Abstract
Enhanced oil recovery (EOR) techniques have garnered the attention of the oil industry because they promise to extend the life of hydrocarbon reservoirs that have passed their peak of productivity. The heat and momentum transfer equations are the main governing equations for the thermal EOR processes. Thermal EOR, in particular, helps improve the extraction yield by injecting heat into the reservoir and altering the physical properties of porous rocks (e.g., permeability) and fluids therein (e.g., viscosity). Simulation modelling can help improve our understanding of the underlying thermal processes, but its prediction accuracy largely depends on the use of the correct energy balance equation as the governing equation. This study considered two established energy balance equations: the first without porosity (case I); and the second with porosity in terms of thermal energy transported by heat conduction (case II). We numerically solved the model equations and compare the porosity anomalies in both cases. Our results show that the inclusion of porosity in the heat conduction term of the energy balance equation has a significant impact on the temperature profile of the reservoir and dimensionless numbers in heat transfer because it increases the fluid storage capacity of the porous medium. Therefore, one should carefully select the appropriate energy balance equation for the defined porous medium. It is also established that Hossain–Abu-Khamsin numbers \(N_{HA3} \) and \(N_{HA4} \) become important when porosity is considered. Porosity plays a significant role in temperature distribution. In addition, the study showed that the thermal conductivity of fluids has more influence on the temperature profile than that of solid rocks. The findings have important implications for improving the yield and efficiency of thermal EOR.
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Abbreviations
- A :
-
Cross sectional area of rock perpendicular to the flow of flowing fluid (m\(^{2}\))
- \(c_\mathrm{f} \) :
-
Total fluid compressibility of the system (1/Pa)
- \(c_\mathrm{s} \) :
-
Formation rock compressibility of the system (1/Pa)
- \(c_\mathrm{t} \) :
-
Total compressibility of the system (1/Pa)
- \(c_\mathrm{pf} \) :
-
Total specific heat capacity of the formation fluids (kJ/kg K)
- \(c_\mathrm{pg} \) :
-
Specific heat capacity of gas (kJ/kg K)
- \(c_\mathrm{po} \) :
-
Specific heat capacity of oil (kJ/kg K)
- \(c_\mathrm{ps} \) :
-
Specific heat capacity of solid rock matrix (kJ/kg K)
- \(c_\mathrm{pw} \) :
-
Specific heat capacity of water (kJ/kg K)
- \(h_\mathrm{c} \) :
-
Convection heat transfer coefficient (kJ/h m\(^{2}\) K)
- k :
-
Reservoir permeability (m\(^{2}\))
- \(k_\mathrm{e} \) :
-
Overall effective thermal conductivity of fluid saturated porous media (kJ/h m K)
- \(k_\mathrm{eff} \) :
-
Local effective thermal conductivity of fluid saturated porous media (kJ/h m K)
- \(k_\mathrm{f} \) :
-
Absolute thermal conductivity of fluid within the porous rock matrix (kJ/h m K)
- \(k_\mathrm{g} \) :
-
Thermal conductivity of gas (kJ/h m K)
- \(k_\mathrm{o} \) :
-
Thermal conductivity of oil (kJ/h m K)
- \(k_\mathrm{s} \) :
-
Absolute thermal conductivity of solid rock matrix (kJ/h m K)
- \(k_\mathrm{w} \) :
-
Thermal conductivity of water (kJ/h m K)
- L :
-
Distance between production and injection well along x direction (m)
- \(L_\mathrm{c} \) :
-
Characteristic length related to pore-throat diameter (m)
- \(L^{*}\) :
-
Dimensionless length of the reservoir
- M :
-
Average system heat capacity (kJ/m\(^{3}\) K)
- \(N_\mathrm{PeL}\) :
-
\(L_\mathrm{c} \rho _\mathrm{f} c_\mathrm{pf} u_x /k_\mathrm{e}=\) local Peclet number (dimensionless)
- p :
-
Pressure of the system (Pa)
- \(p_\mathrm{i} \) :
-
Initial pressure of the system (Pa)
- \(p_0 \) :
-
A reference pressure of the system (Pa)
- \(q_\mathrm{i} \) :
-
Au = initial volume production rate (m\(^{3}\)/s)
- \(q_x \) :
-
Fluid mass flow rate per unit area in x-direction (kg/m\(^{2}\) s)
- \(q^{*}\) :
-
Dimensionless volume production rate
- \(q_\mathrm{inj} \) :
-
Injection volume flow rate of steam = Au (m\(^{3}\)/s)
- \(q_\mathrm{prod} \) :
-
Production volume flow rate of oil = Au (m\(^{3}\)/s)
- \(r_\mathrm{pt} \) :
-
Pore-throat radius (microns)
- \(S_\mathrm{g} \) :
-
Gas saturation (volume fraction)
- \(S_\mathrm{o} \) :
-
Oil saturation (volume fraction)
- \(S_\mathrm{w} \) :
-
Water saturation (volume fraction)
- \(S_\mathrm{wi} \) :
-
Initial water saturation (volume fraction)
- t :
-
Time (s)
- T :
-
Temperature (K)
- \(T_\mathrm{f} \) :
-
Temperature of injected fluid (K)
- \(T_\mathrm{s} \) :
-
Average temperature of solid rock matrix (K)
- \(T^{*}\) :
-
Dimensionless temperature
- \(T_\mathrm{HW} \) :
-
Temperature of injected fluid (K)
- \(T_\mathrm{HW}^*\) :
-
Dimensionless temperature of injected fluid
- \(T_\mathrm{i} \) :
-
Initial reservoir temperature (K)
- \(t^{*}\) :
-
Dimensionless time
- \(u_x \) :
-
Fluid velocity in porous media in the direction of x axis (m/s)
- \(u^{*}\) :
-
Dimensionless velocity
- x :
-
Flow dimension at any point along x-direction (m)
- \(x^{*}\) :
-
Dimensionless distance
- \(\alpha \) :
-
Fractional order of differentiation, dimensionless
- \(\alpha _\mathrm{H}\) :
-
\(k/\phi \mu c_\mathrm{t} =\) hydraulic diffusivity of the fluid-saturated porous medium (m\(^{2}\)/s)
- \(\Delta T\) :
-
Temperature difference (K)
- \(\upsilon \) :
-
\(\mu /\rho _\mathrm{f} =\) kinematic viscosity (ratio of absolute or dynamic viscosity to density) (m\(^{2}\)/s)
- \(\mu \) :
-
Fluid dynamic viscosity (Pa s)
- \(\rho \) :
-
Fluid density (kg/m\(^{3}\))
- \(\Delta \rho \) :
-
\(\rho _\mathrm{w} -\rho _\mathrm{o} =\) density difference of fluids (water and oil) (kg/m\(^{3}\))
- \(\phi \) :
-
Porosity of the rock matrix, volume fraction
- \(\rho _\mathrm{f} \) :
-
Weighted-average density of the formation fluids (kg/m\(^{3}\))
- \(\rho _\mathrm{g} \) :
-
Density of gas (kg/m\(^{3}\))
- \(\rho _\mathrm{o} \) :
-
Density of oil (kg/m\(^{3}\))
- \(\rho _\mathrm{s} \) :
-
Density of solid rock matrix (kg/m\(^{3}\))
- \(\rho _\mathrm{w} \) :
-
Density of water (kg/m\(^{3}\))
- API:
-
American petroleum institute
- rb:
-
Reservoir barrels
- stb:
-
Standard barrels
- scf:
-
Standard cubic feet
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Hossain, M.E. Role of Porosity on Energy Transport with Equal Rock-Fluid Temperatures During Thermal EOR Process. Arab J Sci Eng 42, 1621–1631 (2017). https://doi.org/10.1007/s13369-016-2343-8
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DOI: https://doi.org/10.1007/s13369-016-2343-8