Guidelines for Numerically Modeling Co- and Counter-current Spontaneous Imbibition


We present guidelines for accurately simulating both co- and counter-current spontaneous imbibition (SI) phenomenon in 1D systems. We first consider several cases for this study, which involve strongly water-wet, weakly water-wet and mixed-wet wettability states, to simulate co- and counter-current SI in an oil–water system. We create two separate 1D models on a numerical simulator to simulate and obtain saturation profiles for the different cases. We then match simulation results with saturation profiles obtained through the capillary dominated flow semi-analytical solution proposed by Schmid et al. (Water Resour Res 47(2), 2011, SPE J 21:2–308, 2016). The numerical study evaluates the effect of model orientation and co-ordinate system on the saturation profiles. Moreover, we perform grid sensitivity analysis to choose the optimal number of grid cells, as well as the optimal time steps for the model. We find that capturing 0.25% of core volume in each grid cell is sufficient to numerically model an SI experiment within the acceptable margin of error of 5%. Simulations are performed for 23 different cases based on the SI mode, wettability and mobility ratios. The simulation results in saturation profiles have a mean absolute percentage error from the profile obtained from the semi-analytical solution between 0.14 and 5.41% for counter-current SI for the different wettability states. For most wettability states for the co-current SI, however, we do not get a close match, indicating that the semi-analytical solution does not hold for co-current SI. The paper lists some useful guidelines for simulating SI phenomenon, such as selecting the optimum number of grid cells for the SI model and accounting for capillary backpressure, which could be extended to be applied for simulating coreflooding experiments. This paper also discusses current limitations of the semi-analytical solution. These calibration and sensitivity studies can significantly improve the accuracy of the simulation results.

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  1. 1.


F :

Capillary dominated fractional flow


First derivative of F


Second derivative of F

f :

Buckley–Leverett fractional flow

ɸ :


C :

Imbibition constant (m/√s)

D :

Capillary dispersion coefficient (m2/s)

k :

Permeability (m2)

λ w :

Water mobility (1/Pa s)

λ o :

Oil mobility (1/Pa s)

λ t :

Total mobility (1/Pa s)

dP c /dx :

Capillary pressure gradient (Pa/m)

q w :

Water flow rate (m3/s)

q o :

Oil flow rate (m3/s)

k rw :

Water relative permeability

k ro :

Oil relative permeability

k rw max :

Maximum water relative permeability

k ro max :

Maximum oil relative permeability

S w :

Water saturation

S *w :

Water saturation when capillary pressure is zero

S o :

Oil saturation

S wi :

Initial water saturation

S or :

Residual oil saturation

P c :

Capillary pressure (Pa)


Recovery factor

P c :

Capillary pressure (Pa)

P entry :

Entry capillary pressure (Pa)

P cb :

Capillary back pressure (Pa)

n :

Wetting phase exponent

m :

Non-wetting phase exponent

l :

Capillary pressure exponent

A t :

Actual value

F t :

Forecast value

n i :

Total number of data points

t :

Time (s)

x :

Distance in the core (m)

ω :

Scaling factor (m/√s)


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We would like to thank Texas A&M University at Qatar for funding this project. In addition, we would like to thank Prof. Martin J. Blunt for his insightful comments and suggestions.

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Correspondence to Nayef Alyafei.

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Khan, A.S., Siddiqui, A.R., Abd, A.S. et al. Guidelines for Numerically Modeling Co- and Counter-current Spontaneous Imbibition. Transp Porous Med 124, 743–766 (2018).

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  • Spontaneous imbibition
  • Capillary dominated flow
  • Numerical simulation
  • Semi-analytical solution
  • Wettability