# Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media
## 5 Numerical Investigations  
In this section, we present error estimations for the time dependent variables, and introduce two numerical examples to check the validity and the efficiency of the proposed scheme. The rate of injection on the inlet boundary is used to calculate the normal velocity, \(\mathbf{u}_{i}=\phi d_{i}\) (PVI), here PVI is the pore volume injection and \(d_{i}\) is inlet diameter. Table 1 shows the physical parameters definitions and values. The error estimates for saturation and temperature are presented in Table 2 for different values of the time steps number (TSN) and iterations number (ITN). A domain of size 1.0 \(\times\) 0.5 m discretized by 40 \(\times\) 30 is used in this error estimation. Non-iterative calculations are carried out at 2000, 1000, 500, and 100 time steps. In addition, we carried out some calculations for 10, 30, and 50 iterations with 100 time steps. As there is no exact solution for such a complicated problem, we created a reference case of highly dense mesh and a very small time step which is very close to the exact solution. It is \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Units** \\ \hline \(T_{in}\) & 360 & K \\ \(T_{o}\) & 7.18 & W/(m/K) \\ \(h_{w}\) & 0.6 & W/(m/K) \\ \(h_{n}\) & 0.2 & W/(m/K) \\ \(\rho_{s}\) & 2500 & kg/m\({}^{3}\) \\ \(\rho_{w}\) & 1000 & kg/m\({}^{3}\) \\ \(\rho_{n}\) & 660 & kg/m\({}^{3}\) \\ \(c_{p,s}\) & 800 & J/kg\(\cdot\)K \\ \(c_{p,u}\) & 4000 & J/kg\(\cdot\)K \\ \(c_{p,n}\) & 2300 & J/kg\(\cdot\)K \\ \(B_{w}\) & 0.005 & K\({}^{-1}\) \\ \(\beta_{n}\) & 0.003 & K\({}^{-1}\) \\ \(S_{ur},S_{nr}\) & 0.001 & – \\ \(\phi\) & 0.3 & – \\ \(\mu_{w}\) & 1 & cP = 1.0 \(\times\) 10\({}^{-3}\) Pa\(\cdot\)s \\ \(\mu_{u}\) & 0.45 & cP \\ \(k_{r\
u 0},k_{r\
u 0}\) & 1 & – \\ \(P_{e}\) & 50 & bar = 1.0 \(\times\) 10\({}^{5}\) Pa \\ \hline \end{tabular}
\end{table} 
Table 1: Values of the physical parameters. under the same conditions but as mentioned above the non-iterative case is just a reference. Table 2 illustrates that the error deceases as the number of time steps and number of iterations increase. It is also clear that the temperature converges faster than saturation. Moreover, in Table 3, the comparison of error estimates of saturation with a previous work [15] are presented for 40 \(\times\) 30 of spatial discretization. The reference case is \(\approx\) 0.9565 is calculated at the point (0.06, 0.053) for non-iterative dense mesh of 200 \(\times\) 150 for the domain 0.6 \(\times\) 0.4 m over a total of 2000 time steps. The iterative case is taken after 100 time steps without iteration (ITN = 1), at ITN = 10, and ITN = 30. The comparison shows a good agreement. In the following, two examples are provided and discussed for heterogeneous media. The first example is a real field high-heterogeneous permeability media [27]. The spatial discretization is taken as \(120\times 50\) of a uniform rectangular mesh. A hot wetting fluid is injecting at a flow rate of 0.1 pore volume injection (PVI) into the left bottom corner of the domain. Four cases of injections are taken into consideration, i.e., continue until reaches 0.35, 0.5, 0.75, and 0.85 PV (pore volume). In this example, we used a time step size of 0.01 and a maximum number of iterations of 50. Figure 1 shows the contours of the saturation of the wetting phase after 0.35, 0.5, 0.75, and 0.85 PV. It is interesting to observe discontinuities in the saturation distribution caused by the heterogeneity. Moreover, the contours of temperature at the heterogenous medium are plotted in Figure 2. It seems to be that the heterogeneity of the medium has no effect on the temperature distribution. In the second example, the medium is artificial and heterogeneous, such that the it consists of two subdomains with different permeability distribution (0.1 md for the barrier and 100 md for the rest of the domain); a similar configuration to the first example. The spatial discretization is \(120\times 50\) of uniform rectangular mesh.