Capillary pressure for the sand–CO2–water system under various pressure conditions. Application to CO2 sequestration

Accurate modeling of storage of carbon dioxide (CO₂) in heterogeneous aquifers requires experiments of the capillary pressure as function of temperature and pressure. We present a method with which static drainage and imbibition capillary pressures can be measured continuously as a function of saturation at various temperature (T) and pressure (P) conditions. The measurements are carried out at (T, P) conditions of practical interest. Static conditions can be assumed as small injection rates are applied. The capillary pressure curves are obtained for the unconsolidated sand–distilled water–CO₂ system. The experimental results show a decrease of drainage and imbibition capillary pressure for increasing CO₂ pressures and pronounced dissolution rate effects for gaseous CO₂. Significant capillary pressure fluctuations and negative values during imbibition are observed at near critical conditions. The measurement procedure is validated by a numerical model that simulates the experiments.     

Scale dependent dynamic capillary pressure effect for two-phase flow in porous media

Causes and effects of non-uniqueness in capillary pressure and saturation (P^c–S) relationship in porous media are of considerable concern to researchers of two-phase flow. In particular, a significant amounts of discussion have been generated regarding a parameter termed as dynamic coefficient (τ) which has been proposed for inclusion in the functional dependence of P^c–S relationship to quantify dynamic P^c and its relation with time derivative of saturation. While the dependence of the coefficient on fluid and porous media properties is less controversial, its relation to domain scale appears to be dependent on artefacts of experiments, mathematical models and the intra-domain averaging techniques. In an attempt to establish the reality of the scale dependency of the τ–S relationships, we carry out a series of well-defined laboratory experiments to determine τ–S relationships using three different sizes of cylindrical porous domains of silica sand. In this paper, we present our findings on the scale dependence of τ and its relation to high viscosity ratio (μ_r) silicone oil–water system, where μ_r is defined as the viscosity of non-wetting phase over that of the wetting phase. An order of magnitude increase in the value of τ was observed across various μ_r and domain scales. Also, an order of magnitude increase in τ is observed when τ at the top and the bottom sections in a domain are compared. Viscosity ratio and domain scales are found to have similar effects on the trend in τ–S relationship. We carry out a dimensional analysis of τ which shows how different variables, e.g., dimensionless τ and dimensionless domain volume (scale), may be correlated and provides a means to determine the influences of relevant variables on τ. A scaling relationship for τ was derived from the dimensionless analysis which was then validated against independent literature data. This showed that the τ–S relationships obtained from the literature and the scaling relationship match reasonably well.     

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.     

Prediction of capillary air-liquid interfacial area vs. saturation function from relationship between capillary pressure and water saturation

This short communication investigates if the capillary air-liquid interfacial area vs. saturation relationship A_lv(S) can be predicted from the capillary pressure vs. saturation relationship S(h), using the theoretical sample scale model of Diamantopoulos and Durner (2013, 2015). We selected three published experimental datasets, where S(h) and A_lv(S) relationships had been measured for the same porous media. The sample scale model was fitted to the retention curve S(h) of each porous medium and then used to predict the air-liquid interfacial area A_lv(S). We also included in the analysis the thermodynamic models of Leverett (1941) and Grant and Gerhard (2007). For two sandy materials and especially for high saturation values, the model predicted the capillary A_lv(S) successfully, which was in one case given by a pore-network model simulation (Kibbey and Chen, 2012) and in the other case experimentally determined (Brusseau et al., 2006). For glass bead experiments, the contact angle needed to be fitted to properly describe the experimental A_lv(S) curve.     

Theoretical estimation of free and entrapped nonwetting–wetting fluid interfacial areas in porous media

Fluid–fluid interfacial areas play important roles in numerous subsurface processes such as dissolution, volatilization, and adsorption. Integral expressions have been derived to estimate both entrapped (discontinuous) and free (continuous) nonwetting fluid–wetting fluid specific interfacial areas in porous media. The expressions, compatible with widely used capillary head-saturation and entrapment models, require information on capillary head-saturation relation parameters, porosity, and fluid-pair interfacial tension. In addition, information on the maximum entrapped nonwetting fluid saturation as well as the main drainage branch reversal point for water and total liquid saturations is necessary to estimate entrapped fluid interfacial areas. Implementation of the interfacial area equations in continuum-based multifluid flow simulators is straightforward since no additional parameters are needed than those required by the simulators to complete the multifluid flow computations. A limited sensitivity analysis, based on experimentally obtained parameter values, showed that imposed variations resulted in logical and consistent changes in predicted specific interfacial areas for both entrapped and free nonwetting fluid–wetting fluid systems. A direct comparison with published experimental work to test the derived expressions was limited to free air–water systems and yielded reasonable results. Such comparisons are often not possible because of the lack of information given on retention parameters, and variables used to determine nonwetting fluid entrapment. This contribution is dedicated to John W. Cary.     

Crossover from capillary fingering to compact invasion for two-phase drainage with stable viscosity ratios

Motivated by a wide range of applications from enhanced oil recovery to carbon dioxide sequestration, we have developed a two-dimensional, pore-level model of immiscible drainage, incorporating viscous, capillary, and gravitational effects. This model has been validated quantitatively, in the very different limits of zero viscosity ratio and zero capillary number; flow patterns from modeling agree well with experiment. For a range of stable viscosity ratios (μ_injected/μ_displaced ? 1), we have increased the capillary number, N_c, and studied the way in which the flows deviate from capillary fingering (the fractal flow of invasion percolation) and become compact for realistic capillary numbers. Results exhibiting this crossover from capillary fingering to compact invasion are presented for the average position of the injected fluid, the fluid–fluid interface, the saturation and fractional flow profiles, and the relative permeabilities. The agreement between our results and earlier theoretical predictions [Blunt M, King MJ, Scher H. Simulation and theory of two-phase flow in porous media. Phys Rev A 1992;46:7680–99; Lenormand R. Flow through porous media: limits of fractal patterns. Proc Roy Soc A 1989;423:159–68; Wilkinson D. Percolation effects in immiscible displacement. Phys Rev A 1986;34:1380–90; Xu B, Yortsos YC, Salin D. Invasion Percolation with viscous forces. Phys Rev E 1998;57:739–51] supports the validity of these general theoretical arguments, which were independent of the details of the porous media in both two and three dimensions.     

Shear viscosity of rhyolite-vapor emulsions at magmatic temperatures by concentric cylinder rheometry

The viscosity of natural rhyolitic melt from Lipari, Aeolian Islands and melt-bubble emulsions (30–50 vol% porosity) generated from Lipari rhyolite have been measured in a concentric cylinder rheometer at temperatures and shear rates in the range 925–1150°C and 10⁻³–10^−1.2 s⁻¹, respectively, in order to better understand the dependence of emulsion shear viscosity on temperature and shear rate in natural systems. Bubble-free melt exhibits Newtonian–Arrhenian behavior in the temperature range 950–1150°C with an activation energy of 395±30 kJ/mol; the shear viscosity is given by log η_m=−8.320+20624/T. Suspensions were prepared from natural rhyolite glass to which small amounts of Na₂SO₄ were added as a ‘foaming agent’. Reasonably homogeneous magmatic mixtures with an approximate log-normal distribution of bubbles were generated by this technique. Suspension viscosity varied from 10^6.1 to 10^8.37 Pa s and systematically correlates with temperature and porosity in the shear stress range (10^4.26–10^5.46 Pa) of the experiments. The viscosity of melt-bubble emulsions is described in terms of the relative viscosity, η_r=η_e/η_m where η_e is the emulsion viscosity and η_m is the viscosity of melt of the same composition and temperature. The dependence of relative viscosity on porosity for magmatic emulsions depends on the magnitude of the capillary number Ca≡G/(σr_b⁻¹η_m⁻¹), the ratio of viscous forces acting to deform bubbles to interfacial forces resisting bubble deformation. For inviscid bubbles in magmatic flows three regimes may be identified. For Ca<0.1, bubbles are nearly spherical and relative viscosity is an increasing function of porosity. For dilute systems, η_r=1+φ given by the classical result of Taylor [Proc. R. Soc. London A 138 (1932) 41–48]. For Ca in the range 0.1<Ca<10, emulsions behave as power law fluids and the relative viscosity depends on shear rate (or Ca) as well as porosity. At high Ca (Ca>10) an asymptotic regime is reached in which relative viscosity decreases with increasing porosity and is independent of Ca. Our experiments were carried out for 30<Ca<925 in order to quantify the maximal effect of bubbles in reducing the viscosity of magmatic emulsions relative to single-phase melt at identical conditions of shear rate and temperature. The viscosity of a 50 vol% emulsion is a factor of five smaller than that of melt alone. Rheometric measurements obtained in this study are useful in constraining models of magma transport and volcanic eruption mechanics relevant to transport of volatile-saturated magma in the crust and upper mantle.     

Effects of fractional wettability on capillary pressure–saturation–relative permeability relations of two-fluid systems

Capillary pressure (P_c)–saturation (S)–relative permeability (k_r) relationships must be quantified to accurately predict non-aqueous phase liquid (NAPL) distribution in the subsurface. Several experimental techniques are presented here for two-fluid P_c–S–k_r relationships for various saturation paths to better define the effect of fractional wettability on these relationships. During the primary drainage path of the P_c–S curves, the air–water system showed no distinct trend as a function of the fraction of sand treated by organosilane (S) to render it non-water wetting. In a NAPL–water system, however, a consistent decrease of capillary pressure with increase of the fraction of non-water wetting sands was observed. The much lower contact angle for air–water (a–w) system may result in the observed insensitivity of the a–w P_c–S curves to fractional wettability, at least for the PD pathway. For the main imbibition path of NAPL–water system, capillary pressure decreased as the fraction of the S component increased, requiring forced imbibition (negative capillary pressures) for a certain range of saturations. Systems with an increasing percentage of the S component also exhibited a higher water k_r and lower NAPL or air k_r at a given saturation for the primary drainage and main imbibition paths in both air–water and NAPL–water systems. The increase of water k_r with increase of the fraction of the S component can be explained by the ability of water to occupy larger and highly conductive pores in such a system. Experimental k_r–S data for the primary drainage path of NAPL–water system presented here were used to test the Bradford et al. [Bradford SA, Abriola LM, Leij FJ. Wettability effects on two- and three-fluid relative permeabilities. J Contam Hydrol 1997;28:171–91] model and the modified Mualem model for estimating the k_r–S curves from measured P_c–S data as a function of fractional wettability. Both models predicted significantly less variation in the k_r–S curves than measured indicating that they did not adequately represent the system under investigation.     

Co-sequestration of SO2 with supercritical CO2 in carbonates: An experimental study of capillary trapping,relative permeability,and capillary pressure

In this study we performed three categories of steady- and unsteady-state core-flooding experiments to investigate capillary trapping, relative permeability, and capillary pressure, in a scCO₂ + SO₂/brine/limestone system at elevated temperature and pressure conditions, i.e., 60 °C and 19.16 MPa. We used a Madison limestone core sample acquired from the Rock Springs Uplift in southwest Wyoming. We carried out two sets of steady-state drainage-imbibition relative permeability experiments with different initial brine saturations to study hysteresis. We found that the final scCO₂ + SO₂ drainage relative permeability was very low, i.e., 0.04. We also observed a rapid reduction in the scCO₂-rich phase imbibition relative permeability curve, which resulted in a high residual trapping. The results showed that between 62.8% and more than 76% of the initial scCO₂ + SO₂ at the end of drainage was trapped by capillary trapping mechanism (trapping efficiency). We found that at higher initial brine saturations, the trapping efficiency was higher. The maximum initial and residual scCO₂-rich phase saturations at the end of primary drainage and imbibition were 0.525 and 0.329, respectively. Each drainage-imbibition cycle was followed by a dissolution process to re-establish S_w = 1. The dissolution brine relative permeabilities for both cycles were also obtained. We characterized the scCO₂ + SO₂/brine capillary pressure hysteresis behavior through unsteady-state primary drainage, imbibition, and secondary drainage experiments. We observed negative imbibition capillary pressure curve indicative of possible wettability alteration throughout the experiments due to contact with scCO₂ + SO₂/brine fluid system. The trapping results were compared to those reported in literature for other carbonate core samples. We noticed slightly more residual trapping in our sample, which might be attributed to heterogeneity, different viscosity ratio, and pore-space topologies. The impact of dynamic effects, i.e., high brine flow rate imbibition tests, on trapping of the scCO₂-rich phase was also explored. We performed two imbibition experiments with relatively high brine flow rates. The residual scCO₂ saturation dropped to 0.291 and 0.262 at the end of the first and second imbibition tests, i.e., 11.5% and 20.4%, respectively, compared to 0.329 under capillary-dominated regime.     

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.     

Modeling the velocity field during Haines jumps in porous media

When nonwetting fluid displaces wetting fluid in a porous rock many rapid pore-scale displacement events occur. These events are often referred to as Haines jumps and any drainage process in porous media is an ensemble of such events. However, the relevance of Haines jumps for larger scale models is often questioned. A common counter argument is that the high fluid velocities caused by a Haines jump would average-out when a bulk representative volume is considered. In this work, we examine this counter argument in detail and investigate the transient dynamics that occur during a Haines jump. In order to obtain fluid–fluid displacement data in a porous geometry, we use a micromodel system equipped with a high speed camera and couple the results to a pore-scale modeling tool called the Direct HydroDynamic (DHD) simulator. We measure the duration of a Haines jump and the distance over which fluid velocities are influenced because this sets characteristic time and length scales for fluid–fluid displacement. The simulation results are validated against experimental data and then used to explore the influence of interfacial tension and nonwetting phase viscosity on the speed of a Haines jump. We find that the speed decreases with increasing nonwetting phase viscosity or decreasing interfacial tension; however, for the same capillary number the reduction in speed can differ by an order of magnitude or more depending on whether viscosity is increased or interfacial tension is reduced. Therefore, the results suggest that capillary number alone cannot explain pore-scale displacement. One reason for this is that the interfacial and viscous forces associated with fluid–fluid displacement act over different length scales, which are not accounted for in the pore-scale definition of capillary number. We also find by analyzing different pore morphologies that the characteristic time scale of a Haines jump is dependent on the spatial configuration of fluid prior to an event. Simulation results are then used to measure the velocity field surrounding a Haines jump and thus, measure the zone of influence, which extends over a distance greater than a single pore. Overall, we find that the time and length scales of a Haines jump are inversely proportional, which is important to consider when calculating the spatial and temporal averages of pore-scale parameters during fluid–fluid displacement.     

Regional coda magnitudes of underground nuclear explosions

The concept of determining magnitudes, M_τ, of regional events (Δ < 1000 km) by means of coda-duration measurements is re-examined by using short-period vertical-component seismograms from Nevada Test Site explosions. The duration is specified as the time interval between the expected arrival of the S-wave and the time when coda waves fall and stay below 80 μm peak-to-peak ground displacement. The suggested procedure requires that for M_τ = 4.0 the coda duration at a distance of 100 km is 120 s. The adaptivity of the method is examined in terms of the single-station magnitude scatter, and with respect to the potential accuracy of the yield estimation of the explosions.The derived magnitude formula for underground nuclear explosions in granite is of the form: M_τ =0.18+0.001Δ+1.79 log τ+C_s, where C_s is a station correction coefficient for non-WWNSS stations.     

Dynamics of a spherical viscoelastic shell: Implications to a criterion for fragmentation/expansion of bubbly magma

This paper investigates dynamics of a spherical bubble surrounded by a viscoelastic fluid. The purpose of the study is to understand the parameters which control expansion and fragmentation of bubbly magma by decompression. In particular, we focus on which occurs first, fragmentation or expansion. Supposing that rupture of the bubble wall occurs in a critical stress condition, we calculate the change of the bubble radius and tensile stress at the bubble wall for various decompression rates. Conditions in which tensile stress is stored in the shell are represented in terms of dimensionless parameters. The results are interpreted as follows: when magma viscosity is larger than a critical value, and the decompression time is shorter than viscous expansion time, tensile stress is stored before expansion; when magma viscosity is smaller than the critical value, tensile stress is not stored, no matter how rapid the decompression. Although it is a generally accepted theory that fragmentation is effected by stress conditions and decompression time, exactly how decompression time (t₁) effects the fragmentation is not yet fully understood. This study demonstrates that the stress condition is controlled by the length of the decompression time not relative to the viscoelastic relaxation time (t₁ / τ), but relative to the viscous expansion time (t₁ / τ^l_rlx). As suggested by recent experimental studies, the decompression time relative to viscoelastic relaxation time (t₁ / τ) is also significant to the fragmentation process itself. It indicates that the decompression time effects the fragmentation not through the stress condition. However more work must be completed to fully understand the particular relationship between the decompression time and relaxation time in terms of its influence on fragmentation.     

The distribution of 210Pb and 210Po in the surface waters of the Pacific Ocean

By modelling the observed distribution of²¹⁰Pb and²¹⁰Po in surface waters of the Pacific, residence times relative to particulate removal are determined. For the center of the North Pacific gyre these are τ_Po = 0.6years andτ_Pb = 1.7years. The surface ocean τ_Pb is determined by particulate transport rather than plankton settling. The fact that it is about two orders of magnitude smaller than τ_Pb for the deep ocean implies a sharp change in the adsorptive quality of particles during descent through the water column.     

Continuum percolation theory for pressure–saturation characteristics of fractal soils: extension to non-equilibrium

Systematic experimental deviations from theoretical predictions derived for water retention characteristics of fractal porous media have previously been interpreted in terms of continuum percolation theory (at low moisture contents, below the critical volume fraction of water, α_c capillary flow ceases). In other work, continuum percolation theory was applied to find the hydraulic conductivity as a function of saturation for saturations high enough to guarantee percolation of capillary flow. Now these two problems are further linked, using percolation theory to estimate non-equilibrium water retention at matric potential values such that the equilibrium water content is too low for percolation of capillary flow paths. In particular, a procedure for developing a time-dependent moisture content is developed for experimental time scales long enough that film flow can provide an alternate mechanism for equilibrating when continuous capillary flow is not possible. The time scales are defined in terms of moisture-dependent length scales and film flow and capillary flow hydraulic conductivities. Imbibition is treated in the extreme case of no film-flow contribution to equilibration. In another application at higher matric potentials, recursive relations are derived for the water content of porous media during drying when external pressures are changed at rates too rapid for equilibrium to be attained by capillary flow.     

Comparison of relative permeability–fluid saturation–capillary pressure relations in the modelling of non-aqueous phase liquid infiltration in variably saturated,layered media

The characteristic functions relating relative permeabilities and capillary pressures to fluid saturations (k_r–S–P_c models) are of great importance for the modelling of sub-surface multi-phase flow and transport. In order to test their performance and to identify their important parameters, four well-known three-phase k_r–S–P_c models have been tested against published experimental data on non-aqueous phase liquid (NAPL) migration in the unsaturated zone. Both homogenous systems and systems with embedded heterogeneities have been analysed.     

Pore-scale imbibition experiments in dry and prewetted porous media

Two-phase imbibition behavior of immiscible fluids was studied in dry and prewetted porous media using a laser-induced fluorescence technique. Imbibition was first investigated in two-dimensional (2-D) systems under conditions comparable to those for a study of drainage [Ovdat H, Berkowitz B. Pore-scale study of drainage displacement under combined capillary and gravity effects in index-matched porous media. Water Resources Research 2006;42:W06411. doi:10.1029/2005WR004553] in the capillary-dominated regime. The effect of initial wetting saturation (IWS) was then explored in 2-D and 3-D porous media under the combined effect of gravity, capillary and viscous forces, within and outside the capillary-dominated regime. Parameters that describe maximum vertical advance, volumetric fraction, total surface area and specific surface area of the invading fluid were used to quantify the behavior. Comparison of 2-D drainage and imbibition patterns demonstrates significant qualitative differences under analogous viscosity ratio, buoyancy number, and capillary number values. However, quantitative analyses show strong pore-scale similarities between these patterns. Invasion structures in 3-D, prewetted (IWS ≈ 8% of the pore volume) porous media are ramified, with lateral branching and regions containing trapped residual fluid. These structures are qualitatively and quantitatively different from the compact, branchless structures that develop in dry (IWS = 0) porous media.     

Comparison of CO2 trapping in highly heterogeneous reservoirs with Brooks-Corey and van Genuchten type capillary pressure curves

Geological heterogeneities affect the dynamics of carbon dioxide (CO₂) plumes in subsurface environments in important ways. Previously, we showed how the dynamics of CO₂ plumes are influenced by the multiscaled sedimentary architecture in deep brine fluvial-type reservoirs. The results confirm that representing small-scale features and the corresponding heterogeneity in saturation functions, along with hysteresis in saturation functions, are all critical to understanding capillary trapping processes. Here, we show that when heterogeneity and hysteresis are represented, the two conventional approaches for defining saturation functions, Brooks-Corey and van Genuchten, represent fundamentally different physical systems. The Brooks-Corey approach represents heterogeneity in entry pressures, and leads to trapping by capillary pinning. The van Genuchten approach represents a network of pores transporting the nonwetting fluid, across rock types, with negligible capillary entry pressure, and leads to capillary retardation. These differences significantly affect the large-scale characteristics of CO₂ plumes (i.e., their mass, shape, and position).     

Latitudinal dependence of the seasonal variation of particulate extinction in the UTLS over the Indian longitude sector during volcanically quiescent period based on lidar and SAGE-II observations

The altitude profiles of particulate extinction in the upper troposphere and lower stratosphere (UTLS) obtained from SAGE-II in the latitude region 0–30°N over the Indian longitude sector (70–90°E) are used to study the latitudinal variation of its annual pattern in this region during the volcanically quiescent period of 1998–2003. The SAGE-II data is compared with the lidar measurements from Gadanki (13.5°N, 79.2°E) when the satellite had an overhead occultation pass over a small geographical grid centered at this location. The particulate optical depth (τ_p) in the UT region shows a general decrease with increase in latitude and a pronounced summer–winter contrast with relatively low values during winter and high values during summer. In general, these variations are in accordance with the latitudinal variation of convective available potential energy (CAPE) and thunderstorm activity, which are good representative indices of tropospheric convection. While the particulate extinction (and τ_p) in the 18–21 km (LS₁) region is relatively low in the equatorial region up to 15°N, it shows an increase in the off-equatorial region, beyond 15°N. While the annual variation of τ_p in the LS₁ region is almost insignificant near the equator, it is rather well pronounced in latitude region between 10 and 15°N with relatively high values during winter and low values during summer. Beyond 20°N, this shows a prominent peak during summer. At a higher altitude, the 21–30 km (LS₂) region, the latitude variation of τ_p shows a different pattern with high values near the equator and low values in the off-equatorial region confirming the existence of a stratospheric aerosol reservoir. Low values of τ_p at lower regime (LS₁) near the equator could be due to rapid transport of particulates from the near equatorial region to higher latitudes, while the equatorial high at upper regime (LS₂) could be due to lofting and subsequent accumulation.     

Transient and steady-state relative permeabilities from two-phase flow experiments in planar pore networks

The water k_rw and oil k_ro relative permeability curves of a glass-etched planar pore network are estimated with history matching from transient displacement experiments performed at varying values of the capillary number, Ca  , for two fluid systems: one of intermediate and one of strong wettability. The transient _krw,_kro$k_{\text{rw}}\text{,}{k}_{\text{ro}}$ are compared to corresponding ones measured with the steady-state method on the same porous medium [Avraam DG, Payatakes AC. Flow regimes and relative permeabilities during steady-state two-phase flow in porous media. J Fluid Mech 1995;293:207–36; Avraam DG, Payatakes AC. Generalized relative permeability coefficients during steady-state two-phase flow in porous media and correlation with the flow mechanisms. Transport Porous Med 1995;20:135–68; Avraam DG, Payatakes AC. Flow mechanisms, relative permeabilities, and coupling effects in steady-state two-phase flow through porous media. The case of strong wettability. Ind Eng Chem Res 1999;38:778–86.], and potential differences from them are interpreted in the light of the differences between the transient growth pattern, and the steady-state two-phase flow regime. For intermediate wettability, the transient k_ro and k_rw exceed the corresponding steady-state functions at low Ca values and have the tendency to become smaller than the steady-state ones at high Ca values. For strong wettability, the transient k_ro and k_rw are increasing functions of Ca, the transient k_ro is higher than the steady-state one, whereas the transient k_rw decreases substantially and becomes lower than the steady-state one at low Ca values. The dynamic capillary pressure estimated from transient experiments is a decreasing function of Ca in agreement with previous theoretical and experimental studies.