Discovering accurate relationships between pore structure and elastic properties of porous rocks is a long standing problem in geophysics. Understanding the interaction between rock, pore space and fluids and how they control rock properties is crucial to better interpretation of geophysical measurements. Expressions that relate elastic moduli to porosity, pore-fluid compressibility and fluid saturation form the basis for reservoir assessment and monitoring procedures. They are used to infer porosity from well logs, as well as in-situ indicators of pore fluid type.
Properties of porous rocks depend primarily on the morphology of the pore space and solid phase(s). Relevant aspects of the rock structure include porosity, pore shape and size and the type and frequency of interconnections between pore and solid regions. These features, some of which unfortunately lack precise definition, comprise the morphology of the rock. Accurately predicting properties from microstructural information requires: an accurate quantitative description of the complex microstructure of the medium, and the ability to solve for mechanical properties on large three-dimensional grids. In the absence of a full structural characterisation, past attempts to relate the elastic properties of rocks to porosity have been limited to empirical relationships [Han 1986], effective medium theories [Berryman 1980], rigorous bounding methods [Hashin and Shtrikman 1962, Milton 1981] and simple deterministic models [Wyllie 1956,Raymer 1980]. None of these is entirely satisfactory.
Typically, empirical formulae are obtained statistically from experimental data sets. They provide a simple and convenient, but deceptive form of summarizing extensive experimental data. Lacking a rigorous connection with microstructure, these formulae do not offer predictive or interpretive power, seldom carry physical insight, and often fail when applied to a wider range of rock types. In effective medium theory, the microstructure corresponding to a specific model is not realistic; agreement or disagreement with data can neither confirm nor reject a particular model. A clear advantage of bounds is that they incorporate microstructural information and can be applied to arbitrarily complex structures. Bounds are extremely useful if the constituent materials have similar properties. For materials like porous sedimentary rocks, the bounds are quite far apart due to the large contrast in elastic properties between pore fluid and rock matrix which severely limits their predictive power. Simple deterministic models attempt to find a meaningful explanation for experimental observations. The best known example is Wyllie's equation [Wyllie 1956]. This equation is based on the observation that for clean sandstones the compressional wave velocity has a strong linear correlation with porosity. Raymer [Raymer 1980] modified this formula by suggesting different laws for different porosity ranges. Nur also used this method in suggesting a critical porosity model [Nur 1995]. These models work for certain classes of rock types, but do not have general applicability.
An alternative approach is to computationally solve the equations of elasticity directly on digitized models of microstructure [Roberts and Garboczi 2000]. Computer memory and processing speed now make it possible to handle the large three-dimensional models and number of computations needed to obtain useful results. As input to these methods, statistical models have been proposed for reconstructing 3D porous materials [Joshi 1974, Quiblier 1984, Adler 1990, Adler 1992, Roberts 1997, Yeong and Torquato 1998]. Complete characterization of the effective morphology however requires knowledge of an infinite set of n-point statistical correlation functions. In practice only lower order morphological information is available; common methods [Joshi 1974] are based on matching the first two moments (volume fraction and two-point correlation function) of the binary phase function to a random model. Random 3D models are then generated which match the measured statistical properties. It is widely recognised [Adler 1990, Adler 1992, Roberts 1997, Yeong and Torquato1998] that although the two-point correlation functions of a reference and a reconstructed system are in good agreement, this does not ensure that the structures of the two systems will match well. Adler used this technique to reconstruct Fontainebleau sandstone and found that computations of permeability [Adler 1990] and conductivity [Adler 1992] were consistently lower than experimental data, a result most likely due to percolation differences between model and real materials [Bentz and Martys 1994, Roberts and Knackstedt1996].
Direct techniques which provide a detailed 3D description of the pore structure were initially limited to sets of 2D serial sections imaged and combined to build the 3D image [Lin and Cohen 1982]. However this method is extremely tedious and time consuming. Direct measurement of a 3D structure is now readily available from synchotron and micro x-ray computed microtomography [Flannery 1987, Dunsmuir 1991, Spanne 1994] and laser confocal microscopy [Fredrich 1995]. These techniques provide the opportunity to directly measure the complex morphology of the pore space of sedimentary rock in three dimensions at resolutions down to a few microns. In parallel, computational techniques have progressed to the point where material properties such as diffusivity, elasticity and conductivity can be calculated on large three dimensional digitised images containing up to one billion 10003 voxels. With the development of these experimental and computational methods it is possible to replace synthetic images derived from statistical models with actual images and base calculations directly on the measured three-dimensional microstructure. This has been done previously [Spanne 1994, Schwartz 1994, Auzerais 1996] for the geometric and transport properties of sandstones. Their calculations showed good agreement with laboratory measurements for porosity and pore-volume-to-surface ratio. Their calculations of transport properties were less successful. In a recent paper [Arns 2001b] we showed that it is possible to accurately predict transport properties from digitized images by estimating and minimizing sources of numerical error. In the present paper we calculate for the first time the elastic properties of a tomographic image of sandstone. We consider the elastic properties of the digitized images under dry, water- and oil-saturated conditions. Numerical predictions are in excellent agreement with available experimental data. The observed change in the elastic properties due to fluid substitution is consistent with the exact Gassmann's equations [Gassmann1951, Berryman1999]. This shows both the feasibility and the accuracy of combining microtomographic images with elastic calculations to predict petrophysical properties of individual rock morphologies. We compare the numerical predictions to various empirical, effective medium, deterministic and rigorous approximations used to relate the elastic properties of rocks to porosity.
The paper is organized as follows. We first briefly describe the experimental acquisition method and define morphological measures used to choose an appropriate window size for simulation. Numerical methods used to derive the linear elastic properties are described in detail along with a discussion of potential numerical errors. We then give the predictions for the suite of Fontainebleau samples used in this study, and compare to experiment and to theory. Finally, we compare the numerical predictions with commonly used empirical methods.