Patricia A. Berge

Critique of two explicit schemes for estimating elastic properties of multiphase composites

Abstract Explicit schemes of Mori-Tanaka and Kuster-Toksoz for estimating elastic properties of multiphase composites are compared and contrasted. Both methods are known to have a limited range of validity. Reexamination of the theories and comparisons to experiment leads to the conclusion that these explicit schemes should only be used to estimate properties of systems in which the host material occupies about 70–80% or more of the total volume. By introducing the concept of a reference material, we find a mutually consistent pair of equations for effective stiffness and compliance from which many of the standard approximation schemes may be easily derived. A unified analysis of the various approximation schemes results.

Using two-point correlation functions to characterize microgeometry and estimate permeabilities of sandstones and porous glass

We have developed an image processing method for characterizing the microstructure of rock and other porous materials and to provide a quantitative means for understanding the dependence of physical properties on the pore structure. Our method is based upon the statistical properties of the microgeometry as observed in scanning electron micrograph (SEM) images of cross sections of porous materials. This method uses a simple statistical measure of microstructure called the spatial correlation function. We formulate a two-point spatial correlation function and show how it can be used to estimate porosity, specific surface area, and other microstructural features such as pore and grain sizes. The porosity and specific surface area are of special interest as they can be used in a Kozeny-Carman relation to predict permeability of porous materials. We explore the Kozeny-Carman relation and show how it incorporates a characteristic microstructural length parameter similar to that used in other analyses of permeability. We analyze SEM images of several different porous glasses and natural sandstones using the two-point correlation function, discuss the importance of image resolution, and show for the sandstones studied here how the appropriate choices of image resolution can be made so the measured parameters are consistent with those used in a simple flow model for computation of permeability. Estimates of permeabilities for several different porous glasses and natural sandstones are presented. Comparison of these estimates to laboratory measurements shows good qualitative agreement and quantitative agreement within about a factor of 2 for most samples and 3 for all samples.

Ultrasonic velocity-porosity relationships for sandstone analogs made from fused glass beads

Using fused glass beads, we have constructed a suite of clean sandstone analogs, with porosities ranging from about 1 to 43 percent, to test the applicability of various composite medium theories that model elastic properties. We measured P‐ and S‐wave velocities in dry and saturated cases for our synthetic sandstones and compared the observations to theoretical predictions of the Hashin‐Shtrikman bounds, a differential effective medium approach, and a self‐consistent theory known as the coherent potential approximation. The self‐consistent theory fits the observed velocities in these sandstone analogs because it allows both grains and pores to remain connected over a wide range of porosities. This behavior occurs because this theory treats grains and pores symmetrically without requiring a single background (host) material, and it also allows the composite medium to become disconnected at a finite porosity. In contrast, the differential effective medium theory and the Hashin‐Shtrikman upper bound overest...

Velocity‐porosity relationships in the upper oceanic crust: Theoretical considerations

We consider here the application of rock physics theories to investigate relationships between seismic velocities and porosities in the shallow oceanic crust. Classical Hashin-Shtrikman limits ignore void shapes and are too broad to provide useful constraints on velocities and porosities. Making some assumptions about the distribution of void shapes improves the constraints. Theories which ignore crack-crack interactions underestimate the effects of porosities on velocities, thus providing upper bounds on velocities and porosities. “Self-consistent” theories overestimate crackcrack interactions and so provide lower bounds. At the high porosities required to reduce basalt from a P velocity of 7km/s in massive form to the 2.2km/s observed in zero-age oceanic crust, however, the bounds are too far apart to be useful. The theories are strictly valid only for very small porosities. Using an algorithm proposed by Cheng for iteratively building up porosity to create a highly porous medium, analogous to differential computation methods traditionally used to improve upon the self-consistent approach, we have devised two hybrid theories, which we term extended Walsh and extended Kuster-Toksoz. These two theories remain approximately valid at the high porosities of oceanic crustal layer 2A to provide useful upper and lower bounds on velocity for a given porosity and pore aspect ratio distribution. We attempt the inverse problem, determining porosity from a given velocity, using on-bottom refraction data collected on the flank of the East Pacific Rise. For 120ka material with a P velocity of 2.5km/s, if our assumptions regarding the aspect ratio distribution are correct, porosity lies somewhere between 24 and 34%. Resolution on slower, zero-age crust (2.2km/s) is poorer: there we predict a porosity between 26 and 43%. Use of shear-wave information would tighten these bounds.

Influence of microstructure on rock elastic properties

Depending on details of the composite microstructure, different theories may be needed to obtain good agreement with measured elastic properties. This observation is especially pertinent whenever the composite is porous, as is normally true for rocks. Predictions of three theories are compared to data for porous glass samples. The differential effective medium (DEM) theory and Hashins composite spheres assemblage (H) do a good job of predicting elastic behavior of a porous foam composed of glass. The self-consistent (SC) effective medium theory does equally well at predicting behavior of a sintered glass-bead sample. The realizable microstructure of each theoretical model is a good analog of the microstructure for one or the other of these two very different porous glasses. Velocities of granular rocks such as sandstones may be estimated accurately using the SC theory, whereas velocities of rocks such as basalts having isolated cracks and pores may be better estimated using either the DEM theory or Hashins model.

Differential effective medium modeling of rock elastic moduli with critical porosity constraints

Rocks generally have a percolation porosity at which they lose rigidity and fall apart. Percolation behavior is a purely geometrical property, independent of any physical properties, and is a powerful constraint on any valid velocity-porosity relation. We show how the conventional Differential Effective Medium (DEM) theory can be modified to incorporate percolation of elastic moduli in rocks by taking the material at the critical porosity as one of the constituents of a two-phase composite. Any desired percolation porosity can be specified as an input. In contrast, the conventional DEM model always predicts percolation at a porosity of either 0 or 100 percent. Most sedimentary rocks however have intermediate percolation porosities and are therefore not well represented by the conventional theory. The modified DEM model incorporates percolation behavior, and at the same time is always consistent with the Hashin-Shtrikman bounds. The predictions compare favorably with laboratory sandstone data.

Estimating rock porosity and fluid saturation using only seismic velocities

Evaluation of the fluid content in deep earth reservoirs or fluid contaminants in shallow earth environments has required the use of geophysical imaging methods such as seismic reflection prospecting. Interpretation of seismic velocities and amplitudes is based on theories of fluid‐saturated and partially saturated rocks that have been available since the 1950s. Here we present a new synthesis of the same physical concepts that uses compressional‐wave velocities together with shear‐wave velocities in a scheme that is much simpler to understand and apply yet yields detailed information about porosity and fluid saturation magnitudes and spatial distribution. The key idea revolves around the fact that the density and the Lame elastic parameter λ are the only two parameters determining seismic velocities that also contain information about fluid saturation. At low enough frequencies, Gassmanns well‐known equations show that the shear modulus is independent of the fluid saturation level. We use these facts to...

Realizability of negative pore compressibility in poroelastic composites

For elastic materials containing fluid-saturated porosity, the pore compressibility is a measure of the deformation of a unit pore volume in response to a change in fluid pressure. Rather than being measured, this quantity has been routinely set equal to an effective solid compressibility, since this equality is exact whenever a single solid component is present. However, we show that the pore compressibility and solid compressibility may be uncorrelated in general. In certain special circumstances they do not even share the same sign. Although thermodynamic and mechanical stability constraints cause solid and drained-frame bulk moduli of a porous composite to be positive and bounded by component properties, the pore compressibility is unconstrained and, therefore, can have negative values. For special realizable model materials, the value of the pore compressibility can be found using an exact expression valid for a composite made up of one fluid and two solid components, i.e., two porous components. In order to quantify how various factors affect the sign and magnitude of the pore compressibility, pore compressibilities were calculated for models that used two porous components having the micro geometry of an assemblage of concentric spheres. This model implicitly assumes the pores are on a much smaller length scale than the concentric spheres. Modeling results show that with the stiffer porous material forming the outer shells of the concentric spheres, the pore compressibility of such materials is negative when solid component bulk moduli differ by at least a factor of 5, if in addition, the porosities and drained frame moduli of the two porous components are relatively low. Negative pore compressibilities were found for realizable models whose two porous constituents had the properties of silicon nitride and either sandstone or clay. For models using combinations of alumina and glass foam properties, pore compressibilities were non-negative but smaller than the compressibilities of the solid components.

Pore pressure buildup coefficient in synthetic and natural sandstones

Abstract We present laboratory measurements of Skemptons pore pressure buildup coefficient B for synthetic and natural sandstones, and use our results to estimate the unjacketed pore compressibility 1/Ko. Measured values of B are always near unity for differential pressures below ∼2 MPa, regardless of porosity, grain composition, or the presence of small amounts of clay in the samples. We observed B near 0.7–0.8 for differential pressures of about 8–21 MPa in a clay-free synthetic sandstone. Similar values are expected for natural sandstones with low crack densities, whereas sandstones containing greater concentrations of microcracks apparently have values of B near 0.6. Our results suggest that the pore compressibility may have values close to the pore-fluid compressibility rather than the grain compressibility.

Transformation of Seismic Velocity Data to Extract Porosity and Saturation Values for Rocks

For wave propagation at low frequencies in a porous medium, the Gassmann-Domenico relations are well-established for homogeneous partial saturation by a liquid. They provide the correct relations for seismic velocities in terms of constituent bulk and shear moduli, solid and fluid densities, porosity and saturation. It has not been possible, however, to invert these relations easily to determine porosity and saturation when the seismic velocities are known. Also, the state (or distribution) of saturation, i.e., whether or not liquid and gas are homogeneously mixed in the pore space, is another important variable for reservoir evaluation. A reliable ability to determine the state of saturation from velocity data continues to be problematic. It is shown how transforming compressional and shear wave velocity data to the (rho/lambda, mu/lambda)-plane (where lambda and mu are the Lame parameters and rho is the total density) results in a set of quasi-orthogonal coordinates for porosity and liquid saturation that greatly aids in the interpretation of seismic data for the physical parameters of most interest. A second transformation of the same data then permits isolation of the liquid saturation value, and also provides some direct information about the state of saturation. By thus replotting the data in the (lambda/mu, rho/mu)-plane, inferences can be made concerning the degree of patchy (inhomogeneous) versus homogeneous saturation that is present in the region of the medium sampled by the data. Our examples include igneous and sedimentary rocks, as well as man-made porous materials. These results have potential applications in various areas of interest, including petroleum exploration and reservoir characterization, geothermal resource evaluation, environmental restoration monitoring, and geotechnical site characterization.