Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity

Zhaoyun Zong; Fubin Chen; Xingyao Yin; Reza Rezaee; Théo Le Gallais

doi:10.1111/1365-2478.13608

E-ISSN: 1365-2478

Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity
Authors Zhaoyun Zong^1,2, Fubin Chen^1,2,3, Xingyao Yin^1,2, Reza Rezaee⁴ and Théo Le Gallais⁴
View Affiliations Hide Affiliations

¹ State Key Laboratory of Deep Oil and Gas, China University of Petroleum (East China), Qingdao, P. R. China ² Laoshan Laboratory, Qingdao, P. R. China ³ Centre for Exploration Geophysics, Curtin University, Perth, Western Australia, Australia ⁴ Western Australia School of Mines: Minerals, Energy and Chemical Engineering, Curtin University, Perth, Western Australia, Australia
Publisher: European Association of Geoscientists & Engineers
Source: Geophysical Prospecting, Volume 72, Issue 9, Oct 2024, p. 3230 - 3246
DOI: https://doi.org/10.1111/1365-2478.13608
- Received: 22 Apr 2024
- Accepted: 02 Sep 2024
- Published online: 29 Sep 2024

Abstract

Accurate characterization for effective elastic moduli of porous solids is crucial for better understanding their mechanical behaviour and wave propagation, which has found many applications in the fields of engineering, rock physics and exploration geophysics. We choose the spheroids with different aspect ratios to describe the various pore geometries in porous solids. The approximate equations for compressibility and shear compliance of spheroid pores and differential effective medium theory constrained by critical porosity are used to derive the asymptotic solutions for effective elastic moduli of the solids containing randomly oriented spheroids. The critical porosity in the new asymptotic solutions can be flexibly adjusted according to the elastic moduli – porosity relation of a real solid, thus extending the application of classic David‐Zimmerman model because it simply assumes the critical porosity is one. The asymptotic solutions are valid for the solids containing crack‐like oblate spheroids with aspect ratio $α$\alpha $$ < 0.3, nearly spherical pores (0.7 < $α$\alpha $$ < 1.3) and needle‐like prolate pores with $α$\alpha $$ > 3, instead of just valid in the limiting cases, for example perfectly spherical pores ( $α$\alpha $$ = 1) and infinite thin cracks ( $α→$\alpha \ \to $$ 0). The modelling results also show that the accuracies of asymptotic solutions are weakly affected by the critical porosity $ϕc${{\phi }_{\mathrm{c}}}$$ and grain Poisson's ratio $v0${{v}_0}$$ , although the elastic moduli have appreciable dependency of $ϕc${{\phi }_{\mathrm{c}}}$$ and $v0${{v}_0}$$ . We then use the approximate equations for pore compressibility and shear compliance as inputs into the Mori–Tanaka and Kuster–Toksoz theories and compare their calculations to our results from differential effective medium theory. By comparing the published laboratory measurements with modelled results, we validate our asymptotic solutions for effective elastic moduli.

Article metrics loading...

/content/journals/10.1111/1365-2478.13608

2024-10-11

2026-02-11

From This Site

/content/journals/10.1111/1365-2478.13608

dcterms_title,dcterms_subject,pub_keyword

-contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution

10

5

Full text loading...

References

Berryman, J.G., Pride, S.R. & Wang, H.F. (2002) A differential scheme for elastic properties of rocks with dry or saturated cracks. Geophysical Journal International, 151, 597–611.
[Google Scholar]
Carcione, J.M. (2007) Wave fields in real media: theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media, 2nd ed.Amsterdam: Elsevier Science Press.
[Google Scholar]
Carvalho, F.C.S. & Labuz, J.F. (1996) Experiments on effective elastic modulus of two‐dimensional solids with cracks and holes. International Journal of Solids Structures, 33, 4119–4130.
[Google Scholar]
Chen, F.B., Zong, Z.Y. & Yin, X.Y. (2022) Acoustothermoelasticity for joint effects of stress and thermal fields on wave dispersion and attenuation. Journal of Geophysical Research: Solid Earth, 127(4), e2021JB023671.
[Google Scholar]
Chen, F.B., Zong, Z.Y., Yin, X.Y., Yang, Z.F. & Yan, X.F. (2023) Pressure and frequency dependence of elastic moduli of fluid‐saturated dual‐porosity rocks. Geophysical Prospecting, 71(8), 1599–1615.
[Google Scholar]
David, E.C. & Zimmerman, R.W. (2011a) Compressibility and shear compliance of spheroidal pores: exact derivation via the Eshelby tensor, and asymptotic expressions in limiting cases. International Journal of Solids Structures, 48, 680–686.
[Google Scholar]
David, E.C. & Zimmerman, R.W. (2011b) Elastic moduli of solids containing spheroidal pores. International Journal of Engineering Science, 49, 544–560.
[Google Scholar]
Eshelby, J.D. (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society A, 241, 376–396.
[Google Scholar]
Fabricius, I.L., Bächle, G.T. & Eberli, G.P. (2010) Elastic moduli of dry and water‐saturated carbonates—effect of depositional texture, porosity, and permeability. Geophysics, 75, N65–N78.
[Google Scholar]
Han, D.H., Nur, A. & Morqan, D. (1986) Effects of porosity and clay content on wave velocities in sandstones. Geophysics, 51, 2093–2107.
[Google Scholar]
Hashin, Z. & Shtrikman, S. (1961) Note on a variational approach to the theory of composite elastic materials. Journal of the Franklin Institute, 271, 336–341.
[Google Scholar]
Kuster, G.T. & Toksoz, M.N. (1974) Velocity and attenuation of seismic waves in two‐phase media: part I. Theoretical formulations. Geophysics, 39, 587–606.
[Google Scholar]
Mavko, G., Mukerji, T. & Dvorkin, J. (2020) The rock physics handbook. Cambridge: Cambridge University Press.
[Google Scholar]
Mori, T., & Tanaka, K. (1973) Average stress in matric and average elastic energy of materials with misfitting inclusions. Acta Materialia, 21, 571–574.
[Google Scholar]
Mukerji, T., Berryman, J., Mavko, G. & Berge, P. (1995) Differential effective medium modeling of rock elastic moduli with critical porosity constraints. Geophysical Research Letters, 22(5), 555–558.
[Google Scholar]
Norris, A.N. (1985) A differential scheme for the effective moduli of composites. Mechanics of Materials, 4, 1–16.
[Google Scholar]
Nur, A. (1992) Critical porosity and the seismic velocities in rocks (abstract). EOS Transactions of the American Geophysical Union, 73, 43–66.
[Google Scholar]
Nur, A., Mavko, G., Dvorkin, J. & Galmudi, D. (1998) Critical porosity: a key to relating physical properties to porosity in rocks. The Leading Edge, 17(3), 357–362.
[Google Scholar]
O'Connell, R.J. & Budiansky, B. (1974) Seismic velocities in dry and saturated cracked solids. Journal of Geophysical Research, 79, 5412–5426.
[Google Scholar]
O'Connell, R.J. & Budiansky, B. (1977) Viscoelastic properties of fluid‐saturated cracked solids. Journal of Geophysical Research, 82, 5719–5735.
[Google Scholar]
Saenger, E.H., Kruger, O.S. & Shapiro, S.A. (2006) Effective elastic properties of fractured rocks: dynamic vs. static considerations. International Journal of Fracture, 139, 569–576.
[Google Scholar]
Sun, Y.Y. & Gurevich, B. (2020) Modeling the effect of pressure on the moduli dispersion in fluid‐saturated rocks. Journal of Geophysical Research: Solid Earth, 125, e2019JB019297.
[Google Scholar]
Walsh, J.B. (1965) The effect of cracks on the compressibility of rocks. Journal of Geophysical Research, 70, 381–389.
[Google Scholar]
Wu, T.T. (1966) The effect of inclusion shape on the elastic moduli of a two‐phase material. International Journal of Solids and Structures, 2, 1–8.
[Google Scholar]
Zhang, J.J., Yin, X.Y. & Zhang, G.Z. (2020) Rock physics modelling of porous rocks with multiple pore types: a multiple‐porosity variable critical porosity model. Geophysical Prospecting, 68(3), 955–967.
[Google Scholar]
Zimmerman, R.W. (1985) The effect of microcracks on the elastic moduli of brittle materials. Journal of Materials Science Letters, 4, 1457–1460.
[Google Scholar]
Zimmerman, R.W. (1991) Compressibility of sandstones. Amsterdam: Elsevier.
[Google Scholar]

/content/journals/10.1111/1365-2478.13608

Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity

Geophysical Prospecting 72, 3230 (2024); https://doi.org/10.1111/1365-2478.13608

/content/journals/10.1111/1365-2478.13608

Data & Media loading...

Article Type: Research Article

Keyword(s): critical porosity; differential effective medium theory; rock physics

Analytic solutions for effective elastic moduli of isotropic solids containing oblate spheroid pores with critical porosity

Abstract

From This Site

Most Read This Month

Most Cited This Month Most Cited RSS feed

The natural combination of full and image‐based waveform inversion

Poststack diffraction imaging using reverse‐time migration

Characterizing the effect of elastic interactions on the effective elastic properties of porous, cracked rocks

Fracture detection by Gaussian beam imaging of seismic data and image spectrum analysis

Laboratory measurements of guided‐wave propagation within a fluid‐saturated fracture

Anisotropic elastic full‐waveform inversion of walkaway vertical seismic profiling data from the Arabian Gulf

Frequency‐dependent multi‐offset phase analysis of surface waves: an example of high‐resolution characterization of a riparian aquifer

A comparison of helicopter‐borne electromagnetic systems for hydrogeologic studies

A fully unsupervised and highly generalized deep learning approach for random noise suppression

Full waveform inversion of SH‐ and Love‐wave data in near‐surface prospecting