## 3 Numerical Analysis  
### Numerical Model  
In this section, numerical simulation is performed to investigate the bottom-up assisted pressure drive process. As shale barriers are widely present in oil sand reservoirs, a numerical simulation model with multiple shale layers is developed. The simulation model is built with a commercial simulator (CMG STARS, Version 2019, Computer Modeling Group Ltd.). A two-dimensional Cartesian reservoir model is built to represent a typical oil sand formation. The width and thickness of the numerical model are 75 m and 30 m, respectively. A typical grid size of 1 m \(\times\) 1 m is used in the numerical model. There are 75 grid blocks in the x-direction and 30 grid blocks in the vertical direction. Typical Athabasca oil sand reservoir properties are used in this work, as presented in Table 1[32; 33]. Bitumen from the Athabasca region is used in this work, and its properties are listed in Table 2[34]. The bitumen viscosity as a function of temperature is depicted in Figure 2. In this work, bitumen is treated as a pseudo-component. This work provides a preliminary evaluation of bottom-up assisted pressure drive. The same bitumen properties are used to compare the performance between bottom-up assisted pressure drive and SAGD processes. It should be noted that bitumen properties are complex [35; 36]. More detailed compositional analysis will be performed in future work. Relative permeability curves derived from an underground test facility (UTF) are used in this work [5; 32]. In this work, both SAGD and bottom-up assisted pressure drive processes are simulated. The typical horizontal wells are included in both processes, and the horizontal well length is 800 m. In the SAGD process, a pair of horizontal wells consisting of an injection well and a production well are included in the model, and the vertical distance between the injector and producer is 5 m. In the bottom-up assisted pressure drive process, the injector and producer are positioned near the bottom of the reservoir at the same depth. The horizontal distance between the injector and the producer is 37 m. In order to be comparable with the simulation results of the SAGD process, one injector and two half-producers are included in the simulation model of the bottom-up assisted pressuredrive. The well fractions of these two producers are set as 0.5 in the simulation. In this way, the number of wells in both processes is equivalent. The same well constraints are set in both processes, in which the injection pressure is 2300 kPa, and the steam rate in the production well is set as 10 m\({}^{3}\)/day. This is the same as those in the literature [33]. In the numerical model, multiple shale barriers are also considered. The properties of the shale barriers are listed in Table 3, which is the same as that in the literature [37,38]. Although shale barriers are complex in real reservoirs, a simplified representation of shale configuration is applied in the simulation. The schematic of the grid systems for the two processes is presented in Figure 3 with respect to the well locations and shale barrier distributions. In order to simulate the dilated zone, a one-meter thickness dilated zone is set at the bottom of the formation connecting the injector and producer. The permeability and porosity of this dilated zone are 10 Darcy and 0.5, respectively (as shown in Figure 3b). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Unit** & **Value** \\ \hline Porosity & - & 0.3 \\ Horizontal permeability & mD & 5000 \\ Vertical permeability & mD & 1000 \\ Formation depth & m & 280 \\ Initial temperature & \({}^{\circ}\)C & 12 \\ Reference pressure & kPa & 1300 \\ Initial water saturation & - & 0.17 \\ Initial oil saturation & - & 0.83 \\ Rock heat capacity & kJ/(m\({}^{3}\)-C) & 1200 \\ Rock thermal conductivity & kJ/(m\(\cdot\)d\(\cdot\)C) & 650 \\ Over/underburden heat capacity & kJ/(m\({}^{3}\)-C) & 2370 \\ Over/underburden thermal conductivity & kJ/(m\(\cdot\)d\(\cdot\)C) & 202 \\ \hline \hline \end{tabular}
\end{table}