## 2 Methodology  
A synthetic reservoir model is constructed with the typical properties of the oil sands reservoirs in Athabasca, AB, Canada. The 2D homogeneous reservoir is assumed to eliminate the effects of mass transfer along to the direction of horizontal well, i.e., the \(j\) direction, and also those of heterogeneous rock properties affecting the fluid flow between the water-bearing zone and the oil sands deposit. Table 1 summarizes the reservoir properties and Table 2 shows the molecular weight and viscosities of the oil with temperature. The molecular weight of nitrogen is 28.013 kg/kg-mol. The critical pressure and temperature of nitrogen are 3394 kPa and \(-\)146.95 \({}^{\circ}\)C (degrees Celsius). The solubility of nitrogen in liquid is assumed that its value is small enough to be ignored. The effect of solution gas dissolved in oil is neglected so that the gas phases are assumed as steam and nitrogen. The gas-liquid \(K\) value of nitrogen is assumed as a function of pressure and temperature (see Equation (1)). \[K=\frac{a}{P}\exp(\frac{b}{T-c}) \tag{1}\]  
In Equation (1), \(K\) denotes the \(K\) value of nitrogen. \(P\) represents the pressure (kPa) and \(T\) is temperature (\({}^{\circ}\)C). The coefficients \(a\), \(b\), and \(c\) are 416,360, \(-\)588.72, and \(-\)266.55, respectively. The mole fraction (mol %) of the injected-steam amount determines the nitrogen volume. The temperatures of the steam and nitrogen are both 250 \({}^{\circ}\)C. The horizontal permeability (\(i\) and \(j\) directions; lateral direction) is 2.5 Darcy, and the vertical permeability is 1.25 Darcy. Table 3 shows the operational constraints of the SAGP process. Figure 1 depicts the top water-bearing zones and an oil sands deposit, wherein Figure 1a shows a finite aquifer, and Figure 1b illustrates an infinite aquifer. The upper area of both aquifers is assumed as impermeable. One injector is installed at 255 mTVD (true vertical depth), and a producer is located at 5 m below the injector. The vertical thickness of the bitumen formation is 15 m, and that of the top water-bearing zone is 13 m. The horizontal distance of the bitumen layer is 123 m. The grid system of the bitumen layer is rectangular; the grid size comprises \((\Delta i,\Delta j,\Delta k)\) = (1 m, 20 m, 0.5 m), and its grid number is \((N_{i},N_{j},N_{k})\) = (123, 1, 30). Both of the formation-volume factors of the oil and water are assumed as 1. The stock tank oil initially in place (_STOIIP_) is calculated as 8821.8 m\({}^{3}\). The grid organizations of the finite aquifer (see Figure 1a) are \((\Delta i,\Delta j,\Delta k)\) = (1 m, 20 m, 1.3 m) and \((N_{i},N_{j},N_{k})\) = (123, 1, 10). In the infinite aquifer, the aquifer size is increased so that its grid numbers are \((N_{i},N_{j},N_{k})\) = (137, 1, 10) (see Figure 1b). To demonstrate the constant pressure boundaries and the continuous mass transfer in the infinite aquifer, four image wells are located at the edges of the aquifer boundaries, as illustrated in Figure 1b. In consideration of the gravity force, the pressures of the image wells are set as 1524 kPa for the injector and 1528 kPa for the producer to prevent a natural downward flow. The SAGP is modeled using the thermal simulator, STARS (Computer Modelling Group, Calgary, AB, Canada) [22]. The following three parameters are implemented to evaluate the energy efficiency of the SAGP operation: steam to oil ratio (_SOR_; Equation (2)), cumulative steam to oil ratio (_cSOR_; Equation (3)), and simple thermal efficiency parameter (_STEP_; Equation (4)). \[SOR=\frac{V_{s,t}}{V_{o,t}} \tag{2}\]  
\[cSOR=\frac{\sum\limits_{t=0}^{n}V_{s,t}}{\sum\limits_{t=0}^{n}V_{o,t}} \tag{3}\]