Showing papers in "Transport in Porous Media in 2020"

Linking Morphology of Porous Media to Their Macroscopic Permeability by Deep Learning

Abstract: Flow, transport, mechanical, and fracture properties of porous media depend on their morphology and are usually estimated by experimental and/or computational methods. The precision of the computational approaches depends on the accuracy of the model that represents the morphology. If high accuracy is required, the computations and even experiments can be quite time-consuming. At the same time, linking the morphology directly to the permeability, as well as other important flow and transport properties, has been a long-standing problem. In this paper, we develop a new network that utilizes a deep learning (DL) algorithm to link the morphology of porous media to their permeability. The network is neither a purely traditional artificial neural network (ANN), nor is it a purely DL algorithm, but, rather, it is a hybrid of both. The input data include three-dimensional images of sandstones, hundreds of their stochastic realizations generated by a reconstruction method, and synthetic unconsolidated porous media produced by a Boolean method. To develop the network, we first extract important features of the images using a DL algorithm and then feed them to an ANN to estimate the permeabilities. We demonstrate that the network is successfully trained, such that it can develop accurate correlations between the morphology of porous media and their effective permeability. The high accuracy of the network is demonstrated by its predictions for the permeability of a variety of porous media.

The Sensitivity of Estimates of Multiphase Fluid and Solid Properties of Porous Rocks to Image Processing

Abstract: X-ray microcomputed tomography (X-ray μ-CT) is a rapidly advancing technology that has been successfully employed to study flow phenomena in porous media. It offers an alternative approach to core scale experiments for the estimation of traditional petrophysical properties such as porosity and single-phase flow permeability. It can also be used to investigate properties that control multiphase flow such as rock wettability or mineral topology. In most applications, analyses are performed on segmented images obtained employing a specific processing pipeline on the greyscale images. The workflow leading to a segmented image is not straightforward or unique and, for most of the properties of interest, a ground truth is not available. For this reason, it is crucial to understand how image processing choices control properties estimation. In this work, we assess the sensitivity of porosity, permeability, specific surface area, in situ contact angle measurements, fluid–fluid interfacial curvature measurements and mineral composition to processing choices. We compare the results obtained upon the employment of two processing pipelines: non-local means filtering followed by watershed segmentation; segmentation by a manually trained random forest classifier. Single-phase flow permeability, in situ contact angle measurements and mineral-to-pore total surface area are the most sensitive properties, as a result of the sensitivity to processing of the phase boundary identification task. Porosity, interfacial fluid–fluid curvature and specific mineral descriptors are robust to processing. The sensitivity of the property estimates increases with the complexity of its definition and its relationship to boundary shape.

Foam Generation, Propagation and Stability in Porous Medium

Abstract: There have been several foam field applications in recent years. Foam treatments targeting gas mobility control in injectors as well as gas blocking in production wells have been performed without causing operational problems. The most widely used injection strategy of foam has been injecting alternating slugs of surfactant in brine with gas injection. This procedure seems to be beneficial as injection is easy to perform and control below fracturing pressure. Simultaneous injection of surfactant solution and gas may give difficulties, especially with interpretation of the tests, if fracturing pressure are exceeded during the injection period. This paper reviews critical aspects of foam for reservoir applications and intends to motivate for further field trials. Key parameters for qualification of foam are: foam generation, propagation in porous medium, foam strength and stability of foam. Stability is discussed, especially in the presence of oil at reservoir conditions. Data on each of these topics are included, as well as extracted summary of relevant literature. Experimental studies have shown that foam is generated at low surfactant concentration even below the CMC (critical micelle concentration). Results indicate that in situ foam generation in porous medium may depend on available nucleation sites. In situ generation of foam is complex and has been found to be especially difficult in oil wet carbonate rocks. Foam propagation in porous medium has been summarized, and propagation rate for a given experiment seems to be constant with time and distance. Laboratory studies confirm a propagation rate of 1–3 m/day. Field tests performed have not given reliable information of foam propagation rate, and future field pilots are encouraged to include observation wells in order to gain information of field-scale foam propagation. Foam strength is generally high with all gases. The exception is CO2 at high pressure where CO2 becomes supercritical. Stability of foam has been studied in laboratory and field tests, and has confirmed long-term stability of foam.

Study on Permeability Anisotropy of Bedded Coal Under True Triaxial Stress and Its Application

Abstract: Anisotropy is a very typical observation in the intrinsic bedding structure of coal. To study the influence of anisotropy of coal structure and stress state on the evolution of permeability, a newly developed multifunctional true triaxial geophysical apparatus was used to carry out mechanical and seepage experiments on bedded coal. The permeability and deformation of three orthogonal directions in cubic coal samples were collected under true triaxial stress. It has detected the significant permeability anisotropy, and the anisotropy is firmly determined by the bedding direction and stress state of coal. Based on the true triaxial mechanical and seepage test results, the coal with bedding was simplified to be represented by a cubic model, and the dynamic anisotropic (D-A) permeability model was derived by considering the influence of bedding and stress state. The rationality of the permeability model was verified by the experimental data. Comparing the permeability model with Wang and Zang (W–Z) model, Cui and Bustin (C–B) model and Shi and Durucan (S–D) model, it is found that the theoretical calculated values of the D-A permeability model are in better agreement with the experimental measured values, reflecting the superiority of the D-A permeability model. Based on incorporating the model of D-A permeability under the concept of multiphysics field coupling, the numerical simulation experiments of coal seam gas extraction with different initial permeability anisotropic ratios were carried out by using COMSOL multiphysics simulator. The influence of initial permeability anisotropy ratio on gas pressure distribution in coal seam during gas extraction was explored, which provides theoretical guidance for the optimization of borehole layout for gas extraction in coal mine.

Pore Network Investigation of Trapped Gas and Foam Generation Mechanisms

Abstract: The mobility of gas is greatly reduced when the injected gas is foamed. The reduction in gas mobility is attributed to the reduction in gas relative permeability and the increase in gas effective viscosity. The reduction in the gas relative permeability is a consequence of the larger amount of gas trapped when foam is present while the increase in gas effective viscosity is explicitly a function of foam texture. Therefore, understanding how foam is generated and subsequent trapped foam behavior is of paramount importance to modeling of gas mobility. In this paper, we push the envelope to enlighten our decisions of which descriptions are most physical to foam flow in porous media regarding both the flowing foam fraction and the rate of generation. We use a statistical pore network interwoven with the invasion percolation with memory algorithm to model foam flow as a drainage process and investigate the dependence of the flowing foam fraction on the pressure gradient and to shed light on foam generation mechanisms. A critical snap-off probability is required for strong foam to emerge in our network. The pressure gradient and, hence, the gas mobility reduction are very low below this critical snap-off probability. Above this snap-off probability threshold, we find that the steady-state flowing lamellae fraction scales as $$( abla \tilde{p})^{0.19}$$ in 2D lattices and as $$( abla \tilde{p})^{0.32}$$ in 3D lattices. Results obtained from our network were convolved with percolation network scaling ideas to compare the probabilities of snap-off and lamella division mechanisms in the network during the initial gas displacement at the leading edge of the gas front. At this front, during strong foam flow, lamella division is practically nonexistent in 2D lattices. In 3D lattices, lamella division occurs, but the probability of snap-off is always greater than the probability of lamella division.

Accelerating Reactive Transport Modeling: On-Demand Machine Learning Algorithm for Chemical Equilibrium Calculations

Abstract: During reactive transport modeling, the computing cost associated with chemical equilibrium calculations can be 10 to 10,000 times higher than that of fluid flow, heat transfer, and species transport computations. These calculations are performed at least once per mesh cell and once per time step, amounting to billions of them throughout the simulation employing high-resolution meshes. To radically reduce the computing cost of chemical equilibrium calculations (each requiring an iterative solution of a system of nonlinear equations), we consider an on-demand machine learning algorithm that enables quick and accurate prediction of new chemical equilibrium states using the results of previously solved chemical equilibrium problems within the same reactive transport simulation. The training operations occur on-demand, rather than before the start of the simulation when it is not clear how many training points are needed to accurately and reliably predict all possible chemical conditions that may occur during the simulation. Each on-demand training operation consists of fully solving the equilibrium problem and storing some key information about the just computed chemical equilibrium state (which is used subsequently to rapidly predict similar states whenever possible). We study the performance of the on-demand learning algorithm, which is mass conservative by construction, by applying it to a reactive transport modeling example and achieve a speed-up of one or two orders of magnitude (depending on the activity model used). The implementation and numerical tests are carried out in Reaktoro (reaktoro.org), a unified open-source framework for modeling chemically reactive systems.

Conjugate Phase Change Heat Transfer in an Inclined Compound Cavity Partially Filled with a Porous Medium: A Deformed Mesh Approach

Abstract: In this paper, the melting process of a PCM inside an inclined compound enclosure partially filled with a porous medium is theoretically addressed using a novel deformed mesh method. The sub-domain area of the compound enclosure is made of a porous layer and clear region. The right wall of the enclosure is adjacent to the clear region and is subject to a constant temperature of Tc. The left wall, which is connected to the porous layer, is thick and thermally conductive. The thick wall is partially subject to the hot temperature of Th. The remaining borders of the enclosure are well insulated. The governing equations for flow and heat transfer, including the phase change effects and conjugate heat transfer at the thick wall, are introduced and transformed into a non-dimensional form. A deformed grid method is utilized to track the phase change front in the solid and liquid regions. The melting front movement is controlled by the Stefan condition. The finite element method, along with Arbitrary Eulerian–Lagrangian (ALE) moving grid technique, is employed to solve the non-dimensional governing equations. The modeling approach and the accuracy of the utilized numerical approach are verified by comparison of the results with several experimental and numerical studies, available in the literature. The effect of conjugate wall thickness, inclination angle, and the porous layer thickness on the phase change heat transfer of PCM is investigated. The outcomes show that the rates of melting and heat transfer are enhanced as the thickness of the porous layer increases. The melting rate is the highest when the inclination angle of the enclosure is 45°. An increase in the wall thickness improves the melting rate.

A Study of the Role of Microfractures in Counter-Current Spontaneous Imbibition by Lattice Boltzmann Simulation

Abstract: During waterflooding, spontaneous imbibition is a fundamental recovery mechanism in fractured reservoirs. A large number of numerical and experimental studies have been devoted to understand the interaction mechanism between the matrix and the fractures at the core scale under various boundary conditions. Little attention has been paid, however, to the effect of microfractures on pore-scale spontaneous imbibition. In this study, five models with various types of microfractures, embedded in the same porous matrix, are used to investigate their role in counter-current spontaneous imbibition, using an optimized color-gradient lattice Boltzmann method. During the entire counter-current imbibition, the influence of the microfractures on the macro-recovery of the non-wetting fluid and the two-fluid interfaces, including the initial and local dynamics of the interfaces, and the evolution of the interface morphology, is analyzed in detail. The results indicate that microfractures have little influence on both the interfacial dynamics at the initial stage and the local interface dynamics in the matrix. The evolution of the interface morphology is, however, controlled by the geometric shape of the microfractures. In addition, the microfractures improve significantly the recovery of oil. The length of a single microfracture and the bifurcation angle of a microfracture bifurcated into two other microfractures affect significantly the recovery curve by influencing the evolution of the two-phase interface.

Modeling Geometric State for Fluids in Porous Media: Evolution of the Euler Characteristic

Abstract: Multiphase flow in porous media is strongly influenced by the pore-scale arrangement of fluids. Reservoir-scale constitutive relationships capture these effects in a phenomenological way, relying only on fluid saturation to characterize the macroscopic behavior. Working toward a more rigorous framework, we make use of the fact that the momentary state of such a system is uniquely characterized by the geometry of the pore-scale fluid distribution. We consider how fluids evolve as they undergo topological changes induced by pore-scale displacement events. Changes to the topology of an object are fundamentally discrete events. We describe how discontinuities arise, characterize the possible topological transformations and analyze the associated source terms based on geometric evolution equations. Geometric evolution is shown to be hierarchical in nature, with a topological source term that constrains how a structure can evolve with time. The challenge associated with predicting topological changes is addressed by constructing a universal geometric state function that predicts the possible states based on a non-dimensional relationship with two degrees of freedom. The approach is validated using fluid configurations from both capillary and viscous regimes in ten different porous media with porosity between 0.10 and 0.38. We show that the non-dimensional relationship is independent of both the material type and flow regime. We demonstrate that the state function can be used to predict history-dependent behavior associated with the evolution of the Euler characteristic during two-fluid flow.

Study of Nanoparticle-Stabilized Foams in Harsh Reservoir Conditions

Abstract: Many oil reservoirs are at high temperatures and contain brines of high salinity and hardness. The focus of this work is to develop robust foams stabilized by a mixture of nanoparticles and surfactants for such reservoirs. Two types of silica nanoparticles (Si-NP1, Si-NP2) with different grafted low molecular weight ligands/polymers were used. First, aqueous stability analysis of these nanoparticle dispersions were conducted at high-temperature (80 °C) and high-salinity conditions (API Brine; 8 wt% NaCl and 2 wt% CaCl2). The screened nanoparticles were used in combination with an anionic surfactant. Second, bulk foam and emulsion stability tests were performed to investigate their performance in stabilizing the air–water and oil–water interface, respectively. Third, foam flow experiments in the absence of oil were performed to characterize the foam rheology. Finally, oil displacement experiments were conducted in an in-house, custom-built 2D sand pack with flow visualization. The sand pack had two layers of different mesh size silica sand which yielded a permeability contrast of 6:1. Brine floods followed by foam floods (80% quality) were conducted, and foam flow dynamics were monitored. The grafting of low molecular weight polymers/ligands on silica nanoparticle surfaces resulted in steric stabilization under high-temperature and high-salinity conditions. Foam flow experiments revealed a synergy between Si-NP2 and surfactant in stabilizing foam in the absence of crude oil. In the oil displacement experiments in the layered sand packs, the waterflood recoveries were low (~ 33% original oil in place) due to channeling in the top high-permeability zone, leaving the bottom low-permeability zone completely unswept. Foam flooding with just the surfactant leads to a drastic improvement in sweep efficiency. It resulted in an incremental oil recovery as high as 43.3% OOIP. Different cross-flow behaviors were observed during foam flooding. Significant cross-flow of oil from low-permeability zone to high-permeability zone was observed for the case of surfactant. Conversely, the Si-NP2-surfactant blend resulted in no cross-flow from the low-permeability region with complete blocking of the high-permeability region due to the formation of in situ emulsion. Such selective plugging of high-perm zones using nanoparticles with tailored surface coating and concentration has significant potential in recovering oil from heterogeneous reservoirs.

Predicting Resistivity and Permeability of Porous Media Using Minkowski Functionals

Abstract: Permeability and formation factor are important properties of a porous medium that only depend on pore space geometry, and it has been proposed that these transport properties may be predicted in terms of a set of geometric measures known as Minkowski functionals. The well-known Kozeny–Carman and Archie equations depend on porosity and surface area, which are closely related to two of these measures. The possibility of generalizations including the remaining Minkowski functionals is investigated in this paper. To this end, two-dimensional computer-generated pore spaces covering a wide range of Minkowski functional value combinations are generated. In general, due to Hadwiger’s theorem, any correlation based on any additive measurements cannot be expected to have more predictive power than those based on the Minkowski functionals. We conclude that the permeability and formation factor are not uniquely determined by the Minkowski functionals. Good correlations in terms of appropriately evaluated Minkowski functionals, where microporosity and surface roughness are ignored, can, however, be found. For a large class of random systems, these correlations predict permeability and formation factor with an accuracy of 40% and 20%, respectively.

A Comparative Study of Gas Flooding and Foam-Assisted Chemical Flooding in Bentheimer Sandstones

Abstract: A laboratory study of principal immiscible gas flooding schemes is reported. Very well-controlled experiments on continuous gas injection, water-alternating-gas (WAG) and alkaline–surfactant–foam (ASF) flooding were conducted. The merits of WAG and ASF compared to continuous gas injection were examined. The impact of ultra-low oil–water (o/w) interfacial tension (IFT), an essential feature of the ASF scheme along with foaming, on oil mobilisation and displacement of residual oil to waterflood was also assessed. Incremental oil recoveries and related displacement mechanisms by ASF and WAG compared to continuous gas injection were investigated by conducting CT-scanned core-flood experiments using n-hexadecane and Bentheimer sandstone cores. Ultimate oil recoveries for WAG and ASF at under-optimum salinity (o/w IFT of 10−1 mN/m) were found to be similar [60 ± 5% of the oil initially in place (OIIP)]. However, ultimate oil recovery for ASF at (near-)optimum salinity (o/w IFT of 10−2 mN/m) reached 74 ± 8% of the OIIP. Results support the idea that WAG increases oil recovery over continuous gas injection by drastically increasing the trapped gas saturation at the end of the first few WAG cycles. ASF flooding was able to enhance oil recovery over WAG by effectively lowering o/w IFT (< 10−1 mN/m) for oil mobilisation. ASF at (near-)optimum salinity increased clean oil fraction in the production stream over under-optimum salinity ASF.

On the Modelling of Immiscible Viscous Fingering in Two-Phase Flow in Porous Media

Abstract: Viscous fingering in porous media is an instability which occurs when a low-viscosity injected fluid displaces a much more viscous resident fluid, under miscible or immiscible conditions. Immiscible viscous fingering is more complex and has been found to be difficult to simulate numerically and is the main focus of this paper. Many researchers have identified the source of the problem of simulating realistic immiscible fingering as being in the numerics of the process, and a large number of studies have appeared applying high-order numerical schemes to the problem with some limited success. We believe that this view is incorrect and that the solution to the problem of modelling immiscible viscous fingering lies in the physics and related mathematical formulation of the problem. At the heart of our approach is what we describe as the resolution of the “M-paradox”, where M is the mobility ratio, as explained below. In this paper, we present a new 4-stage approach to the modelling of realistic two-phase immiscible viscous fingering by (1) formulating the problem based on the experimentally observed fractional flows in the fingers, which we denote as $$ f_{\rm w}^{*} $$ , and which is the chosen simulation input; (2) from the infinite choice of relative permeability (RP) functions, $$ k_{\rm rw}^{*} $$ and $$ k_{\rm ro}^{*} $$ , which yield the same $$ f_{\rm w}^{*} $$ , we choose the set which maximises the total mobility function, $$ \lambda_{\text{T}}^{{}} $$ (where $$ \lambda_{\text{T}}^{{}} = \lambda_{\text{o}}^{{}} + \lambda_{\text{w}}^{{}} $$ ), i.e. minimises the pressure drop across the fingering system; (3) the permeability structure of the heterogeneous domain (the porous medium) is then chosen based on a random correlated field (RCF) in this case; and finally, (4) using a sufficiently fine numerical grid, but with simple transport numerics. Using our approach, realistic immiscible fingering can be simulated using elementary numerical methods (e.g. single-point upstreaming) for the solution of the two-phase fluid transport equations. The method is illustrated by simulating the type of immiscible viscous fingering observed in many experiments in 2D slabs of rock where water displaces very viscous oil where the oil/water viscosity ratio is $$ (\mu_{\text{o}} /\mu_{\text{w}} ) = 1600 $$ . Simulations are presented for two example cases, for different levels of water saturation in the main viscous finger (i.e. for 2 different underlying $$ f_{\rm w}^{*} $$ functions) produce very realistic fingering patterns which are qualitatively similar to observations in several respects, as discussed. Additional simulations of tertiary polymer flooding are also presented for which good experimental data are available for displacements in 2D rock slabs (Skauge et al., in: Presented at SPE Improved Oil Recovery Symposium, 14–18 April, Tulsa, Oklahoma, USA, SPE-154292-MS, 2012. https://doi.org/10.2118/154292-MS , EAGE 17th European Symposium on Improved Oil Recovery, St. Petersburg, Russia, 2013; Vik et al., in: Presented at SPE Europec featured at 80th EAGE Conference and Exhibition, Copenhagen, Denmark, SPE-190866-MS, 2018. https://doi.org/10.2118/190866-MS ). The finger patterns for the polymer displacements and the magnitude and timing of the oil displacement response show excellent qualitative agreement with experiment, and indeed, they fully explain the observations in terms of an enhanced viscous crossflow mechanism (Sorbie and Skauge, in: Proceedings of the EAGE 20th Symposium on IOR, Pau, France, 2019). As a sensitivity, we also present some example results where the adjusted fractional flow ( $$ f_{\rm w}^{*} $$ ) can give a chosen frontal shock saturation, $$ S_{\rm wf}^{*} $$ , but at different frontal mobility ratios, $$ M(S_{\rm wf}^{*} ) $$ . Finally, two tests on the robustness of the method are presented on the effect of both rescaling the permeability field and on grid coarsening. It is demonstrated that our approach is very robust to both permeability field rescaling, i.e. where the (kmax/kmin) ratio in the RCF goes from 100 to 3, and also under numerical grid coarsening.

A Triple Pore Network Model (T-PNM) for Gas Flow Simulation in Fractured, Micro-porous and Meso-porous Media

Abstract: In this study, a novel triple pore network model (T-PNM) is introduced which is composed of a single pore network model (PNM) coupled to fractures and micro-porosities. We use two stages of the watershed segmentation algorithm to extract the required data from semi-real micro-tomography images of porous material and build a structural network composed of three conductive elements: meso-pores, micro-pores, and fractures. Gas and liquid flow are simulated on the extracted networks and the calculated permeabilities are compared with dual pore network models (D-PNM) as well as the analytical solutions. It is found that the processes which are more sensitive to the surface features of material, should be simulated using a T-PNM that considers the effect of micro-porosities on overall process of flow in tight pores. We found that, for gas flow in tight pores where the close contact of gas with the surface of solid walls makes Knudsen diffusion and gas slippage significant, T-PNM provides more accurate solution compared to D-PNM. Within the tested range of operational conditions, we recorded between 10 and 50% relative error in gas permeabilities of carbonate porous rocks if micro-porosities are dismissed in the presence of fractures.

A Dual-Permeability Approach to Study Anomalous Moisture Transport Properties of Cement-Based Materials

Abstract: Anomalous moisture transport in cement-based materials is often reported in the literature, but the conventional single-porosity moisture transport models generally fail to provide accurate simulation results. Previous studies suggested that the anomalous moisture transport could be caused by different moisture transport velocity in large and small pores. Based on this concept, the present study proposes a continuous dual-permeability model for cement-based material. The proposed model includes the transport contribution of both liquid water and water vapor, which are governed by liquid advection and vapor diffusion, respectively. We explicitly consider that moisture transport in the large pore region is faster than the small pore region. The volumetric fraction of each region is determined when fitting the measured sorption isotherms by using a bimodal equation. The validation with experimental data shows that the dual-permeability model can well simulate both the “normal” and the anomalous moisture transport. The applicability of the proposed model implies that the “dual-porosity property” could be one of reasons that cause anomalous moisture transport in cementitious materials. In addition, results show that vapor diffusion can be neglected for moisture transport in both porosities at high relative humidity (RH), while at low RH, vapor diffusion must be considered.

Characterizing the Impact of Fractured Caprock Heterogeneity on Supercritical CO $$_2$$2 Injection

Abstract: We present a set of multiphase flow simulations where supercritical CO$$_2$$ (scCO$$_2$$) displaces water at hydrostatic conditions within three-dimensional discrete fracture networks that represent paths for potential leakage through caprock above CO$$_2$$ storage reservoirs. The simulations are performed to characterize and compare the relative impact of hydraulic and structural heterogeneity in fractured media on the initial movement of scCO$$_2$$ through these caprock formations. In one scenario, intrinsic fracture permeabilities are varied stochastically within a fixed network structure. In another scenario, we generate multiple independent, identically distributed network realizations with varying fracture network densities to explore a wide range of geometric and topological configurations. Analysis of the simulations indicates that network structure, specifically connectivity and the presence of hanging fractures, plays a larger role in controlling the displacement of water by scCO$$_2$$ than variations in local hydraulic properties. We identify active surface area of the network as a single-phase feature that could provide a lower bound on the percentage of the network surface area reached by scCO$$_2$$.

Coupling Between Poromechanical Behavior and Fluid Flow in Tight Rock

Abstract: Proper characterization of the mechanical and flow properties of participating rock formations is crucial for subsurface geo-energy projects, including hydrocarbon extraction, geologic carbon storage, and enhanced geothermal systems. Application of mechanical and hydraulic pressures changes the porosity of rock and modifies flow paths. For low-permeable or “tight” rock that mainly contains nanoscale pores and serves as the confining layer for underground storage operations, a significant change in permeability may occur due to a small change in porosity. The pore volume changes in nanoporous geomaterials are extremely difficult to measure directly, but can be assessed from the knowledge of the hydro-mechanical response. Experimental methods to measure the stress-dependent permeability and poroelastic parameters of fluid-saturated tight rock are introduced. Eau Claire shale, Opalinus clay (claystone), and Charcoal granite are selected as representative materials for tight rock and their pore structure and material properties are carefully investigated. The porosity–permeability relationship for tight rock is established by adopting a power-law dependence with the exponent value in the range of 15–17, thus being significantly larger than that for a porous reservoir rock. Consequently, even small perturbations of porosity can cause orders of magnitude changes in permeability possessing a risk on the sealing capacity of the tight formations.

Küppers–Lortz Instability in the Rotating Brinkman–Bénard Problem

Abstract: We investigate the Kuppers–Lortz (KL) instability in the rotating Brinkman–Benard convection problem by assuming that there is local thermal non-equilibrium (LTNE) between the Newtonian liquid and the high-porosity medium that it has occupied to the point of saturation. The effects of local thermal non-equilibrium parameters on the threshold value of the Taylor number and the angle between the rolls at which KL-instability sets in are presented. The four routes through which the local thermal equilibrium situation can be approached are presented with the help of asymptotic analyses. The corresponding results of the rotating Darcy–Benard problem are extracted as a limiting case from the present problem with the help of another asymptotic analysis. The problem identifies the specific range of values of parameters within which LTNE effect is discernible and also clearly shows that the onset of KL-instability is delayed by the ratio of thermal conductivities. The heat transfer coefficient, however, has a dual effect on $${\mathrm{Ta}}_{\mathrm{c}}$$. Such a dual nature is seen, perhaps, due to the heat transport equations being of the hyperbolic type when local thermal non-equilibrium effect is significant. The results show that LTNE in the presence of rotation favors hexagonal pattern.

Pore-Network Stitching Method: A Pore-to-Core Upscaling Approach for Multiphase Flow

Abstract: Pore-network modeling is a widely used predictive tool for pore-scale studies in various applications that deal with multiphase flow in heterogeneous natural rocks. Despite recent improvements to enable pore-network modeling on simplified pore geometry extracted from rock core images and and its computational efficiency compared to direct numerical simulation methods, there are still limitations to modeling a large representative pore-network for heterogeneous cores. These are due to the technical limits on sample size to discern void space during X-ray scanning and computational limits on pore-network extraction algorithms. Thus, there is a need for pore-scale modeling approaches that have the natural advantages of pore-network modeling and can overcome these limitations, thereby enabling better representation of heterogeneity of the 3D complex pore structure and enhancing the accuracy of prediction of macroscopic properties. This paper addresses these issues with a workflow that includes a novel pore-network stitching method to provide large-enough representative pore-network for a core. This workflow uses micro-CT images of heterogeneous reservoir rock cores at different resolutions to characterize the pore structure in order to select few signature parts of the core and extract their equivalent pore-network models. The space between these signature pore-networks is filled by using their statistics to generate realizations of pore-networks which are then connected together using a deterministic layered stitching method. The output of this workflow is a large pore-network that can be used in any flow and transport solver. We validate all steps of this method on different types of natural rocks based on single-phase and two-phase flow properties such as drainage relative permeability curves of carbon dioxide and brine flow. Then, we apply the stochastic workflow on two large domain problems, connecting distant pore-networks and modeling a heterogeneous core. We generate multiple realizations and compare the average results with properties from a defined reference pore-network for each problem. We demonstrate that signature parts of a heterogeneous core, which are a small portion of its entire volume, are sufficient inputs for the developed pore-network stitching method to construct a representative pore-network and predict flow properties.

Suitability of 2D modelling to evaluate flow properties in 3D porous media

Abstract: The employment of 2D models to investigate the properties of 3D flows in porous media is ubiquitous in the literature. The limitations of such approaches are often overlooked. Here, we assess to which extent 2D flows in porous media are suitable representations of 3D flows. To this purpose, we compare representative elementary volume (REV) scales obtained by 2D and 3D numerical simulations of flow in porous media. The stationarity of several quantities, namely porosity, permeability, mean and variance of velocity, is evaluated in terms of both classical and innovative statistics. The variance of velocity, strictly connected to the hydrodynamic dispersion, is included in the analysis in order to extend conclusions to transport phenomena. Pore scale flow is simulated by means of a Lattice Boltzmann model. The results from pore scale simulations point out that the 2D approach often leads to inconsistent results, due to the profound difference between 2D and 3D flows through porous media. We employ the error in the evaluation of REV as a quantitative measure for the reliability of a 2D approach. Moreover, we show that the acceptance threshold for a 2D representation to be valid strongly depends on which flow/transport quantity is sought.

Study of the Factors Influencing Diffusive Tortuosity Based on Pore-Scale SPH Simulation of Granular Soil

Abstract: Tortuosity quantitatively reflects the complexity of porous media. For the diffusion process in porous media, diffusive tortuosity is inevitably an important research topic. A simulation experimental method is proposed to study diffusive tortuosity in granular soil based on 3D pore-scale simulation by smoothed particle hydrodynamics (SPH). On the basis of the simulation results, the relationships between diffusive tortuosity and microstructural parameters are discussed. Simulation experiments are implemented on 3D granular soil columns generated by PFC software and soil layers formed by periodic expansion of the soil columns. The accuracy of the pore-scale SPH diffusion model is verified by the analytic solution for the particular case of a pure water column. The results show that the dimensionality does affect the diffusive tortuosity. 2D profiles cannot represent the original 3D medium in terms of the diffusion characteristics. Within the limited range of the variation of porosity during soil compression, the relationship between porosity and diffusive tortuosity can be considered to be linear. In addition, for the case of uniform granule size, diffusive tortuosity is almost linearly related to the specific surface area. Tortuosity values for soil columns are larger than those for the soil layer, which denotes that the effect of the sidewall on diffusion cannot be overlooked for a centimeter-scale soil column.

Contact Angle Measurements Using Sessile Drop and Micro-CT Data from Six Sandstones

Abstract: 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 CO2 behavior in carbon storage reservoirs.

Petrophysical Correlations for the Permeability of Coquinas (Carbonate Rocks)

Abstract: The pore structure of many carbonate formations is known to be very complex and heterogeneous. Heterogeneity is manifested by the presence of different types, sizes, and shapes of pores resulting from sedimentation and diagenetic actions. These complexities greatly increase uncertainties in estimated rock hydraulic properties in that different permeability values may occur for samples having similar porosities. In order to understand the effects of pore structure and heterogeneity, petrophysical analyses were performed on coquina samples from the Morro do Chaves Formation (Barremian, Sergipe-Alagoas Basin), which is an analogue of Brazilian Pre-salt oil reservoirs of Itapema Formation in the Santos Basin. Routine core analyses, and NMR and MICP measurements were carried out to obtain pore body and pore throat distributions. Obtained T2 relaxation times were converted to pore size radii by matching the NMR and MICP curves. Pore-scale imaging and pore network modelling were performed using microCT scans and the PoreFlow software, respectively. Calculated permeabilities using PoreFlow showed excellent agreement with the routine laboratory measurements. Samples having pore bodies with a higher coordination number showed much larger permeabilities at similar porosities. This study includes a statistical analysis of various features that caused the observed differences in permeability of the coquinas, including the role of connectivity of the entire porous system. Limitations and challenges of the various techniques, and the imaging and pore-scale flow simulations, are discussed.

Admissible Parameters for Two-Phase Coreflood and Welge–JBN Method

Abstract: The Welge–JBN method for determining relative permeability from unsteady-state waterflood test is commonly used for two-phase flows in porous media. We discuss the theoretical criteria that limits application of the basic Buckley–Leverett model and Welge–JBN method and the operational criteria of the accuracy of measurements during core waterflood tests. The objective is determination of the waterflood test parameters (core length, flow velocity and effluent sampling frequency) that fulfil the theoretical and operational criteria. The overall set of criteria results in five inequalities in three-dimensional Euclidian space of these parameters. For known rock and fluid properties, a formula for minimum core length to fulfil Welge–JBN criteria is derived. For cases where the core length is given, formulae for test’s flow velocity and sampling period are provided to satisfy the test admissibility conditions. The application of the proposed methodology is illustrated by two coreflood tests.

Laboratory Investigation of Liquid Injectivity in Surfactant-Alternating-Gas Foam Enhanced Oil Recovery

Abstract: Surfactant-alternating-gas (SAG) is a favored method of foam injection, in part because of excellent gas injectivity. However, liquid injectivity is usually very poor in SAG. We report a core-flood study of liquid injectivity under conditions like those near an injection well in SAG application in the field, i.e., after a prolonged period of gas injection following foam. We inject foam [gas (nitrogen) and surfactant solution] into a 17-cm-long Berea core at temperature of 90 °C with 40 bar back pressure. Pressure differences are measured and supplemented with CT scans to relate water saturation to mobilities. Liquid injectivity directly following foam is very poor. During prolonged gas injection following foam, a collapsed-foam region forms near the inlet and slowly propagates downstream, in which water saturation is reduced. This decline in liquid saturation reflects in part liquid evaporation, also pressure-driven flow and capillary effects on the core scale. In the collapsed-foam region, liquid mobility during subsequent liquid injection is much greater than downstream, and liquid sweeps the entire core cross section rather than a single finger. Mobility in the region of liquid fingering is insensitive to the quality of foam injected before gas and the duration of the period of gas injection. This implies that at the start of liquid injection in a SAG process in the field, there is a small region very near the well, crucial to injectivity, substantially different from that further out, and not described by current foam models. The results can guide the development of a model for liquid injectivity based on radial propagation of the various banks seen in the experiments.

Resolution Effect: An Error Correction Model for Intrinsic Permeability of Porous Media Estimated from Lattice Boltzmann Method

Abstract: In digital rock physics, the intrinsic permeability of a porous rock sample can be evaluated from its micro-computed tomography ($$\upmu$$-CT) image through lattice Boltzmann method (LBM) simulation. The LBM permeability evaluation has been increasingly adopted by the oil and gas industries, especially when the access to core samples is limited. In order to accurately evaluate the permeability of porous media, this digital approach requires high-quality $$\upmu$$-CT images with sufficient resolution and size. In practice, however, the LBM simulation is often performed using images of reduced resolution, due to limitations in computing power and simulation time. As a result, the permeability results obtained are often compromised with significant errors, known as the resolution effect. In this study, the resolution effect is quantitatively investigated to identify the primary causes of error, based on which an error correction model for the LBM permeability evaluation is proposed. The model uses such geometric attributes as connected porosity, specific surface area and diffusion tortuosity to quantify the resolution effect and achieve error correction. Demonstrated on various types of porous media including sandstone, carbonate rock, sand pack, synthesis silica, etc., the proposed error correction model can effectively correct the errors in LBM permeability evaluation due to the resolution effect. Our error correction model makes image resolution reduction more meaningful and creditable for LBM permeability evaluation of porous media, thereby supporting its adoption in practical applications.

The Porosity and Permeability of Binary Grain Mixtures

Abstract: The processes that control binary mixing of two sizes of grains have been investigated theoretically and validated by comparison with experimental data. These seemingly simple experiments are difficult to carry out with the degree of precision needed to test the models. We have developed a methodology allowing porosity and permeability to be measured to within ± 4.415% and ± 4.989% (at a flow rate of 5.13 cm3/s) of each value, respectively. Theoretical considerations recognise mixing processes: (1) an interstitiation process whereby small grains fit between larger grains and (2) a replacement process whereby large grains replace smaller grains and the porosity associated with them. A major result of this work is that the theoretical models describing these two processes are independent of grain size and grain shape. The latter of these two findings infers that the models developed in this work are applicable to any shape of grain or type of packing, providing that a representative porosity of each size of grain pack is known independently, either experimentally or theoretically. Experimental validation has shown that the newly developed relationships for porosity described measurements of porosity for near-ideal binary mixtures extremely well, confirming that porosity is always reduced by binary mixing, and that the degree of reduction depends upon the size of the ratio between the two grain sizes. Calculation of permeability from the packing model has also been done. Six different permeability estimation methods have been used. It was found that the most accurate representations of the experimental permeability were obtained (1) when the exact RGPZ (Revil, Glover, Pezard, Zamora) method was used with the porosity mixing models developed in this work and (2) when the exact RGPZ method was used with the weighted geometric mean to calculate a representative grain size.

Pore-scale Water Vapor Condensation Behaviors in Shales: An Experimental Study

Abstract: Water condensation in shales impacts its hydro-mechanical response. A mechanistic understanding of the pore-water system is made more challenging by significant anisotropy of pore architecture and nano-scale heterogeneity of pore surfaces. We probe the condensation response in two contrasting shales exposed to a vapor of contrast-matching water, as characterized by in situ ultra-small/small-angle neutron scattering (USANS/SANS) techniques under various relative humidities. One shale with a higher content of both kerogen and clay has rougher surfaces and higher anisotropy than the other shale (less clay and no kerogen) over length scales from 2.5 to 250 nm. Scanning electron microscopy with energy-dispersive spectrometry (SEM–EDS) analysis also confirms that the organic-rich shale presents more anisotropic microfabrics and higher heterogeneity compared to the other shale with less clay and no kerogen. USANS/SANS results show that water condensation effectively narrows the pore volume in the way of reducing the aspect ratio of non-equiaxed pores. For the shale with less clays and no kerogen under a relative humidity of 83%, a wetting film uniformly covers the pore-matrix interface over a wide range of length scale (1 nm–1.9 µm) without smoothing the surface roughness. In contrast, for the organic-rich and clay-rich shale with a strong wetting heterogeneity, condensation occurs at strongly curved hydrophilic asperities (1–10 nm) and smoothens the surface roughness. This is consistent with water vapor condensation behavior in a Vosges sandstone by Broseta et al. (Phys. Rev. Lett. 86:5313, 2001). Though well representing the condensation behavior of water vapor in mesopores/macropores (radii > 1 nm), USANS/SANS techniques could underestimate total water adsorption due to potential cation hydration and clay swelling in micropores (radii < 1 nm).

CNN-PFVS: Integrating Neural Network and Finite Volume Models to Accelerate Flow Simulation on Pore Space Images

Abstract: Direct numerical simulations of flow on micro-computed tomography (micro-CT) images are extensively used in many disciplines of science and engineering. Recently, we have developed a pore-scale finite volume solver (PFVS) to directly solve for flow on micro-CT images and predict permeability of digital cores. The solver assigns a local conductivity to each voxel based on geometrical and topological constraints. The local conductivity term in PFVS is conventionally calculated by an iterative local scanning algorithm, where the number of iterations depends on the size of the largest flow channel. This can increase the computation time of PFVS significantly if the largest flow channel is reasonably large. In this paper, we apply convolutional neural networks (CNN) to predict local conductivity for each voxel, thus bypassing the iterative algorithm while also preserving the mass conservation in the system by still solving for flow using conventional methods. The network is trained to convert segmented binary images of rocks into a numerical map required for flow simulation by the use of paired image-to-image translation using a ResNet-Style architecture. Comparison of the generated and original coefficient maps shows that the average error is within 1% over the 3D pore geometries used in this study. Then, we compare the absolute permeability results obtained from the original PFVS and the CNN-PFVS and the errors are within 20% with the average of 13.8%. Machine learning improves the computation time significantly especially on the images with large domain size and flow channels. On the samples tested, the speedup factor is 10 times using CNN compared to iterative calculations.

Calibrating and Scaling Semi-empirical Foam Flow Models for the Assessment of Foam-Based EOR Processes (in Heterogeneous Reservoirs)

Abstract: Models for simulating foam-based displacements fall into two categories: population-balance (PB) models that derive explicitly foam texture or bubble size from pore-level mechanisms related to lamellas generation and coalescence, and steady-state semi-empirical (SE) models that account implicitly for foam texture effects through a gas mobility reduction factor. This mobility reduction factor has to be calibrated from a large number of experiments on a case-by-case basis in order to match the physical effect of parameters impacting foam flow behaviour such as fluids saturation and velocity. This paper proposes a methodology to set up steady-state SE models of foam flow on the basis of an equivalence between SE model and PB model under steady-state flow conditions. The underlying approach consists in linking foam mobility and foam lamellas density (or texture) data inferred from foam corefloods performed with different foam qualities and velocities on a series of sandstones of different permeabilities. Its advantages lie in a deterministic non-iterative transcription of flow measurements into texture data and in a separation of texture effects and shear-thinning (velocity) effects. Then, scaling of foam flow parameters with porous medium permeability is established from the analysis of calibrated foam model parameters on cores of different permeability, with the help of theoretical representations of foam flow in a confined medium. Although they remain to be further confirmed from other well-documented experimental data sets, the significance of those scaling laws is great for the assessment of foam-based enhanced oil recovery (EOR) processes because foam EOR addresses heterogeneous reservoirs. Simulations of foam displacement in a reservoir cross section demonstrate the necessity to scale foam SE models with respect to facies heterogeneity for reliable evaluation.