# Sensitivity Analysis for Dynamic Spontaneous Imbibition with Variable Inlet Saturation and Interfacial Coupling Effects

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## Abstract

Dynamic spontaneous imbibition (DSI) plays an important role in oil reservoir characterization. Conventional equations used to characterize DSI consider neither interfacial coupling effects (ICE) nor variable inlet saturation (*S**) for DSI. Yazzan et al. (Transp Porous Media 87(1):309–333, 2011a; 86(3):705–717, 2011b) developed a set of equations, and a numerical solution scheme, to take into account ICE and variable *S** for DSI. Based on these, a graphical user interface (GUI) simulator was built. A sensitivity analysis has been conducted to study the effect of the fluid and rock properties on DSI. The results reveal that including a variable *S** has no significant impact; however, neglecting ICE results in an overestimation of the imbibition flow rate. Moreover, it is shown that the capillary and relative permeability curves determine the type of frontal advance, and that the imbibition recovery is proportional to the square root of time.

## Keywords

Dynamic spontaneous imbibition Interfacial coupling Transport equations Two-Phase flow Immiscible flow## List of symbols

## Variables

*a*Experimentally determined parameter in definition for

*R*_{12}*a*_{i},*b*_{i},*c*_{i},*d*_{i}Fitting coefficients

*i*= 0, 1, 2*c*Experimentally determined parameter which controls amount of viscous coupling

- \({\hat{A}}\)
Slope obtained from plot of \({\overline Q}\) versus \({\sqrt{\tau}}\)

*A*_{c}Area under the capillary pressure curve

*C*(*S*)Capillary function

*F*_{1}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*_{r1}Relative permeability to the wetting phase

*k*_{r2}Relative permeability to the non-wetting phase

*L*Length of porous medium

*M*_{r}End point mobility ratio

*N*_{g}Gravity number, \({N_{\rm g} = \frac{\Delta \rho^{\prime}g {\rm sin} (\theta )L}{A_{\rm c}}}\)

*P*_{c}Capillary pressure

- \({\overline{q}}\)
Dimensionless normalized imbibition flow rate

- \({\overline{q}}\)
Dimensionless normalized cumulative imbibition production

*R*_{12}Hydrodynamic factor,

*R*_{12}= 1−*a*(1 −*S*)*S*Normalized wetting phase saturation

*S**Normalized variable inlet saturation

## Greek symbols

*α*_{i}Interfacial coupling factor,

*α*_{ i }=*α*_{c}*i*·*α*_{v}*i*,*i*= 1, 2*α*_{ci}Capillary coupling parameters,

*i*= 1, 2;*α*_{c1}=*α*_{c2}=*α*_{c}= 1 −*φ**α*_{vi}Viscous coupling parameters,

*i*= 1, 2; \({\alpha _{v1} = 1 - \frac{c}{R_{12}} \frac{\lambda _2^0}{\lambda _m^0}}\) ; \({\alpha _{v2} = 1 - cR_{12} \frac{\lambda _1^0}{\lambda _m^0}}\)*Δρ′*Difference in density without hydrodynamic effect

*λ*(*S*)Average coupling effect, \({\lambda (S) = ( {\frac{\alpha _1+\alpha _2}{2}})}\)

- \({\lambda_i^o }\)
Mobilities measured in a steady-state co-current (SSCO) flow experiment,

*i*= 1, 2*λ*_{1r}Effective mobility to the wetting phase at its residual saturation to the non-wetting phase

*λ*_{2r}Effective mobility to the non-wetting phase at its irreducible saturation to the wetting phase

- \({\lambda_m^0 }\)
Parameter to ensure dimensional consistency, \({\lambda _m^0 = S\lambda _{1r}^0 + (1 - S)\lambda _{2r}^0 }\)

*ξ*Normalized distance measured from the inlet surface of a core

*π*_{c}(*S*)Dimensionless capillary pressure

*θ*Angle of inclination of the core to the horizontal

*ρ*Fluid density

*τ*Dimensionless time

*φ*Porosity of the porous medium

## Superscripts

- 0
Denotes the parameter is for co-current flow

## Subscripts

- 1
Wetting phase

- 2
Non-wetting phase

- nw
Non-wetting phase

- w
Wetting phase

- r
Residual

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