Consolidating rock-physics classics: A practical take on granular effective medium models

Abstract

Granular effective medium (GEM) models rely on the physics of a random packing of spheres. Although the relative simplicity of these models contrasts with the complex texture of most grain-based sedimentary rocks, their analytical form makes them easier to apply than numerical models designed to simulate more complex rock structures. Also, unlike empirical models, they do not rely on data acquired under specific physical conditions and can therefore be used to extrapolate beyond available observations. In addition to these practical considerations, the appeal of GEM models lies in their parameterization, which is suited for a quantitative description of the rock texture. As a result, they have significantly helped promote the use of rock physics in the context of seismic exploration for hydrocarbon resources by providing geoscientists with tools to infer rock composition and microstructure from sonic velocities. Over the years, several classic GEM models have emerged to address modeling needs for different rock types such as unconsolidated, cemented, and clay-rich sandstones. We describe how these rockphysics models, pivotal links between geology and seismic data, can be combined into extended models through the introduction of a few additional parameters (matrix stiffness index, cement cohesion coefficient, contact-cement fraction, and laminated clays fraction), each associated with a compositional or textural property of the rock. A variety of real data sets are used to illustrate how these parameters expand the realm of seismic rock-physics diagnostics by increasing the versatility of the extended models and facilitating the simulation of plausible geologic variations away from the wells. Introduction Based on the physics of a random packing of identical spheres, granular effective medium (GEM) models are an obvious analog for sedimentary rocks made of an aggregate of rounded mineral grains. They are a standard option for modeling sandstones made of quartz and feldspar grains or limestones formed from ooids, spherical calcite, or aragonite grains. Their simplicity of use is such that they are often applied beyond their original scope and associated assumptions when no better alternatives are available. Most GEM models are hybrid models that combine several geologic concepts implemented through a mix of rigorous theories and heuristic techniques. A compaction trend captures the evolution of the elastic properties of a packing of grains during burial. Various contact models developed by Brandt (1955), Digby (1981), Walton (1987), and Jenkins et al. (2005) relate the elastic properties of a random aggregate of identical spheres to the normal and tangential stiffness of their grain-to-grain contacts. All derive from the original theory developed by Hertz (1882) and Mindlin (1949) to model the elastic behavior of two identical spheres in contact. An optional cementation trend formalized by Dvorkin et al. (1991) describes Fabien Allo1 the stiffening effect of cement deposited at grain contacts. An additional sorting trend defines the effect of porosity reduction due to the introduction of solid material in the interstitial pore space. This sorting trend is usually modeled through elastic bounds such as the ones defined by Hashin and Shtrikman (1963). To obtain saturated rock properties, the effect of the fluid present in the pore space is added through a transformation model such as the standard fluid substitution relations introduced by Gassmann (1951). From a practical standpoint, the way these concepts are combined as well as the set of model parameters that can be adjusted are as important as the underlying theories and models. Parameters directly linked to rock composition and microstructure make the calibration to field data more intuitive and the quality control of predictions more objective. A geologically oriented parameterization also simplifies the integration of textural data obtained from cores, which mitigates part of the uncertainty inherent to the calibration process. We present how, through the introduction of a few physically meaningful parameters (matrix stiffness index [MSI], cement cohesion coefficient [CCC], contact-cement fraction [CCF], and laminated clays fraction [LCF]), classic GEM models can be combined into extended models suitable for improved rock-physics diagnostics over a broader range of formations. Particular attention is paid to the impact these parameters have on the modeled elastic rock properties. Often limited to rock-physics experts, this knowledge is key to achieving successful calibrations to field data. The potential for rock-physics diagnosis based on the extended models is demonstrated using a variety of real data sets. All illustrations focus on compressional wave velocity (VP), but the extended models also provide shear wave velocity (VS), which in combination with VP is key to an accurate interpretation of lithology, pore fluid, and pore pressure from seismic data. Unconsolidated sandstone models Dvorkin and Nur (1996) introduce the friable-sand or soft sandstone model to describe porosity-velocity relationships of poorly consolidated formations from the North Sea. This model uses a modified Hashin-Shtrikman lower bound to interpolate the bulk and shear moduli of the dry rock frame between the pure mineral and critical porosity end members. These bounds are defined by the moduli of the mineral grains and the pressure-dependent moduli of the dry pack of grains as given by the Hertz-Mindlin theory, respectively. The stiff sandstone model described by Mavko et al. (1998) shares the same end members but uses a modified HashinShtrikman upper bound to interpolate between them. Combining these two models into an extended unconsolidated sandstone model is achieved by using a weighted modified Hashin-Shtrikman bound. The weighted bound allows linear interpolation between the moduli derived from the upper bound (Mstiff) and the lower bound (Msoft): 1CGG, Calgary, Alberta, Canada. E-mail: [email protected]. https://doi.org/10.1190/tle38050334.1.