Skip to main content
SpringerLink
Log in

Influence of Equation Nonlinearity on Pulse-Decay Permeability Measurements of Tight Porous Media

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

Abstract

The pulse-decay method is believed to be more suitable than the steady-state method for permeability measurements on tight porous media because it records pressure variations instead of flow rates and does not require the establishment of a steady state. Most of the previous analytical solutions for the pulse-decay process are based on a linearized governing equation, which may be inapplicable to measurements with large differential pressures. In this study, a nonlinear governing equation is derived through mass conservation and Darcy’s law. For rigid porous media such as the sedimentary rock samples, by comparing the magnitude of the pressure sensitivity of the physical properties for both the testing gas and the core sample, we found that only the gas compressibility and the apparent permeability have to be regarded as pressure-dependent, while the others can be regarded as constant. The perturbation method and the eigenfunction expansion method are combined to derive the general solution of the nonlinear governing equation. The results show that in the plot of logarithmic differential pressure versus time, a straight line can be obtained at the late-time stage and its slope value can be used to evaluate the apparent permeability. We further estimate the error in permeability evaluation, induced by selecting mean pore pressure as the characteristic pressure. The theoretical analysis has been verified by both numerical simulation and experimental measurements.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

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

Similar content being viewed by others

Abbreviations

\(A\) :

Cross-sectional area of cylindrical core sample, m2

\(L\) :

Length of cylindrical core sample, m

\(P\) :

Pore pressure, Pa

\(P_{D}\) :

Dimensionless pore pressure

\(P_{D}^{\left( i \right)}\) :

The ith term in the asymptotic series

\(P_{D} \left( \infty \right)\) :

Dimensionless equilibrium pore pressure

\(P_{c}\) :

Confining pressure, Pa

\(P_{{{\text{char}}}}\) :

Characteristic pressure, Pa

\(P_{d} \left( 0 \right)\) :

Initial downstream pressure, Pa

\(P_{{{\text{eq}}}}\) :

Equilibrium pore pressure, Pa

\(P_{{{\text{mean}}}}\) :

Mean pore pressure, Pa

\(P_{u} \left( 0 \right)\) :

Initial upstream pressure, Pa

\(T\) :

Temperature, K

\(V_{u}\) :

Volume of upstream reservoir, m3

\(V_{d}\) :

Volume of downstream reservoir, m3

\(Z\) :

Gas compressibility factor

\(a\) :

Volume ratio of sample’s pore space to upstream reservoir

\(b\) :

Volume ratio of sample’s pore space to downstream reservoir

\(b_{s}\) :

Klinkenberg slippage factor, Pa

\(f\) :

Intercept in plot of dimensionless differential pressure versus time

\(k_{{{\text{app}}}}\) :

Apparent permeability, m2

\(k_{{{\text{error}}}}\) :

Permeability coefficient evaluated with mean pore pressure as characteristic pressure, m2

\(k_{{{\text{int}}}}\) :

Intrinsic permeability, m2

\(k_{{{\text{real}}}}\) :

Permeability coefficient evaluated with equilibrium pore pressure as characteristic pressure, m2

\(t\) :

Time, s

\(t_{D}\) :

Dimensionless time

\(v\) :

Flow rate, m·s-1

\(w\) :

Weight coefficient to determine characteristic pressure

\(x\) :

Displacement, m

\(x_{D}\) :

Dimensionless displacement

\({\varGamma}\) :

Dimensionless duration of a pulse-decay test

\({\Delta }P_{D}\) :

Dimensionless differential pressure

\(\alpha\) :

Slope in plot of dimensionless differential pressure versus time

\(\beta_{Z}\) :

Pressure sensitivity of gas compressibility factor, Pa−1

\(\beta_{{b_{s}}}\) :

Pressure sensitivity of slippage factor, Pa−1

\(\beta_{{k_{{{\text{app}}}} }}\) :

Pressure sensitivity of apparent permeability, Pa−1

\(\beta_{{k_{{{\text{int}}}} }}\) :

Pressure sensitivity of intrinsic permeability, Pa−1

\(\beta_{\mu }\) :

Pressure sensitivity of viscosity, Pa−1

\(\beta_{\rho }\) :

Gas compressibility, Pa−1

\(\beta_{\phi }\) :

Pressure sensitivity of porosity, Pa−1

\(\delta\) :

Relative error induced by misusing of characteristic pressure

\(\varepsilon\) :

Strength of nonlinearity

\(\theta_{i}\) :

The ith positive solution to Eq. (37)

\(\rho\) :

Gas density, kg·m−3

\(\mu\) :

Dynamic viscosity of testing gas, Pa·s

\(\phi\) :

Porosity of core sample

References

  • Abdelmalek, B., Karpyn, Z., Liu, S., Yoon, H., Dewers, T.: Gas permeability measurements from pressure pulse decay laboratory data using pseudo-pressure and pseudo-time transformations. J. Pet. Explor. Prod. Technol. 8(3), 839–847 (2018)

    Article  Google Scholar 

  • Akkutlu, I.Y., Fathi, E.: Multiscale gas transport in shales with local kerogen heterogeneities. SPE J. 17(04), 1002–1011 (2012)

    Article  Google Scholar 

  • Alnoaimi, K.R., Kovscek, A.R.: Influence of microcracks on flow and storage capacities of gas shales at core scale. Transp. Porous Media 127(1), 53–84 (2019)

    Article  Google Scholar 

  • Bender, C.M., Orszag, S.A.: Advanced mathematical methods for scientists and engineers I: asymptotic methods and perturbation theory. Springer Science & Business Media, Berlin (2013)

    Google Scholar 

  • Bhandari, A.R., Flemings, P.B., Polito, P.J., Cronin, M.B., Bryant, S.L.: Anisotropy and stress dependence of permeability in the Barnett shale. Transp. Porous Media 108(2), 393–411 (2015)

    Article  Google Scholar 

  • Boulin, P., Bretonnier, P., Gland, N., Lombard, J.-M.: Contribution of the steady state method to water permeability measurement in very low permeability porous media. Oil Gas Sci. Technol.-Revue d’IFP Energ. Nouv 67(3), 387–401 (2012)

    Article  Google Scholar 

  • Brace, W.F., Walsh, J., Frangos, W.: Permeability of granite under high pressure. J. Geophys. Res. 73(6), 2225–2236 (1968)

    Article  Google Scholar 

  • Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics. Wiley, New York (1998)

  • Chen, T., Feng, X.-T., Cui, G., Tan, Y., Pan, Z.: Experimental study of permeability change of organic-rich gas shales under high effective stress. J. Nat. Gas Sci. Eng. 64, 1–14 (2019)

    Article  Google Scholar 

  • Chenevert, M., Sharma, A.: Permeability and effective pore pressure of shales. SPE Drill. Complet. 8(01), 28–34 (1993)

    Article  Google Scholar 

  • Civan, F., Rai, C.S., Sondergeld, C.H.: Determining shale permeability to gas by simultaneous analysis of various pressure tests. SPE J. 17(03), 717–726 (2012)

    Article  Google Scholar 

  • Clarkson, C., Jensen, J., Chipperfield, S.: Unconventional gas reservoir evaluation: what do we have to consider? J. Nat. Gas Sci. Eng. 8, 9–33 (2012)

    Article  Google Scholar 

  • Crank, J., Nicolson, P.: A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Philos. Soc. 43(1), 50–67 (1947)

  • Cui, X., Bustin, A., Bustin, R.M.: Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 9(3), 208–223 (2009)

    Article  Google Scholar 

  • Darcy, H.: Les fontaines publiques de la ville de Dijon. Dalmont, Paris (1856)

  • Davudov, D., Moghanloo, R.G., Lan, Y.: Evaluation of accessible porosity using mercury injection capillary pressure data in shale samples. Energy Fuels 32(4), 4682–4694 (2018)

    Article  Google Scholar 

  • Dicker, A.I., Smits, R.M.: A practical approach for determining permeability from laboratory pressure-pulse decay measurements. In: International Meeting on Petroleum Engineering. SPE 17578 (1988).

  • Dong, J.-J., Hsu, J.-Y., Wu, W.-J., Shimamoto, T., Hung, J.-H., Yeh, E.-C., Wu, Y.-H., Sone, H.: Stress-dependence of the permeability and porosity of sandstone and shale from TCDP Hole-A. Int. J. Rock Mech. Min. Sci. 47(7), 1141–1157 (2010)

    Article  Google Scholar 

  • Fink, R., Krooss, B., Amann-Hildenbrand, A.: Stress-dependence of porosity and permeability of the Upper Jurassic Bossier shale: an experimental study. Geol. Soc. Lond. Spec. Publ. 454(1), 107–130 (2017)

    Article  Google Scholar 

  • Gaus, G., Amann-Hildenbrand, A., Krooss, B.M., Fink, R.: Gas permeability tests on core plugs from unconventional reservoir rocks under controlled stress: a comparison of different transient methods. J. Nat. Gas Sci. Eng. 65, 224–236 (2019)

    Article  Google Scholar 

  • Ghanizadeh, A., Amann-Hildenbrand, A., Gasparik, M., Gensterblum, Y., Krooss, B.M., Littke, R.: Experimental study of fluid transport processes in the matrix system of the European organic-rich shales: II. Posidonia Shale (Lower Toarcian, northern Germany). Int. J. Coal Geol. 123, 20–33 (2014a)

    Article  Google Scholar 

  • Ghanizadeh, A., Gasparik, M., Amann-Hildenbrand, A., Gensterblum, Y., Krooss, B.M.: Experimental study of fluid transport processes in the matrix system of the European organic-rich shales: I. Scandinavian Alum Shale. Mar. Pet. Geol. 51, 79–99 (2014b)

    Article  Google Scholar 

  • Hannon, M.J.: Alternative approaches for transient-flow laboratory-scale permeametry. Transp. Porous Media 114(3), 719–746 (2016)

    Article  Google Scholar 

  • Haskett, S.E., Narahara, G.M., Holditch, S.A.: A method for simultaneous determination of permeability and porosity in low-permeability cores. SPE Form. Eval. 3(03), 651–658 (1988)

    Article  Google Scholar 

  • Hatami, M., Bayless, D., Sarvestani, A.: Poroelastic effects on gas transport mechanisms and influence on apparent permeability in shale. Int. J. Rock Mech. Min. Sci. 153, 105102 (2022)

    Article  Google Scholar 

  • Heller, R., Vermylen, J., Zoback, M.: Experimental investigation of matrix permeability of gas shales. AAPG Bull. 98(5), 975–995 (2014)

    Article  Google Scholar 

  • Hinch, E.J.: Perturbation methods. In: Cambridge texts in applied mathematics. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  • Hsieh, P.A., Tracy, J.V., Neuzil, C.E., Bredehoeft, J.D., Silliman, S.E.: A transient laboratory method for determining the hydraulic properties of ‘tight’ rocks—I, Theory. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 18(3), 245–252 (1981)

  • Jones, S.C.: A rapid accurate unsteady-state Klinkenberg permeameter. Soc. Petrol. Eng. J. 12(05), 383–397 (1972)

    Article  Google Scholar 

  • Jones, S.: A technique for faster pulse-decay permeability measurements in tight rocks. SPE Form. Eval. 12(01), 19–26 (1997)

    Article  Google Scholar 

  • Katsube, T.: Shale permeability and pore-structure evolution characteristics. Natural Resources Canada, Geological Survey of Canada, Ottawa (2000)

    Book  Google Scholar 

  • Klinkenberg, L. J.: The permeability of porous media to liquids and gases, API Drilling Prod. Pract., 200–213 (1941)

  • Lasseux, D., Jannot, Y., Profice, S., Mallet, M.: The “Step Decay”: a new transient method for the simultaneous determination of intrinsic permeability, Klinkenberg coefficient and porosity on very tight rocks. In: International Symposium of the Society of Core Analysts. SCA2012–25 (2012)

  • Lemmon, E.W., McLinden, M.O., Friend, D.G.: Thermophysical properties of fluid systems. In: Linstrom, P.J., Mallard, W.G. (eds.) NIST Chemistry WebBook, NIST Standard Reference Database Number 69. National Institute of Standards and Technology, Gaithersburg (2023).

  • Letham, E., Bustin, R.: Klinkenberg gas slippage measurements as a means for shale pore structure characterization. Geofluids 16(2), 264–278 (2016)

    Article  Google Scholar 

  • Liang, Y., Price, J.D., Wark, D.A., Watson, E.B.: Nonlinear pressure diffusion in a porous medium: approximate solutions with applications to permeability measurements using transient pulse decay method. J. Geophys. Res. Solid Earth 106(B1), 529–535 (2001)

    Article  Google Scholar 

  • Lin, Y.-Y., Myers, M.T.: Impact of non-linear transport properties on low permeability measurements. J. Nat. Gas Sci. Eng. 54, 328–341 (2018)

    Article  Google Scholar 

  • Lin, W.: Compressible fluid flow through rocks of variable permeability. In. California Univ., Livermore (USA). Lawrence Livermore Lab., (1977)

  • Liu, M.-M., Chen, Y.-F., Wei, K., Zhou, C.-B.: Interpretation of gas transient pulse tests on low-porosity rocks. Geophys. J. Int. 210(3), 1845–1857 (2017)

    Article  Google Scholar 

  • Loney, N.W.: Applied mathematical methods for chemical engineers. CRC Press, Boca Raton (2006)

    Book  Google Scholar 

  • Lyu, Q., Shi, J., Gamage, R.P.: Effects of testing method, lithology and fluid-rock interactions on shale permeability: a review of laboratory measurements. J. Nat. Gas Sci. Eng. 78, 103302 (2020)

    Article  Google Scholar 

  • Ma, Y., Pan, Z., Zhong, N., Connell, L.D., Down, D.I., Lin, W., Zhang, Y.: Experimental study of anisotropic gas permeability and its relationship with fracture structure of Longmaxi Shales, Sichuan Basin. China. Fuel 180, 106–115 (2016)

    Article  Google Scholar 

  • Metwally, Y.M., Sondergeld, C.H.: Measuring low permeabilities of gas-sands and shales using a pressure transmission technique. Int. J. Rock Mech. Min. Sci. 48(7), 1135–1144 (2011)

    Article  Google Scholar 

  • Nolte, S., Fink, R., Krooss, B.M., Amann-Hildenbrand, A., Wang, Y., Wang, M., Schmatz, J., Klaver, J., Littke, R.: Experimental investigation of gas dynamic effects using nanoporous synthetic materials as tight rock analogues. Transp. Porous Media 137(3), 519–553 (2021)

    Article  Google Scholar 

  • Pan, Z., Ma, Y., Connell, L.D., Down, D.I., Camilleri, M.: Measuring anisotropic permeability using a cubic shale sample in a triaxial cell. J. Nat. Gas Sci. Eng. 26, 336–344 (2015)

    Article  Google Scholar 

  • Pang, Y., Soliman, M.Y., Deng, H., Emadi, H.: Analysis of effective porosity and effective permeability in shale-gas reservoirs with consideration of gas adsorption and stress effects. SPE J. 22(06), 1739–1759 (2017)

    Article  Google Scholar 

  • Richtmyer, R.D., Morton, K.W.: Difference methods for initial-value problems. Interscience Publishers, New York (1994)

  • Riley, K.F., Hobson, M.P., Bence, S.J.: Mathematical Methods for Physics and Engineering. Cambridge University Press, Cambridge (1999)

  • Rushing, J., Newsham, K., Lasswell, P., Cox, J., Blasingame, T.: Klinkenerg-corrected permeability measurements in tight gas sands: steady-state versus unsteady-state techniques. In: SPE annual technical conference and exhibition. SPE 89867 (2004)

  • Sander, R., Pan, Z., Connell, L.D.: Laboratory measurement of low permeability unconventional gas reservoir rocks: a review of experimental methods. J. Nat. Gas Sci. Eng. 37, 248–279 (2017)

    Article  Google Scholar 

  • Tinni, A., Fathi, E., Agarwal, R., Sondergeld, C.H., Akkutlu, I.Y., Rai, C.S.: Shale permeability measurements on plugs and crushed samples. In: SPE canadian unconventional resources conference, SPE 162235 (2012)

  • Trimmer, D.A.: Design criteria for laboratory measurements of low permeability rocks. Geophys. Res. Lett. 8(9), 973–975 (1981)

    Article  Google Scholar 

  • Walder, J., Nur, A.: Permeability measurement by the pulse-decay method: Effects of poroelastic phenomena and non-linear pore pressure diffusion. Int. J. Rock Mech. Min. Sci. Geomech. Abst. 3, 225–232 (1986)

  • Wang, Y., Liu, S., Elsworth, D.: Laboratory investigations of gas flow behaviors in tight anthracite and evaluation of different pulse-decay methods on permeability estimation. Int. J. Coal Geol. 149, 118–128 (2015)

    Article  Google Scholar 

  • Wang, Y., Tian, Z., Nolte, S., Amann-Hildenbrand, A., Krooss, B.M., Wang, M.: Reassessment of transient permeability measurement for tight rocks: the role of boundary and initial conditions. J. Nat. Gas Sci. Eng. 95, 104173 (2021)

    Article  Google Scholar 

  • Wang, Y., Tian, Z., Nolte, S., Krooss, B.M., Wang, M.: An improved straight-line method for permeability and porosity determination for tight reservoirs using pulse-decay measurements. J. Nat. Gas Sci. Eng. 105, 104708 (2022)

    Article  Google Scholar 

  • Yang, Z., Sang, Q., Dong, M., Zhang, S., Li, Y., Gong, H.: A modified pressure-pulse decay method for determining permeabilities of tight reservoir cores. J. Nat. Gas Sci. Eng. 27, 236–246 (2015).

    Article  Google Scholar 

  • Zhang, R., Ning, Z., Yang, F., Wang, X., Zhao, H., Wang, Q.: Impacts of nanopore structure and elastic properties on stress-dependent permeability of gas shales. J. Nat. Gas Sci. Eng. 26, 1663–1672 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

This work was financially supported by NSFC grants (No. U1837602, 11761131012) and the DFG grant AM 423/1-1 (Project number: 392108477; Joint Sino-German Research Project: “NanGasPor”).

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    Yue Wang, Zhiguo Tian & Moran Wang

  2. Institute of Geology and Geochemistry of Petroleum and Coal, Energy and Mineral Resources Group (EMR), RWTH Aachen University, Lochnerstr. 4-20, 52056, Aachen, Germany

    Steffen Nolte & Bernhard Krooss

Authors
  1. Yue Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiguo Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Steffen Nolte
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bernhard Krooss
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Moran Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moran Wang.

Ethics declarations

Conflict of interest

The authors declare no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Y., Tian, Z., Nolte, S. et al. Influence of Equation Nonlinearity on Pulse-Decay Permeability Measurements of Tight Porous Media. Transp Porous Med 148, 291–315 (2023). https://doi.org/10.1007/s11242-023-01939-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-023-01939-z

Keywords

Search

Navigation