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.     

Analytical solutions to steady state unsaturated flow in layered,randomly heterogeneous soils via Kirchhoff transformation

In this study, we derive analytical solutions of the first two moments (mean and variance) of pressure head for one-dimensional steady state unsaturated flow in a randomly heterogeneous layered soil column under random boundary conditions. We first linearize the steady state unsaturated flow equations by Kirchhoff transformation and solve the moments of the transformed variable up to second order in terms of σ_Y and σ_β, the standard deviations of log hydraulic conductivity Y=ln(K_s) and of the log pore size distribution parameter β=ln(α). In addition, we also give solutions for the mean and variance of the unsaturated hydraulic conductivity. The analytical solutions of moment equations are validated via Monte Carlo simulations.     

Propagation of seismic waves through liquefied soils

To predict the earthquake response of saturated porous media it is essential to correctly simulate the generation, redistribution, and dissipation of excess pore water pressure during and after earthquake shaking. To this end, a reliable numerical tool requires a dynamic, fully coupled formulation for solid–fluid interaction and a versatile constitutive model. Presented in this paper is a 3D finite element framework that has been developed and utilized for this purpose. The framework employs fully coupled dynamic field equations with a u–p–U formulation for simulation of pore fluid and solid skeleton interaction and a SANISAND constitutive model for response of solid skeleton. After a detailed verification and validation of the formulation and implementation of the developed numerical tool, it is employed in the seismic response of saturated porous media. The study includes examination of the mechanism of propagation of the earthquake-induced shear waves and liquefaction phenomenon in uniform and layered profiles of saturated sand deposits.     

Field scale impacts of spatially correlated relative permeability in heterogeneous multiphase systems

Two-dimensional numerical simulations of two-phase (DNAPL-water) flow in spatially correlated random fields demonstrate the influence of nonwetting phase (NWP) relative permeability–saturation (k_r,N–S_W) relationships correlated to porous media intrinsic permeability (k). Both the volume of porous media invaded by the NWP and the length of time during which the NWP is migrating are under predicted if k_r,N–k correlation is not accounted for in the model formulation. Not accounting for the k_r,N–k correlation resulted in under predicting the volume of porous media invaded by up to approximately 10%, which is likely not significant for many practical applications. However, not accounting for the k_r,N–k correlation resulted in under predicting field scale migration times by up to a factor of 4, which is likely significant in that the migration times are on the order of years to several decades for the DNAPL (1,2-DCE) considered in this study. The under prediction of migration times was greater for lower permeability aquifers.     

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.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved     

Wave-induced dynamic response of saturated multi-layer porous media: Analytical solutions and validity regions of various formulations in non-dimensional parametric space

In this paper, dynamic response of saturated-layered porous media under harmonic waves is evaluated through a semi-analytical solution. The coupled differential equations governing the dynamics of saturated or nearly saturated porous media such as soils containing all the inertial terms of solid and fluid phases are presented for a multi-layer system. Possible simplifications of the equations which are called formulations are introduced based upon the presence of inertial terms associated with the phases. The semi-analytical solutions to the response of multiple layers for all the formulations are presented in terms of pore water pressure and stress variations considering a set of non-dimensional parameters and their respective ratios. Validity of the formulations is presented in a non-dimensional parametric space. The maximum discrepancies in the pore pressure response of the formulations leading to validity regions are illustrated for typical dynamic problems. Subsequently, the effects of layering and drainage conditions on these regions are also presented. The proposed semi-analytical solution may be served as a benchmark one for validating the coupled numerical solutions, which can be used to deal with real scientific and geo-engineering problems in the emerging field of computational geomechanics.     

Inverse modelling techniques for determining hydraulic properties of coarse-textured porous media by transient outflow methods

A series of multi-step outflow experiments was carried out to identify the unsaturated hydraulic properties of two homogeneous coarse-textured porous media (glass beads and sand). Because of the measured sharp fronts of water content decrease during these experiments the hydraulic functions are assumed to be represented by the complete van Genuchten–Mualem closed-form expressions with variable coefficients α, n, m and θ_r. The values of θ_s and K_s were measured directly. A sensitivity analysis with respect to α, n, and m shows that conditions of local identifiability are satisfied if measurements of water content at some inner points inside the column are considered. The inverse modelling technique consists of two steps: first, computation of objective function values based on water content data responses to obtain initial parameter estimations, and second, a more detailed parameter determination using a Levenberg–Marquardt scheme. In both steps a numerical model incorporating the hydraulic functions is utilized to simulate theoretical pressure head and water content distributions along the column. For both porous media unique solutions of the inverse problem could be obtained, and afterwards, the corresponding hydraulic functions were verified from additional drainage experiments.©1998 Elsevier Science Limited. All rights reserved     

Analysis of interfacial error in saturated–unsaturated flow models

Interfacial error results from estimation of interblock conductivities related to the saturated–unsaturated interface. Both interfacial conductivity error (IE_K) and interfacial pressure error (IE_h) were analyzed under the arithmetic mean scheme while IE_K was numerically investigated under the averaging schemes arithmetic, geometric and harmonic. IE_K, dependent on the media pore size, is regularly less than zero while IE_h, associated with the height of capillary fringe, may be greater than zero. An interfacial discretization technique was developed to add two complementary equations into the saturated–unsaturated model with respect to the interface. The proposed interfacial approach may eliminate interfacial error from the approximations of interblock conductivities. Underestimation of the water-table response to infiltration is related to the negative IE_K. The water-table response error reaches −5.13% in our investigation, which is an accumulated result from IE_K.     

Hydrodynamics of coupled flow above and below a sediment–water interface with triangular bedforms

The hydrodynamics of a system where there is a coupled flow above and below a sediment–water interface (SWI) are not completely understood. We numerically simulate mean two-dimensional, unidirectional, steady, viscous flow in these systems using a sequentially coupled formulation. Simulations were conducted to determine fundamental relationships between bedform geometry, Reynolds number for the water-column flow (Re), interfacial exchange zone depth (d_z) in the sediments, and flux through the SWI (q_int); the latter two parameters play a significant role in biogeochemical and aquatic-life processes across the SWI. d_z and Re are functionally related through an asymptotic growth-curve model while q_int and Re follow a power function. These relationships are dynamically explained by the manner in which pressure gradients along the SWI develop due to current–bedform interactions at different Res and by Darcy’s Law. We found that the coupling between water column and exchange zone flow is controlled by the behavior of the water-column eddy. The eddy detaches at or near the point of minimum pressure along the interface, and reattaches near the point of maximum pressure. These two critical points determine the pressure gradient along the bed surface that controls the exchange zone flow field. Moreover, the reattachment point corresponds to flow divides within the sediments. Lastly, pore-water velocities drop with depth below the SWI, and are larger below the bedform crests than below the troughs.     

Low-frequency dilatational wave propagation through fully-saturated poroelastic media

The strong coupling of applied stress and pore fluid pressure, known as poroelasticity, is relevant to a number of applied problems arising in hydrogeology and reservoir engineering. The standard theory of poroelastic behavior in a homogeneous, isotropic, elastic porous medium saturated by a viscous, compressible fluid is due to Biot, who derived a pair of coupled partial differential equations that accurately predict the existence of two independent dilatational (compressional) wave motions, corresponding to in-phase and out-of-phase displacements of the solid and fluid phases, respectively. The Biot equations can be decoupled exactly after Fourier transformation to the frequency domain, but the resulting pair of Helmholtz equations cannot be converted to partial differential equations in the time domain and, therefore, closed-form analytical solutions of these equations in space and time variables cannot be obtained. In this paper we show that the decoupled Helmholtz equations can in fact be transformed to two independent partial differential equations in the time domain if the wave excitation frequency is very small as compared to a critical frequency equal to the kinematic viscosity of the pore fluid divided by the permeability of the porous medium. The partial differential equations found are a propagating wave equation and a dissipative wave equation, for which closed-form solutions are known under a variety of initial and boundary conditions. Numerical calculations indicate that the magnitude of the critical frequency for representative sedimentary materials containing either water or a nonaqueous phase liquid is in the kHz–MHz range, which is generally above the seismic band of frequencies. Therefore, the two partial differential equations obtained should be accurate for modeling elastic wave phenomena in fluid-saturated porous media under typical low-frequency conditions applicable to hydrogeological problems.     

Intra-seasonal and inter-annual variations in phytoplankton biomass,primary production and bacterial production at North West Cape,Western Australia: Links to the 1997–1998 El Niño event

Phytoplankton biomass, community and size structure, primary production and bacterial production were measured at shelf and continental slope sites near North West Cape, Western Australia (20.5°S–22.5°S) over two summers (October–February 1997–1998 and 1998–1999), and in April 2002. The North West Cape region is characterized by upwelling-favorable, southwesterly winds throughout the summer. Surface outcropping of upwelled water is suppressed by the geostrophic pressure gradients and warm low-density surface waters of the southward flowing Leeuwin Current. Strong El Niño (ENSO) conditions (SOI <0) prevailed through the summer of 1997–1998 which resulted in lower sea levels along the northwestern Australian coast and a weaker Leeuwin Current. La Niña conditions prevailed during the 1998–1999 summer and in April 2002. During the summer of 1997–1998, the North West Cape region was characterized by a shallower thermocline (nutricline), resulting in larger euphotic zone stocks of inorganic nitrogen and silicate over the continental slope. There was evidence for episodic intrusions of upper thermocline waters and the sub-surface chlorophyll maximum onto the outer continental shelf in 1997–1998, but not in 1998–1999. Pronounced differences in phytoplankton biomass, community size structure and productivity were observed between the summers of 1997–1998 and 1998–1999 despite general similarities in irradiance, temperature and wind stress. Phytoplankton primary production and bacterial production were 2- to 4-fold higher during the summer of 1997–1998 than in 1998–1999, while total phytoplankton standing crop increased by<2-fold. Larger phytoplankton (chiefly diatoms in the >10 μm size fraction) made significant contributions to phytoplankton standing crop and primary production during the summer of 1997–1998, but not 1998–1999. Although there were no surface signs of upwelling, primary production rates near North West Cape episodically reached levels (3–8 g C m⁻² day⁻¹) characteristic of eastern boundary Ekman upwelling zones elsewhere in the world. Bacterial production (0.006–1.2 g C m⁻² day⁻¹) ranged between 0.6 and 145 percent (median=19 percent) of concurrent primary production. The observed differences between years and within individual summers suggest that variations in the Leeuwin Current driven by seasonal or ENSO-related changes in the Indonesian throughflow region may have episodic, but significant influences on pelagic productivity along the western margin of Australia.     

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.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

On the effective hydraulic conductivity in mean vertical unsaturated steady flows

Water flow in partially saturated heterogeneous porous formations is modelled by regarding the hydraulic parameters as stationary random space functions (RSFs). As a consequence, the flow variables are also RSFs, and we aim to develop a procedure to derive the effective hydraulic conductivity (EHC). The methodology relies on a perturbation approach which regards the variances of the hydraulic parameters as small quantities. By using the Gardner’s [Gardner WR. Some steady state solutions of unsaturated moisture flow equations with application to evaporation from a water table. Soil Sci 1958;85:228–32] two-parameters (K_s, α) model for the local unsaturated conductivity, we obtain the EHC for any dimensionality d of the flow domain, and arbitrary correlation functions of the input RSFs. Unlike previous studies [e.g. Yeh T-CJ, Gelhar J, Gutjahr A. Stochastic analysis of unsaturated flow in heterogenous soils. 1. Statistically isotropic media. Water Resour Res 1985;21;447–56, Yeh T-CJ, Gelhar J, Gutjahr A. Stochastic analysis of unsaturated flow in heterogenous soils. 2. Statistically anisotropic media with variable α. Water Resour Res 1985:21:457–64], the EHC is represented here as product between the local scale conductivity valid for a domain of mean parameters, and a correction function κ¹ which depends on the medium heterogeneity structure and the mean pressure head.Generally, the correction function κ¹ is expressed by d-fold quadrature. These quadratures are further reduced after adopting specific (i.e. exponential and Gaussian) structure for the (cross) correlation functions involved in the computation of κ¹. We have also focused on some particular formation structures which are relevant for the applications, and permit simplification of the computational aspect, as well.We investigate effects of the heterogeneity formation properties as well as the mean head on the structure of κ¹. Overall, results suggest that, given the formation statistics, the impact of the heterogeneity upon κ¹ is enhanced as the medium becomes drier. This is particularly so when the variability of the fluctuation of Y = ln K_s is small compared with that of ζ = ln α. Conversely, when the heterogeneity of Y is prevalent upon that of ζ, κ¹ is influenced solely by the anisotropic structure of the formation unless the horizontal correlation scales are much greater than the vertical ones.     

On the dynamic response of a suspended cable

Linearized equations are derived to account for the additional tensions and deflections induced by dynamically exciting a suspended cable. Wave-type and modal solutions are presented to explore the influence of a fundamental geometric and elastic parameter, λ².     

A new solution of transient confined–unconfined flow driven by a pumping well

A new solution of transient confined–unconfined flow driven by a pumping well is developed and compared to previous approximate solutions of Moench and Prickett [Moench AF, Prickett TA. Radial flow in an infinite aquifer undergoing conversion from artesian to water table conditions. Water Resour Res 1972;8:494–9] and Hu and Chen [Hu L, Chen C. Analytical methods for transient flow to a well in a confined–unconfined aquifer. Ground Water 2008;46(4):642–6]. The problem is rewritten in dimensionless form with the Boltzmann transform. The nonlinear equation for flow in the unconfined zone is solved with the Runge–Kutta method. Position of the conversion interface is determined with an iteration scheme. This study shows that the confined–unconfined flow depends on three dimensionless parameters that represent the confined–unconfined storativity ratio (a_D), the ratio of the initial hydraulic head over the aquifer thickness (f_i), and the dimensionless pumping rate (q_D). The rate of expansion of the unconfined zone increases with q_D, but decreases with a_D and f_i. Differences between the two previous approximate solutions and the new solution of this study are observable in the estimated position of the conversion interface and the drawdown–time curves. The new solution can be applied to estimate the time for confined–unconfined conversion to occur (critical conversion time), and the time when the pumping well becomes dry (critical drying time). The critical conversion time is found to be very sensitive to the initial hydraulic head. The critical drying time is often much larger than the critical conversion time and may never be observed during a finite pumping period.     

Large-scale cenozoic tectonics of Central and South-central Asia: Products of continental collision

Three basic tectonic units, the Eurasian continent, the Rigid Boundary Zone and the Indian continent, are recognized. The Rigid Boundary Zone is identified with aseismic lineaments on an epicenter map of the Himalayas and their vicinity. The origin of the Himalayan syntaxes, their re-entrant character, and the observation that most of the Himalayas are made up of rocks belonging geologically to the Indian peninsular shield, are explained by deformation of the Indian continent within this Rigid Boundary Zone since the continental collision. The focal mechanism solutions of six earthquakes in the Indian Peninsula support the inference derived from the model that the high stresses generated by the continental collision may be very extensive spatially, and that the entire Indian Peninsula may be under a state of left lateral shear along NNE vertical planes.     

Particular solutions to the problem of horizontal flow of water and air through porous media near a water table

Particular solutions to the problem of horizontal flow of water and air through porous media near a water table are derived and regularity properties of the solutions are presented. It is found that a singularity occurs in the solutions at the water table and the water table can be interpreted as an acceleration wave of the nth order in terms of either air or water flow where n is a positive integer. Effects of air flow on water flow are discussed.