# Experimental and Analytical Investigation of Spontaneous Imbibition in Water-Wet Carbonates

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## Abstract

We perform co-current spontaneous imbibition ambient-condition experiments in three carbonates with a wide range of permeability under strongly water-wet conditions. We measure water saturation profiles as a function of distance and time in air-filled rocks with no initial water saturation using X-ray CT scanning. We demonstrate that the saturation profiles are functions of distance divided by the square root of time. We also demonstrate that the profiles are consistent with analytical solutions for imbibition in one dimension, and using reasonable estimates of relative permeability and capillary pressure, we can match the experimental results. We discuss how, in combination with conventional measurements of relative permeability (steady-state or using Buckley–Leverett theory in an unsteady-state experiment), the capillary pressure can be determined, or how the relative permeability can be determined from the spontaneous imbibition experiment and the capillary pressure.

## Keywords

Spontaneous imbibition Analytical solution Relative permeability Capillary pressure Capillary-dominated flow Carbonates## List of Symbols

- \(\alpha \)
Capillary pressure exponent

- \(\lambda _\mathrm{{nw}}\)
Mobility of the non-wetting phase

- \(\lambda _\mathrm{{t}}\)
Total mobility

- \(\lambda _\mathrm{{w}}\)
Mobility of the wetting phase

- \(\rho _\mathrm{{nw}}\)
Density of the non-wetting phase

- \(\rho _\mathrm{{w}}\)
Density of the wetting phase

- \(\mu _\mathrm{{nw}}\)
Viscosity of the non-wetting phase

- \(\mu _\mathrm{{w}}\)
Viscosity of the wetting phase

- \(\phi \)
Porosity

- \(\omega \)
Imbibed distance over square root of time

*C*Proportionality constant used in the analytical solution

- \({{D}}(S_\mathrm{{w}}\))
Capillary dispersion coefficient

*f*Fractional flow for viscous-dominated flow

- \(f^{\prime }\)
First derivative of

*f**F*Fractional flow for capillary-dominated flow

- \(F^{\prime }\)
First derivative of

*F*- \(F^{\prime \prime }\)
Second derivative of

*F*- \(g_x\)
Gravitational acceleration

*k*Permeability

- \(k_\mathrm{{r}}\)
Relative permeability

- \(k_\mathrm{{rg}}\)
Gas relative permeability

- \(k_{\mathrm{{rg}}, \mathrm{{max}}}\)
Maximum gas relative permeability

- \(k_\mathrm{{rw}}\)
Water relative permeability

- \(k_\mathrm{{rw,max}}\)
Maximum water relative permeability

*m*Corey gas exponent

*n*Corey water exponent

- \(P_\mathrm{{c}}\)
Capillary pressure

- \(P_\mathrm{{c,entry}}\)
Entry capillary pressure

- \(P_\mathrm{{nw}}\)
Non-wetting phase pressure

- \(P_\mathrm{{w}}\)
Wetting phase pressure

- \(q_\mathrm{{t}}\)
Total Darcy velocity

- \(q_\mathrm{{nw}}\)
Non-wetting phase Darcy velocity

- \(q_\mathrm{{w}}\)
Wetting phase Darcy velocity

- \(S_\mathrm{{gr}}\)
Residual gas saturation

- \(S_\mathrm{{w}}\)
Water saturation

- \(S_\mathrm{{wi}}\)
Initial water saturation

- \(S_\mathrm{{wir}}\)
Irreducible water saturation

*t*Time

- \(v_\mathrm{{sh}}\)
Shockfront moving speed

- \(v_\mathrm{{shD}}\)
Dimensionless shockfront moving speed

*x*Distance

## Notes

### Acknowledgments

We would like to acknowledge funding from the Qatar Carbonates and Carbon Storage Research Centre, QCCSRC, which is supported jointly by Qatar Petroleum, Shell and the Qatar Science & Technology Park.

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