Pore-scale Simulation of Water Alternate Gas Injection

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower during water injection in the presence of gas due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase.     

A Parametric Model for Predicting Relative Permeability-Saturation-Capillary Pressure Relationships of Oil–Water Systems in Porous Media with Mixed Wettability

A parametric two-phase, oil–water relative permeability/capillary pressure model for petroleum engineering and environmental applications is developed for porous media in which the smaller pores are strongly water-wet and the larger pores tend to be intermediate- or oil-wet. A saturation index, which can vary from 0 to 1, is used to distinguish those pores that are strongly water-wet from those that have intermediate- or oil-wet characteristics. The capillary pressure submodel is capable of describing main-drainage and hysteretic saturation-path saturations for positive and negative oil–water capillary pressures. At high oil–water capillary pressures, an asymptote is approached as the water saturation approaches the residual water saturation. At low oil–water capillary pressures (i.e. negative), another asymptote is approached as the oil saturation approaches the residual oil saturation. Hysteresis in capillary pressure relations, including water entrapment, is modeled. Relative permeabilities are predicted using parameters that describe main-drainage capillary pressure relations and accounting for how water and oil are distributed throughout the pore spaces of a porous medium with mixed wettability. The capillary pressure submodel is tested against published experimental data, and an example of how to use the relative permeability/capillary pressure model for a hypothetical saturation-path scenario involving several imbibition and drainage paths is given. Features of the model are also explained. Results suggest that the proposed model is capable of predicting relative permeability/capillary pressure characteristics of porous media mixed wettability.     

Co-history Matching: A Way Forward for Estimating Representative Saturation Functions

Core-scale experiments and analyses would often lead to estimation of saturation functions (relative permeability and capillary pressure). However, despite previous attempts on developing analytical and numerical methods, the estimated flow functions may not be representative of coreflood experiments when it comes to predicting similar experiments due to non-uniqueness issues of inverse problems. In this work, a novel approach was developed for estimation of relative permeability and capillary pressure simultaneously using the results of “multiple” corefloods together, which is called “co-history matching.” To examine this methodology, a synthetic (numerical) model was considered using core properties obtained from pore network model. The outcome was satisfactorily similar to original saturation functions. Also, two real coreflood experiments were performed where water at high and low rates were injected under reservoir conditions (live fluid systems) using a carbonate reservoir core. The results indicated that the profiles of oil recovery and differential pressure (dP) would be significantly affected by injection rate scenarios in non-water wet systems. The outcome of co-history matching could indicate that, one set of relative permeability and capillary pressure curves can reproduce the experimental data for all corefloods.     

Study of the effect of capillary pressure on the permeability of porous media embedded with a fractal-like tree network

In this paper, the analytical expressions for permeability of (both saturated and unsaturated) porous media embedded with a fractal-like tree network are presented based on fractal theory and technique when the capillary pressure is taken into account. Both the dimensionless effective permeability and the relative permeability of the composites, which are defined as porous media embedded with a fractal-like tree network in this work, are derived and found to be a function of saturation, the capillary pressure and microstructural parameters of the networks. The relative permeabilities predicted by the present fractal model are compared with the available experimental data and a fair agreement between them is found.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part I. Development of Interacting Capillary Bundle Model

An interacting capillary bundle model is developed for analysing immiscible displacement processes in porous media. In this model, pressure equilibration among the capillaries is stipulated and capillary forces are included. This feature makes the model entirely different from the traditional tube bundle models in which fluids in different capillaries are independent of each other. In this work, displacements of a non-wetting phase by a wetting phase at different injection rates were analysed using the interacting capillary bundle model. The predicted evolutions of saturation profiles were consistent with both numerical simulation and experimental results for porous media reported in literature which cannot be re-produced with the non-interacting tube bundle models.     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

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.     

Numerical Simulation of a CT-Scanned Counter-Current Flow Experiment

Counter-current flow occurs in many reservoir processes and it is important to understand and model these processes in order to operate them effectively. Both drainage and imbibition processes exist simultaneously during counter-current flow. It has thus proven difficult to model this type of flow using conventional techniques because of the impossibility of assigning a single capillary pressure curve applicable over the entire sample. In the current paper, a new saturation-history-dependent approach has been developed to simulate a counter-current flow experiment done with an X-ray CT scanner. Hysteresis in both capillary pressure and relative permeabilities is considered during simulation. Capillary hysteresis loop and relative permeabilities are extracted through history matching and a family of scanning curves is constructed connecting the two branches of the capillary hysteresis loop. Each gridblock of the sample is assigned a different scanning curve according to the local saturation history. History-dependent modeling of the experiment reproduced two-dimensional saturation distributions over time with good accuracy, which cannot be obtained with traditional simulation using only one capillary pressure curve.     

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.     

Enhancing Immiscible Fluid Displacement in Porous Media by Capillary Pressure Discontinuities

Fluid displacement in porous media plays an important role in many industrial applications, including biological filtration, carbon capture and storage, enhanced oil recovery, and fluid transport in fuel cells. The displacement front is unstable, which evolves from smooth into ramified patterns, when the mobility (ratio of permeability to viscosity) of the displacing fluid is larger than that of the displaced one; this phenomenon is called viscous fingering. Viscous fingering increases the residual saturation of the displaced fluid, considerably impairing the efficacy of fluid displacement. It is of practical importance to develop suitable methods to improve fluid displacement. This paper presents an experimental study on applying the discontinuity of capillary pressure to improve immiscible fluid displacement in drainage for which the displacing fluid (air) wets the porous media less preferentially than does the displaced fluid (silicone oil). The concept involves using a heterogeneous packing system, where the upstream region features large pores and small capillary pressure, and the downstream region features small pores and large capillary pressure. The increase in capillary pressure prevents fingering from directly crossing the media interface, thus enhancing the displacement. The experimental apparatus was a linear cell comprising porous media between two parallel plates, and glass beads of 0.6 and 0.125 mm diameter were packed to compose the heterogeneous porous media. The time history of the finger flow was recorded using a video camera. Pressure drops over the model from the inlet to the outlet were measured to compare viscous pressure drops with capillary pressures. The results show that the fluid displacement was increased by the capillary discontinuities. The optimal displacement was determined through linear regression by adjusting the relative length of the large- and small-pore region. The results may assist in the understanding of fingering flow across the boundaries of different grain-sized bands for the gas and oil reservoir management, such as setting the relative location of the injection and production wells. The findings may also serve as a reference for industrial applications such as placing the grain bands in an adequate series to improve the displacement efficacy in biological filtration.     

A Multiscale Network Model for Simulating Moisture Transfer Properties of Porous Media

A multiscale network model is presented to model unsaturated moisture transfer in hygroscopic capillary-porous materials showing a broad pore-size distribution. Both capillary effects and water sorption phenomena, water vapour and liquid water transfer are considered. The multiscale approach is based on the concept of examining the porous space at different levels of magnification. The conservation of the water vapour permeability of dry material is used as scaling criterion to link the different pore scales. A macroscopic permeability is deduced from the permeabilities calculated at the different levels of magnification. Each level of magnification is modelled using an isotropic nonplanar 2D cross-squared network. The multiscale network simulates the enhancement of water vapour permeability due to capillary condensation, the hysteresis phenomenon between wetting and drying, and the steep increase of moisture permeability at the critical moisture saturation level. The calculated network permeabilities are compared with experimental data for calcium silicate and ceramic brick and a good agreement is observed.     

Water removal from porous media by gas injection: experiments and simulation

The flow of a saturated gas through a porous medium, partially occupied by a liquid phase, causes evaporation due to gas expansion. This process, referred to as flow-through drying, is important in a wide variety of natural and industrial applications, such as natural gas production, convective drying of paper, catalysts, fuel cells and membranes. X-ray imaging experiments were performed to study the flow-through drying of water-saturated porous media during gas injection. The results show that the liquid saturation profile and the rate of drying are dependent on the viscous pressure drop, the state of saturation of the gas and the capillary characteristics of the porous medium. During the injection of a completely saturated gas, drying occurs only due to gas expansion. Capillary-driven flow from regions of high saturation to regions of low saturation lead to more uniform saturation profiles. During the injection of a dry gas, a drying front develops at the inlet and propagates through the porous medium. The experimental results are compared with numerical results from a continuum model. A good agreement is found for the case of sandstone. The comparison is less satisfactory for the experiments with limestone.     

In-Situ Capillary Trapping of CO<Subscript>2</Subscript> by Co-Injection

Co-injection of water with CO₂ is an effective scheme to control initial gas saturation in porous media. A fractional flow rate of water of approximately 5–10% is sufficient to reduce initial gas saturations. After water injection following the co-injection, most of the gas injected in the porous media is trapped by capillarity with a low fractional volume of migrating gas. In this study, we first derive an analytical model to predict the gas saturation levels for co-injection with water. The initial gas saturation is controlled by the fractional flow ratio in the co-injection process. Next, we experimentally investigate the effect of initial gas saturation on residual gas saturation at capillary trapping by co-injecting gas and water followed by pure water injection, using a water and nitrogen system at room temperature. Depending on relative permeability, initial gas saturation is reduced by co-injection of water. If the initial saturation in the Berea sandstone core is controlled at 20–40%, most of the gas is trapped by capillarity, and less than 20% of the gas with respect to the injected gas volume is migrated by water injection. In the packed bed of Toyoura standard sand, the initial gas saturation is approximately 20% for a wide range of gas with a fractional flow rate from 0.50 to 0.95. The residual gas saturation for these conditions is approximately 15%. Less than approximately 25% of the gas migrates by water injection. The amount of water required for co-injection systems is estimated on the basis of the analytical model and experimental results.     

Measurement of Capillary Pressure–Saturation Relationships Under Defined Compression Levels for Gas Diffusion Media of PEM Fuel Cells

Measurements of capillary pressure–saturation relationships under defined levels of compression for gas diffusion layers (GDL) of polymer electrolyte membrane fuel cells as thin, mixed-wettable porous media have been carried out in a newly constructed device. This article lines out the construction principle of the cell and the preconditioning procedure of the sample to measure the capillary pressure–saturation relationships under well-defined conditions and loads of compression. Three commercial GDLs (Freudenberg H2315T10A, H2315T10AC1, and SGL Carbon BA24) have been examined and a compression-depending hysteresis of the capillary pressure–saturation relationship has been measured and identified.     

A New Model for Immiscible Displacement in Porous Media

Immiscible displacement is regarded as the superposition of forward flows of both water and oil, due to injection of water into the medium, and of additional forward flow of water coupled with reverse flow of oil, caused by the existence of capillary pressure gradients. The model has been evaluated numerically for the prediction of the evolution of saturation profiles in waterfloods covering a wide range of water injection rates. In agreement with experimentation, saturation profiles ranging from a completely flat shape to piston-shape, depending on the injection rate, have been obtained. Also in agreement with experimentation, numerical evaluation of the model for the case of a closed system with an initial step-function saturation profile has predicted a gradual spreading of the piston front into S-shaped profiles with an increasing variance. The final profile corresponds to uniform saturation everywhere in the medium.     

Relative Permeability Analysis of Tube Bundle Models,Including Capillary Pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcy's law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.     

Modeling the Formation of Fluid Banks During Counter-Current Flow in Porous Media

Fluid banks sometimes form during gravity-driven counter-current flow in certain natural reservoir processes. Prediction of flow performance in such systems depends on our understanding of the bank-formation process. Traditional modeling methods using a single capillary pressure curve based on a final saturation distribution have successfully simulated counter-current flow without fluid banks. However, it has been difficult to simulate counter-current flow with fluid banks. In this paper, we describe the successful saturation-history-dependent modeling of counter-current flow experiments that result in fluid banks. The method used to simulate the experiments takes into account hysteresis in capillary pressure and relative permeabilities. Each spatial element in the model follows a distinct trajectory on the capillary pressure versus saturation map, which consists of the capillary hysteresis loop and the associated capillary pressure scanning curves. The new modeling method successfully captured the formation of the fluid banks observed in the experiments, including their development with time. Results show that bank formation is favored where the p_c-versus-saturation slope is low. Experiments documented in the literature that exhibited formation of fluid banks were also successfully simulated.     

Determination of Capillary Pressure Function from Resistivity Data

A model has been derived theoretically to correlate capillary pressure and resistivity index based on the fractal scaling theory. The model is simple and predicts a power law relationship between capillary pressure and resistivity index (P _c = p _e · I ^β) in a specific range of low water saturation. To verify the model, gas-water capillary pressure and resistivity were measured simultaneously at a room temperature in 14 core samples from two formations in an oil reservoir. The permeability of the core samples ranged from 0.028 to over 3000 md. The porosity ranged from less than 8 to over 30. Capillary pressure curves were measured using a semi-permeable porous-plate technique. The model was tested against the experimental data obtained in this study. The results demonstrated that the model could match the experimental data in a specific range of low water saturation. The experimental results also support the fractal scaling theory in a low water saturation range. The new model developed in this study may be deployed to determine capillary pressure from resistivity data both in laboratories and reservoirs, especially in the case in which permeability is low or it is difficult to measure capillary pressure.     

Investigating the Fracture Network Effects on Sweep Efficiency during WAG Injection Process

In this study, the main recovery mechanisms behind oil/water/gas interactions during the water-alternating-gas (WAG) injection process, in a network of matrix/fracture, were fundamentally investigated. A visual micromodel was utilized to provide insights into the potential applications of WAG process in fractured oil-wet media as well as the possibility of observing microscopic displacement behavior of fluids in the model. The model was made of an oil-wet facture/matrix network system, comprised of four matrix blocks surrounded with fractures. Different WAG injection scenarios, such as slug arrangements and the effects of fluid injection rates on oil recovery were studied. A new equation representing the capillary number, considering the fracture viscous force and matrix capillary force, was developed to make the experimental results more similar to a real field. In general, WAG tests performed in the fractured model showed a higher oil recovery factor compared with the results of gas and water injection tests at their optimum rates. The results showed that the presence of an oil film, in all cases, was the main reason for co-current drainage and double displacement of oil under applied driving forces. Furthermore, the formation of oil liquid bridges improved the recovery efficiency, which was greatly influenced by the size of fracture connecting the two matrix blocks; these connecting paths were more stable when there was initial water remaining in the media. Analyzing different recovery curves and microscopic view of the three phases in the transparent model showed that starting an injection mode with gas (followed by repeated small slugs of water and gas), could considerably improve oil recovery by pushing water into the matrix zone and increasing the total sweep efficiency.