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Published laboratory elastic-wave velocity versus porosity data in carbonate rocks exhibit significant scatter even at a fixed mineralogy. This scatter is usually attributed to the strong variability in the rock-frame or pore-space geometry, which, in turn, is driven by the richness and complexity of diagenetic alteration in these very reactive sediments. Yet, by examining wireline data from oil-bearing high-to-medium porosity chalk deposits, we find surprisingly tight velocity–porosity trends. Moreover, these trends are continued into the low-porosity domain by data from a location thousands of miles away from the chalk field. This congruence implies a universality of diagenetic trends, at least in the massive deposits under examination. We also find that the elastic bulk and shear moduli of the pure-calcite end member are somewhat smaller than such values reported in the literature. Using the end-member elastic constants relevant to the data under examination, we establish a theoretical rock physics model to match and generalize these data. [ABSTRACT FROM AUTHOR]

ABSTRACT: We explore a package of parallel porous layers, each filled with a different fluid. Assume that this package is sampled by an elastic wave with the wavelength much larger than the thickness of an individual layer. Also assume that the layers are hydraulically isolated from each other, meaning that the diffusion length is smaller than that of the individual layer. This assumption is relevant to a patchy saturation scenario. Suppose that we wish to conduct the fluid substitution operation on this package treated as a single porous elastic body. What is the effective bulk modulus of the pore fluid to be used in this operation that will result in the same elastic modulus as computed by Backus averaging the individual moduli of the layers? We address this question analytically by assuming that the porosity, dry frame, and the mineral matrix properties of the individual layers are the same for all layers. The only difference between the layers is the pore fluid. We find that the resulting effective bulk modulus of the fluid thus derived falls between the arithmetic and harmonic averages of the fluid bulk moduli in the layers. It can be approximated by a linear combination of these two bounds where the weights are 0.50 and 0.50 or 0.75 for the arithmetic average and 0.25 for the harmonic average, depending on the elastic moduli of the dry frame, the mineral, and the pore fluids. This solution also provides a relation between the effective bulk modulus of the pore fluid in the system under examination and water saturation to be used in the fluid substitution operation at a coarse spatial scale. [ABSTRACT FROM AUTHOR]

We compute the porosity and specific surface area of a two-dimensional (2D) pore-scale segmented image of a natural rock sample, add existing relations linking tortuosity to porosity, and then use the Kozeny-Carman equation to obtain estimates of the absolute permeability of the three-dimensional (3D) host sample. These estimates are very close to the true permeability, measured or computed, of the host sample in rocks with well-connected pore space, including high-porosity sand, glass bead packs, low-to-medium porosity sandstones, and some carbonates. They are higher than the true permeability in carbonates with relatively large pores connected by constricted conduits. They fail in impermeable synthetically constructed samples with disconnected pores. Still, this extremely simple method proves to be viable for a large class of natural rocks. Moreover, it can be used to construct a permeability-porosity trend from a microscopic 2D image by selecting subsections of this image and computing their permeability-porosity pairs. Because the measured or computed permeability-porosity pair of the host 3D sample falls upon such a trend, we argue that this trend may be applicable at a spatial scale much coarser that of the pore-scale image. • We use the Kozeny-Carman equation to obtain estimates of the absolute permeability of the three-dimensional (3D) host sample. • These estimates are compared to the permeabilities of the host 3D samples obtained by the lattice-Boltzmann flow simulations. • These estimates are close to the true permeability, measured or computed, of the host sample in rocks with well-connected pore space, including high-porosity sand, glass bead packs, low-to-medium porosity sandstones, and some carbonates. [ABSTRACT FROM AUTHOR]

We micro-CT-scanned several granular packs, including glass beads of two different particle sizes and their mixtures, a natural dune sand, as well as two sieved samples of this sand and their mixtures. After segmenting these digital volumes into grains and pores, we computed their porosity (φ) and specific surface area (S). Moreover, we subsampled these segmented volumes and computed the respective φ and S of the subvolumes as well. The resulting data pairs exhibited fairly tight S versus φ trends. These trends, as observed among different samples, present a somewhat inconsistent picture. In some granular digital samples, S increases with increasing φ , while in others the opposite behavior is observed. To explain these behaviors, we analytically model the evolution of the pore-space geometry as that of a binary mixture of large and small particles. Where the small particles gradually fill the pores of an undisturbed large-particle frame, S becomes larger as φ reduces. The opposite behavior takes place where the disparate large particles are embedded into the continuum of the small particles. The smaller the number of the large particles the higher the porosity and the larger the specific surface area. Where the particles in a pack are of uniform size, S reduces with increasing φ. This theory quantitatively supports the observed experimental relations. The observed consistency between the experiment and theory means that the behaviors observed on microscopic samples are applicable at a much larger spatial scale. • Specific surface area versus porosity transforms in granular samples depend on grain sorting. • These transforms are obtained from digital images on several granular samples, glass beads and sand. • Simple theoretical equations describe the phenomena observed in binary mixture and match the data. [ABSTRACT FROM AUTHOR]

By computing the total porosity φ and specific surface area S for a number of segmented digital volumes of natural sandstones, carbonates, and granular samples, as well as their subvolumes, we observe fairly tight trends between those two variables. The emerging picture is different for low-to-medium porosity rocks from that for high-porosity granular samples. In the former, S increases with increasing φ , while the behavior is opposite in the particulates. We explain these trends by invoking simple theoretical derivations where the consolidated low-to-medium porosity samples are modeled as solids with inclusions, while the particulates are represented by packs of grains. While in the former S is linearly proportional to φ , it is linearly proportional to 1 − φ in the latter. The digital data are fairly accurately matched by the respective theoretical curves with the pore- and grain-size statistics extracted from the digital volumes. This fact arguably means that the trends obtained here from microscopic digital volumes are valid at a much coarser core and reservoir scale. • Specific surface area versus porosity transforms depend on rock type, consolidated versus granular. • These transforms are obtained from digital pore-scale images on a large number of natural rock samples. • Simple theoretical equations describe the phenomena observed and match the data. [ABSTRACT FROM AUTHOR]

Matrix acidizing is vital to enhance the production of tight carbonate formations. In weak carbonate rocks or acid over-dissolution, the rock is prone to failure upon developing wormholes. Strengthening agents can play a significant role in mitigating failure and supporting more extended productivity. In this study, we tested 0.1 M and 0.2 M zinc sulfate (ZnSO4) and 0.5 M diammonium phosphate ((NH4)2HPO4) as strengthening agents, which change the dominant host mineral into harder ones. Both agents were tested on two different types of acidized (i.e., wormholed) carbonate rocks, specifically Austin chalk and Indiana limestone. The impulse hammer and acoustics measurements evaluated the carbonate rocks' mechanical properties before and after the treatment. In addition, scanning electron microscopy energy-dispersive X-ray spectroscopy (SEM–EDS) and scanning electron microscopy (SEM) were used to affirm the change of rock mineralogy into harder minerals after the treatment. The chemical treatment caused no noticeable change in the wormhole size and shape, as revealed by the scanned micro-computed tomography (micro-CT) images. The chemical treatment of the samples was conducted by core flooding the plugs at high pressure and temperature with 4–5 pore volumes of the chemicals and leaving them in the treatment solution for 72 h. As a result of the formation of harder minerals, the mechanical properties after the treatment showed improvement. Zinc sulfate raised the surface rock hardness of limestone by 16%, while diammonium phosphate showed a 30% increase when evaluated with the same technique. The most noticeable improvement happened to chalk after treatment with diammonium phosphate. Chalk impulse hammer hardness almost quadrupled with the diammonium phosphate. The acoustics measurements showed a similar result to the impulse hammer through an increase in the dynamic Young's modulus and a decrease in Poisson's ratio. [ABSTRACT FROM AUTHOR]

The question posed in this work is whether and how elemental data obtained on small and often sub-representative fragments of natural rock can be exploited to generate a physically valid transform between two physical properties useable at a coarser spatial scale, such as the reservoir scale. The pair of such properties targeted here is the porosity and electrical formation factor. We use the process-based upscaling to prove, by example, that a coarse-scale transform between these two variables can be obtained from a dataset computed on microscopic digital volumes of natural rock, including unconsolidated sand, medium-porosity sandstones, and low- and high-porosity carbonates. This upscaling method is based on constructing a coarse-scale (effective) rock volume from the elemental volumes with a subsequent simulation of a relevant physical process in this effective object, specifically the electrical current. A numerical simulation of Laplace equation governing electrical current is used to compute the effective resistivity. The effective porosity is simply the arithmetically averaged porosity of the elements. By using hundreds of random realizations of the effective object, we generate tight formation factor versus porosity transforms for each of the natural rock samples under examination. Because the proof offered here is by-example, it is not general. However, the method offered is general and, as such, can be used to address the same question as posed for various situations. • Formation factor versus porosity relations from digital rock physics show consistent trends. • These transforms are obtained from digital pore-scale images on several natural rock samples. • Process-based upscaling is used to show that these transforms can be used at coarser spatial scales. [ABSTRACT FROM AUTHOR]

Abstract: This is the second and final part of our digital rock physics (DRP) benchmarking study. We use segmented 3-D images (one for Fontainebleau, three for Berea, three for a carbonate, and one for a sphere pack) to directly compute the absolute permeability, the electrical resistivity, and elastic moduli. The numerical methods tested include a finite-element solver (elastic moduli and electrical conductivity), two finite-difference solvers (elastic moduli and electrical conductivity), a Fourier-based Lippmann-Schwinger solver (elastic moduli), a lattice-Boltzmann solver (hydraulic permeability), and the explicit-jump method (hydraulic permeability and electrical conductivity). The set-ups for these numerical experiments, including the boundary conditions and the total model size, varied as well. The results thus produced vary from each other. For example, the highest computed permeability value may differ from the lowest one by a factor of 1.5. Nevertheless, all these results fall within the ranges consistent with the relevant laboratory data. Our analysis provides the DRP community with a range of possible outcomes which can be expected depending on the solver and its setup. [Copyright &y& Elsevier]

Abstract: The key paradigm of digital rock physics (DRP) “image and compute” implies imaging and digitizing the pore space and mineral matrix of natural rock and then numerically simulating various physical processes in this digital object to obtain such macroscopic rock properties as permeability, electrical conductivity, and elastic moduli. The steps of this process include image acquisition, image processing (noise reduction, smoothing, and segmentation); setting up the numerical experiment (object size and resolution as well as the boundary conditions); and numerically solving the field equations. Finally, we need to interpret the solution thus obtained in terms of the desired macroscopic properties. For each of these DRP steps, there is more than one method and implementation. Our goal is to explore and record the variability of the computed effective properties as a function of using different tools and workflows. Such benchmarking is the topic of the two present companion papers. Here, in the first part, we introduce four 3D microstructures, a segmented Fontainebleau sandstone sample (porosity 0.147), a gray-scale Berea sample; a gray-scale Grosmont carbonate sample; and a numerically constructed pack of solid spheres (porosity 0.343). Segmentation of the gray-scale images by three independent teams reveals the uncertainty of this process: the segmented porosity range is between 0.184 and 0.209 for Berea and between 0.195 and 0.271 for the carbonate. The implications of the uncertainty associated with image segmentation are explored in a second paper. [Copyright &y& Elsevier]

Natural rock is heterogeneous at all scales. This challenges the digital rock physics concept since the images of natural rock are mm-sized, while the results should be applicable at much larger spatial scales. To address this challenge we subsample a digital volume and compute the permeability of its subvolumes, as well as that of the entire volume, using the lattice-Boltzmann method (LBM). We reassemble the resulting values in 3D and use Darcy's flow simulator to re-compute the permeability of the entire host volume. In spite of significant heterogeneity of the subvolume properties, the Darcy-upscaled entire-volume permeability falls very close to that LBM-computed. Next, by using random reconstructions of the subvolume permeability matrix, we obtain physically meaningful permeability versus porosity transforms that can arguably be used at a scale much greater than that of a digital CT-scan image. This simple principle "to see a trend in a grain of sand" is the novel contribution of our work. Arguably, it can be implemented using images of drill cuttings collected within an interval in a well. We discuss examples of this principle for three basic rock types: high-porosity unconsolidated sand, medium-porosity sandstone, and medium-porosity carbonate. • Permeability-porosity transforms can be obtained from a single micro-CT scan. • Elemental permeabilities are computed using the Lattice-Boltzmann method. • Upscaled transforms are produced using Darcy's flow simulation. • Process-based averaging concept enables upscaling. [ABSTRACT FROM AUTHOR]

A patchy saturated two‐layered porous rock, with each layer filled with a different fluid, is examined. We assume that this system is probed by an elastic wave having a wavelength much larger than the layers' thicknesses. We also assume that the diffusion length is smaller than the thickness of an individual layer, which implies hydraulic disconnection between layers. In the context of Gassmann's fluid substitution theory, we analytically derived an exact expression for the effective fluid bulk modulus assuming that both layers have the same porosity, dry frame, and mineral matrix properties. In addition we derived an approximate solution that works well at relatively high porosities. Both solutions are expressed as a weighted average of the arithmetic and harmonic averages of individual bulk moduli of the pore fluid. These weights are explicitly given as functions of the porosity, the fractional thicknesses of both layers, and the elastic moduli of the constituents. For the approximate solution, one does not require explicit knowledge of the shear modulus of the rock. The comparison with laboratory data showed that, in the case where a porous, isotropic rock is filled with water and gas, the approximate solution can be used to model the measured data for high values of water saturation. Key Points: We derive exact and approximate analytic solutions to model the effective fluid bulk modulus of patchy saturated porous rocksWe compare and validate the solutions with stochastic simulation and laboratory data [ABSTRACT FROM AUTHOR]

Sand and glass bead samples were monotonically loaded from 0 to 50 MPa and then monotonically unloaded, all under hydrostatic stress conditions. Sand deformed irreversibly and its porosity loss was permanent. In contrast, the glass bead deformation was almost fully reversible with the initial porosity recovered. During loading, the static bulk modulus in all sand samples increased from 0 to about 0.5 GPa between 0‐ and 15‐MPa stress and remained constant for the remainder of the loading. Such behavior, somewhat unexpected in a granular system, means that sand deforms as a linear elastic body during loading from 15 to 50 MPa. The unloading behavior was very different—the static bulk modulus almost linearly decreased from about 4 GPa at 50 MPa to zero as the stress was reduced. Once again, this behavior was very different from that observed in glass beads, where the static bulk modulus during loading was essentially the same as during unloading and appeared to be a unique function of stress, independent of the direction of its variation. Moreover, the static and dynamic bulk moduli in glass beads, the latter computed from Vp, Vs, and density measured during the loading/unloading cycle, were very close to each other. In contrast, the dynamic bulk modulus in dry sand exceeded the static modulus during unloading by a factor of 1.5 to 2.0. We used a quantitative theory based on contact mechanics and the concept of stress heterogeneity in particulates to analytically match the observed results. Key Points: Measurements of static moduli of granular materialsDeformation of sand and glass beadsTheoretical explanation of results [ABSTRACT FROM AUTHOR]

Ultrasonic elastic‐wave velocities, as well as the stress‐strain behavior, were measured on dry dune sand from Saudi Arabia under hydrostatic loading from 0 to 50 MPa. The samples came from the dune's crest, limb, and base. The experiments included several loading and unloading cycles. All samples exhibited irreversible deformation and porosity reduction during the tests due to grain repacking in low‐stress regimes and grain breakage at higher stress. The loading behavior was plastic, while the unloading/reloading behavior was elastic. In spite of noticeable porosity variations during the experiments, the elastic‐wave velocities mainly depended on the stress rather than on porosity. Our velocity versus stress data is very close to earlier published results, although the provenance of the sand is very different, pointing at a universality of such behavior in loose sand. Our results were matched by the contact Hertz‐Mindlin theory with varying coordination number and shear‐stiffness adjustment. We also offer a new theoretical model for the static bulk modulus measured in these experiments. This model is based on the concept of stress heterogeneity (stress chains or patches) in a granular pack and uses elastic bound averaging of the respective moduli of stressed and unstressed parts of the pack. The same model is used to describe the velocity versus stress behavior. These results and theory constitute a new installment into the published sparse data of the mechanical properties of unconsolidated sand as well as theoretical analyses thereof and are the first ever such data generated in Saudi Arabia. Key Points: New dataset on dune sand, velocity and deformationNew theory for elastic‐wave velocityNew theory for static bulk modulus [ABSTRACT FROM AUTHOR]

The pore structure is an important parameter to understand the transport and storage capacities in shale. There is very limited research reported on the controlling factors of pore development in lacustrine shales to date. This study investigated pore developments and hydrocarbon generation potential of lacustrine Roseneath and Murteree shale formations. Although these shale formations are proven shale gas reservoir in the Cooper Basin, Australia, pore development studies are still rare. The geochemical properties measured on core samples include total organic contents (TOC), thermal maturity, hydrocarbon generation potential, and mineral composition by X-ray diffraction (XRD). Field-emission scanning electron microscopy (FE-SEM), Focused Ion Beam milling scanning electron microscopy (FIB-SEM), and low-pressure nitrogen adsorption-desorption isotherm experiments were used to quantify and qualify pore structure. Furthermore, the factors were analyzed that control pore development through statistical analysis. The results indicate that organic contents for the Roseneath shales (0.9%–1.74% R o %) and Murteree shales (1%–1.7% R o %) range between 0.81% and 5.36% and 0.51%–6.69%, respectively. The dominant mineral phases are quartz, clay (Illite, Kaolinite), and siderite, with variable quantities of the studied formations. Rock-Eval pyrolysis suggests Kerogen Type III and implies mostly late oil generation to dry gas generation potential in both shale formations. In Roseneath shale, total surface area (SSA) and total pore volume (TPV) varies from 2.25 cm2/g to 5 cm2/g and 0.005 cm3/g – 0.02 cm3/g, respectively. In Murteree shale, SSA and TPV vary from 3.5 cm2/g to 6.35 cm2/g and 0.005 cm3/g to 0.032 cm3/g, respectively. The pores exhibit unimodal to bimodal distribution with peaks at 2–4 nm, 90–120 nm, and 170–180 nm. The mesopores are dominant in studied formations, followed by macropores and micropores. Mesopores have the largest contribution to specific surface area and pore volume. The desorption hysteresis loop on the N 2 adsorption-desorption curve indicates the presence of permeable and impermeable pores, as most of the samples belong to H2 and H3 type hysteresis loop. The pore size of intraparticle and interparticle various greatly; however, the pores with a diameter of 2.5 nm, 18 nm and 49 nm are common in both formations. The organopores contribute to micropores' development, and most organic pores have a pore diameter of 1.5 nm and 1.8 nm. The total pore volume (TPV) and specific surface area (SSA) are positively correlated with organic matter contents, quartz, and thermal maturity in both shale formations. However, clay minerals are positively correlated with SSA and TPV, which indicate that clay minerals have a significant contribution to Roseneath shale's pore development. The siderite mineral contributes to intraparticle pores development and positively correlated with SSA and TPV in Murteree shale. The thermal maturity significantly enhances pore's development by creating organopores and inorganic pores in both shales, as evidenced through statistical analysis and image analysis. The study has significant importance in understanding the pore structure and gas storage mechanism in shale. • Hydrocarbon generation potential based on geochemical properties. • Pore characteristics through low-pressure adsorption and image analysis • Factors controlling the pore development in shale. [ABSTRACT FROM AUTHOR]

Matrix acidizing is a well stimulation method that is applied to enhance productivity. The success of a stimulation job depends mainly on the efficiency of the acid systems that are employed. This study investigated a new and efficient acid system that is biodegradable, not corrosive, and safe to handle. The system efficiency was compared to that of HCl and chelating agents. Acid coreflooding has been applied to chalk, dolomite, and limestone carbonate rock samples to measure the efficiency in terms of injected pore volume to breakthrough (PVBT). The rock samples were characterized in terms of porosity, permeability, composition using X-ray diffraction (XRD), and pore distribution using nuclear magnetic resonance (NMR). The effluent was collected at different times and analyzed using induced coupled plasma (ICP). Medical computed tomography (MCT) was used to visualize the interior of the rock samples after acidizing. The new acid system was more efficient in terms of acid spending than straight HCl acid and chelating agents in Indiana limestone. At the optimum injection rate, chelating agents had an optimum PVBT value of about 15 compared to 0.85 for plain HCl acid. Using the new acid system resulted in an optimum PVBT value of only 0.41, which is less than half of HCl acid. A numerical wormhole simulator was implemented to match the experimental PVBT and assess the reactivity of the acid system. • A novel acid system has been identified to be better than aqueous HCl. • A modelling approach reproduces the wormholing. • Different lithologies react (systematically) different. [ABSTRACT FROM AUTHOR]

New experimental tests on a suite of tight gas sandstone samples show a strong dependence of both P- and S-wave velocities, V p and V s , on hydrostatic confining stress in the range from almost zero to 50 MPa. The Klinkenberg-corrected absolute gas permeability of these samples also exhibited a dramatic stress dependence. It decreased by one to two orders of magnitude as the stress increased. A new approach is used to consolidate the 18-sample dataset at each confining stress and arrive at the effective permeability and elastic moduli. For the former, we utilize a randomly generated permeability matrix with its elements drawn from the 18 datapoints. For the latter, we use the elastic bounds. As a result, we obtain permeability-stress, porosity-stress, and velocity-stress relations characteristic of the class of rock under examination. Theoretical simulations based on the inclusion-based differential effective medium theory mimic the experimental velocity-stress results. We also derive a permeability versus acoustic impedance and versus stress relation to be used for inferring permeability variations as a function of the measured impedance versus stress changes. This relation can serve to estimate the permeability from the elastic property variations during time-lapse monitoring, as well as during fracking of this rock. • Laboratory tests on tight gas sandstone samples measuring the elastic properties and absolute permeability. • Consolidating the 18-sample dataset. • Permeability-stress, porosity-stress, and velocity-stress relations. • Permeability versus acoustic impedance and versus stress relation. [ABSTRACT FROM AUTHOR]

A carbonate sample extracted from the depth of about 10 kft was subjected to uniaxial loading while the confining stress remained constant. Post-experiment inspection of the sample showed an inclined crack at an angle less than 20° to the horizontal. This subhorizontal crack orientation was contrary to the expected 45° inclination, the plane of the maximum shear stress. Coincidentally, as shown by CT-scan prior to loading, there was a boundary between two layers of different density inside the sample located almost exactly where the crack appeared. This density difference has arguably translated into the contrast in the elastic properties at the boundary. The hypothesis is that because of this elastic heterogeneity, an incipient crack developed at the boundary due to the unavoidable tensile stressing of the sample as it was brought to the benchtop from its original state of high confining stress at depth. Controlled uniaxial compression made the sample slip along this crack, which then developed into a prominent feature. This assumption was corroborated by a numerical experiment showing a strong von Mises stress concentration at the elastic contrast boundary during hydrostatic tensile loading. Another sample, from the same formation, but without strong density heterogeneity, exhibited a classic 45° crack after uniaxial loading. These results provide a novel and important insight into the mechanics, breakage, and strength of natural rock. [ABSTRACT FROM AUTHOR]

The shapes of particles play a crucial role in the transport and mechanical behavior of granular media. How densely particles of a specific shape can be arranged in a given volume has an enormous practical significance for many scientific and industrial fields. We developed a numerical workflow to investigate the effects of irregularly shaped frictionless, convex particles combined with the polydispersity of particle sizes on the porosity, permeability, and elastic bulk modulus of granular porous media. The workflow is based on the computationally generated three‐dimensional granular assemblies of both regular and irregularly shaped particles and pore‐scale simulation of physics to estimate the absolute permeability and elastic bulk modulus of the generated digital granular porous media. Our numerical results show that the porosity decreases by 14–25%, absolute permeability decreases by 45–76%, and elastic bulk modulus increases by 20–66% as we deviate from the regular‐shaped particle with the specified particle size distribution. How rapidly and to what degree these properties of granular assemblies of irregularly shaped particles change depends on particle size distribution. However, once the maximal densest packings of irregularly shaped particles with the prescribed aspect ratio are generated, any further change in the shape of particles has either a minimal impact or an opposite effects are observed. Overall, our numerical observations suggest that the irregular shape of particles has the greatest effect on permeability and the least effect on porosity in both monodisperse and polydisperse granular media. Key Points: We developed a numerical workflow to investigate the effects of irregularly shaped frictionless, convex particles combined with the polydispersity of particle sizes on the physical properties of granular porous mediaHow rapidly and to what degree the properties of granular packings of irregularly shaped particles change depends on the particle size distributionThe irregular shape of particles has the greatest impact on permeability and the least effect on porosity in both monodisperse and polydisperse granular assemblies [ABSTRACT FROM AUTHOR]