With a detailed microscopic image of a rock sample, one can determine the corresponding 3-D grain geometry, forming a basis to calculate the elastic properties numerically. The issues which arise in such a calculation include those associated with image resolution, the registration of the digital numerical grid with the digital image, and grain anisotropy. Further, there is a need to validate the numerical calculation via experiment or theory. Because of the geometrical complexity of the rock, the best theoretical test employs the Hashin–Shtrikman result that, for an aggregate of two isotropic components with equal shear moduli, the bulk modulus is uniquely determined, independent of the micro-geometry. Similarly, for an aggregate of two isotropic components with a certain combination of elastic moduli defined herein, the Hashin–Shtrikman formulae give a unique result for the shear modulus, independent of the micro-geometry. For a porous, saturated rock, the solid incompressibility may be calculated via an “unjacketed” test, independent of the micro-geometry. Any numerical algorithm proposed for digital rock physics computation should be validated by successfully confirming these theoretical predictions. Using these tests, we validate a previously published staggered-grid finite difference damped time-stepping algorithm to calculate the static properties of digital rock models.
Cwebs beyond three loops in multiparton amplitudes