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Estimating flow and transport properties of porous media that undergo deformation as a result of applying an external pressure or force is important to a wide variety of processes, ranging from injecting a fracking liquid into shale formations, to CO 2 sequestration in spent oil reservoirs. We propose a novel model for estimating the effective flow and transport properties of such porous media. Assuming that the solid matrix of a porous medium undergoes elastic deformation, and given its initial porosity before deformation, as well as the Young’s modulus of its grains, the model uses an extension of the Hertz–Mindlin theory of contact between grains to compute the new PSD that results from applying an external pressure P to the medium, and utilizes the updated PSD in the effective-medium approximation (EMA) to estimate the effective flow and transport properties at pressure P . In the present part of this series, we use the theory to predict the effective permeability as a function of the applied pressure. Comparison between the predictions and experimental data for twenty-four types of sandstones indicates excellent agreement between the two.

In Part I of this series, we presented a new theoretical approach for computing the effective permeability of porous media that are under deformation by a hydrostatic pressure P . Beginning with the initial pore-size distribution (PSD) of a porous medium before deformation and given the Young’s modulus and Poisson’s ratio of its grains, the model used an extension of the Hertz–Mindlin theory of contact between grains to compute the new PSD that results from applying the pressure P to the medium and utilized the updated PSD in the effective-medium approximation (EMA) to estimate the effective permeability. In the present paper, we extend the theory in order to compute the electrical conductivity of the same porous media that are saturated by brine. We account for the possible contribution of surface conduction, in order to estimate the electrical conductivity of brine-saturated porous media. We then utilize the theory to update the PSD and, hence, the pore-conductance distribution, which is then used in the EMA to predict the pressure dependence of the electrical conductivity. Comparison between the predictions and experimental data for twenty-six sandstones indicates agreement between the two that ranges from excellent to good.

Numerous sessile drop and micro-computed tomography (micro-CT) studies have been conducted to quantify geologic carbon storage formation wettability by measuring static contact angles ( θ ); however, the influence of pore geometry remains unknown. In this work, six sandstones (Bandera Brown, Berea, Bentheimer, Mt. Simon, Navajo, and Nugget) are used to measure θ using the two aforementioned experimental methods at identical testing conditions (45 °C and 12.41 MPa). The range of θ measured at in situ conditions (micro-CT) exceeds the range at ex situ (sessile drop method) conditions for all sandstones. However, when droplets with more representative in situ diameters are analyzed, θ averages show ex situ θ exceed those of in situ θ . Pore geometry does influence local θ , but the size of ex situ droplets relative to pore size appears to influence θ . This is important to consider for future sessile drop studies used for analysis of CO 2 behavior in carbon storage reservoirs.

The permeability of complex porous materials is of interest to many engineering disciplines. This quantity can be obtained via direct flow simulation, which provides the most accurate results, but is very computationally expensive. In particular, the simulation convergence time scales poorly as the simulation domains become less porous or more heterogeneous. Semi-analytical models that rely on averaged structural properties (i.e., porosity and tortuosity) have been proposed, but these features only partly summarize the domain, resulting in limited applicability. On the other hand, data-driven machine learning approaches have shown great promise for building more general models by virtue of accounting for the spatial arrangement of the domains’ solid boundaries. However, prior approaches building on the convolutional neural network (ConvNet) literature concerning 2D image recognition problems do not scale well to the large 3D domains required to obtain a representative elementary volume (REV). As such, most prior work focused on homogeneous samples, where a small REV entails that the global nature of fluid flow could be mostly neglected, and accordingly, the memory bottleneck of addressing 3D domains with ConvNets was side-stepped. Therefore, important geometries such as fractures and vuggy domains could not be modeled properly. In this work, we address this limitation with a general multiscale deep learning model that is able to learn from porous media simulation data. By using a coupled set of neural networks that view the domain on different scales, we enable the evaluation of large ( > 512 3 ) images in approximately one second on a single graphics processing unit. This model architecture opens up the possibility of modeling domain sizes that would not be feasible using traditional direct simulation tools on a desktop computer. We validate our method with a laminar fluid flow case using vuggy samples and fractures. As a result of viewing the entire domain at once, our model is able to perform accurate prediction on domains exhibiting a large degree of heterogeneity. We expect the methodology to be applicable to many other transport problems where complex geometries play a central role.

To investigate the energy evolution characteristics of rock materials under uniaxial compression, the single-cyclic loading–unloading uniaxial compression tests of four rock materials (Qingshan granite, Yellow sandstone, Longdong limestone and Black sandstone) were conducted under five unloading stress levels. The stress–strain curves and failure characteristics of rock specimens under the single-cyclic loading–unloading uniaxial compression tests basically corresponded with those of under uniaxial compression, which indicates that single-cyclic loading–unloading has minimal effects on the variations in the loading–deformation response of rocks. The input energy density, elastic energy density and dissipated energy density of four rocks under five unloading stress levels were calculated using the graphical integration method, and variation characteristics of those three energy density parameters with different unloading stress levels were explored. The results show that all three energy density parameters above increased nonlinearly with increasing unloading stress level as quadratic polynomial functions. Meanwhile, both the elastic and dissipated energy density increased linearly when the input energy density increased, and the linear energy storage and dissipation laws for rock materials were observed. Furthermore, a linear relationship between the dissipated and elastic energy density was also proposed. Using the linear energy storage or dissipation law, the elastic and dissipated energy density at any stress levels can be calculated, and the internal elastic (or dissipated) energy density at peak compressive strength (the peak elastic and dissipated energy density for short) can be obtained. The ratio of the elastic energy density to dissipated energy density with increasing input energy density was investigated using a new method, and the results show that this ratio tends to be constant at the peak compressive strength of rock specimens.

A large number (1253) of high-quality streaming potential coefficient ( C sp ) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity ( C f ) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10 - 5 and 4.5 mol/dm 3 ) upon which accurate measurements of their electrical conductivity and pH were taken. These C sp measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The C sp measurements follow a pH-sensitive power law behaviour with respect to C f at medium salinities ( C sp = - 1.44 × 10 - 9 C f - 1.127 , units: V/Pa and mol/dm 3 ) and show the effect of rock microstructure on the low salinity C sp clearly, producing a smaller decrease in C sp per decade reduction in C f for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The C sp measurements include 313 made at C f > 1  mol/dm 3 , which confirm the limiting high salinity C sp behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential ( ζ ) was calculated from each C sp measurement. It was found that ζ is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to C f at low to medium salinities ( ζ = 0.01133 log 10 C f + 0.003505 , units: V and mol/dm 3 ) and a limiting zeta potential (zeta potential offset) at high salinities of ζ o = - 17.36 ± 5.11  mV in the pH range 6–8, which is also pH dependent. The sensitivity of both C sp and ζ to pH and of C sp to rock microstructure indicates that C sp and ζ measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.

Little is known of the properties of the sarsen stones (or silcretes) that comprise the main architecture of Stonehenge. The only studies of rock struck from the monument date from the 19th century, while 20th century investigations have focussed on excavated debris without demonstrating a link to specific megaliths. Here, we present the first comprehensive analysis of sarsen samples taken directly from a Stonehenge megalith (Stone 58, in the centrally placed trilithon horseshoe). We apply state-of-the-art petrographic, mineralogical and geochemical techniques to two cores drilled from the stone during conservation work in 1958. Petrographic analyses demonstrate that Stone 58 is a highly indurated, grain-supported, structureless and texturally mature groundwater silcrete, comprising fine-to-medium grained quartz sand cemented by optically-continuous syntaxial quartz overgrowths. In addition to detrital quartz, trace quantities of silica-rich rock fragments, Fe-oxides/hydroxides and other minerals are present. Cathodoluminescence analyses show that the quartz cement developed as an initial <10 μm thick zone of non-luminescing quartz followed by ~16 separate quartz cement growth zones. Late-stage Fe-oxides/hydroxides and Ti-oxides line and/or infill some pores. Automated mineralogical analyses indicate that the sarsen preserves 7.2 to 9.2 area % porosity as a moderately-connected intergranular network. Geochemical data show that the sarsen is chemically pure, comprising 99.7 wt. % SiO2. The major and trace element chemistry is highly consistent within the stone, with the only magnitude variations being observed in Fe content. Non-quartz accessory minerals within the silcrete host sediments impart a trace element signature distinct from standard sedimentary and other crustal materials. 143Nd/144Nd isotope analyses suggest that these host sediments were likely derived from eroded Mesozoic rocks, and that these Mesozoic rocks incorporated much older Mesoproterozoic material. The chemistry of Stone 58 has been identified recently as representative of 50 of the 52 remaining sarsens at Stonehenge. These results are therefore representative of the main stone type used to build what is arguably the most important Late Neolithic monument in Europe.