Vertically fractured transversely isotropic media: dimensionality and deconstruction

A vertically fractured transversely isotropic (VFTI) elastic medium is one in which any number of sets of vertical aligned fractures (each set has its normal lying in the horizontal x₁, x₂‐plane) pervade the medium and the sets of aligned fractures are the only features of the medium disturbing the axi‐symmetry about the x₃‐axis implying that in the absence of fractures, the background medium is transversely isotropic (TI). Under the assumptions of long wavelength equivalent medium theory, the compliance matrix of a fractured medium is the sum of the background medium's compliance matrix and a fracture compliance matrix. For sets of parallel rotationally symmetric fractures (on average), the fracture compliance matrix is dependent on 3 parameters − its normal and tangential compliance and its strike direction. When one fracture set is present, the medium is orthorhombic and the analysis is straightforward. When two (non‐orthogonal) or more sets are present, the overall medium is in general elastically monoclinic; its compliance tensor components are subject to two equalities yielding an 11 parameter monoclinic medium. Constructing a monoclinic VFTI medium with n embedded vertical fracture sets, requires 5 TI parameters plus 3×n fracture set parameters. A deconstruction of such an 11 parameter monoclinic medium involves using its compliance tensor to find a background transversely isotropic medium and several sets of vertical fractures which, in the long wavelength limit, will behave exactly as the original 11 parameter monoclinic medium. A minimal deconstruction, would be to determine, from the 11 independent components, the transversely isotropic background (5 parameters) and two fracture sets (specified by 2 × 3 = 6 parameters). Two of the background TI medium's compliance matrix components are known immediately by inspection, leaving nine monoclinic components to be used in the minimal deconstruction of the VFTI medium. The use of the properties of a TI medium, which are linear relations on its compliance components, allows the deconstruction to be reduced to solving a pair of non‐linear equations on the orientations of two fracture sets. A single root yielding a physically meaningful minimum deconstruction yields a unique minimal representation of the monoclinic medium as a VFTI medium. When no such root exists, deconstruction requires an additional fracture set and uniqueness is lost. The boundary between those monoclinic media that have a unique minimal representation and those that do not is yet to be determined.     

DARCY'S COEFFICIENT OF PERMEABILITY AS SYMMETRIC TENSOR OF SECOND RANK

Synopsis

The dynamic equation of motion that governs the laminar flow of water through soils is the empirical equation of Darcy. According to Darcy's equation the velocity of the flowing water is proportional to the hydraulic gradient under which the water is flowing, with the constant of proportionality being the coefficient of permeability. The interesting question arising is whether or not the coefficient of permeability is a scalar quantity (having only a magnitude) or a vector (having both magnitude and direction). It is proved, in the present paper, that the permeability coefficient is neither a scalar nor a vector but a symmetric tensor of second rank. The fact that the permeability tensor is symmetric gives rise to great simplifications and permits a simple graphical construction of the tensor ellipsoid. Having the tensor ellipsoid, the determination of the direction at which the water will flow under a known imposed hydraulic gradient can be found graphically. In case of isotropic soils (the permeability coefficient has the same value along any direction) the ellipsoid reduces to a sphere and the tensor becomes a scalar. In the general case of anisotropic soils the permeability tensor is an entity with nine elements, six of which are independent representing pure extension or contraction along the three principal coordinate axes, thus transforming the permeability sphere into an ellipsoid and vice versa. It should be noted that in anisotropic soils the only directions along which the flow takes place in the direction of the hydraulic gradient are those of the principal axes of the tensor ellipsoid.

Permeability tests were conducted on anisotropic sandstone samples taken at different directions with respect to rectangular coordinates. The permeability coefficient values plotted on a two-dimensional polar coordinate graph paper give rise to an ellipse substantiating therefore the tensor concept of the permeability coefficient. The graphical construction of the tensor ellipse and the use of it in order to obtain the direction of flow by knowing the direction of the hydraulic gradient is also shown.     

Hydrodynamic modelling for groundwater flow through permeable reactive barriers

Hydrodynamic modelling for analysis of groundwater flow through permeable reactive barriers (PRBs) is addressed in this paper. Permeable reactive barriers constitute an emerging technology for in situ remediation of groundwater contamination and have many advantages over the traditional ex situ treatment methods. The transport domains during groundwater flow through PRBs often may involve free‐flow or non‐porous sections. To model the fluid mobility efficiently in such situations, the free and porous flow zones (PRBs) must be studied in conjunction with each other. The present paper is devoted to the analysis of groundwater flow through combined free flow domains and PRBs. The free‐flow regime is modelled using the Navier–Stokes equations whereas the permeable barriers are simulated by either the Darcy or the Brinkman equation. In order to couple the governing equations of motions, well‐posed mathematical formulations of matching boundary conditions are prescribed at the interface between the free‐groundwater‐flow zones and the permeable barriers. Combination of the Navier–Stokes equations with the Brinkman equation is more straightforward owing to their analogous forms. However, the Navier–Stokes and Darcy equations are incompatible mathematically and cannot be linked directly. The problem is resolved in this paper by invoking validated hydrodynamical expressions for describing the flow behaviour at the interfaces between free‐flow and porous zones. Three schemes for the analyses of fluid flow in combined domains are applied to the case of groundwater flow through permeable reactive barriers and different model results are compared. Copyright © 2002 John Wiley & Sons, Ltd.     

Effects of variations of river stage and hydraulic conductivity on temporal scaling of groundwater levels: numerical simulations

It was found in previous studies that groundwater levels may fluctuate as a temporal fractal. In this study numerical simulations of groundwater level fluctuations in an unconfined aquifer near a river were conducted to investigate the effects of aquifer heterogeneity and river stage variations on the fractal behavior of the water levels, h(t). Groundwater recharge was taken to be a white-noise process. The aquifer heterogeneity was simulated with a second-order stationary field of hydraulic conductivity (K) with an exponential variogram model. The results showed that groundwater levels fluctuate as a temporal fractal in both homogeneous and heterogeneous aquifers as long as K is less than 10 m/d. Most aquifers may indeed act as a fractal filter which takes a random non-fractal recharge inputs and produces a fractal responses of groundwater level fluctuations. A crossover in temporal scaling of h(t) may appear in more permeable aquifers. Fluctuations of the groundwater level in a homogeneous aquifer are dominated by the recharge process when the river stage is constant or by the river stage variations when the river stage varies in highly permeable aquifers. Heterogeneity plays an important role in the temporal scaling of h(t) in more permeable aquifers: the stronger the heterogeneity, the stronger the temporal scaling of h(t).     

Anisotropic permeability in fractured reservoirs from frequency‐dependent seismic Amplitude Versus Angle and Azimuth data

Attempts have previously been made to predict anisotropic permeability in fractured reservoirs from seismic Amplitude Versus Angle and Azimuth data on the basis of a consistent permeability‐stiffness model and the anisotropic Gassmann relations of Brown and Korringa. However, these attempts were not very successful, mainly because the effective stiffness tensor of a fractured porous medium under saturated (drained) conditions is much less sensitive to the aperture of the fractures than the corresponding permeability tensor. We here show that one can obtain information about the fracture aperture as well as the fracture density and orientation (which determines the effective permeability) from frequency‐dependent seismic Amplitude Versus Angle and Azimuth data. Our workflow is based on a unified stiffness‐permeability model, which takes into account seismic attenuation by wave‐induced fluid flow. Synthetic seismic Amplitude Versus Angle and Azimuth data are generated by using a combination of a dynamic effective medium theory with Rüger's approximations for PP reflection coefficients in Horizontally Transversely Isotropic media. A Monte Carlo method is used to perform a Bayesian inversion of these synthetic seismic Amplitude Versus Angle and Azimuth data with respect to the parameters of the fractures. An effective permeability model is then used to construct the corresponding probability density functions for the different components of the effective permeability constants. The results suggest that an improved characterization of fractured reservoirs can indeed be obtained from frequency‐dependent seismic Amplitude Versus Angle and Azimuth data, provided that a dynamic effective medium model is used in the inversion process and a priori information about the fracture length is available.     

Hydromechanical Heterogeneities of a Mature Fault Zone: Impacts on Fluid Flow

In this paper, fluid flow is examined for a mature strike‐slip fault zone with anisotropic permeability and internal heterogeneity. The hydraulic properties of the fault zone were first characterized in situ by microgeophysical (V_P and σ_c) and rock‐quality measurements (Q‐value) performed along a 50‐m long profile perpendicular to the fault zone. Then, the local hydrogeological context of the fault was modified to conduct a water‐injection test. The resulting fluid pressures and flow rates through the different fault‐zone compartments were then analyzed with a two‐phase fluid‐flow numerical simulation. Fault hydraulic properties estimated from the injection test signals were compared to the properties estimated from the multiscale geological approach. We found that (1) the microgeophysical measurements that we made yield valuable information on the porosity and the specific storage coefficient within the fault zone and (2) the Q‐value method highlights significant contrasts in permeability. Fault hydrodynamic behavior can be modeled by a permeability tensor rotation across the fault zone and by a storativity increase. The permeability tensor rotation is linked to the modification of the preexisting fracture properties and to the development of new fractures during the faulting process, whereas the storativity increase results from the development of micro‐ and macrofractures that lower the fault‐zone stiffness and allows an increased extension of the pore space within the fault damage zone. Finally, heterogeneities internal to the fault zones create complex patterns of fluid flow that reflect the connections of paths with contrasting properties.     

Morphological controls in sandy estuaries: the influence of tidal flats and bathymetry on sediment transport

The morphodynamics of shallow, vertically well-mixed estuaries, characterised by tidal flats and deeper channels, have been investigated. This paper examines what contributes to flood/ebb-dominant sediment transport in localised regions through a 2D model study (using the TELEMAC modelling system). The Dyfi Estuary in Wales, UK has been used as a case study and, together with idealised estuary shapes, shows that shallow water depths lead to flood dominance in the inner estuary whilst tidal flats and deep channels cause ebb dominance in the outer estuary. For medium sands and with an artificially ‘flattened’ bathymetry (i.e. no tidal flats), the net sediment transport switches from ebb-dominant to flood-dominant where the parameter a/h (local tidal amplitude ÷ local tidally averaged water depth) exceeds 1.2. Sea level rise will reduce this critical value of a/h and also reduce the ebb-directed sediment transport significantly, leading to a flood-dominated estuarine system. A similar pattern, albeit with greater transport, was simulated with tidal flats included and also with a reduced grain size. This suggests that analogous classifications for flood/ebb asymmetry of the tide in estuaries as a whole may not represent the local sediment transport in sufficient detail. Through the Dyfi simulations, the above criterion involving a/h is shown to be complicated further by augmented flow past a spit at the estuary mouth which gives rise to a self-maintaining scour hole. Simulations of one year of bed evolution in an idealised flat-bottomed estuary, including tidal flow past a spit, recreate the flood/ebb dominance on either side of the spit and the formation of a scour hole in between. The erosion rate at the centre of the hole is reduced as the hole deepens, suggesting the establishment of a self-maintaining equilibrium state.     

Human‐induced hazardous debris flows in Carrara marble basins (Tuscany,Italy)

In Carrara marble basins, the long and intensive quarrying activities (which began in the first century BC ) have produced extensive dump deposits, locally known as ravaneti. Ravaneti are of such large dimensions and diffusion as to make them a widespread landform of the Apuane Alps (Tuscany). In recent years these quarry dump deposits have been affected by frequent debris flows, more than 50 in 1996/97. This phenomenon is the most significant currently active geomorphological process in this landscape. The evolution of quarrying techniques produced a variety of sedimentological compositions of ravaneti. The debris flows analysed involve only the surface layers where debris is mixed with fine material with a lower permeability (active ravaneti) than the coarser underlying debris (older ravaneti). The presence of different permeability layers causes a wetting front to move downwards. Source area observations indicate a soil slip movement in the initial phases of the failure. The transformation of landslides into debris flow occurs by means of both soil contractive failure and an increase of granular temperature. The morphological and sedimentological analyses of depositional lobes resulted in a classification of three types of lobe, based on fabric–morphometry relationships allowing the identification of different flow dynamics: (1) type A lobe (platy form), matrix‐supported and well developed fabric with general tendency of ab clast plane strikes to lie generally parallel to flow direction as a consequence of laminar flow; (2) type B lobe (elongated form), clast‐supported and random fabric as a consequence of both turbulent flow and coarser composition of starting material; (3) type C lobe, intermediate type A–B morphometry, tendency for ab clast plane to lie in a semi‐circle around the main flow direction determined by the presence of secondary flow lines divergent from it in the stopping phase. In Carrara marble basins, the anomalous frequency with which debris flows tend to be triggered by medium‐intensity rainstorms (about 30 mm h⁻¹ rainfall) is due to the recent increases in silt dump produced by modern quarrying techniques. This represents a significant example of debris flows as an environmental problem in major anthromorphized landscapes. Copyright © 2000 John Wiley & Sons, Ltd.     

Testing the ‘two water worlds’ hypothesis under variable preferential flow conditions

Widespread observations of ecohydrological separation are interpreted by suggesting that water flowing through highly conductive soil pores resists mixing with matrix storage over periods of days to months (i.e., two ‘water worlds’ exist). These interpretations imply that heterogeneous flow can produce ecohydrological separation in soils, yet little mechanistic evidence exists to explain this phenomenon. We quantified the separation between mobile water moving through preferential flow paths versus less mobile water remaining in the soil matrix after free-drainage to identify the amount of preferential flow necessary to maintain a two water world's scenario. Soil columns of varying macropore structure were subjected to simulated rainfall of increasing rainfall intensity (26 mm h⁻¹, 60 mm h⁻¹, and 110 mm h⁻¹) whose stable isotope signatures oscillated around known baseline values. Prior to rainfall, soil matrix water δ²H nearly matched the known value used to initially wet the pore space whereas soil δ¹⁸O deviated from this value by up to 3.4‰, suggesting that soils may strongly fractionate ¹⁸O. All treatments had up to 100% mixing between rain and matrix water under the lowest (26 mm h⁻¹) and medium (60 mm h⁻¹) rainfall intensities. The highest rainfall intensity (110 mm h⁻¹), however, reduced mixing of rain and matrix water for all treatments and produced significantly different preferential flow estimates between columns with intact soil structure compared to columns with reduced soil structure. Further, artificially limiting exchange between preferential flow paths and matrix water reduced bypass flow under the most intense rainfall. We show that (1) precipitation offset metrics such as lc-excess and d-excess may yield questionable interpretations when used to identify ecohydrological separation, (2) distinct domain separation may require extreme rainfall intensities and (3) domain exchange is an important component of macropore flow.     

Moment tensor inversion and source rupture process of the September 27, 2003 M S=7.9 earthquake occurred in the border area of China, Russia and Mongolia

We conducted moment tensor inversion and studied source rupture process for M _S=7.9 earthquake occurred in the border area of China, Russia and Mongolia on September 27 2003, by using digital teleseismic P-wave seismograms recorded by long-period seismograph stations of the global seismic network. Considering the aftershock distribution and the tectonic settings around the epicentral area, we propose that the M _S=7.9 earthquake occurred on a fault plane with the strike of 127°, the dip of 79° and the rake of 171°. The rupture process inversion result of M _S=7.9 earthquake shows that the total rupture duration is about 37 s, the scalar moment tensor is M ₀=0.97×10²⁰ N·m. Rupture mainly occurred on the shallow area with 110 km long and 30 km wide, the location in which the rupture initiated is not where the main rupture took place, and the area with slip greater than 0.5 m basically lies within 35 km deep middle-crust under the earth surface. The maximum static slip is 3.6 m. There are two distinct areas with slip larger than 2.0 m. We noticed that when the rupture propagated towards northwest and closed to the area around the M _S=7.3 hypocenter, the slip decreased rapidly, which may indicate that the rupture process was stopped by barriers. The consistence of spatial distribution of slip on the fault plane with the distribution of aftershocks also supports that the rupture is a heterogeneous process owing to the presence of barriers.     

On the tensorial nature of advective porosity

Field tracer tests indicate that advective porosity, the quantity relating advective velocity to Darcy flux, may exhibit directional dependence. Hydraulic anisotropy explains some but not all of the reported directional results. The present paper shows mathematically that directional variations in advective porosity may arise simply from incomplete mixing of an inert tracer between directional flow channels within a sampling (or support) volume ω of soil or rock that may be hydraulically isotropic or anisotropic. In the traditional fully homogenized case, our theory yields trivially a scalar advective porosity equal to the interconnected porosity ϕ, thus explaining neither the observed directional effects nor the widely reported experimental finding that advective porosity is generally smaller than ϕ. We consider incomplete mixing under conditions in which the characteristic time t_D of longitudinal diffusion along channels across ω is much shorter than the characteristic time t_H required for homogenization through transverse diffusion between channels. This may happen where flow takes place preferentially through relatively conductive channels and/or fractures of variable orientation separated by material that forms a partial barrier to diffusive transport. Our solution is valid for arbitrary channel Peclet numbers on a correspondingly wide range of time scales t_D ⩽ t ≪ t_H. It shows that the tracer center of mass is advected at a macroscopic velocity which is generally not collinear with the macroscopic Darcy flux and exceeds it in magnitude. These two vectors are related through a second-rank symmetric advective dispersivity tensor Φ. If the permeability k of ω is a symmetric positive-definite tensor, so is Φ. However, the principal directions and values of these two tensors are generally not the same; whereas those of k are a fixed property of the medium and the length-scale of ω, those of Φ depend additionally on the direction and magnitude of the applied hydraulic gradient. When the latter is large, diffusion has negligible effect on Φ and one may consider tracer mass to be distributed between channels in proportion to the magnitude of their Darcy flux. This is made intuitive through a simple example of an idealized fracture network. Our analytical formalism reveals the properties of Φ but is too schematic to allow predicting the latter accurately on the basis of realistic details about the void structure of ω and tracer mass distribution within it. Yet knowing the tensorial properties of Φ is sufficient to allow determining it indirectly on the basis of ω-scale hydraulic and tracer data, including concentrations that represent homogenized samples extracted from (or sensed externally across) an ω-scale plume.     

Hillslope hydrology in tropical rainforest steeplands in Brunei

Many remaining areas of tropical rainforest in south‐east Asia are located on landscapes dominated by deep valleys and very steep slopes. Now that logging activities are extending into these steeplands, it is essential to understand how the natural rainforest system behaves if any kind of realistic assessment of the effects of such disturbance is to be made. This paper examines the hydrological behaviour of an undisturbed rainforest system on steep topography in the Temburong District of Brunei, north‐west Borneo. The physical and hydrological properties of the regolith material are generally typical of tropical residual soils. The regolith has a clay texture and a low dry bulk density beneath a superficial litter/organic horizon. The infiltration capacity of the surface soil was several hundred mm h⁻¹. That of the exposed mineral subsoil was an order of magnitude less, similar to the saturated hydraulic conductivity (K_sat) of around 180 mm h⁻¹ at a depth of 150 cm. There was no indication that K_sat reduced with depth except very near the bedrock interface. Soil tensions were measured using a two‐dimensional array of tensiometers on a 30° slope. During dry season conditions, infiltrating rain‐water contributes to soil moisture, and drying of the soil is dominated by transpiration losses. During wet season conditions, perched water tables quickly develop during heavy rainfall, giving rise to the rapid production of return flow in ephemeral channels. No infiltration excess or saturation overland flow was observed on hillslopes away from channel margins. Subsurface storm flow combined with return flow produce stream flow hydrographs with high peak discharges and very short lag times. Storm event runoff coefficients are estimated to be as high as 40%. It is concluded that the most distinctive feature of the hydrology of this ‘steepland rainforest’ is the extremely ‘flashy’ nature of the catchment runoff regime produced by the combination of thin but very permeable regolith on steep slopes. Copyright © 2000 John Wiley & Sons, Ltd.     

Aerodynamic entrainment thresholds and dislodgement rates on impervious and permeable beds

In the literature it has been suggested that on permeable, granular beds, both the threshold and rate of aerodynamic entrainment may be affected significantly by seepage flows into and out of the bed induced by fluctuating pressures in the overlying turbulent boundary layer. Using a range of grain sizes and flow conditions, the series of laboratory experiments reported here compares directly the aerodynamic entrainment of loose grains overlying fixed permeable sediment beds with that occuring over fixed impervious beds. For a given granular material, no significant differences in entrainment dynamics on the two types of bed were observed and in the range of flow conditions examined both the threshold shear velocity (U _*T) and the aerodynamic entrainment coefficient (k) were found to be independent of bed permeability.     

Dynamic response of a multi-layered poroelastic medium

An exact stiffness matrix method is presented to evaluate the dynamic response of a multi-layered poroelastic medium due to time-harmonic loads and fluid sources applied in the interior of the layered medium. The system under consideration consists of N layers of different properties and thickness overlying a homogeneous half-plane or a rigid base. Fourier integral transform is used with respect to the x-co-ordinate and the formulation is presented in the frequency domain. Fourier transforms of average displacements of the solid matrix and pore pressure at layer interfaces are considered as the basic unknowns. Exact stiffness (impedance) matrices describing the relationship between generalized displacement and force vectors of a layer of finite thickness and a half-plane are derived explicitly in the Fourier-frequency space by using rigorous analytical solutions for Biot's elastodynamic theory for porous media. The global stiffness matrix and the force vector of a layered system is assembled by considering the continuity of tractions and fluid flow at layer interfaces. The numerical solution of the global equation system for discrete values of Fourier transform parameter together with the application of numerical quadrature to evaluate inverse Fourier transform integrals yield the solutions for poroelastic fields. Numerical results for displacements and stresses of a few layered systems and vertical impedance of a rigid strip bonded to layered poroelastic media are presented. The advantages of the present method when compared to existing approximate stiffness methods and other methods based on the determination of layer arbitrary coefficients are discussed.     

Effects of patterned Artemisia capillaris on overland flow velocity under simulated rainfall

Vegetation cover is an important factor for erosion control. Laboratory‐simulated rainfall experiments were conducted to quantify the effectiveness of patchy distributed Artemisia capillaris in retarding overland flow velocity. Simulated storms (60, 90, 120, and 150 mm h^?1) were applied on a bare plot (CK) and four different plant patterns, a banded pattern perpendicular to the slope direction (BP), a single long strip parallel to slope direction (LP), small patches distributed like a checkerboard (SP1), and small patches distributed like a letter “X” (SP2). All treatments had three replicates. Each plot underwent two sets of experiments, intact plant plots and root plots (the above‐ground parts were removed, only roots were reserved), respectively. Results showed that flow velocity increased with rainfall intensity, and the lower slope velocity (V_l) was higher than the upper slope velocity (V_u). The removal of grass shoots increased flow velocity. Compared with bare soil plot, intact plants reduced mean flow velocity by 14%–60%, whereas the reduction declined to <40% for the root plots. BP and both SP treatments performed more effectively than LP in retarding flow velocity, whereas no significant differences were identified between BP and SP. The contributions of A. capillaris shoots and roots to the reductions in flow velocity under different rainfall intensities were different. The shoots made greater contribution of 53%–97% at 60 and 90 mm h^–1, and the roots contributed more (51%–81%) at 120 and 150 mm h^–1. Runoff and sediment rate had significant (p < 0.05) linear correlations with mean flow velocity. Copyright © 2012 John Wiley & Sons, Ltd.     

Factors influencing surface runoff generation in a Mediterranean semi-arid environment: Chicamo watershed,SE Spain

A deeper knowledge of the hydrological response of semi-arid Mediterranean watersheds would be useful in the prediction of runoff production for assessing flood risks and planning flood mitigation works. This study was conducted to identify the runoff generation mechanisms and their controlling factors at the hillslope scale in a Mediterranean semi-arid watershed. Four zero-order microcatchments were selected to measure rainfall and runoff for a three-year period. Two groups of soil were differentiated with respect to the hydrological response. The fine textured, poorly permeable soils of low organic carbon content had a greater runoff coefficient (9%) and lower runoff threshold (3·6 mm) than more permeable, coarser textured soils of medium organic carbon content (<3%, and 8 mm, respectively). The influence of rainfall characteristics on the hydrological response was different. Rain intensity was the major rainfall parameter controlling the runoff response in the microcatchments on fine textured, low infiltrability soils with a poor plant cover, while total rainfall was more closely correlated with runoff in coarser textured, highly permeable soils with a denser plant cover. It can be concluded that there are two runoff generation mechanisms: (i) an infiltration-excess overland flow in the more degraded areas with low organic carbon content (<0·5%) and low infiltrability (>5 mm h⁻¹); and (ii) a saturation-excess overland flow in the less degraded areas with a high organic carbon content (>2%), high infiltrability (>8 mm h⁻¹) and covered by a dense plant cover (>50%). © 1998 John Wiley & Sons, Ltd.     

Bench‐Scaled Nano‐Fe0 Permeable Reactive Barrier for Nitrate Removal

There are many fundamental problems with the injection of nano‐zero‐valent iron (NZVI) particles to create permeable reactive barrier (PRB) treatment zone. Among them the loss of medium porosity or pore blocking over time can be considered which leads to reduction of permeability and bypass of the flow and contaminant plume up‐gradient of the PRB. Present study provides a solution for such problems by confining the target zone for injection to the gate in a funnel‐and‐gate configuration. A laboratory‐scale experimental setup is used in this work. In the designed PRB gate, no additional material from porous media exists. NZVI (d₅₀ = 52 ± 5 nm) particles are synthesized in water mixed with ethanol solvent system. A steady‐state condition is considered for the design of PRB size based on the concept of required contact time to obtain optimum width of PRB gate. Batch experiment is carried out and the results are used in the design of PRB gate width (~50 mm). Effect of high initial NO₃^–‐N concentration, NZVI concentration, and pore velocity of water in the range of laminar groundwater flow through porous media are evaluated on nitrate‐N reduction in PRB system. Results of PRB indicate that increasing the initial NO₃^–‐N concentration and pore velocity has inhibitor effect—against the effect of NZVI concentration—on the process of NO₃^–‐N removal. Settlement velocity (S.V.) of injected NZVI with different concentrations in the PRB is also investigated. Results indicate that the proposed PRB can solve the low permeability of medium in down‐gradient but increasing of the S.V. especially at higher concentration is one of the problems with this system that needs further investigations.     

New formulae for estimating lava flow volumes at Mt. Etna Volcano, Sicily

 A new data set of Etna lava flows erupted since 1868 has been compiled from eight topographic maps of the volcano published at intervals since then. Volumes of 59 flows or groups of flows were measured from topographic difference maps. Most of these volumes are likely to be considerably more accurate than those published previously. We cut the number of flow volumes down to 25 by selecting those examples for which the volume of an individual eruption could be derived with the highest accuracy. This refined data set was searched for high correlations between flow volume and more directly measurable parameters. Only two parameters showed a correlation coefficient of 70% or greater: planimetric flow area A (70%) and duration of the eruption D (79%). If only short duration (<18 days) flows were used, flow length cubed, L³, had a correlation coefficient of 98%. Using combinations of measured parameters, much more significant correlations with volume were found. Dh had a correlation coefficient of 90% (h is the hydrostatic head of magma above the vent), and  , 92% (where W is mean width and E is the degree of topographic enclosure), and a combination of the two , 97%. These latter formulae were used to derive volumes of all eruptions back to 1868 to compare with those from the complete data set. Values determined from the formulae were, on average, lower by 16% (Dh), 7% (, and 19% . Received: 30 November 1998 / Accepted: 20 June 1999     

A low‐dimensional model of bedrock weathering and lateral flow coevolution in hillslopes: 1. Hydraulic theory of reactive transport

This is the first of a two‐part paper exploring the coevolution of bedrock weathering and lateral flow in hillslopes using a simple low‐dimensional model based on hydraulic groundwater theory (also known as Dupuit or Boussinesq theory). Here, we examine the effect of lateral flow on the downward fluxes of water and solutes through perched groundwater at steady state. We derive analytical expressions describing the decline in the downward flux rate with depth. Using these, we obtain analytical expressions for water age in a number of cases. The results show that when the permeability field is homogeneous, the spatial structure of water age depends qualitatively on a single dimensionless number, Hi. This number captures the relative contributions to the lateral hydraulic potential gradient of the relief of the lower‐most impermeable boundary (which may be below the weathering front within permeable or incipiently weathered bedrock) and the water table. A “scaled lateral symmetry” exists when Hi is low: age varies primarily in the vertical dimension, and variations in the horizontal dimension x almost disappear when the vertical dimension z is expressed as a fraction z/H(x) of the laterally flowing system thickness H(x). Taking advantage of this symmetry, we show how the lateral dimension of the advection–diffusion‐reaction equation can be collapsed, yielding a 1‐D vertical equation in which the advective flux downward declines with depth. The equation holds even when the permeability field is not homogeneous, as long as the variations in permeability have the same scaled lateral symmetry structure. This new 1‐D approximation is used in the accompanying paper to extend chemical weathering models derived for 1‐D columns to hillslope domains.