied the condensation process in a vertical downward flow. However, in this work, in addition to the Lee model, the authors provide an extensive list of other phase transition models and the problems to which they have been applied.

However, in most work, the features of the numerical implementation of multiphase flow models, taking into account the phase transition and the difficulties that arise in this case, are poorly covered. In addition, studies that directly simulate steam injection into the reservoir [19,20,21,22,23,24] lack analysis of the influence of the third phase on the phase transition dynamics, and they lack descriptions of the features implemented in numerical schemes in the presence of the third phase.

Thus, many questions remain unanswered regarding the simulation of three-phase flows in oil reservoirs in the presence of a phase transition between steam and water. The purpose of this work is to consider several possible options for a mathematical model of steam–water–oil mixture filtration in a porous medium, taking into account the phase transition between steam and water (under steam injection conditions), and to analyze this model, in terms of both the numerical implementation features and the physical accuracy of the simulation results.

2. Mathematical Model

It should be noted that we consciously made many simplifications in the development of the mathematical model. These simplifications are related to the dimensions of the model, the laws of fluid filtration, and the temperature distribution in porous media. We do not propose to use the geometric model considered below for the direct solution of applied problems that deal with two-dimensional and three-dimensional regions. The main goal of the work is to study the behavior and highlight the features of the numerical implementation of the phase transition model considering three-phase fluid flow in the pore space of the reservoir. In this regard, the use of a geometrically simplified model of a porous medium is appropriate, since it allows one to focus only on the physical processes associated with the phase transition, and not on its combination with other physical factors.

A one-dimensional section of an oil reservoir is considered as a geometric model. This section is bounded at the edges of the injection and production wells (Figure 2).

It is assumed that all the physical and thermophysical properties of the fluids and rock, with the exception of the dynamic viscosity of oil, are homogeneous and constant. The steam–water–oil mixture is incompressible, and its individual components are immiscible with each other. The dynamic viscosity of oil is a function of temperature. The importance of this dependence is justified by the fact that the majority of the oil recovery increase using thermal injection comes from a decrease in oil viscosity with increasing temperature [25,26,27]. Capillary forces are considered implicitly by setting the residual saturation value for each phase in the equations for relative permeability. It is assumed that all fluids and the rock are in a quasi-equilibrium thermal state and have the same temperature. Gravity is also taken into account, since its influence is of great importance for horizontal wells. Thus, the simulation with the gravity term represents the case of horizontal wells, and the simulation without the gravity term represents the case of vertical wells.

To describe the mass transfer in the steam–water–oil mixture, the following continuity equations are used [28,29,30]:

$$\phi \frac{\partial \left({\rho }_s{s}_s\right)}{\partial t}+\nabla \cdot \left({\rho }_s{v}_s\right)=q_s,$$

(1)

$$\phi \frac{\partial \left({\rho }_w{s}_w\right)}{\partial t}+\nabla \cdot \left({\rho }_w{v}_w\right)=q_w,$$

(2)

$$\phi \frac{\partial \left({\rho }_o{s}_o\right)}{\partial t}+\nabla \cdot \left({\rho }_o{v}_o\right)=0,$$

(3)

where $\phi$ is the rock porosity, m³/m³; t is the time, s; $s_s$, $s_w$, and $s_o$ are the steam, water, and oil saturations, respectively, m³/m³; ${\rho }_s$, ${\rho }_w$, and ${\rho }_o$ are the densities of steam, water, and oil, respectively, kg/m³; $v_s$, $v_w$, and $v_o$ are the filtration velocity vectors of steam, water, and oil, respectively, m/s; and $q_s$ and $q_w$ are the mass sources due to the phase transition for steam and water, respectively, kg/(m³∙s).

To ensure mass balance between all the phases, Equations (1)–(3) are supplemented by the mass balance (or mass conservation) equation:

$$s_s+s_w+s_o=1.$$

(4)

Since Equation (4) must always be satisfied, it can be used to find the saturation of one of the phases (hereafter, this phase will be called “dependent”) if the other two saturations are known. In turn, the known saturations for the other two phases (hereafter, these phases will be called “independent”) are calculated using two of the three continuity Equations (1)–(3). The issue of choosing independent phases can be ambiguous due to the presence of a phase transition, so this will be discussed in more detail below.

The macroscopic filtration velocity of each i-th phase is described by the linear Darcy law [31,32]:

$$v_i=-\frac{K{k}_{ri}}{{\mu }_i}\left(\nabla p_i-{\rho }_{i}g\right),$$

(5)

where $i=s,w,o$; K is the absolute permeability of the rock, m²; $k_{ri}$ is the relative permeability of the i-th phase; ${\mu }_i$ is the dynamic viscosity of the i-th phase, Pa∙s; $p_i$ is the pressure of the i-th phase, Pa; $g=g{i}_x$ is the gravity vector, m/s²; $g=9.81$ m/s²; and $i_x$ is the unit vector of the coordinate axis of the computational domain.

The filtration velocity $v_t$ of the steam–water–oil mixture is equal to the sum of the velocities of all phases:

$$v_t=v_s+v_w+v_o.$$

(6)

Since capillary forces are not explicitly taken into account, we can accept the hypothesis of a single pressure:

$$p_s=p_w=p_o=p.$$

(7)

Applying the divergence operator to (6) and using (1)–(3), we obtain the continuity equation for the considered mixture of incompressible fluids:

$$\nabla \cdot v_t=\nabla \cdot v_s+\nabla \cdot v_w+\nabla \cdot v_o=\frac{q_s}{{\rho }_s}+\frac{q_w}{{\rho }_w}.$$

(8)

Taking into account (5), we can rewrite Equation (8) for pressure and obtain an equation for its calculation:

$$\nabla \cdot \left[-\left({\lambda }_s+{\lambda }_w+{\lambda }_o\right)\nabla p+\left({\lambda }_s{\rho }_s+{\lambda }_w{\rho }_w+{\lambda }_o{\rho }_o\right)g\right]=\frac{q_s}{{\rho }_s}+\frac{q_w}{{\rho }_w},$$

(9)

where ${\lambda }_i=\frac{K{k}_{ri}}{{\mu }_i}$ is the mobility of the i-th phase, m²/(Pa∙s).

Relative phase permeabilities are calculated from the Brooks–Corey model [33]:

$$k_{rs}=\{\begin{array}{c}{k}_{rs}^0,s_s\ge 1-s_{wr}-s_{or},\\ k_{rs}^0{\left(\frac{s_s-s_{sr}}{1-s_{sr}-s_{wr}-s_{or}}\right)}^{n_s},s_{sr}<s_s<1-s_{wr}-s_{or},\\ 0,s_{sr}\ge s_s,\end{array}$$

(10)

$$k_{rw}=\{\begin{array}{c}{k}_{rw}^0,s_w\ge 1-s_{sr}-s_{or},\\ k_{rw}^0{\left(\frac{s_w-s_{wr}}{1-s_{sr}-s_{wr}-s_{or}}\right)}^{n_w},s_{wr}<s_w<1-s_{sr}-s_{or},\\ 0,s_{wr}\ge s_w,\end{array}$$

(11)

$$k_{ro}=\{\begin{array}{c}{k}_{ro}^0,s_o\ge 1-s_{sr}-s_{wr},\\ k_{ro}^0{\left(\frac{s_o-s_{or}}{1-s_{sr}-s_{wr}-s_{or}}\right)}^{n_o},s_{or}<s_o<1-s_{sr}-s_{wr},\\ 0,s_{or}\ge s_o,\end{array}$$

(12)

where $s_{ir}$ is the residual saturation of the i-th phase; and $k_{ri}^0$ and $n_i$ are the empirical coefficients for the i-th phase.

Combining Equations (5) and (6), we can rewrite the filtration velocity equation for each phase in terms of the filtration velocity for the mixture:

$$v_i=f_i{v}_t+f_i{\displaystyle \sum _{k\ne i}{\lambda }_k\left({\rho }_i-{\rho }_k\right)}g,$$

(13)

where $f_i=\frac{{\lambda }_i}{{\lambda }_t}$ is the Buckley–Leverett function [34]; and ${\lambda }_t={\lambda }_s+{\lambda }_w+{\lambda }_o$ is the mobility of the steam–water–oil mixture, m²/(Pa∙s).

Taking into account (13), the continuity Equations (1)–(3) can be written in the classical form of the Buckley–Leverett equation for the filtration of immiscible fluids in porous media [35,36]:

$$\phi \frac{\partial \left({\rho }_s{s}_s\right)}{\partial t}+\nabla \cdot \left({\rho }_s{f}_s{v}_t\right)+\nabla \cdot \left({\rho }_s{f}_s\left[{\lambda }_w\left({\rho }_w-{\rho }_s\right)+{\lambda }_o\left({\rho }_o-{\rho }_s\right)\right]g\right)=q_s,$$

(14)

$$\phi \frac{\partial \left({\rho }_w{s}_w\right)}{\partial t}+\nabla \cdot \left({\rho }_w{f}_w{v}_t\right)+\nabla \cdot \left({\rho }_w{f}_w\left[{\lambda }_s\left({\rho }_s-{\rho }_w\right)+{\lambda }_o\left({\rho }_o-{\rho }_w\right)\right]g\right)=q_w,$$

(15)

$$\phi \frac{\partial \left({\rho }_o{s}_o\right)}{\partial t}+\nabla \cdot \left({\rho }_o{f}_o{v}_t\right)+\nabla \cdot \left({\rho }_o{f}_o\left[{\lambda }_s\left({\rho }_s-{\rho }_o\right)+{\lambda }_w\left({\rho }_w-{\rho }_o\right)\right]g\right)=0.$$

(16)

The phase transition is simulated using the Lee model [37,38,39]:

$$q_s=-q_w=\{\begin{array}{c}\phi r_s{s}_s{\rho }_s\frac{T-T_s\left(p\right)}{T_s\left(p\right)},T\le T_s\left(p\right),\\ \phi r_w{s}_w{\rho }_w\frac{T-T_s\left(p\right)}{T_s\left(p\right)},T>T_s\left(p\right),\end{array}$$

(17)

where $r_s$ and $r_w$ are the empirical coefficients of the intensity of condensation and vaporization, respectively, s⁻¹; $T$ is the temperature, °C; and $T_s\left(p\right)$ is the phase transition temperature, °C. The empirical coefficients $r_s$ and $r_w$ are selected by finding a balance between the accuracy of the phase transition simulation and the numerical stability of the solution. It should be noted that the phase transition temperature is not set as a constant value, but as a function of pressure obtained from the Clausius–Clapeyron equation [40]:

$$T_s\left(p\right)={\left[\frac{1}{\left(T_{s0}+273.15\right)}-\frac{R\ln \frac{p}{p_0}}{\Delta H_s}\right]}^{-1}-273.15,$$

(18)

where $T_{s0}$ = 100 °C is the phase transition temperature at normal atmospheric pressure; $R$ = 8.314 J/(mol·°C) is the gas constant; $p_0$ = 101,325 Pa is the normal atmospheric pressure; and $\Delta H_s$ = 40,660 J/mol is the enthalpy of vaporization.

The choice of the Lee model for simulating the phase transition can be argued as follows: (1) the Lee model is the most widely used in simulations of evaporation and condensation processes; (2) although this model is most often used to simulate the physical processes of bubble formation during boiling with small characteristic dimensions of the computational domain, the type of model allows its applications to be extended to computational domains with significantly larger characteristic dimensions (up to tens of meters); (3) the presence of empirical coefficients in the model, on one hand, is a complication, but, on the other hand, this allows the model to be controlled during numerical calculations and its parameters to be selected for better numerical convergence; (4) the form of the model is relatively simple for numerical implementation and analysis.

Since it is assumed that the temperatures of the fluids and rock are in a quasi-equilibrium thermal state and are equal, in this case, heat transfer in a porous medium can be described by the following energy equation [20,41]:

$$\frac{\partial }{\partial t}\left(\left[\left(1-\phi \right){\rho }_m{c}_m+\phi {\displaystyle \sum _i{\rho }_i{c}_i{s}_i}\right]T\right)+{\displaystyle \sum _i\nabla \cdot \left(\phi {\rho }_i{c}_i{s}_i{v}_i\right)}T=\nabla \cdot \left({\chi }_e\nabla T\right)+\Delta h_s{q}_w,$$

(19)

where $c_m$, $c_s$, $c_w$, and $c_o$ are the heat capacities of the rock mass, steam, water, and oil, respectively, J/(kg∙°C); ${\chi }_e=\left(1-\phi \right){\chi }_m+\phi \left({\chi }_s{s}_s+{\chi }_w{s}_w+{\chi }_o{s}_o\right)$ is the effective thermal conductivity, J/(m∙s∙°C); ${\chi }_m$, ${\chi }_s$, ${\chi }_w$, and ${\chi }_o$ are the thermal conductivities of the rock mass, steam, water, and oil, respectively, J/(m∙s∙°C); and $\Delta h_s$ is the specific enthalpy of vaporization, J/kg.

The system of Equations (4), (9), (14)–(16) and (19) is supplemented with the initial and boundary conditions

$$\{\begin{array}{l}{p|}_{x=0}=p_1,{p|}_{x=L}=p_2,\\ {s_s|}_{x=0}=1-s_{sr}-s_{wr}-s_{or},{s_w|}_{x=0}=s_{wr},{s_o|}_{x=0}=1-{s_s|}_{x=0}-{s_w|}_{x=0},\\ {\nabla s_s\cdot n|}_{x=L},{\nabla s_w\cdot n|}_{x=L},{\nabla s_o\cdot n|}_{x=L},\\ {T|}_{x=0}=T_1,{\nabla T\cdot n|}_{x=L}=0,\end{array}$$

(20)

where $p_1$ and $p_2$ are the pressures at the injected and producing wells, respectively, Pa; $T_1$ is the temperature of the injected steam, °C; $n$ is the vector of the outer normal to the considered boundary, m; and $L$ is the length of the computational domain, m.

3. Numerical Implementation

As noted above, the presence of a phase transition in the mathematical model can create difficulty in choosing dependent and independent phases. To address this, a preliminary study was carried out, which included an analysis of three-phase filtration with various combinations of dependent and independent phases. This study was designed to determine (1) whether the solution depends on the chosen combination of phases when gravity is taken into account; (2) the combinations of phases for which the best numerical stability will be achieved and physically accurate results will be obtained for various combinations. Based on the results of the preliminary study, three combinations of Equations (4) and (14)–(16) were selected for further analysis. These combinations are presented in

Table 1. Considered combinations of dependent and independent phases.

Combination No. 1

$\begin{array}{c}\phi \frac{\partial \left({\rho }_s{s}_s\right)}{\partial t}+\nabla \cdot \left({\rho }_s{f}_s{v}_t\right)+\nabla \cdot \left({\rho }_s{f}_s\left[{\lambda }_w\left({\rho }_w-{\rho }_s\right)+{\lambda }_o\left({\rho }_o-{\rho }_s\right)\right]g\right)\\ +\phi \frac{\partial \left({\rho }_o{s}_o\right)}{\partial t}+\nabla \cdot \left({\rho }_o{f}_o{v}_t\right)+\nabla \cdot \left({\rho }_o{f}_o\left[{\lambda }_s\left({\rho }_s-{\rho }_o\right)+{\lambda }_w\left({\rho }_w-{\rho }_o\right)\right]g\right)=q_s,\end{array}$ $\phi \frac{\partial \left({\rho }_w{s}_w\right)}{\partial t}+\nabla \cdot \left({\rho }_w{f}_w{v}_t\right)+\nabla \cdot \left({\rho }_w{f}_w\left[{\lambda }_s\left({\rho }_s-{\rho }_w\right)+{\lambda }_o\left({\rho }_o-{\rho }_w\right)\right]g\right)=q_w,$ $s_o=1-s_s+s_w.$

Combination No. 2

$\phi \frac{\partial \left({\rho }_s{s}_s\right)}{\partial t}+\nabla \cdot \left({\rho }_s{f}_s{v}_t\right)+\nabla \cdot \left({\rho }_s{f}_s\left[{\lambda }_w\left({\rho }_w-{\rho }_s\right)+{\lambda }_o\left({\rho }_o-{\rho }_s\right)\right]g\right)=q_s,$ $s_w=1-s_s+s_o,$ $\phi \frac{\partial \left({\rho }_o{s}_o\right)}{\partial t}+\nabla \cdot \left({\rho }_o{f}_o{v}_t\right)+\nabla \cdot \left({\rho }_o{f}_o\left[{\lambda }_s\left({\rho }_s-{\rho }_o\right)+{\lambda }_w\left({\rho }_w-{\rho }_o\right)\right]g\right)=0.$

Combination No. 3

$\begin{array}{c}\phi \frac{\partial \left({\rho }_s{s}_s\right)}{\partial t}+\nabla \cdot \left({\rho }_s{f}_s{v}_t\right)+\nabla \cdot \left({\rho }_s{f}_s\left[{\lambda }_w\left({\rho }_w-{\rho }_s\right)+{\lambda }_o\left({\rho }_o-{\rho }_s\right)\right]g\right)\\ +\phi \frac{\partial \left({\rho }_w{s}_w\right)}{\partial t}+\nabla \cdot \left({\rho }_w{f}_w{v}_t\right)+\nabla \cdot \left({\rho }_w{f}_w\left[{\lambda }_s\left({\rho }_s-{\rho }_w\right)+{\lambda }_o\left({\rho }_o-{\rho }_w\right)\right]g\right)=0,\end{array}$ $s_w=1-s_s+s_o,$ $\phi \frac{\partial \left({\rho }_o{s}_o\right)}{\partial t}+\nabla \cdot \left({\rho }_o{f}_o{v}_t\right)+\nabla \cdot \left({\rho }_o{f}_o\left[{\lambda }_s\left({\rho }_s-{\rho }_o\right)+{\lambda }_w\left({\rho }_w-{\rho }_o\right)\right]g\right)=0.$

.

It follows from Table 1 that in Combination No. 1, the dependent phase is oil, and the independent phases are steam and water. In Combinations No. 2 and No. 3, the dependent phase is water, and the independent phases are steam and oil. Steam saturation is not calculated from Equation (14), but from the sum of Equations (14) and (16) in the case of Combination No. 1, and from the sum of Equations (14) and (15) in the case of Combination No. 3. The preliminary study showed that this approach makes it possible to eliminate mass balance disturbance when accounting for the phase transition in the case of Combination No. 1. The solution to the sum of the equations provides better convergence of the mass balance between all three phases, since the flows from each phase are explicitly taken into account. Combination No. 3 is considered due to another reason, which consists in checking the possibility of correct simulation of the phase transition by explicitly excluding it from Equation (14). In this case, the phase transition is explicitly taken into account only in pressure (9) and energy (19) equations. It is assumed that such an approach could reduce the requirement for numerical convergence without a significant decrease in the accuracy of evaluating the phase transition.

For the spatial discretization of Equations (4), (9), (14)–(16) and (19), the finite volume method is used [42]. Approximation of the convective terms in the continuity Equations (14)–(16) is performed using a first-order Implicit Hybrid Upwinding scheme [43,44,45]. The significance of this scheme is that it was developed specifically to improve the convergence and ensure the physical accuracy of the results when solving problems involving multiphase flows in porous media, taking into account gravity. Unlike the classical Upwind scheme [42], Implicit Hybrid Upwinding enables the multidirectional motion of the light and heavy phases to be considered when approximating gravitational flows. The classical Upwind scheme was used for the approximation of the convective terms in energy Equation (19). In this equation, there is no need to take into account the multidirectionality of the light and heavy phases in the presence of gravity; only the direction of the total flow is important. Approximation of all the diffusion terms was carried out using the central difference scheme [42,46]. Time discretization was carried out according to the inverse (implicit) Euler scheme [47]. In line with the finite volume method, the computational grid is a set of cells with a unit cross-sectional area and a longitudinal length equal to Δx (Figure 3). Discrete analogues of Equations (4), (9), (14)–(16) and (19) were written for the center of each cell. It should be noted that Figure 3 is schematic, and the actual number of cells in the mesh is much higher and varies from 50 to 200.

As a result, a system of algebraic equations was established to calculate each of the unknown variables $p$, $s_s$, $s_w$, $s_o$, and $T$. Since the convective term in the continuity equations is non-linear (due to the form of the Buckley–Leverett functions), the equations for calculating $s_s$, $s_w$, and $s_o$ are also non-linear, and Newton’s relaxation method is used to linearize them [48]. To solve the linearized algebraic equations, the Wolfram Mathematica computer algebra package was used, and a multifrontal method was applied [49]. Graphical representation and analysis of the results was also carried out using Wolfram Mathematica tools.

4. Results and Discussion

Table 2. Parameters for numerical simulation.

Parameter	Value
Computational area length, m	5
Rock porosity, m3/m3	0.26
Absolute rock permeability, μm2	2.68
$k_{rs}^0$	0.05
$k_{rw}^0$	0.3
$k_{ro}^0$	0.3
$n_s$	3
$n_w$	3.5
$n_o$	3
Residual steam saturation, m3/m3	0
Residual water saturation, m3/m3	0
Residual oil saturation, m3/m3	0.34
Rock density, kg/m3	2100
Steam density, kg/m3	0.5
Water density, kg/m3	997
Oil density, kg/m3	982
Rock thermal conductivity, J/(m∙s∙°C)	2.33
Steam thermal conductivity, J/(m∙s∙°C)	0.023
Water thermal conductivity, J/(m∙s∙°C)	0.58
Oil thermal conductivity, J/(m∙s∙°C)	0.14
Specific heat capacity of rock, J/(kg∙°C)	1050
Specific heat capacity of steam, J/(kg∙°C)	2000
Specific heat capacity of water, J/(kg∙°C)	4200
Specific heat capacity of oil, J/(kg∙°C)	2090
Dynamic viscosity of steam, Pa∙s	0.000016
Dynamic viscosity of water, Pa∙s	0.001
Specific enthalpy of vaporization, J/kg	1,750,000
Injection well pressure, MPa	0.35
Production well pressure, MPa	0.16
Initial reservoir temperature, °C	10
Steam temperature at the inlet, °C	200
Initial oil saturation, m3/m3	0.88
Initial water saturation, m3/m3	0.12

presents the physical parameters and initial and boundary conditions used in the simulation.

The dependence of the dynamic viscosity of oil on temperature is shown in Figure 4a. This dependence is typical for oil from the Yaregskoye field, located in the central part of the Komi Republic on the Timan Ridge [50], and can be described by the following equation:

$${\mu }_o=0.052+\mathrm{Exp}\left(3.28-0.1T\right)$$

(21)

Figure 4b shows the dependence of the phase transition temperature on pressure, plotted according to Equation (18).

For the initial analysis of the simulation results, simulations were performed at a constant phase transition temperature independent of pressure and equal to 100 °C. The task was to find the optimal intensities of condensation $r_s$ and vaporization $r_w$. The calculations were carried out with the following numerical parameters: simulation time—20 days; time step—0.005 days; number of cells in the computational grid—50. The initial intensities of condensation $r_s$ and vaporization $r_w$ were taken equal to 1 day⁻¹. The simulation results using these parameters are presented in Figure 5 (without gravity) and Figure 6 (with gravity).

Figure 5 shows that for the intensities of condensation and vaporization equal to 1 day⁻¹, without accounting for gravity, a complete phase transition does not occur in all combinations. Either there is incomplete evaporation of water in the zone with a temperature above the phase transition temperature, or the steam does not completely condense in the zone with a temperature below the phase transition temperature. This is particularly reflected in the calculated steam distribution for Combinations No. 1 and No. 3, in which the length of the steam zone in front of the phase transition reaches approximately 3.8 m. Despite significant differences in the formulations of the mathematical models for Combinations No. 1 and No. 3, the results obtained for each combination match. The following conclusions were drawn:

• Consideration of the total balance of steam and oil in the first equation for Combination No. 1 mitigates the influence of the mass source on the intense phase transition zone of the steam;
• In the balance equation for water in Combination No. 1 with these parameters, the mass source due to the phase transition has little effect on the phase transition;
• In the calculations for Combinations No. 1 and No. 3, the phase transition pattern is formed mainly due to explicit consideration of the phase transition term in the equations for pressure and temperature. In the equations for Combination No. 3, the phase transition is not taken into account explicitly, but the results of the calculations for this combination correspond to those for Combination No. 1, in which the phase transition term is explicitly considered in both continuity equations.

From Figure 5, a significant difference between the results for Combination No. 2 and those for Combinations No. 1 and No. 3 is notable. We believe that this is due to the use of the initial continuity equations for each independent phase in Combination No. 2; therefore, the influence of the mass source due to the phase transition on the calculation of steam saturation becomes stronger than its influence on the overall solution. From the results for Combination No. 2, a significant increase in water near the phase transition boundary is also notable. This can be explained by the intense condensation of steam and the low rate of water filtration compared to the rate of the incoming mass of condensed steam. In general, this physical situation is typical when hot steam is injected into an oil-bearing formation [7].

When gravity is taken into account (Figure 6), an improvement in the calculated phase transition is observed for Combinations No. 1 and No. 3. This is explained by the direction of gravity, which is opposite to the motion of the phase transition front. Since the steam density is three orders of magnitude lower than the density of water and oil, when gravity is taken into account, it becomes much more difficult for the steam to move ahead of the phase transition, since it is displaced by water and oil. Correspondingly, the opposite is true for the water phase. However, as in the case of the absence of gravity, the results of the calculations for Combinations No. 1 and No. 3 are found to coincide. This again indicates that the formation of the phase transition pattern is largely due to its explicit inclusion in the calculation of pressure and temperature.

According to Figure 5 and Figure 6, using the indicated parameters, Combination No. 2 is generally considered to be the most accurate in accounting for the phase transition. However, before drawing final conclusions, it is necessary to conduct a deeper analysis of each combination in order to (1) evaluate the influence of the time and space steps on the solutions for each combination; and (2) evaluate the sensitivity of the solutions for each combination to changes in the intensities of condensation and vaporization. It is also important to note that regardless of the combination, the three-phase filtration model presented here also reflects the “oil bank” effect [51]. This effect relates to an excess oil saturation value compared to the reservoir value in a certain area with temperatures above that of the reservoir. The effect occurs due to an increase in the mobility of oil with increasing temperature, which leads to the more intensive displacement of oil and its accumulation in the direction of the temperature front.

4.1. Analysis of the Dependence of Combinations on the Time Step and the Number of Calculation Cells

Figure 7, Figure 8 and Figure 9 show the results of calculations for each combination at different time steps, with the number of calculation cells equal to 50, and without accounting for gravity. Figure 10, Figure 11 and Figure 12 show the results of the same calculations, but with gravity taken into account.

From Figure 7, Figure 8 and Figure 9, Combinations No. 1 and No. 3 are observed to be weakly dependent on the change in the time step. However, in the calculations for Combination No. 3, small fluctuations (blue circle in Figure 9b) occur in the water saturation distribution behind the phase transition front, which are difficult to see in Figure 5 due to the superimposed curves.

Combination No. 2 is noticeably affected by the change in the time step. As the time step decreases, the phase transition proceeds more slowly, which is reflected in the increasing amount of steam formed in front of the phase transition. In addition, with a decrease in the time step, the saturation distribution pattern changes.

When gravity is taken into account (see Figure 10, Figure 11 and Figure 12), no time step change effect is observed for Combinations No. 1 and No. 3. For Combination No. 2, the difference between the time steps of 0.005 days and 0.0025 days is relatively insignificant compared to the results of calculations with a time step of 0.00125 days. In this case, the saturation distributions are significantly different, but the curves remain qualitatively similar.

The above analysis of Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 allows the following general conclusions to be drawn regarding the influence of the time step on the calculation results:

• Combinations No. 1 and No. 3 do not depend on the time step. Therefore, we can assume better numerical stability than for Combination No. 2;
• Although the solution for Combination No. 3 does not depend on the time step, small oscillations are observed in the distribution of water saturation in the region behind the front of the phase transition, which do not decrease with a change in the time step;
• Combination No. 2 is the most sensitive to changes in the time step, due to the significant influence of the mass source caused by the phase transition in the continuity equation for steam compared to Combinations No. 1 and No. 3.

Next, an analysis of the influence of the number of cells on the solution for each combination is carried out. Figure 13, Figure 14 and Figure 15 show the dependences of the simulation results on the number of cells with a time step of 0.005 days, without taking gravity into account. Figure 16, Figure 17 and Figure 18 show the same, but accounting for gravity.

From analysis of Figure 13, Figure 14 and Figure 15, it is noted that, unlike the time step, the number of cells affects the solution for each combination. The influence of the spatial step is more significant for Combination No. 2 than for Combinations No. 1 and No. 3. For Combination No. 2, an increase in the number of cells leads to the smearing of the phase transition over a wide zone, which is undesirable. This is already manifested in the deterioration of the process of water evaporation in the zone with a temperature above the phase transition temperature. For Combinations No. 1 and No. 3, minor refinements of the calculated curves are observed. It is notable that for Combination No. 3, oscillations in the distribution of water saturation are minimized with an increase in the cell count.

When gravity is taken into account (see Figure 16, Figure 17 and Figure 18), a slightly greater influence of the number of cells on the simulation results is observed for Combinations No. 1 and No. 3. For Combination No. 2, with the number of calculated cells set at 100 and 200, the saturation distribution pattern is significantly different compared to that with 50 calculated cells. If the difference between 100 and 200 calculated cells is insignificant for steam saturation, then a noticeable difference according to the number of cells is visible for water saturation and oil saturation. In addition, there is both a quantitative and a qualitative difference in the distribution of oil saturation with 200 calculation cells compared to that with 50 and 100 cells.

The following general conclusions may be drawn regarding the influence of the time step and the number of cells on the considered solutions:

• The solutions for Combinations No. 1 and No. 3 do not depend on the time step (in the considered range) and show a weak dependence on the number of calculated cells;
• The solution for Combination No. 3 shows areas of weak non-physical oscillations, which can be minimized by increasing the number of cells;
• The solution for Combination No. 2 has a strong dependence on both the time step and the number of cells, which can be explained by the strong influence of the mass source due to the phase transition in the steam saturation continuity equation;
• The fact that Combination No. 2 has a strong dependence on the time step and the number of computational cells leads to an important recommendation: when using Combination No. 2, it is necessary to select a time step and number of cells for which the simulation results will no longer depend on a further decrease in the time step and an increase in the number of computational cells.

As noted above, Combination No. 2 strongly depends on the time step and the number of computational cells. Therefore, additional modeling was carried out for this combination, with the smallest studied time step of 0.00125 days and the largest number of computational cells set to 200. The simulation results are presented in Figure 19 (not accounting for gravity) and Figure 20 (accounting for gravity).

It can be noted from Figure 19 and Figure 20 that with an increase in accuracy (in terms of both time and space), the overall result changes noticeably, in contrast to all the previous simulation results for Combination No. 2. With an increase in accuracy, the phase transition zone to the left of the purple curve becomes wider. In this area, where the temperature is above the phase transition temperature, a noticeable “tail” of non-evaporated water is observed from the injection well to the front of the phase transition. Therefore, more accurate calculations show that, in this case, the given value of the vaporization intensity is insufficient for the water to evaporate completely in the region with a temperature above the phase transition temperature. It is also notable that with this increased calculation accuracy, the spike in the water saturation near the phase transition front decreases by a factor of approximately 1.5. With a further decrease in the time step and an increase in the number of cells, the water saturation spike does not decrease, but remains at the same level as in Figure 20. These calculations once again confirm that the simulation of the phase transition according to Combination No. 2 requires a more careful approach to selecting the time step and the size of the mesh to avoid obtaining a distorted picture.

4.2. Analysis of the Influence of the Intensity of Condensation and Vaporization on Phase Transition Simulation

Figure 21, Figure 22 and Figure 23 show the calculated saturation distributions for each combination for different intensities of condensation $r_s$ and vaporization $r_w$ without considering gravity. The results of simulations in which gravity is considered are not presented here because, in this case, there is no fundamental difference in simulating the phase transition with and without gravity. The time step for these calculations was taken as equal to 0.0025 days, and the number of cells was 50. The phase transition boundary for these calculations was sometimes significantly shifted with changes in the intensity of the “steam–water” mass transfer. Therefore, the figures present only the position of the phase transition boundary corresponding to values of the condensation and vaporization intensities of 1 day⁻¹. In this case, it is possible to evaluate how the saturation distribution curves were shifted relative to the originating phase transition boundary, and also to determine the displacement of the phase transition boundary itself.

Analysis of Figure 21 and Figure 23 indicates that with an increase in the intensities of condensation and vaporization, the phase transition zone becomes smaller both on the steam side and on the water side for Combinations No. 1 and No. 3. This can be considered as an improvement in the accuracy of the calculation of the phase transition. In this case, there is practically no displacement of the phase transition boundary itself. However, even increasing the intensity by a factor of 9 does not completely eliminate regions with non-condensed steam and non-evaporated water at a distance from the vertical line corresponding to the phase transition temperature.

The solution obtained for Combination No. 2 is highly sensitive to changes in the condensation and vaporization intensities (see Figure 22). An increase in the condensation and vaporization intensities leads to a significant shift in the phase transition boundaries and the formation of non-physical spikes of steam and water near the phase transition boundary (which may indicate numerical instability). In addition, at intensities of 9 day⁻¹, after 10 days, a significant decrease in temperature was observed near the phase transition boundary, and the movement of this boundary was opposite to the general direction of filtration. Therefore, the simulation of steam–water–oil mixture dynamics according to Combination No. 2 requires accurate selection of the intensities $r_s$ and $r_w$, and a balance between an accurate consideration of the phase transition and the overall physicality of the simulation results.

4.3. Simulation Taking into Account the Dependence of the Phase Transition Temperature on Pressure

Based on the results of the above studies, we performed a series of numerical calculations for each combination, taking into account all the physical parameters described in Section 2, including the dependence of the phase transition temperature on pressure. The numerical parameters used and the phase transition parameters for each combination are shown in

Table 3. Numerical parameters for each combination, considering phase transition temperature as a function of pressure.

Parameter	Value (Combination No. 1)	Value (Combination No. 2)	Value (Combination No. 3)
Time step, days	0.00125	0.00125	0.00125
Number of cells	200	200	200
Condensation $r_s$ and evaporation $r_w$ intensities, day−1	30	2	30

.

The results of comparing the numerical solutions for each combination are shown in Figure 24 (without gravity) and Figure 25 (with gravity). In addition to the saturation distributions, the calculated pressure, temperature, and mixture filtration velocity distributions are presented. These figures do not show the position of the phase transition front, because, in this case, it is highly variable.

Figure 24 and Figure 25 indicate that regardless of the combination and differences in the intensities of condensation and vaporization, the resulting pressures and temperatures are almost the same. However, noticeable differences are observed when the mixture filtration velocity and saturation of each phase are considered. It is especially worth noting that when gravity is not taken into account, significantly increased intensities of condensation and vaporization lead to differences between the results for Combinations No. 1 and No. 3, despite the same parameters being set for each. This may indicate that in the continuity equations for Combination No. 1, the mass terms of the phase transition begin to dominate over the additional oil mass balance term. However, when gravity is taken into account, the results for Combinations No. 1 and No. 3 again coincide; therefore, the additional term in the oil balance equation remains dominant.

Nevertheless, it should be noted that despite the noticeable quantitative differences between the results, the qualitative description of the phase transition remains practically unchanged when all combinations are considered. However, it is worth noting the impossibility of Combinations No. 1 and No. 3 to simulate a significant water saturation spike near the phase transition front. This is a significant drawback of these combinations, because such an effect is observed directly in the development of oil fields by thermal methods.

Despite the differences between the combinations, it is possible to use each one for phase transition simulations, keeping in mind the advantages, disadvantages, and limitations. In addition, the most important factor is an experimental basis for adjusting the model. Due to the complexity of the physical processes under consideration, it is difficult to accurately judge the suitability of the simulation results when solving a specific scientific or engineering problem. However, based on the results of this study, when simulating three-phase mixture dynamics with a phase transition, we recommend using a combination that includes the initial continuity equations, but not the sums of the balance equations, for several phases. Hence, the most appropriate combination of those considered is Combination No. 2.

If the system of balance equations is composed in such a way that both steam and water are calculated directly using balance equations, then this could lead to the violation of the mass balance. This is because, due to significant differences in the phase densities, the mass sources of the phase transition provide different contributions to the continuity equations for water and steam. Thus, it is better that only one balance equation explicitly controls the phase transition. These conclusions were attained for the case of constant steam density and do not account for the fact that the steam density can vary depending on pressure and temperature.

5. Conclusions

We have studied three different numerical schemes for the simulation of the one-dimensional filtration of a steam–water–oil mixture in a porous medium. The phase transition between steam and water was simulated using the Lee model. From a practical point of view, the process of pumping hot steam into an oil reservoir was considered.

Over the course of the study, three different numerical implementations of the model were considered, which are the most stable in terms of numerical calculation and the most accurate reflections of the real physics of the process. The differences between these numerical implementations consist of the selection of independent phases and combinations of continuity and mass balance equations. A comparative analysis of the three combinations was carried out, including a study of their dependence on the time step and the number of cells.

The results of the analysis showed that when the saturation of any phase is calculated not from the initial balance equation for the phase but from the sum of two equations, the model of the phase transition itself is less accurate. Vast regions of uncondensed steam and unevaporated water are formed. However, this approach provides the best balance in connectivity between all three phases, is the most numerically stable, and weakly depends on the time step and spatial discretization, allowing it to be used on coarser meshes and with large time steps. The disadvantages of this approach can be partially minimized by increasing the intensities of the condensation and evaporation processes, although this is valid only for the Lee model studied in this work; other phase transition models require additional investigation. However, this does not allow the zones with non-condensed steam and non-evaporated water to be completely disregarded. In addition, increases in the intensities of condensation and vaporization can lead to numerical instability, though this effect can be mitigated by artificial diffusion or an increase in the number of mesh cells.

Simulating three-phase mixture filtration using the original balance equations gives a more physically accurate description of the phase transition, but this approach has disadvantages, namely the strong sensitivity of the solution to the selection of dependent and independent phases, time step, and spatial discretization. In addition, with increased intensities of condensation and vaporization, the numerical stability of this approach can quickly decrease and lead to non-physical effects, which may be expressed as the appearance of oscillations and many spikes in steam and water saturations near the phase transition boundary and a significant decrease in temperature throughout the region. This could cause a shift in the phase transition point in the opposite direction to the total phase filtration. The problem of this sensitivity to the choice of dependent and independent phases is solved by test calculations with various combinations of dependent and independent phases. These calculations enable the selection of the combination that is the most numerically stable and gives physically accurate results. Based on the results of this study, we propose considering two independent phases as follows: the first independent phase (steam or water) participates in the phase transition, and the second independent phase (oil) does not participate in the phase transition. This makes it possible to place the entire phase transition in only one of the phases without violating the mass balance, which occurs if independent phases are considered that both participate in the phase transition. We do not offer a solution to the problem of sensitivity to spatial and temporal discretization, since this is a general problem of numerical simulation. In this case, it is more pronounced due to the influence of the mass source as a result of the phase transition in the continuity equations. It is only possible to select mesh parameters and a time step at which the required accuracy of the physical processes will be achieved with a satisfactory calculation time, taking into account the available computing resources. In addition, it is necessary to correctly select the intensities of condensation and vaporization so that the simulation results remain physically accurate.

The general conclusion for all the considered implementations of the model is that each of the proposed options is suitable for the phase transition simulation. However, from a physical perspective, the most accurate approach uses the initial continuity equations, but not the sums of the balance equations for several phases. In this case, it is necessary to account for the shortcomings and difficulties associated with such an approach to numerical calculations. Of particular importance is the choice of dependent and independent phases. It is also worth noting that the final conclusions on the choice of one or another implementation of the model can be made only after comparing the simulation results with experimental data, after expanding the mathematical model to a two-dimensional or three-dimensional case, and after taking into account more complex laws of medium filtration if this is required by the problem under consideration.

In general, based on the results of this study, we believe that modeling three-phase filtration in porous media while taking into account the phase transition is a non-trivial task, even when considering the one-dimensional case and using a relatively simple phase transition model. In contrast to simulations of two-phase filtration, the numerical simulation of three-phase filtration is sensitive to several factors and requires a more thorough analysis of both the model itself and its numerical implementation before it can be used to solve specific scientific or engineering problems.