Numerical Solution of Equation for Dynamic, Spontaneous Imbibition with Variable Inlet Saturation and Interfacial Coupling Effects
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Abstract
In the oil industry, dynamic spontaneous imbibition plays an important role in several flow processes in porous media. A numerical approach is developed to simulate dynamic spontaneous imbibition with variable inlet saturation and interfacial coupling. The inclusion of interfacial coupling effects invalidates the assumption that the interfaces (fluid/fluid and fluid/solid) act in the same way. The one-dimensional numerical simulation model is developed using a Lagrangian formulation discretized in time and saturation. The solution of the partial differential equations utilizes an iteration process that includes two material balance criteria to ensure the validity of the variable inlet saturation. Furthermore, an error analysis, the validation of the model and a sensitivity study on the optimal number of time steps and saturation grid cells are undertaken. The numerical simulation solution represents an accurate approach to investigate the effect of fluid and rock properties on dynamic spontaneous imbibition.
Keywords
Dynamic spontaneous imbibition Lagrangian formulation Interfacial coupling Variable inlet saturation Counter-current two-phase flow Immiscible flowList of Symbols
- a
Experimentally determined parameter
- A
Cross-sectional area of core
- \({\hat{{\rm A}}}\)
Slope obtained from plotting \({\overline Q }\) versus \({\sqrt{\tau }}\)
- c
Parameter that controls the amount of viscous coupling
- C(S)
Capillary function
- F1
Non-capillary fractional flow function, \({F_1 =\frac{R_{12}\lambda_1^0 }{R_{12}\lambda_1^0 +\lambda_2^0 }}\)
- G(S)
Gravity function
- \({k_{_{r2}}^0}\)
Relative permeability of the non-wetting phase measured in co-current flow
- Mr
End point mobility ratio
- Ng
Gravity number, \({\frac{A\lambda_{1r}^0 \Delta \rho^{\prime}g\sin (\theta)}{q_o}}\)
- \({\overline{{q}}}\)
Dimensionless normalized imbibition flow rate
- \({\overline{{\overline{{q}}}}}\)
Average dimensionless normalized imbibition flow rate
- q0
Characteristic flow rate
- \({\overline Q}\)
Dimensionless normalized cumulative imbibition production
- \({\overline Q_{\rm Analytical}}\)
Dimensionless normalized cumulative imbibition production calculated using Eq. 30
- \({\overline Q_S }\)
Area under the curve ξ (S, τ) versus S
- \({\overline Q_\tau }\)
Area under the curve \({\overline{{q}}({S^{\ast},\tau })}\) versus τ
- R1
Ratio between dimensionless time and dimensionless cumulative imbibed water
- R12
Hydrodynamic effect: function relating the steady-state, potential gradient in phase 1 to that in phase 2, R 12 = 1 − a(1 − S)
- S
Normalized wetting phase saturation
- Soi
Initial oil saturation
- Sro
Residual oil saturation
- Swi
Irreducible water saturation
- S*
Normalized variable inlet saturation
- SFactor
Value determined from the iteration attempts to find S*
- \({S_2^{\ast}}\)
New assumed value of variable inlet saturation
Greek Symbols
- αi
Interfacial coupling factor = α ci · α vi , i = 1, 2
- αci
Capillary coupling parameters, i = 1, 2
- αvi
Viscous coupling parameters, i = 1, 2
- Δ ρ′
Difference in density without hydrodynamic effect
- λ (S)
Average coupling effect, \({\lambda (S)=\left({\frac{\alpha_1 +\alpha_2}{2}}\right)}\)
- \({\lambda_i^o }\)
Mobilities measured in a steady-state co-current (SSCO) flow experiment, i = 1, 2
- \({\lambda_{1r}^0 }\)
Effective mobility to wetting phase at residual saturation of non-wetting phase
- \({\lambda_{2r}^0 }\)
Effective mobility to non-wetting phase at initial saturation of wetting phase
- \({\lambda_m^0 }\)
Parameter to ensure dimensional consistency
- ξ
Normalized distance measured from the inlet surface of a core
- πc (S)
Dimensionless capillary pressure
- ρ
Fluid density
- τ
Dimensionless time
- τ0
Characteristic time
- \({\phi}\)
Porosity of the porous medium
Superscripts
- n
Present time step
- n + 1
Next time step
Subscripts
- 1
Wetting phase
- 2
Non-wetting phase
- i ± z
Saturation grid cell position according to Fig. 1: z = 1/2, 1, 3/2, 2, 5/2
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