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Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations

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Abstract

Reservoir porous rocks usually consist of more than two types of matrix materials, forming a randomly heterogeneous material. The determination of the bulk modulus of such a medium is critical to the elastic wave dispersion and attenuation. The elastic moduli for a simple matrix-inclusion model are theoretically analyzed. Most of the efforts assume a uniform inclusion concentration throughout the whole single-material matrix. However, the assumption is too strict in real-world rocks. A model is developed to estimate the moduli of a heterogeneous bimaterial skeleton, i.e., the host matrix and the patchy matrix. The elastic moduli, density, and permeability of the patchy matrix differ from those of the surrounding host matrix material. Both the matrices contain dispersed particle inclusions with different concentrations. By setting the elastic constant and density of the particles to be zero, a double-porosity medium is obtained. The bulk moduli for the whole system are derived with a multi-level effective modulus method based on Hashin’s work. The proposed model improves the elastic modulus calculation of reservoir rocks, and is used to predict the kerogen content based on the wave velocity measured in laboratory. The results show pretty good consistency between the inversed total organic carbon and the measured total organic carbon for two sets of rock samples.

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References

  1. Gassmann, F, Uber die elastizitat poroser medien. Uber die elastizitat poroser medien 96, 1–23 (1961)

    MathSciNet  MATH  Google Scholar 

  2. Biot, M. A, Theory of propagation of elastic waves in a fluid-saturated porous solid: 2 higher frequency range. Theory of propagation of elastic waves in a fluid-saturated porous solid: 2 higher frequency range 28, 179–191 (1956)

    MathSciNet  Google Scholar 

  3. Biot, M. A, Theory of propagation of elastic waves in a fluid-saturated porous solid: 1 lowfrequency range. Theory of propagation of elastic waves in a fluid-saturated porous solid: 1 lowfrequency range 28, 168–178 (1956)

    Google Scholar 

  4. Biot, M. A, Mechanics of deformation and acoustic propagation in porous media. Mechanics of deformation and acoustic propagation in porous media 33, 1482–1498 (1962)

    MathSciNet  MATH  Google Scholar 

  5. Pride, S. R., Berryman, J. G., and Harris, J. M, Seismic attenuation due to wave-induced flow. Seismic attenuation due to wave-induced flow 109, 59–70 (2004)

    Google Scholar 

  6. Carcione, J. M., Morency, C., and Santos, J. E, Computational poroelasticity: a review. Computational poroelasticity: a review 75, A229–A243 (2010)

    Google Scholar 

  7. Arntsen, B. and Carcione, J. M, Numerical simulation of the Biot slow wave in water-saturated Nivelsteiner sandstone. Numerical simulation of the Biot slow wave in water-saturated Nivelsteiner sandstone 66, 890–896 (2001)

    Google Scholar 

  8. Dvorkin, J., Mavko, G., and Nur, A, Squirt flow in fully saturated rocks. Squirt flow in fully saturated rocks 60, 97–107 (1995)

    Google Scholar 

  9. Mochizuki, S, Attenuation in partially saturated rocks. Attenuation in partially saturated rocks 87, 8598–8604 (1982)

    Google Scholar 

  10. White, J. E, Computed seismic speeds and attenuation in rocks with partial gas saturation. Computed seismic speeds and attenuation in rocks with partial gas saturation 40, 224–232 (1975)

    Google Scholar 

  11. White, J. E., Mihailova, N., and Lyakhovitsky, F, Low-frequency seismic-waves in fluid-saturated layered rocks. Low-frequency seismic-waves in fluid-saturated layered rocks 57, 654–659 (1975)

    Google Scholar 

  12. Dutta, N. C. and Odé, H, Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model): part I, Biot theory. Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model): part I, Biot theory 44, 1777–1788 (1979)

    Google Scholar 

  13. Dutta, N. C. and Odé, H, Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model): part 2, results. Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model): part 2, results 44, 1789–1805 (1979)

    Google Scholar 

  14. Johnson, D. L, Theory of frequency dependent acoustics in patchy-saturated porous media. Theory of frequency dependent acoustics in patchy-saturated porous media 110, 682–694 (2001)

    Google Scholar 

  15. Ba, J., Carcione, J. M., and Nie, J. X, Biot-Rayleigh theory of wave propagation in double-porosity media. Biot-Rayleigh theory of wave propagation in double-porosity media 116, 309–311 (2011)

    Google Scholar 

  16. Mavko, G. and Nur, A, Melt squirt in the asthenosphere. Melt squirt in the asthenosphere 80, 1444–1448 (1975)

    Google Scholar 

  17. Mavko, G. M. and Nur, A.Wave attenuation in partially saturated rocks. Geophysics, 44, 161–178 (1979)

  18. Toms, J., Müler, T. M., Ciz, R., and Gurevich, B, Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks. Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks 26, 548–565 (2006)

    Google Scholar 

  19. Müler, T. M., Gurevich, B., and Lebedev, M, Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks: a review. Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks: a review 75, A147–A164 (2010)

    Google Scholar 

  20. Rubino, J. G. and Holliger, K, Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks. Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks 188, 1088–1102 (2012)

    Google Scholar 

  21. Berryman, J. G. and Wang, H. F, The elastic coefficients of double-porosity models for fluid transport in jointed rock. The elastic coefficients of double-porosity models for fluid transport in jointed rock 100, 24611–24627 (1995)

    Google Scholar 

  22. Pride, S. R. and Berryman, J. G, Linear dynamics of double-porosity dual-permeability materials: I, governing equations and acoustic attenuation. Linear dynamics of double-porosity dual-permeability materials: I, governing equations and acoustic attenuation 68, 141–158 (2003)

    MathSciNet  Google Scholar 

  23. Berryman, J. G. and Wang, H. F, Elastic wave propagation and attenuation in a double-porosity dual-permeability medium. Elastic wave propagation and attenuation in a double-porosity dual-permeability medium 37, 63–78 (2000)

    Google Scholar 

  24. Müler, T. M. and Gurevich, B. A first-order statistical smoothing approximation for the coherent wave field in random porous random media. Journal of the Acoustical Society of America, 117, 1796–1805 (2005)

    Article  Google Scholar 

  25. Müler, T. M. and Gurevich, B, Wave-induced fluid flow in random porous media: attenuation and dispersion of elastic waves. Wave-induced fluid flow in random porous media: attenuation and dispersion of elastic waves 117, 2732–2741 (2005)

    Google Scholar 

  26. Müler, T. M. and Gurevich, B, Effective hydraulic conductivity and diffusivity of randomly heterogeneous porous solids with compressible constituents. Effective hydraulic conductivity and diffusivity of randomly heterogeneous porous solids with compressible constituents 88, 121924 (2006)

    Google Scholar 

  27. Carcione, J. M. and Picotti, S. P-wave seismic attenuation by slow-wave diffusion: effects of inhomogeneous rock properties. Geophysics, 71, O1–O8 (2006)

    Article  Google Scholar 

  28. Ba, J., Carcione, J. M., and Nie, J. X, Biot-Rayleigh theory of wave propagation in double-porosity media. Biot-Rayleigh theory of wave propagation in double-porosity media 116, 309–311 (2011)

    Google Scholar 

  29. Ba, J., Carcione, J. M., and Sun, W, Seismic attenuation due to heterogeneities of rock fabric and fluid distribution. Seismic attenuation due to heterogeneities of rock fabric and fluid distribution 202, 1843–1847 (2015)

    Google Scholar 

  30. Quintal, B., Steeb, H., Frehner, M., and Schmalholz, S. M, Quasi-static finite element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media. Quasi-static finite element modeling of seismic attenuation and dispersion due to wave-induced fluid flow in poroelastic media 116, 200–216 (2011)

    Google Scholar 

  31. Sun, W., Ba, J., Müler, T. M., Carcione, J. M., and Cao, H, Comparison of P-wave attenuation models due to wave-induced flow. Comparison of P-wave attenuation models due to wave-induced flow 63, 378–390 (2014)

    Google Scholar 

  32. Eshelby, J. D, The determination of the elastic field of an ellipsoidal inclusion and related problems. The determination of the elastic field of an ellipsoidal inclusion and related problems 241, 376–396 (1957)

    MathSciNet  MATH  Google Scholar 

  33. Hashin, Z, The elastic moduli of heterogeneous materials. The elastic moduli of heterogeneous materials 29, 143–150 (1962)

    MathSciNet  MATH  Google Scholar 

  34. Hill, R, Elastic properties of reinforced solids: some theoretical principles. Elastic properties of reinforced solids: some theoretical principles 11, 357–372 (1963)

    MATH  Google Scholar 

  35. Brown, W. F, Solid mixture permittivities. Solid mixture permittivities 23, 1514–1517 (1955)

    Google Scholar 

  36. De Loor, G. P, Dielectric properties of heterogeneous mixtures with a polar constituent. Dielectric properties of heterogeneous mixtures with a polar constituent 11, 310–320 (1964)

    Google Scholar 

  37. Hashin, Z. and Shtrikman, S. A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids, 10, 343–352 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hashin, Z. and Shtrikman, S. A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids, 11, 127–140 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  39. Budiansk, B, On elastic moduli of some heterogeneous materials. On elastic moduli of some heterogeneous materials 13, 223–227 (1965)

    Google Scholar 

  40. Hill, R. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213–222 (1965)

    Article  Google Scholar 

  41. Zimmerman, R. W, Elastic-moduli of a solid with spherical pores: new self-consistent method. Elastic-moduli of a solid with spherical pores: new self-consistent method 21, 339–343 (1984)

    Google Scholar 

  42. Mori, T. and Tanaka, K, Average stress in matrix and average elastic energy of materials with misfitting inclusions. Average stress in matrix and average elastic energy of materials with misfitting inclusions 21, 571–574 (1973)

    Google Scholar 

  43. Voigt, W. Lehrbuch der Kristallphysik, B. G. Teubner, Berlin (1910)

    MATH  Google Scholar 

  44. Reuss, A, Berechnung der flie grenze von mischkristallen auf grund der plastizitäts bedingung fu? r einkristalle. Berechnung der flie grenze von mischkristallen auf grund der plastizitäts bedingung fu? r einkristalle 9, 49–58 (1929)

    MATH  Google Scholar 

  45. Hill, R, The elastic behaviour of a crystalline aggregate. The elastic behaviour of a crystalline aggregate 65, 349–354 (1952)

    Google Scholar 

  46. Peselnick, L. and Meister, R, Variational method of determining effective moduli of polycrystals: (A) hexagonal symmetry, (B) trigonal symmetry. Variational method of determining effective moduli of polycrystals: (A) hexagonal symmetry, (B) trigonal symmetry 36, 2879–2884 (1965)

    Google Scholar 

  47. Watt, J. P, Hashin-Shtrikman bounds on the effective elastic-moduli of polycrystals with orthorhombic symmetry. Hashin-Shtrikman bounds on the effective elastic-moduli of polycrystals with orthorhombic symmetry 50, 6290–6295 (1979)

    Google Scholar 

  48. Watt, J. P. and Peselnick, L, Clarification of the Hashin-Shtrikman bounds on the effective elasticmoduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries. Clarification of the Hashin-Shtrikman bounds on the effective elasticmoduli of polycrystals with hexagonal, trigonal, and tetragonal symmetries 51, 1525–1531 (1980)

    Google Scholar 

  49. Budiansky, B. and Oconnell, R. J, Elastic-moduli of a cracked solid. Elastic-moduli of a cracked solid 12, 81–97 (1976)

    MATH  Google Scholar 

  50. Burridge, R. and Keller, J. B, Poroelasticity equations derived from microstructure. Poroelasticity equations derived from microstructure 70, 1140–1146 (1981)

    MATH  Google Scholar 

  51. Xu, S. Y. and White, R. E. A new velocity model for clay-sand mixtures. Geophysical Prospecting, 43, 91–118 (1995)

  52. Hudson, J. A., Liu, E., and Crampin, S, The mechanical properties of materials with interconnected cracks and pores. The mechanical properties of materials with interconnected cracks and pores 124, 105–112 (1996)

    Google Scholar 

  53. Kuster, G. T. and Toksoz, M. N, Velocity and attenuation of seismic waves in two-phase media, part 1: theoretical formulations. Velocity and attenuation of seismic waves in two-phase media, part 1: theoretical formulations 39, 587–606 (1974)

    Google Scholar 

  54. Tang, X. M., Chen, X. L., and Xu, X. K. A cracked porous medium elastic wave theory and its application to interpreting acoustic data from tight formations. Geophysics, 77, D245–D252 (2012)

    Article  Google Scholar 

  55. Chapman, M., Zatsepin, S. V., and Crampin, S, Derivation of a microstructural poroelastic model. Derivation of a microstructural poroelastic model 151, 427–451 (2002)

    Google Scholar 

  56. Christensen, R. M. Mechanics of Composite Materials, Wiley InterScience, New York (1979)

    Google Scholar 

  57. Hashin, Z, Analysis of composite materials—a survey. Analysis of composite materials—a survey 50, 481–505 (1983)

    MATH  Google Scholar 

  58. Hashin, Z, The elastic moduli of heterogeneous materials. The elastic moduli of heterogeneous materials 29, 2938–2945 (1962)

    MathSciNet  MATH  Google Scholar 

  59. Vernik, L. and Nur, A, Ultrasonic velocity and anisotropy of hydrocarbon source rocks. Ultrasonic velocity and anisotropy of hydrocarbon source rocks 57, 727–735 (1992)

    Google Scholar 

  60. Sokolnikoff, I. S. Mathematical Theory of Elasticity, McGraw-Hill, New York (1956)

    MATH  Google Scholar 

  61. Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York (1944)

    MATH  Google Scholar 

  62. Yan, F. and Han, D. H, Measurement of elastic properties of kerogen. Measurement of elastic properties of kerogen 143, 2778–2782 (2013)

    Google Scholar 

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Authors and Affiliations

  1. Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, China

    Weitao Sun

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  1. Weitao Sun
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Correspondence to Weitao Sun.

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Project supported by the Open Project Program of Sinopec Key Laboratory of Multi-Component Seismic Technology (No.GSYKY-B09-33), the National Key Basic Research Program of China (973 Program) (No. 2014CB239006), and the Basic Research Program of Community Networks Program Centers (CNPC) (No. 2014A-3611)

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Sun, W. Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations. Appl. Math. Mech.-Engl. Ed. 38, 15–28 (2017). https://doi.org/10.1007/s10483-017-2157-6

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  • DOI: https://doi.org/10.1007/s10483-017-2157-6

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