Ioannis Chatzis

Network modelling of pore structure and transport properties of porous media

Abstract A methodology is presented for the simultaneous prediction of absolute permeabilities, formation resistivity factors and drainage capillary pressure curves of sandstones by employing a network model of pore structure based on bond-correlated site percolation concepts. The model is a regular cubic lattice consisting of pore throats and pore bodies, having respective pore size distributions. Information about the pore structure, obtained from mercury porosimetry and photomicrographic analysis, is utilized to select the pore throat and pore body size distributions in a manner such that the resulting model (i) matches the porosity of the medium and (ii) satisfactorily simulates the drainage capillary pressure curve of the porous medium under consideration. Assumptions made about the cross-sectional shape of the pore throats and their effect on the network model predictions are discussed. Details of the methodology used in the simulation of transport properties with microscopic pore structure parameters are also presented and discussed. The problem of fluid and electric current flow through the simulated porous medium is reduced to an electric analogue-linear network problem and is numerically solved using a conjugate gradients method for computing the absolute permeability and the formation resistivity factor. Good agreement between the predicted and the measured values is observed for a number of sandstone samples having widely different transport properties.

Simulation of capillary pressure curves using bond correlated site percolation on a simple cubic network

A stochastic approach to network modelling has been used to simulate quasi-static immiscible displacement in porous media. Both number-based and volume-based network saturation results were obtained. Number-based results include: number-based saturation curves for primary drainage, secondary imbibition and secondary drainage, fluid distribution data, and cluster trapping history. Using pore structure data of porous media, it is possible to convert the number-based curves to capillary pressure — saturation relationships. Pore size distribution functions and pore shapes which are thought to closely represent Berea sandstone samples were used to predict the capillary curves. The physical basis of these calculations is a one-to-one correspondence between the cumulative node and bond index fractions in the network analysis, and the cumulative number-based distributions of pore body and pore throat diameters, respectively. The oil-water capillary pressure curve simulated for primary drainage closely resembles those measured experimentally. The agreement between the simulated and the measured secondary imbition and secondary drainage curves is less satisfactory.

Permeability and electrical conductivity of porous media from 3D stochastic replicas of the microstructure

In this paper we report on the application of low-order statistical information (porosity, two-point correlation function), obtained from 2D micrographs of real porous media, to derive stochastic replicas of their 3D pore networks. The main focus is on assessing the usefulness of stochastic reconstruction as a means of relating macroscopic transport coefficients (intrinsic permeability and effective electrical conductivity) to the geometry and topology of the pore space. To this end we employ newly developed algorithms, based on morphological skeletonization, to obtain a comprehensive geometric and topological description of each simulated pore network. Using the skeleton or graph of the pore space as a basis, detailed geometrical measurements are performed to estimate the effective hydraulic and electrical conductance of individual flow paths, identified with skeleton links. In combination with exact knowledge of the network topology, these measurements enable the specification of equivalent resistor-type network models for the calculation of intrinsic permeability and formation factor. Consistently successful predictions of permeability over a wide range of values are obtained for five reservoir rock samples of diverse origin and lithology. Predictions of formation factor are within a factor of three of the experimental values. These predictions are, however, inconsistent, a fact attributed to the inability of the reconstruction method to accurately reflect the contributions of smaller pores and throats to electrical conductivity. Additionally, it is shown that the hydraulic conductance of pore space channels in stochastically simulated pore networks is spatially correlated over distances equal to the characteristic length scale of the two-point correlation function. The effect of spatial resolution and sample size on the prediction of macroscopic properties is also investigated. Finally, the physical meaning of various length scales relating flow permeability to effective electrical conductivity is elucidated.

An experimental study of secondary oil migration

Experiments using long glass columns packed with glass beads or sand have been done to investigate secondary oil migration under hydrostatic conditions. Different combinations of bead sizes, oil densities, oil-water interfacial tensions, and column orientations have been tested. In some experiments, the oil was replaced by air. The observations included the oil migration pathway, the minimum oil column height needed for migration, and the rate of advance of the migration front. Migration was found to take place along restricted pathways and an imbibition front often formed at the bottom of the oil zone. The minimum oil zone height needed for migration can be predicted accurately if the values of the drainage and imbibition capillary pressures are known for the saturations at which the oil just becomes disconnected. In most experiments, the migration front advanced at a constant rate, which depended on the fluid properties, bead size, initial oil height, and pore structure. Migration rate is dependent on buoyant and capillary forces, but the dependence on capillary forces becomes weaker as the oil length increases. Column orientation also has been found to affect migration efficiency.

Electrical Conductivity and Percolation Aspects of Statistically Homogeneous Porous Media

A method of 3-D stochastic reconstruction of porous media based on statistical information extracted from 2-D sections is evaluated with reference to the steady transport of electric current. Model microstructures conforming to measured and simulated pore space autocorrelation functions are generated and the formation factor is systematically determined by random walk simulation as a function of porosity and correlation length. Computed formation factors are found to depend on correlation length only for small values of this parameter. This finding is explained by considering the general percolation behavior of a statistically homogeneous system. For porosities lower than about 0.2, the dependence of formation factor on porosity shows marked deviations from Archies law. This behavior results from the relatively high pore space percolation threshold (∼0.09) of the simulated media and suggests a limitation to the applicability of the method to low porosity media. It is additionally demonstrated that the distribution of secondary porosity at a larger scale can be simulated using stochastic methods. Computations of the formation factor are performed for model media with a matrix-vuggy structure as a function of the amount and spatial distribution of vuggy porosity and matrix conductivity. These results are shown to be consistent with limited available experimental data for carbonate rocks.

Statistical analysis of the porous microstructure as a method for estimating reservoir permeability

Back-scatter scanning electron microscope images of cross-sections of several porous rocks were analyzed to determine the statistical properties of the porous microstructure. For statistically homogeneous media these properties are the porosity and autocorrelation function. A length scale (integral correlation scale), characteristic of the spatial distribution of porosity, was obtained as the integral of the autocorrelation function. The permeability of a wide variety of rock samples, including those investigated by Coskun and Wardlaw (1993), was adequately described by an empirical equation of the form k α φaISb, where φ is the porosity and IS is the integral correlation scale. The results obtained have useful application in the estimation of reservoir permeability from samples not amenable to experimental testing (e.g., drill cuttings) and provide support for the use of statistical methods for the generation of 3-D model porous media.

Macroscopic Percolation Model of Immiscible Displacement: Effects of Buoyancy and Spatial Structure

Percolation theory, commonly used to study quasi-static immiscible displacement at the microscopic scale, is here extended to simulations of gravity-stable drainage and imbibition in three-dimensional porous media with spatially correlated macroscopic properties. The result of this extension is a macroscopic percolation on a regular lattice of sites, where lattice sites represent regions in a porous medium characterized by different macroscopic properties (e.g., absolute permeability, capillary pressure, and relative permeability curves). These properties are assigned to lattice sites by virtue of parametrizations in terms of local permeability. We present a general formulation of macroscopic percolation that accounts for gravitational effects, which can be important in large-scale immiscible displacements with nonzero density difference. In such cases, we find that the local saturation distribution is markedly different from the distribution of saturation under conditions of negligible buoyancy. Displacements with nonzero density difference proceed with the formation of a transition zone of length inversely proportional to a macroscopic Bond number which characterizes the relative importance of capillary and buoyancy forces at the macroscopic scale. Several important features of percolation at the microscopic scale are also manifested at the macroscopic scale. These include the effects of lattice dimensionality and spatial correlation on the macroscopic percolation threshold and accessibility characteristics. In the absence of buoyancy forces, the large-scale capillary pressure and relative permeability behavior of a heterogeneous system is dictated mainly by the structure of the permeability field and can be explained in terms of macroscopic accessibility. Spatial correlation of permeability is found to have pronounced effects on the large-scale drainage relative permeability curves.

VAPEX, Warm VAPEX and Hybrid VAPEX - The State of Enhanced Oil Recovery for In Situ Heavy Oils in Canada

VAPEX, warm VAPEX and hybrid VAPEX rely on a combination of mass and heat transfer to reduce the heavy oils viscosity sufficiently for it to flow via gravity drainage to the production well. In the last couple of years, many combinations of vapour extraxtion and steam-assisted gravity drainage have been proposed for the in situ recovery of heavy oil and bitumen. The question still remains; which technique (VAPEX, warm VAPEX, hybrid VAPEX and/or SAGD) produces more oil with better sweep efficiencies? This paper attempts to define and compare the various enhanced heavy oil recovery techniques, including different solvent choices. The pore-scale mechanisms are identified, key advantages and disadvantages given and the results from simple laboratory experiments are compared to better direct the investigation into the in situ recovery of heavy oil and bitumen. The solvents and solvent mixtures (in combination with non-condensable gases and/or steam) are analyzed based on their physical properties at laboratory and reservoir conditions and the role they play at the pore-scale.

The effect of spatial correlations on the accessibility characteristics of three-dimensional cubic networks as related to drainage displacements in porous media

Percolation theory is used in this paper to study the general accessibility characteristics of bond site correlated networks in the presence or absence of spatial correlations among the sites. Spectral methods are employed to generate networks in which the sites are spatially correlated according to exponential or Gaussian autocovariance functions. Monte Carlo simulations are performed for bond-correlated site percolation with and without trapping, as related to drainage-type displacements of water by oil (or air) and air by mercury, respectively. For the correlation modes studied, it is found that spatial correlations result in significant reduction of the site percolation threshold with a concomitant modification of the site accessibility. However, the bond accessibility characteristics and the bond percolation threshold are not significantly affected. The simulated drainage capillary pressure curves become more gradual and the residual wetting phase saturation associated with oil (or air)-water displacements is significantly decreased when spatial correlations among the pore bodies exist.

A mercury porosimeter for investigating capillary phenomena and microdisplacement mechanisms in capillary networks

Mercury porosimetry was studied in glass-etched micromodels with the aid of an experimental apparatus that enables the accurate measurement of capillary pressures and mercury saturations, as well as the observation of microdisplacement mechanisms at the pore level. The effect of fluid topology, pore size, and pore body to pore throat aspect ratio during quasi-static imbibition for the air—mercury system is demonstrated in terms of experimentally obtained capillary pressure curves. Imbibition is shown to be determined by the interplay of bond-withdrawal (snap-off in throats) and site-withdrawal (withdrawal from pores) processes. Under conditions of small variability in pore body and pre throat size and for relatively small pore body to pore throat aspect ratio, imbibition phenomena are controlled by the fluid topology in a deterministic manner. That is, withdrawal occurs first from pore throats by the snap-off mechanism and proceeds in pore bodies in a manner that preserves the continuity of the nonwetting phase (nwp). Critical capillary pressures were measured for the withdrawal of mercury from pores and throats under various configurations of capillary interfaces. Theoretical calculations were in qualitative agreement with experimental values. For fully saturated capillary networks, snap-off events in pore throats initiate the withdrawal of mercury. The conjecture that the frontal advance mechanism dominates mercury withdrawal over cluster growth at high initial mercury saturation is not valid for conditions whereby all faces of the pore network are exposed to a mercury sink.