Theoretical and numerical modelling of the effect of viscous and viscoelastic fluids on elastic properties of saturated rocks

Abstract

Rock physics is an essential link connecting seismic data to the properties of rocks and fluids in the subsurface. One of the most fundamental questions of rock physics is how to model the effects of pore fluids on rock velocity and density. Contemporary scientific computing allows geophysicists to conduct extremely complex virtual (computational) experiments on realistic digital representations of complex porous media, and thus directly relate the measurable properties of the media to their microstructure and saturation. Computational (digital) rock physics can also serve as an effective tool in examining new and existing rock physics models. The finite element method (FEM) has been proved effective in simulations of the linear elastic properties of porous rock under static conditions. In this thesis, FEM is used to study the effect of patchy saturation on elastic velocities of digital images of rocks. However, FEM belongs to a group of grid methods, and its accuracy is limited by discretization errors. This can cause errors in rock property predictions and needs to be thoroughly examined. In this thesis, a test scenario based on rigorous theories for grid-based methods such as FEM is developed, which allows establishing optimal computational parameters in terms of accuracy of the results and time cost of computations.Gassmann’s equations are the most widely used relations to predict velocity changes resulting from different pore fluid saturations. This problem is also known as fluid substitution. Despite the popularity of Gassmann’s equations and their incorporation in most software packages for seismic reservoir interpretation, important aspects of these equations such as sensitivity to microheterogeneity has not been thoroughly examined. In this thesis, the sensitivity of Gassmann’s equations to microheterogeneity is estimated for different quartz/clay porous mixtures using computational (FEM) simulations. The results of this study suggest that the accuracy of Gassmann’s fluid substitution remains adequate for a wide variety of highly porous rocks even if the contrast between the elastic properties of mineral constituents is large.While Gassmann’s fluid substitution is robust for rocks saturated with Newtonian fluids (brine, gas, light oil), it breaks down for viscoelastic fluids such as heavy oils. An alternative fluid substitution scheme for rocks saturated with viscoelastic fluids based on self-consistent effective medium theory is proposed in this thesis. Comparison with laboratory measurements shows that the scheme realistically estimates the frequency- and temperature dependent properties of heavyoil rocks and can be used for practical applications.A useful tool for modelling and estimation of properties of rocks with arbitrary or unknown microstructure are rigorous bounds on elastic moduli. The common elastic bounding methods such as Hashin-Shtrikman bounds are not applicable for heavy-oil rocks because of viscoelastic rheology of heavy oils. In this work, it is demonstrated that the viscoelastic bounding method of Milton and Berryman for the effective shear modulus of a two phase three-dimensional isotropic composite provides rigorous bounds for dispersion and attenuation of elastic waves in heavy-oil rocks. In particular, computation of these bounds shows that dispersion and attenuation in a rock saturated with a fluid (viscous or viscoelastic) can be much stronger than in the free fluid. This phenomenon is caused by wave-induced fluid flow relative to the solid. At sonic and ultrasonic frequencies, dispersion and attenuation appears to be dominated by the local (pore-scale) flow between pores of different shapes and orientations. The Mavko and Jizba expressions for the so-called unrelaxed frame bulk and shear moduli are one of the most popular quantitative models of squirt dispersion. However, these expressions are limited to liquidsaturated rocks and high frequency. In this thesis, The Mavko-Jizba relations are generalized to gas-saturated rocks. Furthermore, dispersion and attenuation is computed using a new squirt flow model, presented in this thesis. All the parameters in this model can be independently measured or estimated from measurements. The model gives complex frequency- and pressure-dependent effective bulk and shear moduli of a rock consistent with laboratory measurements.Variation of elastic properties of rocks with pressure is often modelled using penny-shaped or spheroidal cracks as idealization of real crack/pore geometry. In this doctorate, the validity of this approach is analysed by extracting the ratios of shear to bulk stress sensitivity coefficients, and normal to tangential compliances from ultrasonic measurements on a number of dry sandstone samples. The ratios show large scatter and, for a large number of dry sandstone samples, are not consistent with spheroidal crack theory. This inconsistency results in significantly different estimates of crack density from bulk and shear moduli, and in deviation of predicted pressure variation of Poisson’s ratio from the measured data.