## 1 Introduction  
Steam-assisted gravity drainage (SAGD), one of the thermal operations for the recovery of extremely viscous oil, releases steam into the native bitumen formation and simultaneously produces the diluted fluid mixture via a production well. A steam chamber, i.e., a steam saturated zone, should grow vertically and spread laterally to obtain the effective gravitational force for successful SAGD operations [1; 2; 3; 4; 5; 6]. Impermeable caprock provides the lateral flow by blocking the vertical movement of the fluid mixture, and also to enlarge steam chambers at thin payzones. A significant amount of bitumen reserves comprises negligible caprock with a top water-bearing zone, e.g., oil sands reservoirs in the Surmont leases, Kearl Lake, and the Wabiskaw-McMurray formation [7; 8; 9].  
The top water-bearing zone near a native bitumen deposit, typically called a heat thief zone, can be problematic since it reduces the energy efficiency of thermal processes, or it interrupts the growth of the steam chamber due to a water influx or steam loss. Compared with the gas-bearing zone underlying or overlying the oil sands formation, the impact of the top water-bearing area on the energy efficiency, whereby a significant water influx into the reservoir and heat loss from the aquifer occur, is significant [10; 11].  
To isolate the top water-bearing zone with the oil sands deposit, the steam and gas push (SAGP) that injects non-condensable gas with SAGD operation has been proposed [12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. The non-condensable gas, e.g., carbon dioxide, nitrogen, and methane, remains as a gas phase within the steam chamber, is accumulated near the board of the steam chamber, and serves as a screen to isolate the human-made steam chamber from the thief zone. However, as the screen is thicker, the growth of the steam chamber is restricted, meaning that the energy efficiency of the thermal process can be reduced even when the blocking effect increases. The non-condensable gas obstructs the heat of the steam from passing through the native bitumen and, consequently, the inclined edges of the steam chamber fail to generate the gravitational forces that draw the optimum downward flow to the producer. An improper high fluid injection can result in a severe steam loss, while the lower pressure within the steam chamber would cause a water invasion from the top aquifer, leading to a significant reduction of the temperature within the steam chamber, thereby resulting in the failure of the enhanced bitumen recovery. A numerical simulation of the steam and non-condensable gas, e.g., nitrogen, push without the top water-bearing area showed that the gas conserved the heat of the steam chamber to accomplish the desired economics [17]. If a fixed-size aquifer exists, another numerical analysis showed that the nitrogen in the SAGD process could improve the oil productivity [19; 20]. The focus of the previous works is the formulation of a way to accomplish energy efficiency during the performance of the thermal operation to overcome the top water zone. The fixed-size thief zone surrounded by a no flow formation, e.g., a lens-shaped water zone, has been researched, and some of the problematic effects of non-condensable gas could be overlooked, such as the aquifer sizes.  
This study analyzes the possibility regarding the use of co-injected nitrogen as a non-condensable gas to overcome the problematic effects of the top water-bearing thief zone. The influence of the aquifer size is investigated in terms of its effect on the SAGP process. Two boundary conditions are evaluated as follows: the finite no-flow boundary and the continuous mass flux. To achieve field applicability, several constraints regarding the well operation are considered, i.e., the injected amount of nitrogen and the operation conditions at a production well (bottom hole pressure, surface liquid rate, and steam rate).