Grain reconstruction of porous media: application to a low-porosity Fontainebleau sandstone

Development of optimal models of porous media by combining static and dynamic data: the porosity distribution

This paper is part of a project, the goal of which is the development of the optimal spatial distributions of the porosity and permeability of a large-scale porous medium by using complementary static and dynamic data for the medium. The data include limited measurements of the porosity, which the method honors (preserves) in the optimal model and utilizes its correlation function, together with the first-arrival (FA) times, at a certain number of receivers, of seismic waves that have propagated in the medium and the time dependence of the pressure of a fluid flowing in the medium. The method uses the simulated-annealing (SA) technique in order to develop the optimal model. In the present paper we utilize the porosity and FA times data in order to develop the optimal spatial distribution of the porosity. This is accomplished by combining the SA method with a simulator that solves for the numerical solution of the acoustic-wave equation from which the FA times are estimated, limited porosity, and FA times data. We show that the optimal model not only honors the data, but also provides accurate estimates of the porosities in the rest of the porous medium. The efficiency of the computations is discussed in detail.

Elastic and transport properties of cellular solids derived from three-dimensional tomographic images

We describe a three-dimensional imaging and analysis study of eight industrial cellular foam morphologies. The foam morphologies were generated by differing industrial processing methods. Tomograms are acquired on an X-ray micro-computed tomography facility at scales of approximately equal to at resolutions down to 7 μm. The image quality is sufficient in all cases to measure local structure and connectivity of the foamed material, and the field of view large enough to calculate a range of material properties. Phase separation into solid and porous components is straightforward.

Three-dimensional structural characteristics are measured directly on the porous and solid phases of the images. A number of morphological parameters are obtained, including pore volume-to-surface-area ratio, connectivity, the pore and solid phase size distributions defined by maximal sphere openings and chord length measurements. We further calculate the pore size distribution associated with capillary pressure via simulating of mercury drainage on the digital images.

The binarized microstructures are used as a basis for calculations of transport properties (fluid permeability, diffusivity and thermal conductivity) and elastic moduli. From the data, we generate property–porosity relationships for the range of foam morphologies imaged and quantitatively analyse the effects of porosity and microstructure on the resultant properties of the foams.

We compare our numerical data to commonly used theoretical and empirical property–porosity relationships. For thermal conductivity, we find that the numerical results agree extremely well with an empirical expression based on experimental data of various foams. The upper Hashin–Shtrikman bound also provides an excellent prediction of the data across all densities. From simulation of the diffusivity, we can define the tortuosity of the pore space within the cellular solid. We find that different processing methods lead to strong variations in the tortuosity of the pore space of the foams. For elastic properties, our results show that for the Young modulus, E , both the differential effective medium theory and the classical correlation give a good correlation. Assuming a constant Poisson's ratio leads to reasonable agreement. The best correlation for is given by assuming a slight variation in as a linear function of porosity. The permeability of the foams varies over three orders of magnitude. Correlations for permeability based on the classical Kozeny–Carman equation lead to reasonable agreement, except at the lowest porosities. Permeability estimations based on mercury porosimetry give excellent agreement for all foams.

Nuclear-magnetic-resonance diffusion simulations in two phases in porous media

Time-dependent diffusion simulations which can be measured by nuclear magnetic resonance (NMR) were numerically performed in consolidated reconstructed porous media saturated by two immobile fluids. The phase distributions were obtained by an immiscible lattice Boltzmann technique which incorporates interfacial tension and wetting. The apparent diffusion coefficient in each fluid was determined by a random walk algorithm. Permeability and conductivity tensors were calculated by finite-difference schemes. The major properties valid for a single phase could be generalized to two phases. First, the characteristic length Lambda introduced by Johnson et al.[Phys. Rev. Lett. 57, 2564 (1986)] is of the order of twice the phase volume to surface ratio. Second, the apparent diffusion coefficients for all porosities, saturations, and phases can be represented by a single dimensionless curve.

Cross-property correlations and permeability estimation in sandstone

We computationally investigate cross-property correlations linking fluid permeability to conductive properties in sedimentary rock for a number of pore size parameters based on three-dimensional digitized rock images. In particular, we focus on correlations based on the pore volume-to-surface-area ratio (V(p)/S), a critical channel diameter (c) associated with mercury porosimetry measurements, length scales associated with the nuclear magnetic resonance relaxation time T2, as well as the mean survival time (tau). Differences between the length scales are discussed. All these correlations yield good agreement with our simulations, but permeability estimates based on the critical diameter (c) are found to be most reliable.

Angstrom-to-millimeter characterization of sedimentary rock microstructure

Backscatter SEM imaging and small-angle neutron scattering (SANS) data are combined within a statistical framework to quantify the microstructure of a porous solid in terms of a continuous pore-size distribution spanning over five orders of magnitude of length scale, from 10 A to 500 microm. The method is demonstrated on a sample of natural sandstone and the results are tested against mercury porosimetry (MP) and nuclear magnetic resonance (NMR) relaxation data. The rock microstructure is fractal (D=2.47) in the pore-size range 10 A-50 microm and Euclidean for larger length scales. The pore-size distribution is consistent with that determined by MP. The NMR data show a bimodal distribution of proton T(2) relaxation times, which is interpreted quantitatively using a model of relaxation in fractal pores. Pore-length scales derived from the NMR data are consistent with the geometrical parameters derived from both the SEM/SANS and MP data. The combined SANS/BSEM method furnishes new microstructural information that should facilitate the study of capillary phenomena in hydrocarbon reservoir rocks and other porous solids exhibiting broad pore-size distributions.

Reconstructing complex materials via effective grain shapes

We introduce a powerful method based on integral geometry and the Kac theorem for the spectrum of the Laplace operator to define the effective shape of an inclusion in a system made up of a distribution of arbitrarily shaped constituents. Reconstructing the microstructure using the effective inclusion shape leads to an excellent match to the percolation thresholds and to the mechanical and transport properties across all phase fractions. Use of the equivalent shape in effective medium formulations leads to good predictions. The method is verified for a sedimentary rock sample.

Study of the microgeometry of porous materials using synchrotron computed microtomography

A series of measurements of the structure of a variety of porous materials has been made using synchrotron computed microtomography (SCMT). The work was carried out at the Brookhaven National Synchrotron Light Source (NSLS), the Argonne Advanced Photon Source (APS) and the European Synchrotron Radiation Facility (ESRF). The experiments at Brookhaven and Argonne were carried out on bending magnet beam lines using area detectors to obtain CT images based on determination of X-ray absorption coefficients. The work at the ESRF used an undulator beam line, a 13 KeV pencil X-ray beam of 2 µm and an energy dispersive X-ray detector to make tomographic sections of trace element distributions by X-ray fluorescence tomography. Most of the work was done with a pixel/voxel size ranging from 0.002 to 0.010 mm. We examined the structure of unconsolidated estuarine sediments, whose structure is relevant to transport of contaminants in rivers and estuaries. Fluorescent tomography with 2–3 µm resolution was used to ascertain whether or not metals were concentrated on the surface or throughout the volume of a single sediment particle. Sandstone samples were investigated to obtain a set of values describing their microstructures that could be useful in fluid flow calculations relevant to petroleum recovery or transport of environmental contaminants. Measurements were also made on sandstone samples that had been subjected to high-pressure compression to investigate the relation between the microgeometry and the magnitude of the applied pressure. Finally, a Wood’s metal-filled sample was scanned for demonstration of resolution enhancement and fluid flow studies.