Equilibrium Capillary Pressure Research Articles

Capillary pressure equilibrium theory mapping of 4D seismic inversion results to predict saturation in a gas-water system

Understanding the effect of fluid concentration in rocks is crucial to characterize a hydrocarbon reservoir. Monitoring of fluid concentration specifically becomes challenging during a sequestration/enhanced oil recovery program because the objective is to understand the amount of initial fluid replaced by liquid/gas injection and identify the probability of leakage with time. The prior assumption of uniform or patchy type of distribution of gas in a rock-physics theory leads to large uncertainty in the prediction of saturation. As a result, we have used the capillary pressure equilibrium theory (CPET) for creating a reservoir model that matches the physics of capillary-induced fluid invasion and avoids the uncertainty related to the type of distribution of gas in pores. We create a CPET model using the reservoir parameters of the clean and unconsolidated sandstone formation of the Sleipner field, North Sea, which is the world’s first industrial-scale CO2-injection project, assuming that there is no significant change in the rock frame throughout the field. The model is then used to analyze the fluid content from the time-lapse seismic inversion results of the Sleipner field. CO2 at higher quantities, according to our research, is analogous to a uniform distribution, whereas CO2 at lower concentrations is mostly in between patchy and uniform distribution or slightly patchy type. We predict maximum CO2 saturation from a quantitative interpretation of six time-lapse seismic data from 1999 to 2010 using the CPET as 75% of the pore space, and the footprint of the CO2 plume in the topmost layer is spreading from zero in 2001 to 7 × 105 m2 in 2010. Our model finds that injected CO2 from all layers below will migrate to the top layer approximately 50 years after the commencement of the injection.

Effects of the Grain Size on Dynamic Capillary Pressure and the Modified Green–Ampt Model for Infiltration

Darcy-scale capillary pressure is traditionally assumed to be constant. By contrast, a considerable gap exists between the measured and equilibrium capillary pressures when the same moisture saturation is considered with a high flow rate, and this gap is called the dynamic effect on the capillary pressure. In this study, downward infiltration experiments of sand columns are performed to measure cumulative infiltration and to calculate the wetting front depth and wetting front velocity in sands with different grain sizes. We estimate the equilibrium capillary pressure head or suction head at the wetting front using both the classical Green–Ampt (GAM) and modified Green–Ampt (MGAM) models. The results show that the performance of MGAM in simulating downward infiltration is superior to that of GAM. Moreover, because GAM neglects the dynamic effect, it systematically underestimates the equilibrium suction head in our experiments. We also find that the model parameters α^ and β of MGAM are affected by the grain size of sands and porosity, and the dynamic effect of the capillary pressure increases with decreasing grain size and increasing porosity.

Pressure Model for Capillary Tree-Shaped Fractal Networks

Capillary tree networks can be efficiently used for several applications. These trees display the same patterns at different levels of scale, which is a property found in fractal objects. This paper combines a thermodynamic formulation with the geometric characteristics of the tree fractal network, to describe the equilibrium capillary pressure. The model proposed here accounts for, in addition to physical properties of the fluids, the fractal dimension, the scale factor between the size of parent and daughter tubes, and the branching level. The effect of each of these properties on the capillary pressure is also investigated.

Experimental study on non-aqueous phase liquid multiphase flow characteristics and controlling factors in heterogeneous porous media

Enhanced understanding of non-aqueous phase liquid (NAPL) infiltration into porous media is important for the effective design of remediation strategies. Six model multi-flow experiments in two-dimensional polymethyl methacrylate tank were carried out to verify the influences of soil initial water content, local low permeability lens, lithologic sharp interface and leakage position on light non-aqueous phase liquid (LNAPL) and dense on non-aqueous phase liquid (DNAPL) transport velocities in soil, with diesel and perchlorethylene as LNAPL and DNAPL, respectively. The evolutions of the plume were recorded through the transparent side of the tank by CCD camera and the contaminant fronts were traced using gel pen on a plastic film coated on tank wall at appropriate intervals. The controlling factors of free NAPL migration paths and speeds were analyzed thoroughly, and capillary pressure equilibrium models of NAPL-water/NAPL-air interface and the partition movement theory were put forward. Results show that the effect of NAPL types: soil initial water content, local low permeability lens, leakage position and lithologic sharp interface on NAPL migration paths and velocities are diverse. The NAPL movement principles should be presented in accordance with partitions as related to the initial water content namely dry soil area, capillary zone and water-saturated zone. The vertical movement and lateral extension of NAPL are interdependent, and the vertical migration is dominant in initial leakage stage. The vertical migration is hindered by lateral growth when meeting with local low permeability lens, lithologic sharp interface or capillary fringe. Vertical migration rate reduction is mainly caused by the pore resistance, NAPL residual and lateral spreading and the causes of corresponding increase of horizontal expansion speed are the decrease of vertical velocity and the increase of buoyancy. The sufficiently short time allows to neglect chemical reactions, dissolution and sorption of NAPL and heterogeneous porous media.

Effect of osmotic pressure on salt extraction by a poultice

Salts are widely recognized as a cause of deterioration of porous building materials. A general approach to slow down salt damage problems is reducing the salt content of the affected object. One of the most common treatments is the application of poultices. Desalination by poultices may result from two different salt transport processes: diffusion and advection. In this study we consider only the advection process. In the case of advection based desalination the efficiency of salt extraction strongly depends on the pore-size range of the substrate and the poultice. For advection to be effective, the poultice materials should be adapted to the pore-size distribution of the substrate. However, the poultice can shrink during drying, which affects the pore-size distribution of the poultice material. The salt transport from the poultice to the substrate by water advection results from a capillary pressure gradient, but upon drying a salt concentration gradient may appear in the poultice/substrate interface. Because of this concentration gradient an osmotic pressure arises, which may change the drying behavior of the poultice/substrate system. We have investigated the effect of the pore size distribution of the poultice and the osmotic pressure on the drying behavior of a poultice/substrate system. To this end, we have measured the time evolution of moisture and salt concentration profiles in a system consisting of kaolin/sand poultice on a fired-clay brick, using a specialized NMR setup. Comparison of the results with a model based on capillary pressure equilibrium at the poultice/substrate interface shows that the accumulation of salt from the substrate in the poultice may improve the salt extraction process, because the osmotic pressure decreases the effective pore size of the poultice.

Seismic attenuation and velocity dispersion in heterogeneous partially saturated porous rocks

SUMMARY Using a numerical approach, we explore wave-induced fluid flow effects in partially saturated porous rocks in which the gas–water saturation patterns are governed by mesoscopic heterogeneities associated with the dry frame properties. The link between the dry frame properties and the gas saturation is defined by the assumption of capillary pressure equilibrium, which in the presence of heterogeneity implies that neighbouring regions can exhibit different levels of saturation. To determine the equivalent attenuation and phase velocity of the synthetic rock samples considered in this study, we apply a numerical upscaling procedure, which permits to take into account mesoscopic heterogeneities associated with the dry frame properties as well as spatially continuous variations of the pore fluid properties. The multiscale nature of the fluid saturation is taken into account by locally computing the physical properties of an effective fluid, which are then used for the larger-scale simulations. We consider two sets of numerical experiments to analyse such effects in heterogeneous partially saturated porous media, where the saturation field is determined by variations in porosity and clay content, respectively. In both cases we also evaluate the seismic responses of corresponding binary, patchy-type saturation patterns. Our results indicate that significant attenuation and modest velocity dispersion effects take place in this kind of media for both binary patchy-type and spatially continuous gas saturation patterns and in particular in the presence of relatively small amounts of gas. The numerical experiments also show that the nature of the gas distribution patterns is a critical parameter controlling the seismic responses of these environments, since attenuation and velocity dispersion effects are much more significant and occur over a broader saturation range for binary patchy-type gas–water distributions. This analysis therefore suggests that the physical mechanisms governing partial saturation should be accounted for when analysing seismic data in a poroelastic framework. In this context, heterogeneities associated with the dry frame properties, which do not play important roles in wave-induced fluid flow processes per se, should be taken into account since they may determine the kind of gas distribution pattern taking place in the porous rock.

Drainage under Nonequilibrium Conditions: Exploring Wettability and Dynamic Contact Angle Effects Using Bundle‐Of‐Tubes Simulations

Numerical simulators often assume that equilibrium capillary pressure–saturation conditions are maintained as changes in fluid saturation are taking place; however, equilibrium conditions may not be maintained in all circumstances. An alternative approach is to describe a nonequilibrium difference between wetting and nonwetting fluid pressures as a function of the rate of change of saturation and a damping coefficient. It has been proposed that this damping coefficient may be a function of multiple fluid and porous medium properties, including wettability. This study used bundle‐of‐tubes simulations to provide insight into the potential effect of increasing equilibrium contact angle, as a measure of wettability, on the magnitude of the damping coefficient. The effect of considering the dynamic contact angle, as a function of wettability, was also investigated. Results showed that when dynamic contact angles were considered, larger damping coefficient values were predicted. These values varied nonmonotonically with equilibrium contact angle and had maximum values near an equilibrium contact angle of 60°. The results also showed that values of the damping coefficient were dependent on the type of pressure boundary condition used. A steadily increasing boundary pressure resulted in larger damping coefficient values that were a function of the equilibrium contact angle, and better represented experimental pressure–saturation observations, than simulations using a series of instantaneous steps. These results suggest that assuming equilibrium conditions may be reasonable under wettability conditions characterized by equilibrium contact angles near 90°, potentially for contact angles near 0°, but not at moderate equilibrium contact angles. Further work is necessary, however, to determine the underlying physical mechanisms that govern nonequilibrium pressure differences and the magnitude of the damping coefficient.

TCAT analysis of capillary pressure in non-equilibrium, two-fluid-phase, porous medium systems

Standard models of flow of two immiscible fluids in a porous medium make use of an expression for the dependence of capillary pressure on the saturation of a fluid phase. Data to support the mathematical expression is most often obtained through a sequence of equilibrium experiments. In addition to such expressions being hysteretic, recent experimental and theoretical studies have suggested that the equilibrium functional forms obtained may be inadequate for modeling dynamic systems. This situation has led to efforts to express relaxation of a system to an equilibrium capillary pressure in relation to the rate of change of saturation. Here, based on insights gained from the thermodynamically constrained averaging theory (TCAT) we propose that dynamic processes are related to changes in interfacial area between phases as well as saturation. A more complete formulation of capillary pressure dynamics is presented leading to an equation that is suitable for experimental study.

Performance equations for a polymer electrolyte membrane fuel cell with unsaturated cathode feed

A mathematical formulation for the cathode of a membrane electrode assembly of a polymer electrolyte membrane fuel cell is proposed, in which the effect of unsaturated vapor feed in the cathode is considered. This mechanistic model formulates the water saturation front within the gas diffusion layer with an explicit analytical expression as a function of operating conditions. The multi-phase flows of gaseous species and liquid water are correlated with the established capillary pressure equilibrium in the medium. In addition, less than fully hydrated water contents in the polymer electrolyte and catalyst layers are considered, and are integrated with the relevant liquid and vapor transfers in the gas diffusion layer. The developed performance equations take into account the influences of all pertinent material properties on cell performance using first principles. The mathematical approach is logical and concise in terms of revealing the underlying physical significance in comparison with many other empirical data fitting models.

An Analytic Approach to Capillary Pressure in Tree-Shaped Networks

This study analyzes the pressure in capillary tree-shaped networks, based on the thermodynamics and the geo- metric description of the network. The presence of equilibrium and non-equilibrium effects on pressure is accounted. In particular, the presence of equilibrium effects is shown to be essential for describing the pressure in the network. An ap- proach to the definition of equilibrium capillary pressure is presented in terms of interfacial energies of the phases but also in terms of the geometry characteristics of the tree-shaped network. Among other, the equilibrium capillary pressure is shown to depend on number of branching levels and number of daughter branches. Finally, a criterion is established to evaluate the importance of equilibrium and non-equilibrium effects on the capillary pressure.

Adsorption of Modified Dextrins on Talc: Effect of Surface Coverage and Hydration Water on Hydrophobicity Reduction

The adsorption of three modified dextrins on the basal plane of talc has been studied using in situ tapping mode atomic force microscopy (TMAFM). The images have been used to determine the layer thickness and coverage of the adsorbed polymers. Adsorption isotherms of the polymers on talc particles were also determined using the depletion technique. Values of the adsorbed amount at equilibrium were compared with the volume of adsorbed material as determined using in situ TMAFM, revealing the presence of significant amounts of hydration water in the adsorbed layer structure. This deduction was confirmed by comparing in and ex situ TMAFM images of the adsorbed dextrins. The effect of layer thickness, coverage, and hydration water content on the contact angle of talc particles treated with polymer was investigated using the Washburn method and the equilibrium capillary pressure (ECP) method. Distinct correlations were observed between adsorbed layer properties and the measured contact angles, with the ECP measurements especially highlighting the effect of the adsorbed polymer layer hydration water. The implications for the performance of the modified dextrins in flotation are discussed.

Dimensional analysis of two-phase flow including a rate-dependent capillary pressure–saturation relationship

The macroscopic modelling of two-phase flow processes in subsurface hydrosystems or industrial applications on the Darcy scale usually requires a constitutive relationship between capillary pressure and saturation, the P c( S w) relationship. Traditionally, it is assumed that a unique relation between P c and S w exists independently of the flow conditions as long as hysteretic effects can be neglected. Recently, this assumption has been questioned and alternative formulations have been suggested. For example, the extended P c( S w) relationship by Hassanizadeh and Gray [Hassanizadeh SM, Gray WG. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resources 1990;13(4):169–86] proposes that the difference between the phase pressures to the equilibrium capillary pressure is a linear function of the rate of change of saturation, thereby introducing a constant of proportionality, the coefficient τ. It is desirable to identify cases where the extended relationship needs to be considered. Consequently, a dimensional analysis is performed on the basis of the two-phase balance equations. In addition to the well-known capillary and gravitational number, the dimensional analysis yields a new dimensionless number. The dynamic number Dy quantifies the ratio of dynamic capillary to viscous forces. Relating the dynamic to the capillary as well as the gravitational number gives the new numbers DyC and DyG, respectively. For given sets of fluid and porous medium parameters, the dimensionless numbers Dy and DyC are interpreted as functions of the characteristic length and flow velocity. The simulation of an imbibition process provides insight into the interpretation of the characteristic length scale. The most promising choice for this length scale seems to be the front width. We conclude that consideration of the extended P c( S w) relationship may be important for porous media with high permeability, small entry pressure and high coefficient τ when systems with a small characteristic length (e.g. steep front) and small characteristic time scale are under investigation.

Capillary pressure in a porous medium with distinct pore surface and pore volume fractal dimensions

The relationship between capillary pressure and saturation in a porous medium often exhibits a power-law dependence. The physical basis for this relation has been substantiated by assuming that capillary pressure is directly related to the pore radius. When the pore space of a medium exhibits fractal structure this approach results in a power-law relation with an exponent of 3-D(v), where D(v) is the pore volume fractal dimension. However, larger values of the exponent than are realistically allowed by this result have long been known to occur. Using a thermodynamic formulation for equilibrium capillary pressure we show that the standard result is a special case of the more general exponent (3-D(v))(3-D(s)) where D(s) is the surface fractal dimension of the pores. The analysis reduces to the standard result when D(s)=2, indicating a Euclidean relationship between a pore's surface area and the volume it encloses, and allows for a larger value for the exponent than the standard result when D(s)>2 .

A theoretical model of the membrane electrode assembly of liquid feed direct methanol fuel cell with consideration of water and methanol crossover

An algebraic model of the membrane electrode assembly of the direct methanol fuel cell is developed, which considers the simultaneous liquid water and methanol crossover effects, and the associated electrochemical reactions. The respective anodic and cathodic polarization curves can be predicted using this model. Methanol concentration profile and flux are correlated explicitly with the operating conditions and water transport rate. The cathode mixed potential effect induced by the methanol crossover is included and the subsequent cell voltage loss is identified. Water crossover is influenced by the capillary pressure equilibrium and hydrophobic property within the cathode gas diffusion layer. The model can be used to evaluate the cell performance at various working parameters such as membrane thickness, methanol feed concentration, and hydrophobicity of the cathode gas diffuser.

Simplified thermodynamic model for equilibrium capillary pressure in a fractal porous medium

Defining a relation for equilibrium pressure in a porous medium has been difficult to do in terms of readily measurable parameters. We present a simplified analysis of this problem using the first law of thermodynamics combined with a fractal description of a porous system. The results show that the variation in fluid interfacial area with fluid volume, and the respective interfacial surface tensions, are dominant factors determining equilibrium capillary pressure. Departures from equilibrium are seen to occur when fluid-solid contact lines are in movement. By describing the pore space as fractal we are able to obtain an expression for the change in fluid interfacial area with respect to its volume, and the resulting model shows a strong fit to pressure data obtained from a capillary rise experiment conducted in a coarse-grained SiO2 sand.

Streaming potentials in two‐phase flow conditions

We report here, for the first time, the dependence of the streaming potential coupling coefficient of two consolidated rock samples (dolomite) in two‐phase flow conditions. We used two carbonate core samples characterized by image analysis and petrophysical measurements including porosity, formation factor, resistivity index at 1 kHz, and the critical water saturation determined from capillary pressure curves. In agreement with the results obtained recently by Guichet et al. [2003], who used an unconsolidated sand, we observed that the relative electrokinetic coupling coefficient scales approximately with the reduced water saturation at capillary pressure equilibrium. This observation is explained with a new model, which makes a clear distinction between Stern and Gouy‐Chapman layer contributions to surface electrical conductivity. The streaming potential coupling coefficient becomes dependent on surface conduction in the Gouy‐Chapman layer where the pore space is not fully saturated.

Investigation of foam and emulsion destruction under the great pressure gradients

The effect of high pressure gradients with the help of the membrane under which a decreased pressure is created and centrifugal forces on a decrease of the content of the dispersion medium in the emulsion and foam and increase in the capillary pressure was studied. In a centrifugal field at low angular velocity of rotation (ω=52.3 s−1) the equilibrium capillary pressure is established in the emulsion and foam. At higher angular velocity the emulsion and foam were collapsed as a certain pressure is reached in Plateau borders of the emulsion and foam; this pressure was smaller than the pressure balancing the forces of the centrifugal field. The capillary pressure in the highly dispersed emulsion reached pressure drop applied to the membrane (Pσ=PT=70 kPa). Emulsions from sodium dodecylsulfate and 9-10 oxyethylated alkylphenol (OP-7) with 0.1 and 0.4 mol/dm3 NaCl adding prepared by shaking with hexane and dodecane destructed at some capillary pressure which were less than the pressure drop supplied to the porous membrane.

Investigation of foam and emulsion destruction under the great pressure gradients

The effect of high pressure gradients with the help of the membrane under which a decreased pressure is created and centrifugal forces on a decrease of the content of the dispersion medium in the emulsion and foam and increase in the capillary pressure was studied. In a centrifugal field at low angular velocity of rotation (ω=52.3 s −1) the equilibrium capillary pressure is established in the emulsion and foam. At higher angular velocity the emulsion and foam were collapsed as a certain pressure is reached in Plateau borders of the emulsion and foam; this pressure was smaller than the pressure balancing the forces of the centrifugal field. The capillary pressure in the highly dispersed emulsion reached pressure drop applied to the membrane ( P σ= P T =70 kPa). Emulsions from sodium dodecylsulfate and 9-10 oxyethylated alkylphenol (OP-7) with 0.1 and 0.4 mol/dm 3 NaCl adding prepared by shaking with hexane and dodecane destructed at some capillary pressure which were less than the pressure drop supplied to the porous membrane.

Behavior of an Emulsion in a Centrifugal Field: Kinetics of an Increase in the Excess Pressure and Destruction of an Emulsion

The effect of centrifugal forces on a decrease in the content of the dispersion medium in an emulsion (increase in its expansion), decrease in the pressure in this medium, and increase in the capillary pressure was conductometrically studied, as well as the effect of these forces on the destruction of the emulsion as a whole. At low rotational speed of the centrifuge (< 500 rpm), the hydrostatically equilibrium capillary pressure and expansion are established in the emulsion. At higher angular velocities, the emulsion column is rapidly destroyed, as a certain “critical” pressure is reached in Plateau borders of the emulsion; this pressure is smaller than the pressure balancing the forces of the centrifugal field.

Capillary Pressure During Immiscible Displacement

Abstract Experiments performed on the immiscible displacement of heptane and mineral oil by water in capillary tubing showed that capillary pressure during drainage and imbibition were constant over a range of velocities from 10−4 to 10−2 m/s. The results obtained were quite different from the equilibrium capillary pressure. The drainage capillary pressure was lower than its equilibrium value by 25﹪ for heptane, and higher by 10﹪ for mineral oil. In the case of imbibition, the capillary pressure for both oils was lower than their respective equilibrium value. Due to the limitation of the apparatus, velocities lower than 10−4 m/s could not be measured. It is, therefore, not clear if capillary pressure during flow would, in fact, reduce to its equilibrium value at low enough velocities. While it is quite adequate to describe the capillary behaviour of two fluids inside a tubing by a single number, one needs to consider capillary pressure as a function of the wetting fluid saturation in the porous medium. The problem of considering the effect of flow on this function is going to be more complicated than the single capillary tubing case. The present result indicates that it is inconsistent to use capillary pressure determined from equilibrium conditions and apply it to flowing conditions. Introduction In the study of fluid flow in porous media, one starts with the material balance equation for each of the fluids. If they are immiscible, then one needs to consider the interaction between them across the interfaces. A common approach(1–4) is to define a capillary pressure to be equal to the difference in pressure of the two fluids in the same representative elemental volume. This pressure is also assumed to be dependent on the saturation of one of thefluids there. In terms of mathematics, the use of these assumptions completes the system of partial differential equations governing the flow of two fluids in porous media. In actual practice, one needs to provide a physically realistic capillary pressure saturation relation in order to make meaningful predictive calculations. Very often, capillary pressure is taken to be zero, as in the Buckley-Leverett solution for immiscible displacement(5), or Muskat's model of water coning(6). However, in some reservoir engineering applications, this may not be feasible. For example, when one needs to build a numerical model with more than one layer in the vertical direction, it is necessary to input a non-zero capillary pressure curve to initialize the fluids in the reservoir. A realistic initialization would be to use the drainage portion of the capillary pressure saturation relation. In most commercially available simulators, the same curve is being used for the subsequent displacement calculations. As the drainage curve is determined while the fluids are at equilibrium, using it for flow calculation implies that capillary pressure is independent of the dynamics. We are not aware of any theoretical justification given in the literature for using this assumption. In fact, there was no discussion given on the physical requirements of the capillary pressure saturation relation given in some of the standard texts on numerical reservoir simulation(7, 8).