# Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media
## 1 Introduction  
In the reservoir simulation, there are two numerical methods for solving the governing equations, namely, implicit pressure, explicit saturation (IMPES) method, and the fully implicit method. For each cell, the solution of these equations provides a way to estimate specific time-dependent changes in pressure, saturation, and other key parameters. Although IMPES is much faster than the fully implicit method, it is much less stable because of the explicit evaluation of mobility terms and capillary pressure at the previous time step \(n\) instead of the current one \(n+1\) which requires smaller time steps. On the other hand, as more equations have to be solved in the fully implicit scheme, it requires more CPU memory. Since the fully implicit method is more stable than IMPES, the simulator can take bigger time steps. The ongoing research into IMPES mainly focuses on the improvement of its stability (such as using iterative schemes) and so keeping it fast and robust. The model of two-phase flow in porous media can be solved numerically using a fully implicit scheme [1; 2; 3], or using an implicit-explicit (IMEX) one [4; 5]. The implicit schemes are unconditionally stable but they are time-consuming and expensive from a computational point of view. Therefore, despite IMEX schemes being conditionally stable, they are usually chosen to solve large-scale systems such as oil reservoir models using the IMPES approach. The IMPES schemes have been improved many times (e.g., [6; 7]).  
The iterative IMPES method is considered to be one of the improved versions of IMPES for two-phase flow [8; 9; 10]. The IMPES approach divides the system of the equation into the pressureequation and saturation equation which are solved sequentially. Iterative IMPES uses IMPES instead of Newton iteration as an iterative scheme for full implicit systems. In some iterative schemes [11; 12], the pressure equation is implicitly solved and for each iteration, an implicit saturation equation with the implicit capillary pressure is constructed and solved. The iterative method was used along with the IMPES scheme by [PERSON] [13] and [PERSON] et al. [14]. In [13], an iterative version of the IMPES scheme was introduced. [PERSON] and [PERSON] [13] linearized the capillary pressure function to be used to couple the implicit saturation equation into the implicit pressure equation. In [14], an iterative implicit pressure, explicit saturation, implicit concentration scheme was developed to solve the model of nanoparticle transport with two-phase flow in porous media.The linearized capillary pressure function was also used to couple the implicit saturation and pressure equations, while the concentration equations were treated implicitly. Recently, [PERSON] et al. [15] introduced a convergence analysis of the nonlinear iterative method for two-phase flow and nanoparticle transport in porous media.  
The non-isothermal two-phase flow in porous media appears in many natural and industrial applications such as steam injection in oil recovery, geothermal reservoirs, and other environmental and engineering applications [16; 17; 18].  
In the current work, we develop an iterative implicit pressure, explicit saturation, implicit temperature (I-IMPES-IMT) scheme to simulate the problem of non-isothermal two-phase flow in porous media. To the best of the author's knowledge, no research develops a similar scheme to simulate the non-isothermal two-phase flow in porous media. The remaining sections of this paper are arranged as follows: in Section 2, we develop a mathematical model consisting of a group of differential equations, algebraic constraints, and initial/boundary conditions. Then, in Section 3, we introduce an iterative scheme for the governing equation. The spatial discretization is presented in Section 4. In Section 5, error estimation and solution verification are discussed and some numerical examples are listed. Finally, the conclusion is drawn in Section 6.