Effective elastic moduli of a heterogeneous oolitic rock containing 3-D irregularly shaped pores

We have characterized the microstructure of a heterogeneous oolitic rock (limestone from Lavoux, France) with X-ray nano-computed tomography. This rock comprises an assemblage of porous grains (oolites), irregularly shaped three-dimensional pores, and inter-oolitic crystals (cement). To model the effect of this microstructure on the macroscopic behavior of the rock, we approximate the porous oolites by spheres, and the irregularly shaped pores by ellipsoids. This approximation is performed based on the principal component analysis PCA, which provides the geometrical properties such as length of semi-axes and orientation of resulting ellipsoids. The sphericity of the approximated oolites was calculated and the value close to 1 allows us to consider oolites as spheres. To verify the approximation in the case of pores, we evaluated the contribution of these irregularly shaped three-dimensional pores to the overall elastic properties. Thus, compliance contribution tensors for 3D irregular pores and their ellipsoidal approximations are calculated using the finite element method. These tensors were compared to the compliance tensors for ellipsoids obtained using analytical solutions based on Eshelby's theory and a relative error is estimated to evaluate the accuracy of the approximation. This error produces a maximum discrepancy of 4.5% between the two solutions respectively of pores and ellipsoids which verifies the proposed approximation procedure based on principal component analysis. The FEA numerical method is verified by comparing the numerical solution of compliance contribution tensors of the ellipsoids to the known analytical solution of these same shapes based on Eshelby's theory. The difference between these two solutions does not exceed 3%. Compliance contribution tensors are finally used to compare effective elastic parameters of a material containing irregular pores via the Maxwell homogenization scheme. These elastic parameters (bulk modulus and shear coefficient) coincide with a maximum deviation of 5% with ones for a material with ellipsoidal pores.