Skip to main content
Springer Nature Link
Log in

An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

Abbreviations

Δρ (= ρ w − ρ o):

Density difference between the water and the oil phase (kg · m−3)

∞:

Infinite time (s)

\({\mathop S\limits^-}\) :

Spatial average normalized water saturation

1D:

One dimensional

A(t), B(t) :

Functions of time

a, b, z :

Exponents

COUCSI:

Counter-current spontaneous imbibition

e :

Euler’s number (2.718282…)

g :

Acceleration due to gravity vertically in downward direction (m · s−2)

\({{J}^{\prime}\left(=\frac{\text d}{\text d{S}_{\rm w}}(J({S}_{\rm w} ))\right)}\) :

Derivative of the dimensionless capillary pressure function with respect to water saturation

J(S w):

Dimensionless capillary pressure function

K :

Permeability (m2)

K ro :

Endpoint oil relative permeability

K rw :

Endpoint water relative permeability

L :

Length (m)

N :

Recovery (m3)

N p :

Ultimate recovery (m3)

P c :

Capillary pressure (Pa)

P o :

Oil pressure (Pa)

P w :

Water pressure (Pa)

R :

Ratio of gravity to capillary forces

S:

Normalized water saturation

S * :

Normalized water saturation after spontaneous imbibition

S or :

Residual oil saturation after forced imbibition

S w :

Water saturation

\({{S}_{{\rm w}}^{\ast}}\) :

Water saturation after spontaneous imbibition

S wi :

Initial water saturation

t :

Time (s)

t D,new :

New dimensionless time

t Ds :

Dimensionless stabilization time

t s :

Stabilization time (s)

W(z) :

Lambert’s W complex function

x :

Distance coordinate oriented positive upward (m)

x 0 :

Distance of imbibition front (m)

x D :

Dimensionless distance

\({{x}_{{\rm D}}^{{\rm 0}}}\) :

Dimensionless distance of imbibition front

\({{x}_{{\rm D,\;max}}^{{\rm 0}}}\) :

Maximum dimensionless distance of imbibition front

α, T :

Rate constant (s−1)

λo :

Oil mobility (Pa−1 · s−1)

λt (= λ w + λ o):

Total mobility (Pa−1 · s−1)

λ w :

Water mobility (Pa−1 · s−1)

μ o :

Oil viscosity (Pa·s)

μ w :

Water viscosity (Pa·s)

ρ o :

Oil density (kg · m−3)

ρ w :

Water density (kg · m−3)

σ :

Interfacial tension (N · m−1)

ø:

Matrix porosity

References

  • Aronofsky J.S., Masse L., Natanson S.G.: A model for the mechanism of oil recovery from the porous matrix due to water invasion in fractured reservoirs. Trans. AIME 213, 17–19 (1958)

    Google Scholar 

  • Babadagli, T., Hatiboglu, C.U., Hamida, T.: Evaluation of matrix-fracture transfer functions for counter-current capillary imbibition. Transp. Porous Media. (2009). doi:10.1007/s11242-009-9337-x

  • Barenblatt G.I., Entov V.M., Ryzhik V.M.: Theory of Fluid Flows Through Natural Rocks. Kluwer, Dordrecht (1990)

    Google Scholar 

  • Barentblatt G.: Filtration of two non-mixing fluids in a homogeneous porous medium, Soviet Academy Izvestia. Mech. Gas Fluids 5, 857–864 (1971)

    Google Scholar 

  • Barenblatt G.I., Vinnichenko A.P.: Non-equilibrium seepage of immiscible fluids. Adv. Mech. 3(3), 35–50 (1980)

    Google Scholar 

  • Barenblatt G.I., Gilman A.: A mathematical model of non-equilibrium countercurrent capillary imbibition. Eng. Phys. J. 53(3), 46–461 (1987)

    Google Scholar 

  • Barenblatt G.I., Patzek T.W., Silin D.B.: The mathematical model of non-equilibrium effects in water-oil displacement. SPEJ 8(4), 409–416 (2002)

    Google Scholar 

  • Barenblatt G.I., Patzek T.W., Silin D.B.: The mathematical model of non-equilibrium effects in water–oil displacement. SPEJ 8(4), 409–416 (2003)

    Article  Google Scholar 

  • Barry D.A., Parlange J.Y., Sander G.C., Sivaplan M.: A class of exact solutions for Richards’ equation. J. Hydrol. 142, 29–46 (1993)

    Article  Google Scholar 

  • Behbahani H.S., Di Donato G., Blunt M.J.: Simulation of counter-current imbibition in water-wet fractured reservoirs. JPSE 50, 21–39 (2006)

    Article  Google Scholar 

  • Chebbi R.: Dynamics of liquid penetration into capillary tubes. J. Colloid Interface Sci. 315, 255–260 (2007)

    Article  Google Scholar 

  • Chilingar G.V., Yen T.F.: Some notes on wettability and relative permeability of carbonate rocks. II. Energy Sources 7, 67–75 (1983)

    Google Scholar 

  • Civan, F.: Quadrature solution for waterflooding of naturally fractured reservoirs. SPE REE (April 1998), 141–147 (1998)

  • Delijani E.B., Pishvaie M.R.: Green Element solution of one-dimensional counter-current spontaneous imbibition in water wet porous media. JPSE 70, 302–307 (2010)

    Google Scholar 

  • Di Donato, G., Lu, H., Tavassoli, Z., Blunt, M.J.: Multirate-transfer dual-porosity modeling of gravity drainage and imbibition. SPEJ (March 2007), 77–88 (2007)

  • Dullien F.A.L.: Porous Media: Fluid Transport and Pore Structure. 2nd edn. Academic Press, San Diego (1992)

    Google Scholar 

  • Fries N., Dreyer M.J.: An analytical solution of capillary rise restrained by gravity. Colloid Interface Sci. 320, 259–263 (2008)

    Article  Google Scholar 

  • Guen S.S.L., Kovscek A.R.: Nonequilibrium effects during spontaneous imbibition. Transp. Porous Media 63, 127–146 (2006)

    Article  Google Scholar 

  • Gupta, A., Civan, F.: An improved model for laboratory measurement of matrix to fracture transfer function parameters in immicible displacement. Paper SPE 28929 presented at the SPE 69th Annual Technical Conference and Exhibition held in New Orleans, LA, USA, 25–28 (1994)

  • Høgnesen E.J., Standnes D.C., Austad T.: Experimental and numerical investigation of high temperature imbibition into preferential oil-wet chalk. JPSE 53, 100–112 (2006)

    Article  Google Scholar 

  • Ingham D.B., Pop I.: Transport Phenomena in Porous Media III. Elsevier, Oxford (2005)

    Google Scholar 

  • Juanes R.: Nonequilibrium effects in models of three-phase flow in porous media. Adv. Water Resour. 31, 661–673 (2009)

    Article  Google Scholar 

  • Kashchiev, D., Firoozabadi, A.: Analytical solutions for 1D countercurrent imbibition in water-wet media. SPEJ (December 2003), 401–408 (2003)

  • Kazemi H., Gilman J.R., Eisharkawy A.M.: Analytical and numerical solution of oil recovery from fractured reservoirs with empirical transfer functions. SPERE 7(2), 219–227 (1992)

    Article  Google Scholar 

  • Li, Y.: Analytical solutions for linear counter-current spontaneous imbibition in the frontal flow period. Transp. Porous Media. (2010). doi:10.1007/s11242-010-9656-y

  • Lukas, D., Soukupova, V.: Recent studies of fibrous materials wetting dynamics. In: INDEX 99 Congress, Geneva, Switzerland 1999

  • Ma S., Morrow N.R., Zhang X.: Generalized scaling of spontaneous imbibition data for strongly water-wet systems. JPSE 18, 165–178 (1997)

    Article  Google Scholar 

  • Mason G., Fischer S., Morrow N.R., Ruth D.W.: Correlation for the effect of fluid viscosities on counter-current spontaneous imbibition. JPSE 72, 195–205 (2010)

    Article  Google Scholar 

  • Mirzaei-Paiaman, A., Masihi, M., Moghadasi, J.: Formation damage through aqueous phase trapping in gas reservoirs. SPE paper 129637, SPE Deep Gas Conference and Exhibition, 24–26 January, Manama, Bahrain. (2010a). doi:10.2118/129637-MS

  • Mirzaei-Paiaman, A., Dalvand, K., Oraki, I., Masihi, M., Moghadasi, J.: Study on key influential factors of gas reservoir’s potential for aqueous phase trapping. Energy Sour. A Recover. Utili. Environ. Eff. (2010b) (in press)

  • Mirzaei-Paiaman A., Masihi M., Moghadasi J.: Formation damage through aqueous phase trapping: a review on the evaluating methods. Petrol. Sci. Technol. 29, 1–10 (2011). doi:10.1080/10916460903551073

    Article  Google Scholar 

  • Morel-Seytoux H.J.: Analytical results for prediction of variable rainfall infiltration. Hydrology 59, 209–230 (1982)

    Article  Google Scholar 

  • Morel-Seytoux, H.J.: Infiltration affected by air, seal, crust, ice and various sources of heterogeneity (special problems). In: Proceedings of ASAE National Conference on Advances in Infiltration, Chicago, 132–146 (1983a)

  • Morel-Seytoux, H.J.: Superiority of two-phase formulation for infiltration. In: Proceedings, ASAE National Conference on Advances in Infiltration, Chicago, 34–47 (1983b)

  • Morrow N.R., Mason G.: Recovery of oil by spontaneous imbibition. Curr. Opin. Colloid Interface Sci. 6, 321–337 (2001)

    Article  Google Scholar 

  • Nield D.A., Bejan A.: Convection in Porous Media. 3rd edn. Springer, New York (2006)

    Google Scholar 

  • Pop I., Ingham D.B.: Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media. Pergamon, Oxford (2001)

    Google Scholar 

  • Ruth, D.W., Arthur, J.K.: A revised analytic solution to the linear displacement problem including capillary pressure effects. Transp. Porous Media. (2010). doi:10.1007/s11242-010-9662-0

  • Silin D.B., Patzek T.W.: On Barenblatt’s model of spontaneous countercurrent imbibition. Transp. Porous Media 54, 297–322 (2004)

    Article  Google Scholar 

  • Standnes D.C.: Spontaneous imbibition of water into cylindrical cores with high aspect ratio: numerical and experimental results. JPSE 50, 151–160 (2006)

    Article  Google Scholar 

  • Standnes D.C.: Scaling spontaneous imbibition of water data accounting for fluid viscosities. JPSE 73, 214–219 (2010a)

    Article  Google Scholar 

  • Standnes D.C.: Scaling group for spontaneous imbibition including gravity. Energy Fuels 24, 2980–2984 (2010b). doi:10.1021/ef901563p

    Article  Google Scholar 

  • Standnes D.C.: A single-parameter fit correlation for estimation of oil recovery from fractured water-wet reservoirs. JPSE 71, 19–22 (2010c)

    Article  Google Scholar 

  • Tavassoli Z., Zimmerman R.W., Blunt M.J.: Analysis of counter-current imbibition with gravity in weakly water-wet systems. JPSE 48, 94–104 (2005)

    Article  Google Scholar 

  • Terez I.E., Firoozabadi A.: Water injection in water-wet fractured porous media: Experiments and a new model with modified Buckley-Leverett theory, SPE 56854. SPEJ 4(2), 134 (1999)

    Article  Google Scholar 

  • Treiber L.E., Archer D.L., Owens W.W.: A laboratory evaluation of the wettability of fifty oil producing reservoirs. SPEJ 12, 531–540 (1971)

    Google Scholar 

  • Vadasz P.: Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008)

    Book  Google Scholar 

  • Vafai K.: Handbook of Porous Media. 2nd edn. Taylor & Francis, New York (2005)

    Book  Google Scholar 

  • Vafai K.: Porous Media: Applications in Biological Systems and Biotechnology. CRC Press, Taylor Francis Group (2010)

    Book  Google Scholar 

  • Viksund, B.G., Morrow, N.R., Ma, S., Wang, W., Graue, A.: Initial water saturation and oil recovery from chalk and sandstone by spontaneous imbibition. SCA 9814. In: Proceedings of the International Symposium of the Society of Core Analysts, The Hague, (1998)

  • Zimmerman R.W., Bodvarsson G.S.: An approximate solution for one-dimensional absorption in unsaturated porous media. Water Resour. Res. 25, 1422–1428 (1989)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Azadi Ave., P.O. Box 11365-9465, Tehran, Iran

    Abouzar Mirzaei-Paiaman & Mohsen Masihi

  2. Statoil ASA Bergen, Sandslihaugen 30, 5020, Bergen, Norway

    Dag Chun Standnes

  3. Department of Petroleum Engineering, National Iranian South Oil Company, Ahvaz, Iran

    Abouzar Mirzaei-Paiaman

Authors
  1. Abouzar Mirzaei-Paiaman
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Mohsen Masihi
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Dag Chun Standnes
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Abouzar Mirzaei-Paiaman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mirzaei-Paiaman, A., Masihi, M. & Standnes, D.C. An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects. Transp Porous Med 89, 49–62 (2011). https://doi.org/10.1007/s11242-011-9751-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-011-9751-8

Keywords

Profiles

  1. Abouzar Mirzaei-Paiaman View author profile