One- and Two-Phase Permeabilities of Vugular Porous Media

The single and double phase macroscopic permeabilities of bimodal reconstructed porous media have been studied. The structure of these bimodal media is characterized by the micro and macroporosities (vug system) and by the micro and macrocorrelation lengths l _p and l _v. For a single phase, if the vugular system does not percolate, it is shown that the absolute permeability K mainly depends on l _p and very little on the other parameters. However, when the vugs percolate, K is also influenced by the density of vugs. For double phase calculations (in strong wettability conditions), it is shown that a vuggy percolating system affects mainly the nonwetting phase permeability. Moreover, the relative permeabilities for a nonpercolating vuggy system are only slightly influenced by the porosity distribution. These predictions are in good agreement with some experimental data obtained with limestones.     

Flow Modeling of Well Test Analysis for Porous–Vuggy Carbonate Reservoirs

Porous–vuggy carbonate reservoirs consist of both matrix and vug systems. This paper represents the first study of flow issues within a porous–vuggy carbonate reservoir that does not introduce a fracture system. The physical properties of matrix and vug systems are quite different in that vugs are dispersed throughout a reservoir. Assuming spherical vugs, symmetrically distributed pressure, centrifugal flow of fluids and considering media that is directly connected with wellbore as the matrix system, we established and solved a model of well testing and rate decline analysis for porous–vuggy carbonate reservoirs, which consists of a dual porosity flow behavior. Standard log–log type curves are drawn up by numerical simulation and the characteristics of type curves are analyzed thoroughly. Numerical simulations showed that concave type curves are dominated by interporosity flow factor, external boundary conditions, and are the typical response of porous–vuggy carbonate reservoirs. Field data interpretation from Tahe oilfield of China were successfully made and some useful reservoir parameters (e.g., permeability and interporosity flow factor) are obtained from well test interpretation.     

Effective Permeability of a Porous Medium with Spherical and Spheroidal Vug and Fracture Inclusions

Vugs and fractures are common features of carbonate formations. The presence of vugs and fractures in porous media can significantly affect pressure and flow behavior of a fluid. A vug is a cavity (usually a void space, occasionally filled with sediments), and its pore volume is much larger than the intergranular pore volume. Fractures occur in almost all geological formations to some extent. The fluid flow in vugs and fractures at the microscopic level does not obey Darcy’s law; rather, it is governed by Stokes flow (sometimes is also called Stokes’ law). In this paper, analytical solutions are derived for the fluid flow in porous media with spherical- and spheroidal-shaped vug and/or fracture inclusions. The coupling of Stokes flow and Darcy’s law is implemented through a no-jump condition on normal velocities, a jump condition on pressures, and generalized Beavers–Joseph–Saffman condition on the interface of the matrix and vug or fracture. The spheroidal geometry is used because of its flexibility to represent many different geometrical shapes. A spheroid reduces to a sphere when the focal length of the spheroid approaches zero. A prolate spheroid degenerates to a long rod to represent the connected vug geometry (a tunnel geometry) when the focal length of the spheroid approaches infinity. An oblate spheroid degenerates to a flat spheroidal disk to represent the fracture geometry. Once the pressure field in a single vug or fracture and in the matrix domains is obtained, the equivalent permeability of the vug with the matrix or the fracture with matrix can be determined. Using the effective medium theory, the effective permeability of the vug–matrix or fracture–matrix ensemble domain can be determined. The effect of the volume fraction and geometrical properties of vugs, such as the aspect ratio and spatial distribution, in the matrix is also investigated. It is shown that the higher volume fraction of the vugs or fractures enhances the effective permeability of the system. For a fixed-volume fraction, highly elongated vugs or fractures significantly increase the effective permeability compared with shorter vugs or fractures. A set of disconnected vugs or fractures yields lower effective permeability compared with a single vug or fracture of the same volume fraction.     

The Permeability of a Representative Carbonate Volume with a Large Vug

There are fundamental challenges in characterizing transport properties of a porous medium with large cavities, or vugs, when the characteristic size of the cavity is larger than or equal to 1 cm. Neither existing flow models nor common laboratory measurements are well suited to deal with such challenges. The present study determines the effective permeability of a representative carbonate volume with an arbitrary connected-through large vug. It also determines the minimum matrix permeability that impacts the flow in vuggy carbonates. The Stokes and Darcy models are used to capture the flows in the vug and in the matrix, respectively. At the interface, the flow models interact through slippage based on the Beavers–Joseph–Saffman model. The developed model is tested with analytical solutions available in the literature and using independent laboratory measurements. The results have major implications for characterizing the effective transport properties of carbonates, which constitute more than half of the world’s hydrocarbon reserves.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Quantitative In Situ Enhanced Oil Recovery Monitoring Using Nuclear Magnetic Resonance

Quantitative in situ monitoring of oil recovery from sedimentary rock is demonstrated for the first time using advanced two-dimensional (2D) nuclear magnetic resonance (NMR) correlation measurements on a low field spectrometer. The laboratory-scale NMR system was chosen to provide a common physics of measurement with NMR well-logging tools. The NMR protocols are used to monitor recovery of a heavy Middle East crude oil from high permeability sandstone plugs using a brine (water) flood followed by chemical enhanced oil recovery agents: polymer and alkaline?Csurfactant?Cpolymer solutions. 2D correlations between relaxation time (T ₁, T ₂) and apparent self-diffusion coefficient D _app are used to obtain simultaneously a volumetric determination of the oil and aqueous fluid-phase saturations present in the porous material. The T ₁ ? T ₂ and D _app ? T ₂ correlations are bulk measurements of the entire rock core-plug; excellent agreement is shown between the measures of remaining oil (from NMR) and recovered oil (from gravimetric assay of the effluent). Furthermore, we introduce the capability to measure spatially resolved T ₂ distributions on a low field spectrometer using a rapid frequency-encoded y ? T ₂ map. A non-uniform distribution of remaining oil is observed due to viscous instabilities in the flowing liquids; the final oil saturation ranges from ${S_{\rm o}^{\rm{(final)}} \approx 0}$ to 20?% along the direction of flow. These results highlight the quantitative nature of the NMR data obtainable in low field NMR core analysis and also the importance of spatially resolved measurements when studying short core-plugs.     

Fluid topology,pore size and aspect ratio during imbibition

Imbibition in glass micromodels for air-mercury and water-oil systems occurs by wetting phase (wp) cluster growth and frontal drive processes. Lower capillary number and higher wetting phase (wp) saturation at the start of imbibition favour cluster growth.Imbibition experiments for both fluid systems show that the rules of nwp withdrawal are related to pore size and to fluid topology as well as to aspect ratio. The emptying of a pore is favoured by small size, small aspect ratio (size rules), and fewer connected throats occupied by nonwetting phase (nwp) (fluid topology rules).The relative importance of fluid topology compared with pore size in determining the sequence of nwp withdrawal from pores is affected by the initial nwp saturation, pore size variability, pore-throat size ratio, pore and throat shape and contact angle. High initial nwp saturation, small variability of pore size and small pore-throat diameter ratio are all factors which increase the effects of fluid topology in determining nwp withdrawal sequence. Under these conditions, nwp displacement efficiency is larger because withdrawal occurs first from dead-end branches without breaking the continuity of the nwp conducting pathways to the nwp sink. The high nwp displacement efficiency obtained in unconsolidated sands may be explained by the importance of topology rules during imbibition in these low aspect ratio media.Roman Letters D effective diameter of pore or throat given by 2 ÷F()(1/x + 1/y),m - F() nondimensional term which varies as a function of cross-sectional shape (Lenormandet al., 1983) - L external dimension of the network (width or length), m - N _ca capillary number in the network, dimensionless - nwp nonwetting phase - P any capillary pressure, Pa - P _I1 apillary pressure for nwp to withdraw from a pore which has one connected throat occupied by nwp, Pa - P _I2 capillary pressure for nwp to withdraw from a pore which has two adjacent connected throats occupied by nwp, Pa - Q total volume flow rate in the network, m³/s - S _wi percent of pore volume occupied by wp at the end of drainage and start of imbibition, dimensionless - S _ni percent of pore volume occupied by nwp at the end of drainage and start of imbibition, i.e. 100 -S _wi, dimensionless - S _nr percent of pore volume occupied by trapped nwp at the end of imbibition, dimensionless - v velocity, m/s - wp wetting phase - X _I diameter ofI2 interface in plan, m - X _P diameter of pore in plan, m - X _T width of throat, m - Y _P depth of pore, m - Y _T depth of throat, m - Z number of throats connected to each pore (coordination number) Greek Letters interfacial tension, N/m - contact angle, degrees - viscosity, Pa · s     

An interferometric investigation of the effect of separation distance and temperature imbalance on natural convection for two horizontal cylinders at moderate Rayleigh numbers

An experimental investigation has been carried out to study the natural convection interaction between a pair of horizontal cylinders located directly one above the other. It is found that the presence of the upper cylinder has negligible effect on the heat transfer rate from the lower cylinder, thus attention has been given on the heat transfer characteristics from the upper cylinder The ratio S/D of the center-to-center separation distance was varied from three to six by a step of one. The temperature imbalance ΔT₁/ΔT₂ was also varied independently. The experiments were conducted with Rayleigh numbers varying from 6300 to 13000. It was found that maximum enhancement of the heat transfer from the upper cylinder for a given ratio S/D depends primarily from the temperature imbalance ΔT₁/ΔT₂ and weakly from the Rayleigh number. The experimental results for the maximum enhancement, can be presented fairly well by the empirical formula (ΔT₁/ΔT₂)_op=0.136(S/D)^0.75 for 2≦S/D≦6, 1000≦Ra≦200000. The findings of this investigation agree closely with those previously reported in the literature.     

Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers

Shape memory polymers (SMPs) are polymers that can demonstrate programmable shape memory effects. Typically, an SMP is pre-deformed from an initial shape to a deformed shape by applying a mechanical load at the temperature T_H>T_g. It will maintain this deformed shape after subsequently lowering the temperature to T_L<T_g and removing the externally mechanical load. The shape memory effect is activated by increasing the temperature to T_D>T_g, where the initial shape is recovered. In this paper, the finite deformation thermo-mechanical behaviors of amorphous SMPs are experimentally investigated. Based on the experimental observations and an understanding of the underlying physical mechanism of the shape memory behavior, a three-dimensional (3D) constitutive model is developed to describe the finite deformation thermo-mechanical response of SMPs. The model in this paper has been implemented into an ABAQUS user material subroutine (UMAT) for finite element analysis, and numerical simulations of the thermo-mechanical experiments verify the efficiency of the model. This model will serve as a modeling tool for the design of more complicated SMP-based structures and devices.     

Three-phase flow and gravity drainage in porous media

We present a theoretical and experimental treatment of three-phase flow in water-wet porous media from the molecular level upwards. Many three-phase systems in polluted soil and oil reservoirs have a positive initial spreading coefficient, which means that oil spontaneously spreads as a layer between water and gas. We compute the thickness and stability of this oil layer and show that appreciable recovery of oil by drainage only occurs when the oil layer occupies crevices or roughness in the pore space. We then analyze the distribution of oil, water and gas in vertical equilibrium for a spreading system, which is governed byα=γ _ow (ρ _o ?ρ _g )/γ _go (ρ _w ?ρ _o ), whereγ _ow andγ _go are the oil/water and gas/oil interfacial tensions respectively, andρ _g ,ρ _o andρ _w are the gas, oil and water densities respectively. Ifα>1, there is a height above the oil/water contact, beyond which connected oil only exists as a molecular film, with a negligible saturation. This height is independent of the structure of the porous medium. Whenα<1, large quantities of oil remain in the pore space and gravity drainage is not efficient. If the initial spreading coefficient is negative, oil can be trapped and the recovery is also poor. We performed gravity drainage experiments in sand columns and capillary tubes which confirmed our predictions.     

Limit Models of Pore Space Structure of Porous Materials for Determination of Limit Pore Size Distributions Based on Mercury Intrusion Data

This paper proposes the application of capillary and chain random models of pore space structure for determination of limit pore diameter distributions of porous materials, based on the mercury intrusion curves. Both distributions determine the range in which the pore diameter distribution of the investigated material occurs and defines the degree of inaccuracy of the method based on the mercury intrusion data caused by the indeterminacy of the sample shape and its pore space architecture. We derived equations describing the quasi-static process of mercury intrusion into the porous layer and porous ball with a random chain pore space structure and analysed the influence of the model parameters on the mercury intrusion curves. It was shown that the distribution of link length in the chain model of the pore space, random location of chain capillaries in the sample and the length distribution of the capillaries do not influence significantly the intrusion process. Therefore, a simple model of the mercury intrusion into the layer is proposed in which chain links of the pore space have random diameters and constant length. This model is used as a basic model of the intrusion process into a sample of any shape and size and with homogeneous and isotropic chain pore space architecture. The thickness of the layer then represents the mean length of chain capillaries in the sample. It was also proved that the capillary and chain models of pore space architecture are limit models of the network model identified in this paper with the pore architecture of the investigated sample. This justifies the application of both models for determination of limit cumulative distributions of pore diameters in porous materials based on the mercury intrusion data.

     

A Numerical Model of Tracer Transport in a Non-isothermal Two-Phase Flow System for {\text{ CO}}_2 Geological Storage Characterization

For the purpose of characterizing geologically stored $\text{ CO}_{2}Air sparging is an in situ soil/groundwater remediation technology, which involves the injection of pressurized air through air sparging well below the zone of contamination. To investigate the rate-dependent flow properties during multistep air sparging, a rule-based dynamic two-phase flow model was developed and applied to a 3D pore network which is employed to characterize the void structure of porous media. The simulated dynamic two-phase flow at the pore scale or microscale was translated into functional relationships at the continuum-scale of capillary pressure?Csaturation (P ^c?CS) and relative permeability??saturation (K _r?CS) relationships. A significant contribution from the air injection pressure step and duration time of each air injection pressure on both of the above relationships was observed during the multistep air sparging tests. It is observed from the simulation that at a given matric potential, larger amount of water is retained during transient flow than that during steady flow. Shorter the duration of each air injection pressure step, there is higher fraction of retained water. The relative air/water permeability values are also greatly affected by the pressure step. With large air injection pressure step, the air/water relative permeability is much higher than that with a smaller air injection pressure step at the same water saturation level. However, the impact of pressure step on relative permeability is not consistent for flows with different capillary numbers (N _ca). When compared with relative air permeability, relative water permeability has a higher scatter. It was further observed that the dynamic effects on the relative permeability curve are more apparent for networks with larger pore sizes than that with smaller pore sizes. In addition, the effect of pore size on relative water permeability is higher than that on relative air permeability.     

Drying Rate Measurements in Convection- and Diffusion-Driven Conditions on a Shaly Sandstone Using Nuclear Magnetic Resonance

Carbon Capture and Storage (CCS) is one of the solutions studied to reduce greenhouse gas accumulation in the atmosphere. Depleted oil and gas reservoirs have been studied for potential storage sites but also saline aquifers that have the advantages of much larger pore volume. In this latter case, injection of large volume of anhydrous carbon dioxide will lead to a strong water desaturation of the near wellbore region because of evaporation mechanisms. Even the capillary trapped water can be removed by thermodynamical transfer of water vapor in the CO₂ phase. The extension in time and space of the dry zone will be controlled by the drying rate induced by the gas flow. Consequences of drying may induce alteration of the injectivity by salt precipitation and/or alteration of the rock fabric itself, especially for shaly sandstones in the case of clay drying. The context of CCS has raised new interests in the understanding of drying kinetic where the water vapor is evacuated by gas convection. In this study, we investigated experimentally the drying rate evolution with time on a shaly sandstone sample in two conditions of drying: convective and diffusive. In convective conditions, air is injected at different flow rates through the porous media in conditions of drying representative of a CO₂ injection site at one million ton per year. In diffusive conditions, no flow is imposed and the water vapor escape by diffusion. Drying rates dynamics in both conditions were measured by Nuclear Magnetic Resonance (NMR) and compared. We varied the temperature and the salinity in diffusive-driven drying and the gas flow rate in convective-driven drying. The water distribution in the pore network and the water saturation profiles were monitored continuously using T₂ relaxation and 1D imaging NMR techniques. For the range of temperature and air flow rate used, we show that drying rates in the two drying conditions are similar but not identical. They both present different periods characteristic of the main mechanisms for water mass transfer. Drying rate has a power law dependence on the temperature, as predicted by thermodynamic, and drying rate was found proportional to the flow rate in convective drying. Presence of salt has a complex effect: an increase of the drying rate at early stage of drying followed by a strong decrease for the remaining time of drying.     

Does ITZ Influence Moisture Transport in Concrete?

Simulations of nuclear magnetic resonance (NMR) signal from fluids contained in porous media (such as rock cores) need to account for both enhanced surface relaxation and the presence of internal magnetic field gradients due to magnetic susceptibility contrast between the rock matrix and the contained fluid phase. Such simulations are typically focussed on the extraction of the NMR T₂ relaxation distribution which can be related to pore size and indirectly to system permeability. Discrepancies between such NMR signal simulations on digital rock cores and associated experimental measurements are however frequently reported; these are generally attributed to spatial variations in rock matric composition resulting in heterogeneously distributed NMR surface relaxivities (ρ) and internal magnetic field gradients. To this end, a range of synthetic sediments composed of variable mixtures of quartz and garnet sands were studied. These two constituents were selected for the following reasons: they have different densities allowing for ready phase differentiation in 3D μCT images of samples to use as simulation lattices and they have distinctly different ρ and magnetic susceptibility values which allow for a rigorous test of NMR simulations. Here these 3D simulations were used to calculate the distribution of internal magnetic field gradients in the range of samples, these data were then compared against corresponding NMR experimental measurements. Agreement was reasonably good with the largest discrepancy being the simulation predicting weak internal gradients (in the vicinity of the quartz sand for mixed samples) which were not detected experimentally. The suite of 3D μCT images and associated experimental NMR measurements are all publicly available for the development and validation of NMR simulation efforts.

     

Direct Observation of Trapped Gas Bubbles by Capillarity in Sandy Porous Media

We investigated the mechanism of residual gas trapping at a microscopic level. We imaged trapped air bubbles in a Berea sandstone chip after spontaneous imbibition at atmospheric pressure. The pore structure and trapped bubbles were observed by microfocused X-ray computed tomography. Distributions of trapped bubbles in Berea and Tako sandstone were imaged in coreflooding at a capillary number of 1.0 × 10⁻⁶. Trapped bubbles are of two types, those occupying the center of the pore with a pore-scale size and others having a pore-network scale size. In low-porosity media such as sandstone, connected bubbles contribute greatly to trapped gas saturation. Effects of capillary number and injected water volume were investigated using a packed bed of glass beads 600μm in diameter, which had high porosity (38%). The trapped N₂ bubbles are stable against the water flow rate corresponding to a capillary number of 1.0 × 10⁻⁴.     

Microtomography and Pore-Scale Modeling of Two-Phase Fluid Distribution

Synchrotron-based X-ray microtomography (micro CT) at the Advanced Light Source (ALS) line 8.3.2 at the Lawrence Berkeley National Laboratory produces three-dimensional micron-scale-resolution digital images of the pore space of the reservoir rock along with the spacial distribution of the fluids. Pore-scale visualization of carbon dioxide flooding experiments performed at a reservoir pressure demonstrates that the injected gas fills some pores and pore clusters, and entirely bypasses the others. Using 3D digital images of the pore space as input data, the method of maximal inscribed spheres (MIS) predicts two-phase fluid distribution in capillary equilibrium. Verification against the tomography images shows a good agreement between the computed fluid distribution in the pores and the experimental data. The model-predicted capillary pressure curves and tomography-based porosimetry distributions compared favorably with the mercury injection data. Thus, micro CT in combination with modeling based on the MIS is a viable approach to study the pore-scale mechanisms of CO₂ injection into an aquifer, as well as more general multi-phase flows.     

Modeling of vibration-dissociation oxygen kinetics at temperatures of 4,000-11,000 K

The time profiles of vibrational molecular oxygen temperature T _v measured earlier in experiments behind a strong shock wave were used for testing the theoretical and empirical models of thermal nonequilibrium dissociation of molecules. To do this, dissociating gas flows behind the strong shock wave front were calculated with account for these models. If the initial gas temperature behind the wave front T ₀ < 6.5×10³ K, the models well describe changing the temperature with time. However, for T ₀ > 7×10³ K neither of the models tested describes the measured temperature profiles satisfactorily. Using the empirical model proposed in the present study made it possible to satisfactorily describe the vibrational temperature evolution observed in experiments at temperatures up to 11×10³ K.     

An Experimental and Numerical Study of Relative Permeability Estimates Using Spatially Resolved $$T_1$$- z NMR

Relative permeability is a key characteristic describing flow properties of petroleum reservoirs, aquifers and water retention of soils. Various laboratory methods, typically categorised as steady-state, unsteady-state and centrifuge are used to measure relative permeability and may lead to different results. In recent years, 1D MRI, NMR \(T_2\) and \(T_1\) profiling have been applied for the characterisation of rock cores. It has been shown that spatially resolved NMR in conjunction with centrifuge technique may provide high-quality capillary pressure curves. Combining Burdine and Brooks–Corey models enables estimation of relative permeability from capillary pressure curves. This approach assumes a strong relationship between capillary pressure and relative permeability known to be complex. Here we compare a generalised approach of Green, which relies on saturation profiles set by various capillary drainage techniques, to a NMR relaxation approach. Comparisons are performed experimentally and numerically using three sandstone rocks to test the influence of rock morphology. The numerical part includes simulation of a centrifuge capillary drainage by applying morphological drainage transforms on high-resolution 3D tomograms. \(T_1\) responses along the sample are simulated using a random walk technique. The NMR relaxation-based approach is then compared to LBM simulated relative permeability and to experiment. The study confirms the applicability of NMR relaxation methods for relative permeability estimation of water-wet rocks and validates a numerical approach against experiment.