Velocity–porosity relationships: Predictive velocity model for cemented sands composed of multiple mineral phases

Computer simulations are used to calculate the elastic properties of model cemented sandstones composed of two or more mineral phases. Two idealized models are considered – a grain-overlap clay/quartz mix and a pore-lining clay/quartz mix. Unlike experimental data, the numerical data exhibit little noise yet cover a wide range of quartz/cement ratios and porosities. The results of the computations are in good agreement with experimental data for clay-bearing consolidated sandstones. The effective modulus of solid mineral mixtures is found to be relatively insensitive to microstructural detail. It is shown that the Hashin–Shtrikman average is a good estimate for the modulus of the solid mineral mixtures. The distribution of the cement phase is found to have little effect on the computed modulus–porosity relationships. Numerical data for dry and saturated states confirm that Gassmann’s equations remain valid for porous materials composed of multiple solid constituents. As noted previously, the Krief relationship successfully describes the porosity dependence of the dry shear modulus, and a recent empirical relationship provides a good estimate for the dry-rock Poisson’s ratio. From the numerical computations, a new empirical model, which requires only a knowledge of system mineralogy, is proposed for the modulus–porosity relationship of isotropic dry or fluid-saturated porous materials composed of multiple solid constituents. Comparisons with experimental data for clean and shaly sandstones and computations for more complex, three-mineral (quartz/dolomite/clay) systems show good agreement with the proposed model over a very wide range of porosities. 1 I N T R O D U C T I O N Developing accurate relationships between porosity and seismic wave velocities in porous rocks has been an important area of interest in rock physics for decades. Such relationships are crucial in relating in situ seismic/sonic measurements to porosity and fluid saturation. Present porosity–velocity relationships are based on theoretical formulae, rigorous bounds or empirical relationships. Theoretical formulae are usually based on idealized morphologies; Wyllie’s time-average equation (Wyllie, Gardner ∗E-mail: [email protected] and Gregory 1963), for example, is based on the assumption that pores and grains are arranged in homogeneous layers perpendicular to the seismic raypath. This equation and its underlying assumptions often lead to poor agreement with experiment. Effective-medium theories were developed to extend exact results for dilute inclusions to higher volume fractions. Some microstructures have been shown a posteriori to have properties that correspond to the theories, but the physical structures are very unusual and do not resemble those of porous rocks (Milton 1984). Bounds are rigorously based on realistic microstructural information and are therefore widely applicable. Variational bounds include the simplest first-order bounds (e.g. Reuss and Voigt) and rigorous bounds based on expansions to third C © 2005 European Association of Geoscientists & Engineers 349 350 M.A. Knackstedt, C.H. Arns and W.V. Pinczewski order (Yeong and Torquato 1998). Bounds are particularly useful when they are narrow. However, when the moduli of two components differ significantly, the span of bounded properties becomes extremely large. Unfortunately this is the case for most porous rocks. Empirical relationships remain most popular because they represent actual laboratory or log measurements. Unfortunately, measured porosity–modulus data usually display a high degree of scatter. This may be due to variations in lithology (pore shape, size, degree of compaction) (Marion et al. 1992) and, in particular, to clay content and distribution (Han 1986). Experimental data sets are therefore usually grouped into lithological types (see e.g. Nur et al. 1995; Wang 2000), such as dolostones, limestones, sandstone, shaly sands and shales. While useful for summarizing lithotypes, empirical relationships are limited by the scatter in the measured data sets and can seldom be applied successfully outside the range of the measured data. In previous papers (Arns, Knackstedt and Pinczewski 2002a; Knackstedt, Arns and Pinczewski 2003), we have shown that it is possible to explore velocity–porosity relationships in a more precise way by computing the elastic properties of digital models of complex microstructures, where the microstructure and mineralogy of the material is known and where it is possible to average over a number of statistically identical samples. The computed data, although based on idealized microstructures, allow a quantitative analysis of the effects of porosity and the distribution of the mineral phases on the elastic properties of the model rocks. Our previous computational studies of modulus–porosity relationships were limited to clean (monomineral) rock systems. However, actual sandstones are rarely clean and often contain minerals other than quartz (e.g. clays), which can affect their elastic properties. Clay can be distributed in a number of ways in the rock framework depending on the conditions at deposition, on compaction, bioturbation and diagenesis. Most empirical relationships (e.g. Castagna, Batzle and Eastwood 1985; Han 1986; Han, Nur and Morgan 1986) and theoretical models (Xu and White 1995) account for the volume of clay present and ignore its distribution. An advantage of numerical models is that they can be used to study complex multiphase materials with physically realistic phase distributions. We consider two model distributions of cemented sandstone composed of two or more mineral phases: a grain-overlap model, where grains of cement are distributed structurally, and a pore-lining cementation model, where the cement phase develops uniformly on the original sand/pore interface. We study model clay/quartz and dolomitic/quartz cemented sands and use computer simulation to determine the elastic properties of the model systems. We generate porosity–moduli relationships for both the dry and water-saturated states for the model cemented sandstone morphologies. Since the pore space and the solid phase microstructure and mineralogy of the model systems are precisely defined, the resultant numerical data sets exhibit very little noise or scatter and it is possible to analyse porosity– modulus relationships quantitatively in the absence of experimental noise. The effective moduli of the solid mineral mixtures show a small dependence on microstructure. We compare the numerically determined moduli of the solid mineral mixtures with effective-medium and empirical estimates and find that the Hashin–Shtrikman average provides a good estimate for the modulus of the solid mineral mixture. Weighted Hashin–Shtrikman fits (Wang, Wang and Cates 2001) are also presented. We find that, for all porosities, the choice of microstructural model for cementation has a minimal effect on the resultant modulus–porosity relationship. Gassmann’s equations are verified for the porous materials made up of multiple solid constituents. Agreement of the predictions with experimental data sets for clay-bearing consolidated sandstones is encouraging. As previously noted for clean sandstone systems (Knackstedt et al. 2003), the Krief empirical relationship (Krief et al. 1990) is found to be particularly successful in describing the porosity dependence of the shear modulus, and a recent empirical method (Arns et al. 2002a) provides a good prediction of the Poisson’s ratio data for dry rock. From the computed results, we propose a new empirical model which describes the full modulus–porosity relationship for isotropic dry and fluid-saturated consolidated sand systems and requires only a knowledge of the mineral constituents and the proportion of each mineral phase present. The effective solid modulus of the mineral phase is given by the averaged Hashin–Shtrikman equation, Meff = (Ml + Mu)/2, (1) where M denotes both the bulk (K) and shear (μ) moduli and Ml and Mu are the Hashin–Shtrikman lower and upper bounds, respectively. The porosity dependence of the cemented sands follows the same empirical method proposed previously for clean sandstones (Krief et al. 1990; Arns et al. 2002a): μ(φ) = μeff(1 − φ) (2) and νdry(φ) = a(φ) + (1 − (2φ))νeff, (3) C © 2005 European Association of Geoscientists & Engineers, Geophysical Prospecting, 53, 349–372 Velocity–porosity relationships 351 where ν is Poisson’s ratio. Gassmann’s equations are subsequently used to determine the fluid-saturated states. Comparisons of the proposed method with the results of numerical simulations for systems containing as many as three individual mineral phases and with a number of experimental studies are encouraging. 2 M O D E L R O C K S Y S T E M S 2.1 Two-mineral-phase microstructure In a previous study we generated idealized single-mineral model rock morphologies over a very wide range of porosities using a simple Boolean overlapping spheres model (Knackstedt et al. 2003). In the present study we extend this idealized model to two-mineral phases in which the second mineral is either randomly dispersed via grain overlap throughout the medium (Fig. 1) or distributed as a uniform pore-lining phase around the first mineral (Fig. 2). A wide range of porosities and mineral ratios are realized by varying the number of spheres and the thickness of the pore-lining phase. We report the ratio of the mineral phases (M1:M2) along with the order of placement, mineral-1 then mineral-2, and the porosity. For example, a quartz/clay grain-overlap model