Applied Mathematics and Mechanics >
Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations
Received date: 2016-04-07
Revised date: 2016-06-16
Online published: 2017-01-01
Supported by
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)
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.
Weitao SUN . Determination of elastic moduli of composite medium containing bimaterial matrix and non-uniform inclusion concentrations[J]. Applied Mathematics and Mechanics, 2017 , 38(1) : 15 -28 . DOI: 10.1007/s10483-017-2157-6
[1] Gassmann, F. Uber die elastizitat poroser medien. Vierteljahrsschr Naturforsch Ges Zürich, 96, 1-23(1961)
[2] Biot, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid:2 higher frequency range. Journal of the Acoustical Society of America, 28, 179-191(1956)
[3] Biot, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid:1 lowfrequency range. Journal of the Acoustical Society of America, 28, 168-178(1956)
[4] Biot, M. A. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33, 1482-1498(1962)
[5] Pride, S. R., Berryman, J. G., and Harris, J. M. Seismic attenuation due to wave-induced flow. Journal of Geophysical Research Atmospheres, 109, 59-70(2004)
[6] Carcione, J. M., Morency, C., and Santos, J. E. Computational poroelasticity:a review. Geophysics, 75, A229-A243(2010)
[7] Arntsen, B. and Carcione, J. M. Numerical simulation of the Biot slow wave in water-saturated Nivelsteiner sandstone. Geophysics, 66, 890-896(2001)
[8] Dvorkin, J., Mavko, G., and Nur, A. Squirt flow in fully saturated rocks. Geophysics, 60, 97-107(1995)
[9] Mochizuki, S. Attenuation in partially saturated rocks. Journal of Geophysical Research, 87, 8598-8604(1982)
[10] White, J. E. Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics, 40, 224-232(1975)
[11] White, J. E., Mihailova, N., and Lyakhovitsky, F. Low-frequency seismic-waves in fluid-saturated layered rocks. Journal of the Acoustical Society of America, 57, 654-659(1975)
[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. Geophysics, 44, 1777-1788(1979)
[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. Geophysics, 44, 1789-1805(1979)
[14] Johnson, D. L. Theory of frequency dependent acoustics in patchy-saturated porous media. Journal of the Acoustical Society of America, 110, 682-694(2001)
[15] Ba, J., Carcione, J. M., and Nie, J. X. Biot-Rayleigh theory of wave propagation in double-porosity media. Journal of Geophysical Research Atmospheres, 116, 309-311(2011)
[16] Mavko, G. and Nur, A. Melt squirt in the asthenosphere. Journal of Geophysical Research, 80, 1444-1448(1975)
[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. Soil Dynamics & Earthquake Engineering, 26, 548-565(2006)
[19] Müler, T. M., Gurevich, B., and Lebedev, M. Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks:a review. Geophysics, 75, A147-A164(2010)
[20] Rubino, J. G. and Holliger, K. Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks. Geophysical Journal International, 188, 1088-1102(2012)
[21] Berryman, J. G. and Wang, H. F. The elastic coefficients of double-porosity models for fluid transport in jointed rock. Journal of Geophysical Research Atmospheres, 100, 24611-24627(1995)
[22] Pride, S. R. and Berryman, J. G. Linear dynamics of double-porosity dual-permeability materials:I, governing equations and acoustic attenuation. Physical Review E, 68, 141-158(2003)
[23] Berryman, J. G. and Wang, H. F. Elastic wave propagation and attenuation in a double-porosity dual-permeability medium. International Journal of Rock Mechanics and Mining Sciences, 37, 63-78(2000)
[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)
[25] Müler, T. M. and Gurevich, B. Wave-induced fluid flow in random porous media:attenuation and dispersion of elastic waves. Journal of the Acoustical Society of America, 117, 2732-2741(2005)
[26] Müler, T. M. and Gurevich, B. Effective hydraulic conductivity and diffusivity of randomly heterogeneous porous solids with compressible constituents. Applied Physics Letters, 88, 121924(2006)
[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)
[28] Ba, J., Carcione, J. M., and Nie, J. X. Biot-Rayleigh theory of wave propagation in double-porosity media. Journal of Geophysical Research, 116, 309-311(2011)
[29] Ba, J., Carcione, J. M., and Sun, W. Seismic attenuation due to heterogeneities of rock fabric and fluid distribution. Geophysical Journal International, 202, 1843-1847(2015)
[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. Journal of Geophysical Research Atmospheres, 116, 200-216(2011)
[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. Geophysical Prospecting, 63, 378-390(2014)
[32] Eshelby, J. D. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proceedings of the Royal Society of London A:Mathematical Physical & Engineering Sciences, 241, 376-396(1957)
[33] Hashin, Z. The elastic moduli of heterogeneous materials. Journal of Applied Mechanics, 29, 143-150(1962)
[34] Hill, R. Elastic properties of reinforced solids:some theoretical principles. Journal of the Mechanics and Physics of Solids, 11, 357-372(1963)
[35] Brown, W. F. Solid mixture permittivities. Journal of Chemical Physics, 23, 1514-1517(1955)
[36] De Loor, G. P. Dielectric properties of heterogeneous mixtures with a polar constituent. Applied Scientific Research, Section B, 11, 310-320(1964)
[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)
[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)
[39] Budiansk, B. On elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13, 223-227(1965)
[40] Hill, R. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, 213-222(1965)
[41] Zimmerman, R. W. Elastic-moduli of a solid with spherical pores:new self-consistent method. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 21, 339-343(1984)
[42] Mori, T. and Tanaka, K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica et Materialia, 21, 571-574(1973)
[43] Voigt, W. Lehrbuch der Kristallphysik, B. G. Teubner, Berlin (1910)
[44] Reuss, A. Berechnung der flie β grenze von mischkristallen auf grund der plastizitäts bedingung f?r einkristalle. Zeitschrift für Angewandte Mathematik und Mechanik, 9, 49-58(1929)
[45] Hill, R. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society, Section A, 65, 349-354(1952)
[46] Peselnick, L. and Meister, R. Variational method of determining effective moduli of polycrystals:(A) hexagonal symmetry, (B) trigonal symmetry. Journal of Applied Physics, 36, 2879-2884(1965)
[47] Watt, J. P. Hashin-Shtrikman bounds on the effective elastic-moduli of polycrystals with orthorhombic symmetry. Journal of Applied Physics, 50, 6290-6295(1979)
[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. Journal of Applied Physics, 51, 1525-1531(1980)
[49] Budiansky, B. and Oconnell, R. J. Elastic-moduli of a cracked solid. International Journal of Solids and Structures, 12, 81-97(1976)
[50] Burridge, R. and Keller, J. B. Poroelasticity equations derived from microstructure. The Journal of the Acoustical Society of America, 70, 1140-1146(1981)
[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. Geophysical Journal International, 124, 105-112(1996)
[53] Kuster, G. T. and Toksoz, M. N. Velocity and attenuation of seismic waves in two-phase media, part 1:theoretical formulations. Geophysics, 39, 587-606(1974)
[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)
[55] Chapman, M., Zatsepin, S. V., and Crampin, S. Derivation of a microstructural poroelastic model. Geophysical Journal International, 151, 427-451(2002)
[56] Christensen, R. M. Mechanics of Composite Materials, Wiley InterScience, New York (1979)
[57] Hashin, Z. Analysis of composite materials-a survey. Journal of Applied Mechanics, 50, 481-505(1983)
[58] Hashin, Z. The elastic moduli of heterogeneous materials. Journal of Applied Mechanics, 29, 2938-2945(1962)
[59] Vernik, L. and Nur, A. Ultrasonic velocity and anisotropy of hydrocarbon source rocks. Geophysics, 57, 727-735(1992)
[60] Sokolnikoff, I. S. Mathematical Theory of Elasticity, McGraw-Hill, New York (1956)
[61] Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York (1944)
[62] Yan, F. and Han, D. H. Measurement of elastic properties of kerogen. SEG Technical Program Expanded Abstracts, 143, 2778-2782(2013)
/
| 〈 |
|
〉 |
Email Alert
RSS