Skip to main content

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

Abstract

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Abbreviations

a :

Pore radius (m)

A :

Cross-sectional area (\(\hbox {m}^{2}\))

\(A_\mathrm{p}\) :

Pore area in a cross section (\(\hbox {m}^{2})\)

\(A_\mathrm{f}\) :

Fracture area in a cross section (\(\hbox {m}^{2}\))

d :

Grain diameter (m)

\(D_\mathrm{f}\) :

Fractal dimension of pore size distribution (–)

\(D_\mathrm{ff}\) :

Fractal dimension of fracture network distribution (–)

\(D_\mathrm{R}\) :

Real dimension (–)

g :

Gravitational acceleration (\(\hbox {m }\hbox {s}^{-2})\)

h :

Width of a hydraulic aperture (\(\upmu \hbox {m}\))

P :

Pressure (Pa)

\(Q_\mathrm{HP}\) :

Volumetric flow rate in a tube (Hagen–Poiseuille law) (\(\hbox {m}^{3}\hbox { s}^{-1})\)

\(Q_\mathrm{D}\) :

Darcy velocity (\(\hbox {m }\hbox {s}^{-1})\)

s :

Radius of leakage area at specified location (m)

u :

Fluid velocity vector (\(\hbox {m }\hbox {s}^{-1})\)

w :

Length of a hydraulic aperture (m)

xyz :

Space coordinates (m)

\(\kappa \) :

Permeability (\(\hbox {m}^{2})\)

\(\mu \) :

Dynamic viscosity of fluid (Pa s)

\(\rho \) :

Fluid density (\(\hbox {kg }\hbox {m}^{-3})\)

\(\sigma \) :

Shear stress (Pa)

\(\tau \) :

Tortuosity (–)

\(\phi _\mathrm{m}\) :

Matrix porosity (–)

\(\phi _\mathrm{f}\) :

Fracture porosity (–)

p:

Pore

f:

Fracture

m:

Matrix

min:

Variable minimum

max:

Variable maximum

References

  • Allouche, J.P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  • Amthor, J.E., Okkerman, J.: Influence of early diagenesis on reservoir quality of Rotliegende Sandstones, Northern Netherlands. Am. Assoc. Petrol. Geol. Bull. 82(12), 2246–2265 (1998)

    Google Scholar 

  • Anderson, G.M., Macqueen, R.W.: Ore Deposit Models-6. Mississippi valley-type lead-zinc deposits. Geosci. Can. 9(2), 108–117 (1982)

  • Appelo, C.A.J., Postma, D.: Geochemistry, Groundwater and Pollution (Second Edition), 2nd edn. Balkema, Amsterdam (2005)

    Book  Google Scholar 

  • Archie, G.E.: Electrical resistivity an aid in core-analysis interpretation. AAPG Bull. 31(2), 350–366 (1947)

    Google Scholar 

  • Bachu, S.: CO2 storage in geological media: Role, means, status and barriers to deployment. Prog. Energy Combust. Sci. 34(2), 254–273 (2008)

    Article  Google Scholar 

  • Barenblatt, G.I., Zheltov, I.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24(5), 1286–1303 (1960)

    Article  Google Scholar 

  • Bayles, G.A., Klinzing, G.E., Chiang, S.-H.: Fractal mathematics applied to flow in porous systems. Particle Particle Syst. Charact. 6(1–4), 168–175 (1989)

    Article  Google Scholar 

  • Bear, J.: Dynamics of Fluids in Porous Media. Dover Publication Inc, New York (1972)

    Google Scholar 

  • Beard, D.C., Weyl, P.K.: Influence of texture on porosity and permeability of unconsolidated sand. AAPG Bull. 57(2), 349–369 (1973)

    Google Scholar 

  • Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  • Bloch, S.: Empirical prediction of porosity and permeability in sandstones. Am. Assoc. Petrol. Geol. Bull. 75(7), 1145–1160 (1991)

    Google Scholar 

  • Bloch, S., McGowen, J.H., Duncan, J.R., Brizzolara, D.W.: Porosity prediction, prior to drilling, in sandstones of the Kekiktuk Formation (Mississippian), North Slope of Alaska. Am. Assoc. Petrol. Geol. Bull. 74(9), 1371–1385 (1990)

    Google Scholar 

  • Bloch, S., Lander, R.H., Bonnell, L.: Anomalously high porosity and permeability in deeply buried sandstone reservoirs: origin and predictability. Am. Assoc. Petrol. Geol. Bull. 86(2), 301–328 (2002)

    Google Scholar 

  • Bogdanov, I.I., Mourzenko, V.V., Thovert, J.F., Adler, P.M.: Effective permeability of fractured porous media with power-law distribution of fracture sizes. Phys. Rev. E Stat. Nonlinear Soft Matter. Phys. 76(3), 1–16 (2007)

    Article  Google Scholar 

  • Bouchelaghem, F.: Flow study in a double porosity medium containing ellipsoidal occluded macro-voids. Math. Geosci. 43(1), 55–73 (2011)

    Article  Google Scholar 

  • Bourbié, T., Coussy, O., Zinszner, B.: Acoustics of Porous Media. Gulf Publishing Co, Paris (1987)

    Google Scholar 

  • Bradbury, M.H.: Geochemical near-field evolution of a deep geological repository for spent fuel and high-level radioactive waste. Nagra Technical Report NTB 12–01. Wettingen, Switzerland (2014)

  • Bratton, T., Gillespie, P., Li, B., Marcinew, R., Ray, S., Nelson, R., Schoderbek, D., Sonneland, L.: The nature of naturally fractured reservoirs. Oilfield Rev. 18(2), 4–23 (2006)

  • Carman, P.C.: Fluid flow through granular beds. Trans. Am. Inst. Chem. Eng. 15, 150–167 (1937)

    Google Scholar 

  • Carman, P.C.: Permeability of saturated sands, soils and clays. J. Agric. Sci. 29, 262–273 (1939)

    Article  Google Scholar 

  • Chilingarian, G.V.: Relationship between Porosity, Permeability and Grain Size Distribution of Sands and Sandstones. In: Van Straaten, J.U. (ed.) Deltaic and Shallow Marine Deposits, pp. 71–75. Elsevier Science Publ. Co., New York (1963)

    Google Scholar 

  • Chilingarian, G.V., Mazzullo, S.J., Rieke, H.H.: Carbonate reservoir characterization: a geologic-engineering analysis, part I., p. 639 (1992)

  • Civan, F.: A multi-purpose formation damage model. In: SPE Formation Damage Control Symposium, Society of Petroleum Engineers (1996)

  • Civan, F.: Fractal formulation of the porosity and permeability relationship resulting in a power-law flow units equation—a leaky-tube model. In: International Symposium and Exhibition on Formation Damage Control, Society of Petroleum Engineers (2002)

  • Civan, F.: Improved permeability equation from the bundle-of-leaky-capillary-tubes model. In: SPE Production Operations Symposium, Society of Petroleum Engineers (2005)

  • Civan, F.: Improved permeability prediction for heterogeneous porous media by bundle-of-leaky-tubes with cross-flow model. In: 5th International Conference on Porous Media and Their Application in Scinece, Engineering and Industry. Kona, Hawaii (2014)

  • Civan, F.: Predictability of porosity and permeability alterations by geochemical and geomechanical rock and fluid interactions. In: SPE International Symposium on Formation Damage Control, Society of Petroleum Engineers(2000)

  • Civan, F.: Scale effect on porosity and permeability: kinetics, model, and correlation. AIChE J. 47(2), 1167–1197 (2001)

    Article  Google Scholar 

  • Corey, A.T.: The interrelation between gas and oil relative permeabilities. Prod. Mon. 19(1), 38–41 (1954)

    Google Scholar 

  • Costa, A.: Permeability-porosity relationship: a reexamination of the Kozeny–Carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett. 33(2), 1–5 (2006)

    Article  Google Scholar 

  • Davies, G.R., Smith, L.B.: Structurally controlled hydrothermal dolomite reservoir facies: an overview. AAPG Bull. 90(11), 1641–1690 (2006)

    Article  Google Scholar 

  • DiPippo, R.: Geothermal Power Plants (Third Edition): Principles, Applications, Case Studies and Environmental Impact, 3rd edn. Elsevier, Tokyo (2012)

    Google Scholar 

  • Dolly, E.D., Mullarkey, J.C.: Hydrocarbon production from low contrast, low resistivity reservoirs. Rocky Mountain and Mid-Continent regions: Log Examples of Subtle Plays. Rocky Mountain Association of Geologists, p. 290 (1996)

  • Ehrenberg, S.N.: Relationship between diagenesis and reservoir quality in sandstones of the Garn Formation, Haltenbanken, Mid-Norwegian Continental Shelf. Am. Assoc. Petrol. Geol. Bull. 74(10), 1538–1558 (1990)

    Google Scholar 

  • Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)

    Book  Google Scholar 

  • Fowler, S.J., Kosakowski, G., Driesner, T., Kulik, D.A., Wagner, T., Wilhelm, S., Masset, O.: Numerical simulation of reactive fluid flow on unstructured meshes. Transp. Porous Media 112(1), 283–312 (2016)

    Article  Google Scholar 

  • Garrison Jr., J.R., Pearn, W.C., Rosenberg, D.U.: The fractal Menger sponge and Sierpinski carpet as models for reservoir rock/pore systems: I.; Theory and image analysis of Sierpinski carpets. In Situ 16, 351–406 (1992)

    Google Scholar 

  • Garrison Jr., J.R., Pearn, W.C., Rosenberg, D.U.: The fractal Menger sponge and Sierpinski carpet as models for reservoir rock/pore systems: II. Image analysis of natural fractal reservoir rocks. In Situ 17, 1–53 (1993)

    Google Scholar 

  • Gaucher, E.C., Blanc, P.: Cement/clay interactions—a review: experiments, natural analogues, and modeling. Waste Manag. 26(7), 776–788 (2006)

    Article  Google Scholar 

  • Gregg, J.M., Bish, D.L., Kaczmarek, S.E., Machel, H.G.: Mineralogy, nucleation and growth of dolomite in the laboratory and sedimentary environment: a review. Sedimentology 62(6), 1749–1769 (2015)

    Article  Google Scholar 

  • Griffiths, L., Heap, M.J., Wang, F., Daval, D., Gilg, H.A., Baud, P., Schmittbuhl, J., Genter, A.: Geothermal implications for fracture-filling hydrothermal precipitation. Geothermics 64, 235–245 (2016)

    Article  Google Scholar 

  • Hogg, A.J.C., Mitchell, A.W., Young, S.: Predicting well productivity from grain size analysis and logging while drilling. Petrol. Geosci. 2(1), 1–15 (1996)

    Article  Google Scholar 

  • Huitt, J.L.: Fluid flow in simulated fractures. AIChE J. 2(2), 259–264 (1956)

    Article  Google Scholar 

  • Ingebritsen, S.E., Gleeson, T.: Crustal permeability: introduction to the special issue. Geofluids 15(1–2), 1–10 (2015)

    Article  Google Scholar 

  • Jefferson, I., Smalley, I.: Soil mechanics in Engineering Practice, 3rd edn. In: Terzaghi, K., Peck, R.B., Mesri, G. (eds.) Engineering Geology. Wiley-Interscience, New York, p. 549, 48(1–2), pp. 149–150 (1997)

  • Jin, Y., Li, X., Zhao, M., Liu, X., Li, H.: A mathematical model of fluid flow in tight porous media based on fractal assumptions. Int. J. Heat Mass Transf. 108, 1078–1088 (2017a)

    Article  Google Scholar 

  • Jin, Y., Wu, Y., Li, H., Zhao, M., Pan, J.: Scientific Report: Definition of Fractal Topography to Essential Understanding of Scale-Invariance. Nature Publishing Group, London (2017b)

    Google Scholar 

  • Jin, Y., Dong, J., Zhang, X., Li, X., Wu, Y.: Scale and size effects on fluid flow through self-affine rough fractures. Int. J. Heat Mass Transf. 105, 443–451 (2017c)

    Article  Google Scholar 

  • Jin, Y., Zhu, Y.B., Li, X., Zheng, J.L., Dong, J.B.: Scaling invariant effects on the permeability of fractal porous media. Transp. Porous Media 109, 433–453 (2015)

    Article  Google Scholar 

  • Katz, A.J., Thompson, A.H.: Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54(12), 1325–1328 (1985)

    Article  Google Scholar 

  • Kirkby, A., Heinson, G.: Three-dimensional resistor network modeling of the resistivity and permeability of fractured rocks. J. Geophys. Res. Solid Earth 122(4), 2653–2669 (2017)

  • Kozeny, J.: Uber kapillare leitung des wassers im boden: Sitzungsber. Acad. Wiss. Wien 136, 271–306 (1927)

    Google Scholar 

  • Krohn, C.E., Thompson, A.H.: Fractal sandstone pores: automated measurements using scanning-electron-microscope images. Phys. Rev. B 33(9), 6366–6374 (1986)

    Article  Google Scholar 

  • Lu, M., Connell, L.D.: A statistical representation of the matrix-fracture transfer function for porous media. Transp. Porous Media 86(3), 777–803 (2011)

    Article  Google Scholar 

  • Lucia, F.J.: Carbonate Reservoir Characterization—An Integrated Approach, 2nd edn. Springer New York, New York (2007)

    Google Scholar 

  • Lucia, F.J.: Rock-fabric/petrophysical classification of carbonate pore space for reservoir characterization 1. AAPG Bull. 79(9), 1275–1300 (1995)

    Google Scholar 

  • Majumdar, A., Bhushan, B.: Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribol. 112(2), 205–216 (1990)

    Article  Google Scholar 

  • Mandelbrot, B.B.: The Fractal Geometry of Nature. W. H. Freeman and Company, San Francisco (1982)

    Google Scholar 

  • Marzano, M.S.: Controls on permeability for unconsolidated sands from conventional core data, offshore Gulf of Mexico. Trans. Gulf Coast Assoc. Geol. Soc. 38, 113–120 (1988)

    Google Scholar 

  • Masch, F.D., Denny, K.J.: Grain size distribution and its effect on the permeability of unconsolidated sands. Water Resour. Res. 2(4), 665–677 (1966)

    Article  Google Scholar 

  • MATLAB.: Multi-paradigm numerical computing environment and fourth-generation programming language. MathWorks, Inc., Natick: MA-USA (2016)

  • Matthäi, S.K., Belayneh, M.: Fluid flow partitioning between fractures and a permeable rock matrix. Geophys. Res. Lett. 31(7), 1–5 (2004)

    Article  Google Scholar 

  • Mavko, G., Nur, A.: The effect of a percolation threshold in the Kozeny–Carman relation. Geophysics 62(5), 1480–1482 (1997)

    Article  Google Scholar 

  • McGregor, R.: The effect of rate of flow on rate of dyeing ii—the mechanism of fluid flow through textiles and its significance in dyeing. J. Soc. Dyers Colour. 81(10), 429–438 (1965)

    Article  Google Scholar 

  • Miller, D.D., McPherson, J.G., Covington, T.E.: Fluviodeltaic reservoir, South Belridge Field, San Joaquin Valley, California: in Sandstone Petroleum Reservoirs, pp. 109–130. Springer, New York (1990)

  • Miller, R.S., Groth, J.L.: Depositional environment and reservoir properties of the lower Tuscaloosa B Sandstone Baywood field, St. Helena Parish, Louisiana. Trans. Gulf Coast Assoc. Geol. Soc. 40, 601–605 (1990)

    Google Scholar 

  • Nelson, P.H.: Permeability-porosity relationships in sedimentary rocks. Log Anal. 35(3), 38–62 (1994)

    Google Scholar 

  • Nelson, P.H., Kibler, J.E.: A Catalog of Porosity and Permeability from Core Plugs in Siliciclastic Rocks: Open-file Report 03–420. Denver, CO - US (2003)

  • Nick, H.M., Matthäi, S.K.: A hybrid finite-element finite-volume method with embedded discontinuities for solute transport in heterogeneous media. Vadose Zone J. 10(1), 299–312 (2011)

    Article  Google Scholar 

  • Noetinger, B.: A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks accounting for matrix to fracture flow. J. Comput. Phys. 283, 205–223 (2015)

    Article  Google Scholar 

  • Panda, M., Lake, L.: A physical model of cementation and its effects on single-phase permeability. Am. Assoc. Petrol. Geol. Bull. 79(3), 431–443 (1995)

    Google Scholar 

  • Panda, M., Lake, L.: Estimation of single-phase permeability from parameters of particle size distrubition. AAPG Bull. 78(7), 1028–1039 (1994)

    Google Scholar 

  • Pape, H., Riepe, L., Schopper, J.R.: A pigeon-hole model for relating permeability to specific surface. Soc. Petrophys. Well-Log Anal. 23(1), 5–13 (1982)

    Google Scholar 

  • Pape, H., Riepe, L., Schopper, J.R.: Interlayer conductivity of rocks—a fractal model of interface irregularities for calculating interlayer conductivity of natural porous mineral systems. Colloids Surf. 27(1–3), 97–122 (1987a)

    Article  Google Scholar 

  • Pape, H., Riepe, L., Schopper, J.R.: Theory of self-similar network structures in sedimentary and igneous rocks and their investigation with microscopical and physical methods. J. Microsc. 148, 121–147 (1987b)

    Article  Google Scholar 

  • Pape, H., Schopper, J.R.: Relations between physically relevant geometrical properties of a multifractal porous system. Stud. Surf. Sci. Catal. 39, 473–482 (1988)

    Article  Google Scholar 

  • Pape, H., Clauser, C., Iffland, J., Krug, R., Wagner, R.: Anhydrite cementation and compaction in geothermal reservoirs: Interaction of pore-space structure with flow, transport, P–T conditions, and chemical reactions. Int. J. Rock Mech. Min. Sci. 42(42), 1056–1069 (2005)

    Article  Google Scholar 

  • Pape, H., Arnold, J., Pechnig, R., Clauser, C., Talnishnikh, E., Anferova, S., Blumich, B.: Permeability prediction for low porosity rocks by mobile NMR. Pure Appl. Geophys. 166, 1125–1163 (2009)

    Article  Google Scholar 

  • Pape, H., Clauser, C., Iffland, J.: Permeability-porosity relationship in sandstone based on fractal pore space geometry. Pure Appl. Geophys. 157, 603–619 (2000)

    Article  Google Scholar 

  • Pape, H., Clauser, C., Iffland, J.: Permeability prediction for reservoir sandstones based on fractal pore space geometry. Geophysics 64(5), 1447–1460 (1999)

    Article  Google Scholar 

  • Pettijohn, F.J., Potter, P.E., Siever, R.: Sand and Sandstone. Springer, New York (1973)

    Book  Google Scholar 

  • Pryor, W.A.: Permeability-porosity patterns and variations in some Holocene sand bodies. Am. Assoc. Petrol. Geol. Bull. 57(1), 162–191 (1973)

    Google Scholar 

  • Raffensperger, J.P., Garven, G.: The formation of unconformity-type uranium ore deposits; 1, coupled groundwater flow and heat transport modeling. Am. J. Sci. 295(5), 581–636 (1995a)

    Article  Google Scholar 

  • Raffensperger, J.P., Garven, G.: The formation of unconformity-type uranium ore deposits; 2, coupled hydrochemical modeling. Am. J. Sci. 295(6), 639–696 (1995b)

    Article  Google Scholar 

  • Ranjbar, E., Hassanzadeh, H., Chen, Z.: One-dimensional matrix-fracture transfer in dual porosity systems with variable block size distribution. Transp. Porous Media 95(1), 185–212 (2012)

    Article  Google Scholar 

  • Saboorian-Jooybari, H., Ashoori, S., Mowazi, G.: Development of an analytical time-dependent matrix/fracture shape factor for countercurrent imbibition in simulation of fractured reservoirs. Transp. Porous Media 92(3), 687–708 (2012)

    Article  Google Scholar 

  • Sarkar, S., Toksoz, M.N., Burns, D.R.: Fluid flow modeling in fractures. Earth Resources Laboratory, MIT Earth, pp. 1–41 (2004)

  • Schlueter, E.: Ph.D. thesis: Predicting the transport properties of sedimentary rocks from microstructure. Report Number: LBL-36900. Lawrence Berkeley Laboratory, University of California (1995)

  • Shepherd, R.G.: Correlations of permeability and grain size. Ground Water 27(5), 633–638 (1989)

    Article  Google Scholar 

  • Sherman, F.S.: Viscous Flow. McGraw-Hill, Santa Monica (1990)

    Google Scholar 

  • Slaughter, W.S.: The Linearized Theory of Elasticity. Birkhaeuser, Boston (2002)

    Book  Google Scholar 

  • Sneider, R.M., Richardson, F.H., Paynter, D.D., Eddy, R.E., Wyant, I.A.: Predicting reservoir rock geometry and continuity in Pennsylvanian reservoirs, Elk City Field, Oklahoma. J. Petrol. Technol. 29, 851–864 (1977)

    Article  Google Scholar 

  • Snow, D.T.: Ph.D. thesis: A parallel plate model of fractured permeable media. University of California, Berkeley (1965)

  • Steefel, C.I., Appelo, C.A.J., Arora, B., Jacques, D., Kalbacher, T., Kolditz, O., Lagneau, V., Lichtner, P.C., Mayer, K.U., Meeussen, J.C.L., Molins, S., Moulton, D., Shao, H., Šimurunek, J., Spycher, N., Yabusaki, S.B., Yeh, G.T.: Reactive transport codes for subsurface environmental simulation. Comput. Geosci. 19(3), 445–478 (2015)

    Article  Google Scholar 

  • Tiab, D., Donaldson, E.C.: Petrophysics. Elsevier, Burlington (2004)

    Google Scholar 

  • Varrato, F., Foffi, G.: Apollonian packings as physical fractals. Mol. Phys. 109(23–24), 2923–2928 (2011)

    Article  Google Scholar 

  • Wang, B., Jin, Y.I., Chen, Q., Zheng, J., Zhu, Y., Zhang, X.: Derivation of permeability-pore relationship for fractal porous reservoirs using series- parallel flow resistance model and lattice boltzmann method. Fractals 22(3) ,(1440005)), pp. 1–15 (2014)

  • Warren, J.: Dolomite: occurrence, evolution and economically important associations. Earth-Sci. Rev. 52(1–3), 1–81 (2000)

    Article  Google Scholar 

  • Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. Society of Petroleum Engineers, pp. 245–255 (1963)

  • White, F.: Fluid Mechanics. McGraw-Hill, New York (2010)

    Google Scholar 

  • Witherspoon, P.A., Wang, J.S.Y., Iwai, K., Gale, J.E.: Validity of cubic law for fluid flow in a deformable rock fracture. Water Resour. Res. 16, 1016–1024 (1980)

    Article  Google Scholar 

  • Xu, P., Yu, B.: Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour. 31(1), 74–81 (2008)

    Article  Google Scholar 

  • Xu, T., Zheng, L., Tian, H.: Reactive transport modeling for CO2 geological sequestration. J. Petrol. Sci. Eng. 78(3–4), 765–777 (2011)

    Article  Google Scholar 

  • Yu, B.: Analysis of flow in fractal porous media. Appl. Mech. Rev. 61(50801), 1–19 (2008)

    Google Scholar 

  • Yu, B., Li, J.: Some fractal characters of porous media. Fractals 9(3), 365–372 (2001)

    Article  Google Scholar 

  • Yu, B., Liu, W.: Fractal analysis of permeabilities for porous media. AIChE J. 50(1), 46–57 (2004)

  • Zheng, Q., Yu, B.: A fractal permeability model for gas flow through dual-porosity media. J. Appl. Phys. 111(24316), 1–7 (2012)

    Google Scholar 

  • Zimmerman, R.W., Bodvarsson, G.S.: Hydraulic conductivity of rock fractures. Transp. Porous Media 23, 1–30 (1996)

    Article  Google Scholar 

Download references

Acknowledgements

This work was done with the support of the EU, ERDF, Flanders Innovation & Entrepreneurship and the Province of Limburg (Grant: 1510487 – SALK WP2: GeoWatt). We thank the anonymous reviewers for their insightful comments, which have improved the quality of this work. The authors would also like to thank Dr. David Lagrou, and Dr. Carlo Mol for their assistance.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Selçuk Erol.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Erol, S., Fowler, S.J., Harcouët-Menou, V. et al. An Analytical Model of Porosity–Permeability for Porous and Fractured Media. Transp Porous Med 120, 327–358 (2017). https://doi.org/10.1007/s11242-017-0923-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-017-0923-z

Keywords

  • Analytical solution
  • Permeability
  • Matrix porosity
  • Fracture porosity
  • Fracture network