Article

# Solvent Diffusion and Dispersion in Partially Saturated Porous Media: An Experimental and Numerical Pore-level Study

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## Abstract

Recent research on unlocking the solvent dispersion as a physical blending process in porous media has been mainly focused on core-scale observations. Pore-level studies of multiphase displacements can help to develop models that correlate rock macro-scale characteristics with its small-scale features, particularly for unsaturated rocks where the intricate fluid arrangements cause dynamic events to be localized in preferential pathways. In this work, to draw new insights into the physics of pore-level interfaces and crystallize the role of small-scale phenomena on the efficiency of solvent-aided bitumen recovery processes, numerical simulations coupled with experiments are conducted in pore-level domains. We simulate fluid flow and transport phenomena through millimeter-sized three-dimensional slabs of consolidated and unconsolidated packings of grains representing the geological rock types of the McMurray formation. We propose a robust numerical workflow for simulation of miscible-floods in unsaturated porous media and investigate the impact of matrix heterogeneity, connate water, cementation, and injection velocity on the longitudinal dispersion coefficient. In particular, primary drainage is simulated at low capillary numbers resulting in two-phase fluid occupancies through pore space domains. Finite element simulations are then carried out in order to solve the mixing advection-diffusion equations within the water-free pore space, and lastly, the effluent history is analyzed to predict the dispersion coefficient in both fully- and partially-saturated conditions and evaluate the efficiency of miscible displacements in the presence of microheterogeneities. A new analytical model for calculation of dispersivity, together with the numerical simulation results, is utilized to adjust a general model for the prediction of longitudinal dispersion coefficient in unsaturated sandy porous media of either uniform or non-uniform grains. Moreover, miscible-flood experiments at low to high injection velocities are conducted in a transparent glass micromodel. According to the results, rock micro-heterogeneities and immobile water both increase the longitudinal dispersion coefficient, and two-phase equilibria control the velocity field by creating connected regions of brine and low resistance oil-filled channels and consequently influence the solute transport and mixing processes. The effect of viscosity contrast on the longitudinal dispersion coefficient is also noteworthy, as the viscous fingering at unfavorable viscosity ratios widens the mixing zone.

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... [14] Recent studies have focused mainly on diffusion measurements and applications. [15][16][17] Yassin and coworkers', [18] for instance, used hydroxypropyl methylcellulose (HPMC) as a model system to implement and develop the current understanding of the role of diffusion in swelling polymers. Using terahertz pulsed imaging, a tool combining high contrast and high spatial resolution between different phases through changing refractive indices and fast timeresolved analysis, they investigated the growth kinetics of the process as well as followed the water diffusion front as it propagated through the porous tablets of HPMC. ...
... In this respect, porosity is used to describe the reduction in the cross-sectional area and quantify the tortuosity. [15] The diffusion rate of the UP resin was measured in the compressed EG tablets, opposing the gravity direction and by vacuum as a driving force (Fig. 3). The diffusion rates of the UP-LV and UP-HV resins through the EG tablets are shown in Fig. 4. The EG1, EG2, and EG3 particles had the densities 0.0038-0.0049 ...
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Taking advantage of the analogy between hydraulic and electrical flows to facilitate the prediction of porous media characteristics is a longstanding practice in engineering and science applications. The relationships between hydraulic and electrical properties are widely used in the characterization of transport phenomena relying on the strong correlation between electric and hydraulic flow conductance. However, due to the lack of small-scale investigations the similarity among their pathways/tortuosities is still unclear. Here a series of direct finite element numerical simulations are conducted within pore-level microstructures to extract and compare the streamlines of both electric and fluid flow currents and examine the accuracy of the hydraulic-electric analogy by predicting the petrophysical characteristics of the case studies. The fluid flow and electric transports are simulated through low-porosity digital unconsolidated packings of polydispersed grains representing the Athabasca oil sands deposit as the second largest oil reserve in the world. The formation factor, porosity, and absolute permeability of the media under consideration are predicted, and consequently, the streamlines of both electric and hydraulic currents are extracted and compared in terms of length, shape, and pathways. According to the results, the fluid flow streamlines are longer and pose differently than the homogeneous electric current streamlines. The average ratio between the hydraulic and electric tortuosities is 1.15 and a local extremum occurs at the porosity of eighteen percent. Pedotransfer functions for tortuosities, dimensionless permeability, and formation factor are proposed underpinning the rigorous relationships between transport processes in porous media.
Article
A stable spreading film between water and gas can extract a significant amount of bypassed non-aqueous phase liquid (NAPL) through immiscible three-phase gas/water injection cycles. In this study, the pore-scale displacement mechanisms by which NAPL is mobilized are incorporated into a three-dimensional pore morphology-based model under water-wet and capillary equilibrium conditions. The approach is pixel-based and the sequence of invasions is determined by the fluids’ connectivity and the threshold capillary pressure of the advancing interfaces. In addition to the determination of three-phase spatial saturation profiles, residuals, and capillary pressure curves, dynamic finite element simulations are utilized to predict the effective permeabilities of the rock microtomographic images as reasonable representations of the geological formations under study. All the influential features during immiscible fluid flow in pore-level domains including wetting and spreading films, saturation hysteresis, capillary trapping, connectivity, and interface development strategies are taken into account. The capabilities of the model are demonstrated by the successful prediction of saturation functions for Berea sandstone and the accurate reconstruction of three-phase fluid occupancies through a micromodel.
Article
Permeability, k, is a property of a porous medium that measures the capacity and ability of the medium to transmit fluids. The permeability of a porous medium (porous rock, soil, packing) can be predicted using empirical relationships, capillary models, statistical models, and hydraulic radius theories. It generally has been recognized that grain size is the fundamental independent variable that controls permeability in unconsolidated sediments [Shepherd, R. G. 1989]. Sub-pore scale modeling is used in this study to evaluate the relation between grain size distribution and permeability of the porous media. In this method the governing equations are applied directly on the porous medium images. The meshing algorithm is applied directly on the pore and grain sections of the images, and the studied mechanism is simulated at the level of pores and grains. Virtual porous media are generated using regular packings of uniforms spheres. The permeability of each packing is calculated computationally and then compared to the predictions of the Kozeny-Carman equation. An in-house developed pattern generator is used to generate spherical grains packing with pre-specified particle size distributions. A sub-pore scale modeling approach is used to calculate permeabilities of generated virtual porous media. The equivalent diameters that provide match from the Kozeny-Carman equation are then predicted.
Article
Enhanced Gas Recovery (EGR) involves CO2 injection into natural gas reservoirs to both increase gas recovery and trap CO2. EGR viability can be determined by reservoir simulations; however, these require a description of fluid dispersion (mixing) between the supercritical CO2 and natural gas. Here, this dispersivity (α) in sandstone rock plugs as a function of residual water fraction is quantified. To ensure the accuracy of such data, a novel core flooding experimental protocol that ensured an even spatial distribution of water was designed, minimized erroneous entry/exit contributions to mixing, and minimized dissolution of the CO2 into the water phase. Dispersivity was found to increase significantly with water content, although the differences in α between sandstones were eliminated upon the inclusion of residual water. This enabled development of a correlation between α and water content and, hence, between the dispersion coefficient and Peclet number that is readily incorporable into reservoir simulations.
Article
Determination of interfacial areas is crucial for accurate identification of multiphase subsurface flow processes, e.g. in situ water treatment, contaminant transport, phase change, convection/ diffusion, and colloid adsorption/ desorption/ migration. Pore-scale imaging and simulation techniques provide an appealing opportunity to explicitly predict such geometrical characteristics of saturated and unsaturated porous media. Here, synthetic unconsolidated sand packs in a wide range of grain arrangements and size distributions are reconstructed to extensively capture all possible heterogeneous pore-level geometries in the McMurray formation, the primary bitumen formation in the Athabasca oil sands deposit. Multi-phase fluid occupancies throughout the partially saturated media during drainage and imbibition are predicted applying a direct pixel-wised pore morphological model, incorporating wetting phase layer covering rock/solid surfaces. The post-processing results are verified using two-phase experimental data points and images, demonstrating a remarkable variation of the interfacial area as a function of saturation profile, rock configuration, and displacement scenarios. Empirical models are proposed to predict the bulk and meniscus areas using average particle diameter and porosity as input parameters.
Article
Assessment of the viability of enhanced gas recovery (EGR), in which CO2 is injected into natural gas reservoirs, requires accurate and appropriate reservoir simulations. These necessitate provision of parameters describing dispersion between the fluids. Here we systematically measure fluid dispersion in various rock cores (sandstones and carbonates), both dry and at irreducible water saturation, at reservoir conditions. In this manner we evaluate the impact of the irreducible water on the miscible displacement processes. As such this represents the first measurement of dispersion as a function of water saturation for supercritical gases in consolidated media. Complementary measurements of water spatial distribution along the rock axis, as well as the pore size distribution occupied by the water were performed using magnetic resonance techniques. Irreducible water was found to increase dispersivity by a factor of up to 7.3. The dispersion coefficient (K) was measured as a function of velocity and the data for both dry and water-containing samples were successfully combined on a K-Péclet number (Pe) plot, enabling ready future inclusion into EGR reservoir models. The power-law dependence of K upon Pe produced an exponent of 1.2 for dry and water-saturated sandstones and 1.4 for dry and water-saturated carbonates, consistent with literature results (Bijeljic et al., 2011; Honari et al., 2015).
Article
Although the previous static experiments provided critical information as to the existence of a critical temperature range that yields the maximum heavy-oil recovery during steam/solvent injection, dynamic experiment are needed to account for the relationship between the solvent introduced into the system and heavy-oil recovery. We conducted a series of dynamic experiments in which liquid (heptane) solvent was injected into a heavy oil saturated rock matrix, surrounded by a fracture with and without pre-thermal injection. Water-wet rock matrix (sandstones) was saturated with heavy oil and placed inside a core holder. Next, the system was placed into an oven and maintained at constant temperature conditions. Then, either hot solvent (superheated to be in vapor phase) or cold solvent was introduced into the system through the fracture at a constant rate. Pressure and temperature was continuously monitored along the core and the properties of oil and liquid condensate from gas produced were measured and analyzed. This scheme was repeated for a wide range of temperature conditions. The first requirement for a successful application is that the solvent should diffuse into matrix effectively before it breaks through and improves gravity drainage of oil by dilution. The second requirement is solvent retrieval. The retrieval of the solvent during solvent injection phase and post-thermal method (steam or hot-water) injection performed at the near-boiling point temperature of the solvent was monitored. Our results and observations indicate that there exists a critical temperature and injection rate that yields a maximized oil recovery and solvent retrieval.
Article
The method used in an earlier paper (Aris 1956) to discuss the dispersion of a solute in a fluid flowing through a tube is applied here to the case in which the solute can also pass into another fluid phase flowing in an annular region around the first. The apparent diffusion coefficient is the sum of the molecular and Taylor diffusion coefficients in the two phases and a term due to the finite rate of partition between them. It is shown how the Taylor diffusion coefficients depend on the ratio of amounts of solute held in the two phases and how this gives a connexion between the coefficient a$^{2}$U$^{2}$/48D found by Taylor (1953) for viscous flow in a circular tube and the 11 a$^{2}$U$^{2}$/48D found by Westhaver (1942) in his analysis of the distillation column. The use of these apparent diffusion coefficients in distillation and partition chromatography is illustrated.
Conference Paper
We simulate flow and transport directly on pore-space images obtained by micro-CT scanning of rock cores. An efficient Stokes solver is used to simulate low-Reynolds number flows. The flow simulator uses a finite-difference method along with a standard predictor-corrector procedure to decouple pressure and velocity. An algebraic multigrid technique solves the linear systems of equations. We then predict permeability and the results are compared with lattice Boltzmann numerical results and available experimental data. For solute transport we apply a streamline-based algorithm that is similar to the Pollock algorithm common in field-scale reservoir simulation, but which employs a novel semi-analytic formulation near solid boundaries to capture, with sub-grid resolution, the variation in velocity near the grains. A random walk method accounts for molecular diffusion. The streamlinebased algorithm is validated by comparison with published results for Taylor-Aris dispersion in a single capillary with a square cross-section. We then accurately predict available experimental data in the literature for longitudinal dispersion coefficient as a function of Peclet number. We introduce a characteristic length based on ratio of volume to pore/grain surface area that can be used for consolidated porous media to calculate Peclet number.
Article
Porous media in which different fluid phases coexist are common in nature (e.g., vadose zone and gas-oil reservoirs). In partially saturated porous media, the intricate spatial distributions of the wetting and nonwetting phases causes their flow to be focused onto preferential paths. Using a novel 2-D experimental setup allowing pore-scale measurement of concentration fields in a controlled unsaturated flow, we highlight mechanisms by which mixing of an invading fluid with the resident fluid is significantly enhanced when decreasing saturation. The mean scalar dissipation rate is observed to decrease slowly in time, while under saturated conditions it decays rapidly. This slow decrease is due to sustained longitudinal solute fingering, which causes concentration gradients to remain predominantly transverse to the average flow. Consequently, the effective reactivity is found to be much larger than under saturated conditions. These results provide new insights into the role that multiphase flows play on mixing/reaction in porous media.
Article
When oil is displaced from a horizontal formation by another fluid of lower density, the latter tends to override the former in the shape of a tongue owing to gravity segregation. This gravity tongue has an adverse influence on the oil recovery. If the fluids are miscible, diffusion (mixing) takes place at the interfacial boundary of the gravity tongue. This mixing should have a favorable effect on oil recovery. The report describes a laboratory study of the magnitude of the mixing zone under various conditions, so as to assess the effect of diffusion on oil recovery both in laboratory experiments and under actual field conditions. The technique used enables visual observation and measurement of the size of the mixing zone in transparent glass-powder packs. The results show that in experiments in small models and cores the width of the mixing zone may well be of the same order of magnitude as the height of the model. In such cases oil recovery is favorably affected by mixing. It can further be concluded that, under conditions prevailing in the field, mixing of the injected fluid with reservoir oil is equal to that caused by molecular diffusion alone, eddy-mixing not taking place to any appreciable extent. A simple calculation, then, shows that molecular diffusion is too small for a beneficial effect to be expected from the injection of miscible fluids in horizontal or nearly horizontal reservoirs unless pay zones are thin. Introduction This paper gives results of experiments made on the mixing which occurs when a miscible displacement is carried out in a horizontal reservoir. This mixing takes place at the interfacial boundary of the gravity tongue formed when the lighter injected fluid overrides the oil present in the reservoir. The object of the experiments was to simulate the field case where, for example, propane with a viscosity of 0.075 cp under reservoir conditions displaces an oil of viscosity 0.6 cp. In the experiments the lighter fluid had a viscosity of about 1 cp and the heavier one a viscosity of about 8 cp, so as to obtain the same viscosity ratio. In order to enable the results to be compared with those published in the literature, a set of experiments with viscosity ratio equal to one was also performed. EXPERIMENTAL TECHNIQUE The technique developed for the purpose enables the width of the mixing zone to be studied as a function of time and place. A glass-powder pack is saturated with a water-glycerine mixture of suitable viscosity, which represents the reservoir fluid. The pack is then rendered transparent by dissolving sufficient ammonium thiocyanate in the mixture to obtain a solution with a refractive index matching that of the glass powder. Alkaline water to which phenolphthalein has been added is used as the lighter and less-viscous displacing fluid. In those places where mixing or diffusion of the two liquids occurs, the slightly acidic ammonium-thiocyanate solution neutralizes the alkali in the water, and the glass pack shows up white against the red-colored invading water. If a black cloth is hung over the back of the apparatus, the transparent part of the glass pack appears black. In this way the position of the two phases and the transition zone between them is clearly visible, as shown in Fig. 1 (where the red-colored invading water is the grey-shaded zone lying uppermost).In all experiments the amount of alkali in the water was chosen such that the upper contour corresponded to a concentration of 5 per cent of the dense liquid. The lower contour is determined by the size of the glass grains and the thickness of the pack and, consequently, varies from experiment to experiment. SPEJ P. 317
Article
Methods for miscible flooding have been researched and field tested since the early 1950's. This paper reviews the technical state of the art and field behavior to date for the major miscible flood processes: first-contact miscible, rich-gas drive, vaporizing-gas drive, and carbon dioxide flooding. Important technological areas selected for review include phase behavior and miscibility, sweepout, unit displacement efficiency, and process design variations. Carbon dioxide flood technology is emphasized, and several technical issues are identified that still need to be resolved. Rules of thumb and ranges of conditions are discussed for applicability of each process. A comparison is made of the incremental recovery and solvent slug effectiveness observed in field trials of the different processes. From the limited data available, processes. From the limited data available, there is no clear-cut evidence that field results on average and for a given slug size have been appreciably better or poorer for one process compared with another. Introduction The search for an effective and economical solvent along with development and field testing of miscible-flood processes has continued since the early processes has continued since the early 1950's. Early focus was on hydrocarbon solvents, and three types of hydrocarbon-miscible processes were developed: the first-contact miscible process; the vaporizing-gas drive process, often called high-pressure gas drive; and the rich-gas drive process, often called condensing-gas drive. First-contact miscible solvents mix directly with reservoir oils in all proportions and their mixtures always remain proportions and their mixtures always remain single phase. Other solvents are not directly miscible with reservoir oils, but under appropriate conditions of pressure and solvent composition these solvents can achieve miscibility in-situ by mass transfer of oil and solvent components through repeated contact with the reservoir oil. miscibility achieved in this manner is termed multiple-contact or dynamic miscibility. The vaporizing-gas drive process achieves dynamic miscibility by in-situ vaporization of intermediate molecular weight hydrocarbons from the reservoir oil into the injected gas. Dynamic miscibility is achieved in the rich-gas drive process by in-situ transfer of intermediate molecular weight hydrocarbons from the injected gas into the reservoir oil. Propane or LPG mixtures typically were the solvents used in first-contact hydrocarbon miscible flooding, whereas natural gas at high pressure and natural gas with appreciable concentrations of intermediate molecular weight hydrocarbons were injection fluids in vaporizing-gas drive and rich-gas drive floods. The high cost of propane, LPG, or rich hydrocarbon gas propane, LPG, or rich hydrocarbon gas dictated that these solvents be injected as slugs which usually were driven with natural gas. Flue gas and nitrogen also have been found to achieve dynamic miscibility at high pressures with some oils by the vaporizing-gas drive mechanism. Hydrocarbon miscible processes have received extensive field testing since the 1950's, primarily in the United States and Canada. Over 100 projects were initiated during this time period. The majority were small-scale pilot tests involving one or at most a few injection wells; however, a number of large projects were undertaken involving several thousand acres or more (more than 4 000 000 m2). A few projects tested flue gas injection.
Article
Fluid mixing plays a fundamental role in many natural and engineered processes, including groundwater flows in porous media, enhanced oil recovery, and microfluidic lab-on-a-chip systems. Recent developments have explored the effect of viscosity contrast on mixing, suggesting that the unstable displacement of fluids with different viscosities, or viscous fingering, provides a powerful mechanism to increase fluid–fluid interfacial area and enhance mixing. However, existing studies have not incorporated the effect of medium heterogeneity on the mixing rate. Here, we characterize the evolution of mixing between two fluids of different viscosity in heterogeneous porous media. We focus on a practical scenario of divergent–convergent flow in a quarter five-spot geometry prototypical of well-driven groundwater flows. We study by means of numerical simulations the impact of permeability heterogeneity and viscosity contrast on the breakthrough curves and mixing efficiency, and we rationalize the nontrivial mixing behavior that emerges from the competition between the creation of fluid-fluid interfacial area and channeling. This article is protected by copyright. All rights reserved.
Article
Heavy oil and bitumen are important hydrocarbon resources that are destined to play an increasingly significant role in the oil supply of the world. The total heavy oil resources are estimated to be about 10 trillion barrels, nearly three times the conventional oil in place in the world. The efficient and economic recovery of heavy oil and bitumen reserves is a crucial technical challenge. Most of these reserves lie deep inside the earth's crust and are not easily recoverable owing to their very high viscosity. Hydrocarbon solvent-based processes are thought to be effective EOR technologies for heavy oil and bitumen production. An important feature of the miscible fluid displacement is the mixing of the solvent and oil. This mixing occurs on the microscopic scale and result from the diffusion and dispersion process. Mixing effect has positive and negative effects on the miscible process. It could exerts a considerable damping effect on the growth of viscous and gravity fingers as an advantage of the dispersion and diffusion and on the other hand mixing of the solvent with oil in a reservoir decreases the effective strength of the solvent, which can have annoying effect on the miscibility and recovery efficiency. So, correct treatment of this mixing effect within the simulation of the miscible recovery process is very important. Mixing of the solvent and oil is governed by the diffusion and dispersion coefficients. For measuring these coefficients, we must depend largely on experimental measurements of them, because no universal theory permits their accurate a priori calculation. There are many influencing parameters on the mixing process during solvent injection into porous media. Designing an experiment to investigate all the effective parameters is extremely difficult (if not impossible) and time consuming. So, based on the sub-pore scale modeling method, a special program has been developed to investigate the mass transfer process in any porous medium, considering all medium properties. A realistic image of the porous medium without any simplification is directly used in the program. Developed program used the porous medium image directly, so, heterogeneity effect could be investigated. Pore and throat size distributions, as an important heterogeneous parameter, has a unique effect on the diffusion and dispersion coefficients. A comprehensive study of the pore and throat size distribution effect on the mixing coefficients will be presented in this paper.
Article
Thermal and miscible methods are commonly used for in-situ recovery of heavy oil and bitumen. Both techniques have their own limitations and associated shortcomings, often times yielding an inefficient process. The most common thermal method is steam injection, which is highly energy intensive. Steam generation costs and water production affect the economics of the thermal technique adversely. On the other hand, miscible methods are energy effective but their economics depends on the solvent retrieval. Various combinations of these two techniques such as co-or alternate injection of steam and solvent have been proposed as a solution, but no optimum method has yet been developed. Thermal and miscible methods can be combined by co-injecting solvent with steam or injecting solvent into a pre-heated reservoir. Current work was undertaken to study the performance of solvents at higher temperatures for heavy oil/bitumen recovery. Glass bead packs and Berea sandstone cores were used in the experiments to represent different types of pore structures, porosity and permeability. After saturating with heavy oil, the samples were exposed to the vapor of paraffinic solvents (propane and butane) at a temperature above the boiling point of the solvent, and a constant pressure of 1500 kPa. A mechanical convection oven was used to maintain constant temperature across the setup. The setup was designed in such a way that a reasonably long sample (up to 30 cm) can be tested to analyze the gravity effect. The oil recovered from each of these experiments was collected using a specifically designed collection system and analyzed for composition, viscosity and asphaltene content. The amount of oil recovered in each case was also analyzed and the quantity and nature of asphaltene precipitated with each of the tested solvents under the prevailing temperature and pressure of the experiment was reported. Optimal conditions for each solvent type were identified for the highest ultimate recovery. It was observed that recovery decreased with increasing temperature and pressure of the system. It was also noticed that butane diluted the oil more than propane which resulted in lower asphaltene content and viscosity of oil produced with butane as a solvent.
Article
This pore-scale modeling study in saturated porous media shows that compound-specific effects are important not only at steady-state and for the lateral displacement of solutes with different diffusivities but also for transient transport and solute breakthrough. We performed flow and transport simulations in two-dimensional pore-scale domains with different arrangement of the solid grains leading to distinct characteristics of flow variability and connectivity, representing mildly and highly heterogeneous porous media, respectively. The results obtained for a range of average velocities representative of groundwater flow (0.1-10 m/day), show significant effects of aqueous diffusion on solute breakthrough curves. However, the magnitude of such effects can be masked by the flux-averaging approach used to measure solute breakthrough and can hinder the correct interpretation of the true dilution of different solutes. We propose, as a metric of mixing, a transient flux-related dilution index that allows quantifying the evolution of solute dilution at a given position along the main flow direction. For the different solute transport scenarios we obtained dilution breakthrough curves that complement and add important information to traditional solute breakthrough curves. Such dilution breakthrough curves allow capturing the compound-specific mixing of the different solutes and provide useful insights on the interplay between advective and diffusive processes, mass transfer limitations, and incomplete mixing in the heterogeneous pore-scale domains. The quantification of dilution for conservative solutes is in good agreement with the outcomes of mixing-controlled reactive transport simulations, in which the mass and concentration breakthrough curves of the product of an instantaneous transformation of two initially segregated reactants were used as measures of reactive mixing.
Article
The enhanced recovery of natural gas by the injection and sequestration of CO2 is an attractive scenario for certain prospective field developments if the risks of gas contamination or early CO2 breakthrough can be assessed reliably. Simulations of enhanced gas recovery (EGR) scenarios require accurate dispersion parameters at reservoir conditions to quantify the size of the miscible CO2–CH4 displacement front; several experimental studies using core-flooding equipment aimed at measuring such parameters have been reported over the last decade. However, such measurements are particularly challenging and the data produced are generally afflicted in their repeatability by limited experimental control and in their accuracy by systematic errors such as gravitational and core-entrance/exit effects. We review here the existing experimental data pertaining to EGR by CO2 sequestration and also report new measurements of longitudinal CO2–CH4 dispersion coefficients at temperatures of 40–80 ◦C, pressures of 8–12 MPa and interstitial velocities of 0.05–1.13 mm s−1 [14.2–320 ft day−1] in 5–10 cm long sandstone cores with permeabilities of 12 and 460 mD. The core-floods were conducted in both a horizontal and vertical orientation, with significant gravitational effects observed for low velocity floods in horizontal cores with high permeabilities. We also analyzed the effects of tubing and core entrance/exit effects on the measurements and found that the latter resulted in apparent dispersion coefficients up to 63% larger than would be due to the core alone. Our results indicate that dispersivities for CO2–CH4 at these supercritical conditions are less than 0.001 m, which indicates that excessive mixing will not occur in EGR scenarios in the absence of conformance effects such as heterogeneity coupled with injection well pattern. Inclusion of such conformance effects is essential for detailed reservoir simulation.
Article
Dispersivity affects the displacement and sweep efficiencies and the required slug size of a displacing fluid. Unfortunately, the dispersivity values estimated from field tests are a few orders of magnitude greater than the values obtained in laboratory tests. This investigation studies the effect of small scale (or core scale) heterogeneities on the effective dispersivity value in a typical gridblock size used in a reservoir simulator. The study is restricted to contact miscible displacements with unit mobility ratio. A finite element simulator is used to investigate these effects. Physical dispersion is explicitly included in the simulator. Results show that the effective dispersivity is affected by the degree of heterogeneity, the average length of heterogeneity, the length of the system, and the manner in which the permeability values are spatially distributed. The effect of heterogeneity becomes significant if the coefficient of variance is greater than 0.4. The effect of dimensionless scale length (ratio of average length of heterogeneity to length of the system) on effective dispersivity is insignificant for dimensionless scale values less than .01; above that value dispersivity increases with an increase in the scale length. The effective dispersivity increases almost linearly with an increase in the length of the system for constant dimensionless scale length. This provides an explanation for the similar trend reported for field dispersivity data. A simple correlation is proposed to calculate the effective dispersivity of the porous medium. The effect of the way in which given permeability values are distributed across the medium on the dispersivity cannot be correlated with the parameters investigated, probably indicating that the effective dispersivity is a unique function of the manner in which permeability values are distributed. This effect was not noted previously in the literature; however, it does not affect the general trends. Introduction Dispersion is important in understanding reservoir performance during enhanced oil recovery processes. Large values of dispersivity reduce the displacement efficiency of multiple contact miscible displacements and increase the sweep efficiency. The net result of dispersion may be to increase the required slug size of a displacing fluid. Unfortunately, the values obtained in the laboratory do not coincide with the values obtained in the field. The reasons for the discrepancy between the field and the lab values may include the effects of heterogeneities in reservoir properties and reservoir stratification. This discrepancy should not arise if all the possible heterogeneities in the reservoir, including the small scale heterogeneities, can be described rigorously. However, this type of description may not be practical. Generally, a reservoir simulator gridblock is the smallest unit by which heterogeneities in the reservoir may be represented. This investigation is restricted to study the effects of permeability variations in porous medium on the effective values of dispersivity. In cases studied, the permeability variations are the heterogeneities with length of variation about the same as that of the core length, (for example, several inches to several feet), sometimes called small scale heterogeneities. The effects of these small scale heterogeneities on the effective dispersivity value in a typical block size in a reservoir simulator are investigated. In other words, we are addressing the question of dispersivity scaling from laboratory to a typical block size in a field scale simulator. The studies are restricted to unit mobility ratio and contact miscible displacements. P. 215
Article
The objective of this paper is to determine the effect of a wetting immobile phase (e.g. connate water) on the diffusion and macroscopic dispersion in homogeneous unconsolidated packs. We measured the effective dispersion coefficients of gases as a function of water saturation, pressure, and velocity in a laboratory set-up. The dispersion coefficients were 2 to 3 times higher than in the dry porous medium. This effect is also studied using a pore network model where the immobile phase blocks a fraction of the pores for fluid flow. A random walk through the network and counting cluster sizes both gave approximately the same results, which were in good agreement with the experimental data.
Article
Tracer displacements were run during steadystate, two-phase flow, where tracers were added to both the wetting and nonwetting phases. Such experiments were conducted in the primary drainage, imbibition, and secondary drainage cycles over a wide range of fractional flows. The effluent tracer profiles from these experiments were fit with a four-parameter, capacitance-dispersion model, which divides each phase into three fractions: flowing, dendritic (branching), and isolated. Mixing in the flowing fraction is described by a dispersion coefficient, and communication between the flowing and dendritic fraction is represented by a single mass transfer coefficient. This model is shown to adequately fit tracer effluent data over a wide range of conditions. In some areas the model is nonunique--thus, more than one solution fits the experimental data equally well. Additional experiments were conducted to differentiate mechanisms in these regions. The effects of core wettability were investigated and shown to be important. The results of all of these experiments were used to formulate a mechanistic picture of steady, two-phase flow in strongly wetted porous media.
Article
Experimental measurements of the dispersion of tracer particles in flow through natural porous media are compared with a percolation model. The experiments show that tracer dispersion is a sensitive function of the width of the pore-size distribution as measured by mercury capillary pressure. Measurements of capillary pressure (or electrical conductivity) are used to estimate the geometric correlation length of the dominant flow path in the rock. Percolation theory is used to derive a power-law relationship between the correlation length and the ratio of the dispersivity to the average grain size. The experimental value of the power-law exponent is in agreement with the theoretical prediction. Measurements on samples containing a residual saturation of wetting epoxy show no significant change in dispersion behavior. This result mediates against dispersion models requiring trapping in dead-end pores. Tracer concentration profiles exhibit anomalous long-time tails in two cases. In carbonate rocks, we associate long-time tails with macroscopic permeability heterogeneities. In sandstones, long-time tails occur in samples with a very narrow pore-size distribution. These samples may have permeability heterogeneities as a result of defects in the packing density. In the limit of low flow velocity, the long-time tail disappears, suggesting a convective mechanism associated with flow heterogeneities at a millimeter-or-larger scale.
Article
The effect of anisotropic dispersion on nonlinear viscous fingering in miscible displacements is examined. The formulation admits dispersion coefficient-velocity field couplings (i.e., mechanical dispersivities) appropriate to both porous media and Hele--Shaw cells. A Hartley transform-based scheme is used to numerically simulate unstable miscible displacement. Several nonlinear finger interactions were observed. Shielding, spreading, tip splitting, and pairing of viscous fingers were observed here, as well as in isotropic simulations. Multiple coalescence and fading were observed in simulations with weak lateral dispersion, but not for isotropic dispersion. Transversely and longitudinally averaged one-dimensional concentration histories demonstrate the rate at which the mixing zone broadens and the increase in lateral scale as the fingers evolve when no tip splitting occurs. These properties are insensitive to both the dispersion anisotropy and the Peclet number at high Peclet number and long times. This suggests the dominance of finger interaction mechanisms that are essentially independent of details of the concentration fields and governed fundamentally by pressure fields.
Article
The soil gas diffusion coefficient (D(p)) and its variations with soil air content (epsilon) and soil water matric potential (psi) control vadose zone transport and emissions of volatile organic chemicals and greenhouse gases. This study revisits the 1904 Buckingham power-law model where D(p) is proportional to E with X characterizing the tortuosity and connectivity of air-filled pore space. One hundred years later, most models linking D(p) (epsilon) to Soil water retention and pore size distribution still assume that the pore connectivity factor, X, is a constant for a given soil. We show that X varies strongly with both epsilon and matric potential [given as pF = log(psi, cm H(2)O)] for individual soils ranging from undisturbed sand to aggregated volcanic ash soils (Andisols). For Andisols with bimodal pore size distribution, the X-pF function appears symmetrical. The minimum X value is typically around 2 and was observed close to psi of -1000 cm H(2)O (pF 3) when inter-aggregate voids are drained. To link D(p) with bimodal pore size distribution, we coupled a two-region van Genuchten soil water retention model with the Buckingham D(p) (epsilon) model, assuming X to vary symmetrically around a given pF. The coupled model well described D(p) as a function of both epsilon and psi for both repacked and undisturbed Andisols and for other soil types. By merely using average values of the three constants in the proposed symmetrical X-pF expression, predictions of D(p) were better than with traditional models.
Article
The transport of gaseous compounds in soil takes place by gas diffusion, advection, and dispersion. Gas transport processes are influenced by tic soil-gas diffusion coefficient (D), air permeability (k(a)) and soil-gas dispersion coefficient (D(H)), respectively. Of three gas transport parameters, DH is the least understood especially hov it is correlated to soil type, moisture conditions, and other transport parameters (i.e., D(P) and k(a)). In this study, a unified measurement system (UMS) that enables sequential measurement of D(P), k(a), and D(H) on the same soil core was developed. The experimental sequence is based on a two-chamber measurement of DH and k(a), followed by a one-chamber measurement of L P. Gaseous oxygen concentration and air pressure sensors are located in inlet and outlet chambers as well as at multiple points along the soil column. Using different particle-size fractions of non-aggregated (Toyoura sand) and aggregated (Nishi-Tokyo loam) soils, the effects of soil structure, particle (aggregate) size, and column scale (5-cm i.d. and 30-cm or 60-cm length) on the three gas transport parameters were investigated. For both soils, D(H) linearly increased with increasing pore-air velocity. For Toyoura sand, gas dispersivity (lambda = D(H)/u(0)) decreased with increasing soil-air content, while for Nishi-Tokyo loam, gas dispersivity decreased with increasing soil-air content to a minimum value when inter-aggregate pores were drained and increased again whether pores inside the soil aggregates started to act as tortuous air-filled pathways. In the arterial bore region (corresponding to the total pore volume for Narita sand and the inter-aggregate pore volume for Nishi-Tokyo loam), a linear relation between tortuosity of the air-filled pore network (T, calculated from D(P)) and the gas dispersivity (lambda) was observed.
Article
Investigations of gas transport and fate processes in packed soil systems require knowledge of the gas diffusion coefficient, D p , as a function of air-filled porosity, e. On the basis of the literature, data from six studies over the porosity range of 0.1 to nearly 1.0, it is reconfirmed that the Marshall (1959) model better predicts D p (e) in completely dry, repacked porous media than do the Penman (1940) and Millington (1959) models. The smaller D p value in wet soil, as compared with dry soil at the same air-filled porosity, is accounted for by introducing a water-induced linear reduction (WLR) term, equal to the ratio of air-filled porosity to total porosity, in the D p (e) model. By adding the WLR term in each of the three D p (e) models for dry porous media, the so-called WLR(Marshall), WLR(Penman), and WLR(Millington) D p (e) models for wet soil are developed. To test the three WLR models, D p was measured at different soil-water contents in six differently textured (6-38% clay) repacked soils. The WLR (Marshall) model accurately and best described D p (e) for all six soils and additional soils from the literature. All three WLR models performed better than previous D p (e) models. This study implies that the smaller D p in a wet soil, which is due to water-induced changes in air-filled pore shape and pore connectivity, can be described by a simple, linear function of relative air-filled porosity. The WLR(Marshall) model represents a conceptual and accurate model to predict D p (e) in sieved, repacked soil.
Article
The paper describes experiments on miscible displacement in various porous media and the results of these experiments. Both glass bead packs and natural cores were used. Bead diameters varied from 0.044 to 0.47 mm, and pack lengths varied from 83 to 678 cm. Natural cores used were Berea and Torpedo sandstone. By taking samples as small as 0.5 cc and using refractive index for analysis, the data on break through curves could be plotted to within ± 0.5 per cent. To plot the data correctly on error function paper, a parameter (Vp - v)/vV was used which allowed for the predicted growth of the front as it moved past the observer. The change in the amount of mixing (length of mixed zone) was studied by varying velocity, length of travel, bead size, viscosity ratio and pack diameter. When the displaced material was less viscous than the displacing material (favorable viscosity ratio), these changes were adequately predicted by theory. When natural cores were used, rather than glass beads, the amount of mixing was greatly increased - also qualitatively predicted by theory. In experiments with favorable viscosity ratios in which the ratio was varied from 0. 175 to 0. 998, it was found that the rate of mixing was changed by a factor of 5. 7. Thus, the rate of mixing is strongly affected by viscosity ratio, even when the theoretical error function relationship for mixing is valid. Experiments using fluids with viscosity ratios near 1.0 showed that the instability effects of even a slightly unfavorable viscosity ratio (1.002) caused disproportionately more elongated breakthrough curves than found with a favorable viscosity ratio (.998). When the viscosity ratio was as high as 5.71 these instability effects were much more pronounced, as evidenced by the shape of the breakthrough curve. The displacements at viscosity ratios above 1.0 no longer followed the theoretical error function curve.
Article
Sir Geoffrey Taylor has recently discussed the dispersion of a solute under the simultaneous action of molecular diffusion and variation of the velocity of the solvent. A new basis for his analysis is presented here which removes the restrictions imposed on some of the parameters at the expense of describing the distribution of solute in terms of its moments in the direction of flow. It is shown that the rate of growth of the variance is proportional to the sum of the molecular diffusion coefficient, D, and the Taylor diffusion coefficient kappa a2U2/D, where U is the mean velocity and a is a dimension characteristic of the cross-section of the tube. An expression for kappa is given in the most general case, and it is shown that a finite distribution of solute tends to become normally distributed.
Article
In this paper, an extensive study is presented on the single-phase flow of xanthan/tracer slugs in a consolidated sandstone. The phenomena studied include polymer/tracer dispersion, excluded/inaccessible-volume effects, polymer adsorption, and viscous fingering. In some floods, there is also evidence of nonequilibrium effects. Macroscopic flow equations are derived that include terms to model all the behaviors listed above. A microscopic approach is also developed that describes certain features of polymer flow in porous media semiquantitatively.
Article
Observations of pore structure in thin-sections are related to the performance of stable, first-contact-miscible displacements in reservoir cores and then to simulations of displacement performance of CO2 corefloods. Results of effluent composition measurements are reported for miscible displacements in seven core samples—three sandstones and four San Andres carbonates from west Texas or eastern New Mexico. Those displacements are interpreted by fitting the measured effluent compositions to the Coats-Smith (C-S) model, which represents the flow as occurring in flowing and stagnant fractions with mass transfer between them. Observations of thin-sections, including measurements of pore-size distributions and a simple measurement of spatial correlation of pore sizes, are also reported. Comparison of displacement results and thin-section data indicates that wide pore-size distributions and preferential flow paths are characterized in the C-S model by high dispersion coefficients and low flowing fractions. Simulations of the interactions of phase behavior and flow in nonuniform pore structures indicate that wide pore-size distributions and preferential flow paths can significantly increase residual oil saturations (ROS’s) in CO2 floods over those for uniform pore structures. Thus, heterogeneities observable at the scale of a thin-section have significant effects in laboratory core but much smaller effects in displacements at field scale. Large-scale heterogeneities present in field floods, however, probably cause similar increases in residual saturation in some fields.
Article
When a soluble substance is introduced into a fluid flowing slowly through a small-bore tube it spreads out under the combined action of molecular diffusion and the variation of velocity over the cross-section. It is shown analytically that the distribution of concentration produced in this way is centred on a point which moves with the mean speed of flow and is symmetrical about it in spite of the asymmetry of the flow. The dispersion along the tube is governed by a virtual coefficient of diffusivity which can be calculated from observed distributions of concentration. Since the analysis relates the longitudinal diffusivity to the coefficient of molecular diffusion, observations of concentration along a tube provide a new method for measuring diffusion coefficients. The coefficient so obtained was found, with potassium permanganate, to agree with that measured in other ways. The results may be useful to physiologists who may wish to know how a soluble salt is dispersed in blood streams.
Article
Inaccurate modeling of reservoir mixing by using large gridblocks in compositional simulation can affect recoveries significantly in miscible gasfloods and lead to inaccurate predictions of recovery performance. Reservoir mixing or dispersion is caused by diffusion of particles across streamlines; mixing can be enhanced significantly if the surface area of contact between the reservoir and injected fluid is increased as fluids propagate through the reservoir. A common way to convert geological models into simulation models is to upscale permeabilities on the basis of reservoir heterogeneity. Upscaling affects the degree of mixing that is modeled, but the importance of reservoir mixing in upscaling is largely ignored. This paper shows how to estimate the level of mixing in a reservoir and how to incorporate mixing into the upscaling procedure. We derive the key scaling groups for first-contact miscible (FCM) flow and show how they have an impact on reservoir mixing. Heterogeneities are assumed to dominate the flow regime so that gravity effects are negligible. We examine only local mixing, not apparent mixing caused by variations in streamline path lengths (convective spreading). Local mixing is important because it affects the strength of the injected fluid and can cause an otherwise multicontact miscible (MCM) flood to become immiscible. More than 1,000 2D numerical simulations are carried out using experimental design to estimate dispersivity as a function of the derived scaling groups. We show that reservoir mixing is enhanced owing to fluid propagation through heterogeneous media. Because mixing is dependent on heterogeneities, upscaling is an iterative process in which the level of mixing in both the longitudinal and transverse directions must be matched from the fine to the coarse scale. The most important groups that affect mixing are the mobility ratio, dispersion number, correlation lengths, and the Dykstra-Parson coefficient. Large dispersion numbers yield greater dispersivities away from the injection well. We show through simulations of both FCM and MCM floods that gridblock size can be increased significantly when reservoir mixing is large. Heterogeneous reservoirs with large longitudinal correlation lengths can be upscaled to larger gridblocks than reservoirs with random permeability fields. This paper shows how to determine a priori the maximum gridblock size allowed in both the x- and z-directions to predict the oil recovery from miscible gasfloods accurately.
Article
The soil gas diffusion coefficient (D(p)) and air permeability (k(a)) and their dependency on soil air content (epsilon) control gas diffusion and advection in soils. This study investigated the effects of average particle size (D(50)) and dry bulk density (rho(b)) on D(p) and k(a) for six sandy soils under variably saturated conditions. Data showed that particle size markedly affects the effective diameter of the drained pores active in leading gas through the sample at -100 cm H(2)O of soil water matric potential (calculated from D(p) and k(a)) as well as the average pore diameter at half saturation (calculated from the water retention curve), both exhibiting similar and exponential relationships with D(50) Under variably saturated conditions, higher D(p) and k(a) in coarser sand (larger D(50)) were observed due to rapid gas diffusion and advection through the less tortuous large-pore networks. In addition, soil compaction (larger rho(b)) simultaneously caused reduced water blockage effects and a reduction of large-pore space, resulting in higher D(p)(epsilon) but lower k(a)(epsilon). Two recent models for D(p)(epsilon) and k(a)(epsilon) were evaluated: the water-induced linear reduction (WLR) model for D(p), and the reference-point power law (RPL) model for k(a), with reference point ka set at -100 cm H(2)O. The performance of both models for the sandy soils (particle size range 0.02-0.9 mm) was improved if the pore connectivity-tortuosity factor and water blockage factors were assumed to be functions of D(50) and rho(b). Water blockage factors, N for the WLR D(p)(epsilon) model and M for the RPL k(a)(epsilon) model, showed a strong nonlinear relationship (R(2) = 0.95) that seems promising for predicting D(p)(epsilon) from the more easily measureable k(a)(epsilon).
Article
Mixing that occurs as on ga displaces another within columns of soil or glass beads was examined by using mathematical model that included mixing by molecular diffusion, mass transport, and gaseous sorption; although apparent diffusion coefficient values were no constant with flow velocity, general behavior of mixing process was adequately described by model.