INFLUENCE OF SPATIAL INTERACTIONS OF INCLUSIONS ON THE EFFECTIVE ELASTIC TENSOR OF CRACKED POROUS MEDIUM

The determination of effective stiffness tensor of microinhomogeneous and, in general, macroscopically homogeneous composite medium is related to so-called problem of many-body interaction. Solution to the problem can be found only as an  approximation. In this paper we consider a solution to such a  problem for porous-cracked medium that is a terrigenous rock  having anisotropic elastic properties. The elastic anisotropy is a  result of many factors including anisotropic properties of clay  minerals and preferential orientation of non-isometric  heterogeneities. Different Effective Medium Theories for calculating  effective stiffness tensor of cracked porous medium use so called  Effective Field Hypothesis (H1, H2 and H3). For example, T-matrix  method, Mori-Tanaka method, General Singular Approximation method, and Effective Field Methods use the Effective Field Hypothesis. Thus, different methods produce similar results.  When constructing models of rock’s effective properties the rock is  treated a composite “made by nature”. In this case of importance is  a proper approximation of the real medium by a parametric model  medium that reflects specific features of rock’s microstructure. The  microstructure is a result of rock evolution. Therefore, the model of  the medium and the model parameters play very important roles in the modelling. To prove this statement, two models of a cracked- porous medium’s properties were created using two different methods: the T-matrix method and General Singular  Approximation Method. The methods were applied for two different  parametric models of one and the same rock. The models were build  based on visual analysis of rock’s thin sections. Each of the  constructed models has different number of parameters. The  parameters are also different. However, a common feature of the two models is that for rocks of this type it is necessary to take into  account a rigidity of contact between mineral grains and organic material. Besides, a connectivity of different heterogeneities should  be also parametrized. For each model a set of parameters was found and a porosity interval where the models produce similar results in terms of elastic wave velocities is determined.

1. Alkhimenkov, Yu.A. 2017, “The T-matrix approach for the mathematical modeling of the effective elastic properties of hydrocarbon reservoirs”. Izv., Phys. Solid Earth. Vol. 53, Issue 3, pp. 477-487. https://doi.org/10.1134/S1069351317030016

2. Bayuk, I.O. 2013, “Main principles of the mathematicall modelling of the macroscopic physical properties of the hydrocarbon collectors”, Tekhnologii Seismorazvedki, no. 4, pp. 5-18

3. Bayuk, I.O., Beloborodov, D.E., Berezina, I.A., Vdovichenko, I.I., Vershinin, A.V., Gilyazetdinova, D.R., Gorbunov, V.N., Zingerman, K.M., Korost, D.V., Krasnova, M.A., Tikhotskiy, S.A., Ulkin, D.A., Fokin, I.V., Yakovlev, M.Ya. & Yalaev, T.R. “Problems of the collector rocks elastic properties upscaling”, Materialy konferencii “Seismicheskie technologii” (Proc. of the conference “Seismic technologies”). Moscow, 2016, pp. 27-30.

4. Morozov, E.M., Levin V.A. & Vershinin, A.V. 2015, Strength Analysis: FIDESYS in engineer hands, URSS, Moscow, 400 pp.

5. Levin, V.A., Lokhin, V.V. & Zingerman, K.M., 1997. “About one way of strength characteristics estimation for the porous bodies suspected to finite deformations”. Izvestya AN. Mechanika tverdogo tela, no. 4, pp. 45-60.

6. Levin, V.A. & Zingerman, K.M., 2002. “About the effective constitutive relations for the porous materials suspected to finite deformations and their overlaing”. Doklady RAN, Vol. 382, no. 4, pp. 482-487.

7. Shermergor, T.D. 1977, Theory of elasticity for micro-inhomogeneous media, Nauka, Moscow, 400 pp.

8. Alkhimenkov, Yu.A. & Bayuk, I.O., 2017. “Analysis of anisotropy parameters of fractured carbonate reservoir”. Proc. 6th EAGE Saint Petersburg International Conference and Exhibition. Sankt-Petersburg, 2016.

9. Alkhimenkov, Y. “Practical Applications of the T-Matrix Approach to Fractured Porous Rocks”. Proc. SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers. Houston, 2015.

10. Bayuk, I. O., Ammerman, M. & Chesnokov E. M., 2007. “Elastic moduli of anisotropic clay”. Geophysics, vol. 72(5), D107-D117.

11. Buryachenko, V.A. 2007, Micromechanics of Heterogeneous Materials. Springer. 687 pp.

12. Buryachenko, V.A., 2010. “On the thermo-elastostatics of heterogeneous materials: I. General integral equation”. Acta mechanica, vol. 213(3), pp. 359- 374.

13. Jakobsen, M., Hudson, J. & Johansen T.A., 2003. “T-matrix approach to shale acoustics”. Geophysical Journal International, vol. 154. pp. 533–558.

14. Kanaun, K.K. & Levin, V.M. 2008. “Self-Consistent Methods for Composites”. Vol. 1, 2. Springer, Dordrecht.

15. Willis, J.R., 1977. “Bounds and self-consistent estimates for the overall properties of anisotropic composites”. Journal of the Mechanic and Physics of Solids, vol. 25, pp.185–202.