Finite Element Modelling of Three-Phase Non-Isothermal Flow in Heavy Oil Reservoirs - SAGD Case Study

Introduction to thermal enhanced oil recovery (TEOR) techniques with a focus on steam injection for heavy oil reservoirs. The study explores the impact of temperature on viscosity using API classification of crude oil. Mathematical models and equations for continuity and momentum in SAGD configurations are discussed.

• Finite Element Modelling
• Non-Isothermal Flow
• Heavy Oil Reservoirs
• SAGD
• Thermal Enhanced Oil Recovery

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1. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Taofik H. Nassan

2. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Introduction Today nearly 50 % of crude oil produced by EOR techniques is attributed to steam injection (EOR methods classification ,By Taofik Nassan) Comsol Conference 2020 Europe 2

3. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Introduction to TEOR by steam injection Temperature effect on viscosity API classification of crude Crude type API Viscosity , cp <100 Remarks Light/Convention al oil Heavy oil Extra-Heavy crude <10 Natural bitumen >22 <22 >100 >10,000 >1000,00 0 <10 also called Tar or Oil sands (source: www.api.org) The empirical relationship of crude oil viscosity and temperature depends on oil composition (Adapted after L.W. Lake 1989) Comsol Conference 2020 Europe 3

4. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Introduction to TEOR by steam injection Steam flooding SAGD Huff &Puff CSS (All figures sketched by Taofik Nassan) SAGD Steam flooding Comsol Conference 2020 Europe 4

5. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD SAGD configuration (SAGD configuration ,By Taofik Nassan) Comsol Conference 2020 Europe 5

6. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Mathematical model Continuity equation ?(?????) ?? ?(?????) ?? ?(?????) ?? + ?. ???? = ??,??? ...........(1?) + ?. ???? = ??,???+ ??,???+ ?????,? .. (1?) + ?. ???? = ??,???+ ??,??? ?????,? .(1?) ? ??????+ ???? + ?. ????+ ???? = ??,???+ ??,??? ??,???+??,??? .(2) ? Momentum equation ??= ??? ??? ????? . ... (3?) ?? ??= ??? ??? ????? . (3?) ?? ??= ??? ??? ????? . (3?) ?? Comsol Conference 2020 Europe 6

7. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Mathematical model Energy equation ? ?? ? ????????? + ?. ???? ? ?. ???? + ??,??? ??,???= 0 ...(4?) ? ????????? + ?. ???? ? ?. ???? + ??,??? ??,???= 0 . (4?) ? ??(1 ?)???? ?. ???? = 0 . .. .(4?) ??????? + ?. ???? ? ?. ???? ??,???= 0 ... .. .(4?) Total energy : ? ?? ?. ???? ??,???+??,??? ??,???+ ??,??? ??,???= 0 .. (5) ???????+ ???????+ ???????+ (1 ?)???? + ?. ???? ?+ ???? ?+ ???? ? Where ??= ??? ??+ ??? ??+ ??? ??+ (1 ?)?? Comsol Conference 2020 Europe 7

8. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Mathematical model ? = ??+ ??+ ?? (6 ) Pressure equation ?? ?? ??? ?? +?? ??? ??+?? ??? ?? ?? ?? ??? ?? ??? ?? ?.? = ? ??? ..(7) ?? ?? ?? ?? Water saturation equation can be obtained from (2) ???? 1 ??? ?? +?? ?? ???? ?? ?.?? 1 ??? ??+????? .(8) + ?.??= ??????? ????? ??? ?? ?? Oil saturation equation for (1a) ???? ??+ ?.??= 1 ??(????) + ????? . .(9) ?? ?? Comsol Conference 2020 Europe 8

9. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Mathematical model Energy equation can be arranged after taking latent heat of steam into consideration as ?? ??+ ???????+ ???????+ ????????.? ????????+ ????????+ ????????+ (1 ?)????? ?? ???? ?? ?. ? ?? +??? +????? ??,???+??,??? ??,???+ ??,??? ??,???= 0 . ..(10) Three-phase Relative permeability is calculated from Stone II [7] ?????,?? = ????+ ??? ????+ ??? ???+ ??? ..(11) Steam water equilibrium temperature depends on pressure: ? = ??? . .. .. . .(12) The relation among phases saturations: ??=??+ ??+ ??= 1 . . ..(13) Comsol Conference 2020 Europe 9

10. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Mathematical model Capillary pressure relations between every two phases reads: ??????,? = ?? ?? .. ...(14) ??????,? = ?? ?? . . . .(15) Assuming that the fractional flow is only dependent on saturations, capillary pressure can be written as: ???= ??????? ??????? .. .(16) Defining the global pressure by: ?? = ???+ ??? . ..(17) For further simplification, the capillary pressure will be neglected which means that the global pressure is the oil pressure: ?? = ??? ... .(18) Comsol Conference 2020 Europe 10

11. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Mathematical model Pressure equation is applied in the general form PDE in Comsol : ??? ???+ ?? ?? ??+ ?.? = ? ... ..(19) ?? ? ??, ? ?? Where ? = ? Consists of the components of velocity uxand uy f is the right hand side of equation (7) Saturation and energy equations are applied in the coefficient form PDE in Comsol : ??? ???+ ?? ?? ??+ ?. ??? ?? + ? + ?.?? + ?? = ? ....(20) ?? Comsol Conference 2020 Europe 11

12. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Results Oil saturation after 4 days Oil saturation after 75 days Comsol Conference 2020 Europe 12

13. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Results Temperature distribution after 75 days Steam saturation after 75 days Comsol Conference 2020 Europe 13

14. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Conclusions - Suitable formulation of three phase non-isothermal flow in porous media is introduced - Finite element method is applied to the model through a simple 2D-SAGD model which is solved efficiently using COMSOL 5.5 - Saturation and heat distribution are shown as results Comsol Conference 2020 Europe 14

15. Finite element modelling of three-phase non-isothermal flow in heavy oil reservoirs- case study SAGD Thank you Comsol Conference 2020 Europe 15