Fontainebleau Sandstone Research Articles

Direct numerical simulation of pore-scale reactive transport: applications to wettability alteration during two-phase flow

We present a finite element – finite volume simulation method for modelling fluid flow and solute transport accompanied by chemical reactions in experimentally obtained 3D pore geometries. The advantage of the proposed methodology with respect to other pore-scale modelling approaches is that no simplifications regarding the geometry of the porous space are required and no approximations to the flow equations are introduced. We apply this method in a proof-of-concept study of a digitised Fontainebleau sandstone sample. We use the calculated velocity profile with the finite volume procedure to simulate pore-scale transport and diffusion of the adsorbing solute. We also demonstrate how analysis of the pore geometry can be used to identify the locations of oil during the two-phase flow and couple this with the reactive transport modeling to show how this procedure can be used to estimate the potential of the enhanced oil recovery techniques.

High-precision synthetic computed tomography of reconstructed porous media

Multiscale simulation of transport in disordered and porous media requires microstructures covering several decades in length scale. X-ray and synchrotron computed tomography are presently unable to resolve more than one decade of geometric detail. Recent advances in pore scale modeling [Biswal, Held, Khanna, Wang, and Hilfer, Phys. Rev. E 80, 041301 (2009)] provide strongly correlated microstructures with several decades in microstructural detail. A carefully calibrated microstructure model for Fontainebleau sandstone has been discretized into a suite of three-dimensional microstructures with resolutions from roughly 128 μm down to roughly 500 nm. At the highest resolution the three-dimensional image consists of 32768^{3}=35184372088832 discrete cubic volume elements with gray values between 0 and 216. To the best of our knowledge, this synthetic image is the largest computed tomogram of a porous medium available at present.

A reconstruction method for three‐dimensional pore space using multiple‐point geology statistic based on statistical pattern recognition and microstructure characterization

SUMMARYTo predict the macroscopic properties (e.g., transport, electromagnetic, and mechanical properties) of porous media, it is necessary to have a three‐dimensional (3D) representation of porous media. We reconstruct the geologically realistic 3D structure of Fontainebleau sandstone based on the two‐dimensional (2D) thin sections by using the multiple‐point statistics method. For this method, the size of template is an important parameter that reflects the perceived scale of spatial structure of porous media. In this paper, we take advantage of entropy method to obtain the appropriate size of the template, which is proven to be correct and feasible. The reconstruction method proposed by us combines successive 2D MPS simulations as well as 3D MPS simulation, which takes account into the pore structure information (e.g., heterogeneity and connectivity) both intralayer and interlayer. This reconstruction method is tested on Fontainebleau sandstone for which 3D images from micro‐CT scanning are available. Applying local percolation theory analysis, this new approach can depict the expected patterns of geological heterogeneities. In addition, it also can well reproduce a high degree of connectivity of the pore space better than other reconstruction methods based on lower‐order statistics. Copyright © 2011 John Wiley & Sons, Ltd.

A New Cell for Electrical Conductivity Measurement on Saturated Samples at Upper Crust Conditions

Electrical resistivity soundings are used by geophysicists to determine the structure and composition of the Earth’s crust and mantle and to explore natural resources (ore, oil, gas, water). Their interpretations in terms of composition and in-situ physical conditions depend mainly on laboratory measurements of electrical conductivity of rocks at simulated crustal conditions of temperature, pressure, saturation and pore pressures. These measurements present a numbers of limitations, in particular, in the case where conductive pore fluids are present, as in the case of deep reservoir conditions, where temperature exceeds 250 °C. Here, we present a new cell capable of measuring electrical conductivity of large saturated samples at confining pressure up to 200 MPa, pore pressure up to 50 MPa, and temperature up to 500 °C. The measurement cell has been developed in a commercial, internally heated, gas pressure apparatus (Paterson press). It is based on the concept of “guard ring” electrode, which is adapted to samples that are jacketed by a very conductive, metallic material. Numerical modeling of the current flow in the electrical cell allowed defining the optimal cell geometry. Calibration tests have been performed on Fontainebleau sandstones saturated with electrolytes of different conductivities, up to 350 °C. The resulting electrical formation factor and temperature dependence of electrical conductivity are in very good agreement with previous studies. This new cell will improve the exploration and exploitation of deep fluid reservoirs, as in unconventional, high enthalpy geothermal fields. In particular, the investigations address possible effects of fluid-rock interactions on electrical resistivity of a reservoir host rock.

Laboratory measurements of porosity, permeability, resistivity, and velocity on Fontainebleau sandstones

The relations among the resistivity, elastic-wave velocity, porosity, and permeability in Fontainebleau sandstone samples from the Ile de France region, around Paris, France were experimentally revisited. These samples followed a permeability-porosity relation given by Kozeny-Carman’s equation. For the resistivity measurements, the samples were partially saturated with brine. Archie’s equation was used to estimate resistivity at 100% water saturation, assuming a saturation exponent, [Formula: see text]. Using self-consistent (SC) approximations modeling with grain aspect ratio 1, and pore aspect ratio between 0.02 and 0.10, the experimental data fall into this theoretical range. The SC curve with the pore aspect ratio 0.05 appears to be close to the values measured in the entire porosity range. The elastic-wave velocity was mea-sured on these dry samples for confining pressure between 0 and [Formula: see text]. A loading and unloading cycle was used and did not produce any significant hysteresis in the velocity-pressure behavior. For the velocity data, using the SC model with a grain aspect ratio 1 and pore aspect ratios 0.2, 0.1, and 0.05 fit the data at [Formula: see text]; pore aspect ratios ranging between 0.1, 0.05, and 0.02 were a better fit for the data at [Formula: see text]. Both velocity and resistivity in clean sandstones can be modeled using the SC approximation. In addition, a linear fit was found between the P-wave velocity and the decimal logarithm of the normalized resistivity, with deviations that correlate with differences in permeability. Combining the stiff sand model and Archie for cementation exponents between 1.6 and 2.1, resistivity was modeled as a function of P-wave velocity for these clean sandstones.

Quantitative analysis of numerical estimates for the permeability of porous media fromlattice-Boltzmann simulations

During the last decade, lattice-Boltzmann simulations have been improved to become anefficient tool for determining the permeability of porous media samples. However,well-known improvements of the original algorithm are often not implemented. Theseinclude, for example, multirelaxation time schemes or improved boundary conditions, aswell as different possibilities to impose a pressure gradient. This paper shows that asignificant difference of the calculated permeabilities can be found unless one uses acarefully selected set-up. We present a detailed discussion of possible simulation set-upsand quantitative studies of the influence of simulation parameters. We illustrate ourresults by applying the algorithm to a Fontainebleau sandstone and by comparingour benchmark studies to other numerical permeability measurements in theliterature.

Permeability and pore connectivity: A new model based on network simulations

The purpose of this paper is to model the effect of pore size heterogeneity and pore connectivity on permeability. Our approach is that of conceptual modeling based on network simulations. We simulated fluid flow through pipe networks with different coordination numbers and different pipe radius distributions. Following a method widely used in percolation theory, we sought “universal” relationships (i.e., independent of lattice type) between macroscopic properties such as permeability k and porosity ϕ, and, pore geometry attributes such as hydraulic radius rH, coordination number z, and so forth. Our main result was that in three‐dimensional simple cubic, FCC, and BCC networks, permeability obeyed “universal” power laws, k ∝ (z − zc)β, where the exponent β is a function of the standard deviation of the pore radius distribution and zc = 1.5 is the percolation threshold expressed in terms of the coordination number. Most importantly, these power law relationships hold in a wide domain, from z close to zc to the maximum possible values of z. A permeability model was inferred on the basis of the power laws mentioned above. It was satisfactorily tested by comparison with published, experimental, and microstructural data on Fontainebleau sandstone.

Efficient flow and transport simulations in reconstructed 3D pore geometries

Upscaling pore-scale processes into macroscopic quantities such as hydrodynamic dispersion is still not a straightforward matter for porous media with complex pore space geometries. Recently it has become possible to obtain very realistic 3D geometries for the pore system of real rocks using either numerical reconstruction or micro-CT measurements. In this work, we present a finite element–finite volume simulation method for modeling single-phase fluid flow and solute transport in experimentally obtained 3D pore geometries. Algebraic multigrid techniques and parallelization allow us to solve the Stokes and advection–diffusion equations on large meshes with several millions of elements. We apply this method in a proof-of-concept study of a digitized Fontainebleau sandstone sample. We use the calculated velocity to simulate pore-scale solute transport and diffusion. From this, we are able to calculate the a priori emergent macroscopic hydrodynamic dispersion coefficient of the porous medium for a given molecular diffusion D m of the solute species. By performing this calculation at a range of flow rates, we can correctly predict all of the observed flow regimes from diffusion dominated to convection dominated.

Continuum reconstruction of the pore scale microstructure for Fontainebleau sandstone

Abstract A stochastic geometrical modeling technique is used to reconstruct a laboratory scale Fontainebleau sandstone with a sidelength of 1.5 cm. The model reconstruction is based on crystallite properties and diagenetic parameters determined from two-dimensional images. The three-dimensional pore scale microstructure of the sandstone is represented by a list of quartz crystallites defined geometrically and placed in the continuum. This allows generation of synthetic μ -CT images of the rock model at arbitrary resolutions. Quantitative microstructure comparison based on Minkowski functionals, two-point correlation function and local porosity theory indicates that this modeling technique can provide more realistic and accurate models of sandstones than many existing techniques used currently. Synthetic μ -CT images at different resolutions from a laboratory scale model of Fontainebleau sandstone are made available to the scientific community for resolution dependent petrophysical analysis.

The brittle deformation regime of water-saturated siliceous sandstones

SUMMARY We present here new experimental data on the mechanical behaviour of water-saturated Bentheim and Fontainebleau sandstones deformed in the brittle failure regime. Bentheim sandstone samples were stressed at room temperature and subjected to confining pressures PC ranging from 12 to 120 MPa and pore pressures PP ranging from 1 to 70 MPa. For all samples the evolution of the volumetric strain first shows compaction eventually reversing to dilation of the pore volume when approaching the peak stress. All samples failed localized on a single shear zone. Critical stresses that is onset of dilatancy and peak stress, can be uniquely defined as a function of the effective pressure Peff = PC – PP. The failure curve parameters, when fitted with the Hoek–Brown criterion or a parabolic envelope, are consistent with previous values obtained on similar material. When compared to data obtained on dry samples of the same rock, the present data show no notable effect of the presence of water on the critical stress levels. The same conclusion holds for the stress–strain curves when dry and water-saturated experiments under equivalent effective confining pressures are compared. When compared to previously published data obtained on various quartzose rocks, which show a quite variable water-weakening effect, our results lead to the conclusion that the quasi-exclusive presence of quartz grains bonded together by a quartzose cement combined with the absence of clayey minerals may explain the absence of this effect in Bentheim sandstone. Moreover, this conclusion is supported by complementary results obtained by Fontainebleau sandstone with similar microstructural and compositional characteristics. A previously developed micromechanical model based on the interaction of pore cracks has been used and modified to take into account the presence of water. When compared to the experimental data, the results of the model show good agreement for low to intermediate effective confining pressures. Critical stresses as well as deformation curves are well reproduced as was the case for the dry rock provided that effective pressure is introduced. At higher effective pressures, a mix of mechanisms including shear-enhanced compaction may explain the progressive discrepancy between the modelled and the experimental curves when approaching the apex of the failure envelope.

Pore-scale simulation of NMR response

The random walk method is used to simulate magnetization decay in porous media. The simulations were performed on images of the pore space obtained using micro-CT scanning and in topologically equivalent networks extracted from these images using a maximal ball algorithm. The simulation results were validated through comparison with experimental measurements of T 2 distribution, absolute permeability and resistivity in two sand packs and from comparing predictions on images and networks of Fontainebleau sandstone. In all cases, the comparisons were good, although the networks gave a slightly narrower T 2 distribution, implying that some fine detail of the pore structure was lost. This work suggests that imaging, network extraction and pore-scale simulation can be used to predict single-phase transport properties successfully. It serves as a validation for pore-network models and the methods used to generate networks.

Seismic stress stimulation mobilizes colloids trapped in a porous rock

Low-frequency (1–500 Hz) stress (seismic) waves can alter flow and mass transport behavior in saturated subsurface porous formations. Numerous physical mechanisms have been proposed to explain this phenomenon. Coupling of dynamic stress to in-situ particle (colloid) mobilization is one such mechanism that can produce changes in pore matrix permeability. This phenomenon has broad-ranging impacts on colloid dynamics and mass transport problems over a scale range of microns to kilometers. Porous-flow experiments in sandstone cores demonstrated that in-situ colloidal particles can be released from pores by applying dynamic mechanical stress stimulation at frequencies below 100 Hz. Due to the lower attenuation at these frequencies, relative to ultrasonic, significant dynamic stress in the Earth can penetrate to distances of a kilometer or more. Thus, seismic waves could affect particle mobility in distant oil formations, aquifers and fault systems. Laboratory results are shown for release of in-situ particles from Fontainebleau sandstone induced by applying stress stimulation at 26 Hz. Although enhanced particle release relative to that induced by flowing de-ionized water alone was observed, the permeability of the core was unchanged, indicating that pore throat fouling was insignificant. The behavior of the post-stimulation particle release was distinctly different than the pre-stimulation behavior in that a cyclical pattern with uniform periodicity was observed. This cyclical behavior was observed to be independent of particle size over the range of 50 to 800 nm and is attributed to stimulation causing a long-term change in the distribution of the rate coefficients for release of particles from the pore space. The rate change is likely not due to alteration of particle–wall interactions which are sensitive to particle size. Size-independent release mechanisms that can explain the stimulated rate change are 1) enhanced flushing or squeezing out of particles trapped in dead-end pores, and 2) forced particle detachment and exposing of new detachment sites on the pore walls. The observations presented here are unique in that they indicate sub-pore-scale particle mobility and transport can be influenced by long-wavelength seismic-band stress. Implications on possible field-scale effects and micro-scale physical mechanisms are discussed.

Axial and Radial Permeability Evolutions of Compressed Sandstones: End Effects and Shear-band Induced Permeability Anisotropy

The influence of hydrostatic and uniaxial stress states on the porosity and permeability of sandstones has been investigated. The experimental procedure uses a special triaxial cell which allows permeability measurements in the axial and radial directions. The core sleeve is equipped with two pressure samplers placed distant from the ends. They provide mid-length axial permeability measure as opposed to the overall permeability measure, which is based on the flow imposed through the pistons of the triaxial cell. The core sleeve is also equipped to perform flows in two directions transverse to the axis of the sample. Two independent measures of axial and complementary radial permeability are thus obtained. Both Fontainebleau sandstone specimens with a porosity of about 5.8% to 8% and low permeability ranging from 2.5 mD to 30 mD and Bentheimer sandstone with a porosity of 24% and a high permeability of 3D have been tested. The initial axial permeability values obtained by each method are in good agreement for the Fontainebleau sandstone. The Bentheimer sandstone samples present an axial mid-length permeability 1.6 times higher than the overall permeability. A similar discrepancy is also observed in the radial direction, also it relates essentially to the shape of flow lines induced by the radial flow. All the tested samples have shown a higher stress dependency of overall and radial permeability than mid-length permeability. The effect of compaction damage at the pistons/sample and radial ports/sample interfaces is discussed. The relevance of directional permeability measurements during continuous uniaxial compression loadings has been shown on the Bentheimer sandstone until the failure of the sample. We can efficiently measure the influence of brittle failure associated to dilatant regime on the permeability: It tends to increase in the failure propagation direction and to decrease strongly in the transverse direction.

Deviation from Archie’s law in partially saturated porous media: Wetting film versus disconnectedness of the conducting phase

We experimentally study the electrical transport in partially water-saturated pore network. The porous medium under investigation is a Fontainebleau sandstone, characterized by x-ray tomography. We show the existence of two electrical conductivity regimes. At high water saturation, the electric resistivity follows a well-known Archie law. Below a water saturation S_{w} approximately 0.2 , a strong negative deviation from this Archie law is observed. We attribute this transition to the existence of "a thick liquid film," assuring the ionic conduction in the low saturation regime. A numerical simulation is proposed to confirm this scenario. Two possible protocols are used to distribute the brine phase in the pore network of a three-dimensional microtomography image. The first one is based on a minimization of the interfacial energy. The second takes into account a quasistatic capillary displacement. The classical random-walk algorithm is used to compute the electric conductivity at various water saturations. Without the "thick film," both of the two fluid-placing protocols show a disconnectedness transition of the brine phase when S_{w}<0.2 . Adding this "film" to solid surface, the electrical continuity is maintained. The bending down trend is correctly reproduced, showing that in this range, the electric response cannot be described by a power law as usual.

Effect of density-driven flow on the through-diffusion experiment

Diffusion is one of the main mechanisms of solute transport in pore water of geologic media. The effective diffusion coefficient of a solute in a rock is usually measured by the through-diffusion experiment. However, in this experiment, the effect of advection, induced by density difference between dense aqueous solution and pure water, has not been considered. To evaluate the effect of density-driven flow, a through-diffusion experiment using Fontainebleau sandstone was conducted for KCl and KI aqueous solutions with various densities. The measured effective diffusion coefficients were positively correlated with the density difference; the effective diffusion coefficient of a 1 M KI solution (density difference, 0.119 g/cm 3) was one order of magnitude larger than that of a 0.1 M KCl solution (density difference, 0.005 g/cm 3). The result is explained by a theoretical model using a diffusion–advection equation including Darcy's law. Based on the theory, a diagram to evaluate the condition at which the measured effective diffusion coefficient does not include the effect of advection is presented.

Cracks in porous rocks: Tiny defects, strong effects

Understanding the physics and mechanics of porous sedimentary rocks is a stimulating challenge. Indeed, these rocks are of great economical interest for two main reasons: they are the places for hydrocarbon resources and for underground storage. Yet the complexity of these natural composites would suggest low expectations for research projects aiming at a fundamental understanding of their behavior. However progress has been observed, not limited to the study of pure quartz sandstones such as Fontainebleau sandstones. Many experiments and models have been developed that deal with more typical, complex sandstones. Even shales, that are certainly the most complex sedimentary rocks, have also been well documented. But why is it of interest to understand better the fundamental physics and mechanics of sedimentary rocks? And how has progress been made possible, given those difficulties?

Network flow modeling via lattice-Boltzmann based channel conductance

Lattice-Boltzmann (LB) computations of single phase, pore-to-pore conductance are compared to models in which such conductances are computed via standard pore body–channel–pore body series resistance (SR), with the conductance of each individual element (pore body, channel) based on geometric shape factor measurements. The LB computations, based upon actual channel geometry derived from X-ray computed tomographic imagery, reveal that the variation in conductance for channels having similar shape factor is much larger than is adequately captured by the geometric models. Fits to the dependence of median value of conductance versus shape factor from the LB-based computations show a power law dependence of higher power than that predicted by the geometric models. We introduce two network flow models based upon the LB conductance computations: one model is based upon LB computations for each pore-to-pore connection; the second is based upon a power law fit to the relationship between computed conductance and throat shape factor. Bulk absolute permeabilities for Fontainebleau sandstone images are computed using the SR-based network models and the two LB-based models. Both LB-based network models produce bulk absolute permeability values that fit published data more accurately than the SR-based models.

Hydromechanical properties of Fontainebleau sandstone: Experimental determination and micromechanical modeling

We measured ultrasonic velocity, hydraulic permeability, and specific storage capacity of eight blocks of Fontainebleau sandstone, a well‐sorted, medium‐grained, almost pure quartz‐sandstone of Oligocene age, covering a range in connected porosity from 3% to 10% and varying significantly in pore geometry for a given porosity. Ultrasonic P‐wave velocity measured on water‐saturated samples covers the full range predicted by variational Hashin‐Shtrikman bounds. Permeability and specific storage were determined by the linear pressurization method at different effective pressures up to 180 MPa and room temperature. The permeability of tested samples varies from 10−13 m2 down to 10−20 m2, depending primarily on porosity and pore geometry, and subordinately on effective pressure. Specific storage capacity always exceeds the contribution of pore fluid compression. The excess corresponds to the contribution of pore deformability controlled by pore geometry and matrix material properties. Since the sandstone samples are composed of a single solid component, quartz, we were able to calculate various poroelastic parameters (Biot‐Willis and Skempton coefficients, drained and undrained bulk moduli) crucial for estimating hydromechanical coupling in fluid‐saturated aggregates. The micromechanical modeling of the effective pressure sensitivity of the hydraulic transport and poroelastic parameters by the contiguity model of Takei (1998) yields good agreement between static and dynamic elastic parameters for saturated samples, partial agreement between the pressure dependence of effective elastic parameters and Hertzian contact mechanics, and an explanation for pressure‐insensitive permeability of samples with the highest porosity.

Nonlinear-nonequilibrium wave propagation in sandstones

The transition from linear to nonlinear-nonequilibrium dynamical elasticity in rocks is of considerable interest in seismic wave propagation as well as in understanding the basic dynamical processes in consolidated granular materials. We have carried out a careful experimental investigation of this transition for Berea and Fontainebleau sandstones. These experiments have showed the existence of two strain regimes. At low strain (10-9 up to 10-7) the rocks behave elastically as classical (Landau-Lifshitz theory) nonlinear materials. At higher strains memory effects due to a driven nonequilibrium state complicate the characterization of the nonlinear behavior. The understanding of this second region is not trivial. The main focus of this work is to review the behavior of the low strain, nonlinear region and to report on new experiments meant to understand more about the behavior in the nonequilibrium regime.

Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications

In the first part of this series of two papers, we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials (media) via two-point correlation functions S(2) and introduced an efficient heterogeneous-medium (re)construction algorithm called the "lattice-point" algorithm. Here we discuss the algorithmic details of the lattice-point procedure and an algorithm modification using surface optimization to further speed up the (re)construction process. The importance of the error tolerance, which indicates to what accuracy the media are (re)constructed, is also emphasized and discussed. We apply the algorithm to generate three-dimensional digitized realizations of a Fontainebleau sandstone and a boron-carbide/aluminum composite from the two-dimensional tomographic images of their slices through the materials. To ascertain whether the information contained in S(2) is sufficient to capture the salient structural features, we compute the two-point cluster functions of the media, which are superior signatures of the microstructure because they incorporate topological connectedness information. We also study the reconstruction of a binary laser-speckle pattern in two dimensions, in which the algorithm fails to reproduce the pattern accurately. We conclude that in general reconstructions using S(2) only work well for heterogeneous materials with single-scale structures. However, two-point information via S(2) is not sufficient to accurately model multiscale random media. Moreover, we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two-point correlation functions.