Shape factor in the drawdown solution for well testing of dual-porosity systems

One of the important parameters in existing commercial dual-porosity reservoir simulators is matrix–fracture shape factor, which is customarily obtained by assuming a constant pressure at the matrix–fracture boundary. In his work, Chang [1] and [2] addressed the impact of boundary conditions at the matrix–fracture interface and presented analytical solutions for the transient shape factor and showed that for a slab-shaped matrix block a constant pressure boundary condition leads to an asymptotic (long-time) shape factor of π²/L², and that a constant volumetric flux leads to an asymptotic shape factor of 12/L². In a recent paper [3], we reconfirmed Chang’s [1] and [2] results using a Laplace transform approach. In this study, we extend our previous analysis and use infinite-acting radial and linear dual-porosity models, where the boundary condition is chosen at the wellbore, as opposed to at the matrix boundary. The coupled equations for fracture and matrix are solved analytically, taking into account the transient exchange between matrix and fracture. The analytical solution that invokes the time dependency of fracture boundary condition under constant rate is then used to calculate the transient shape factors. It is shown that, for a well producing at constant rate from a naturally fractured reservoir, the appropriate value of stabilized shape factor is 12/L². This contrasts with the commonly used shape factor for a slab-shaped matrix block that is subject to a constant pressure boundary condition, which is π²/L². The errors in the matrix–fracture exchange term in a dual-porosity model associated with the use of a shape factor derived based on constant pressure boundary condition at the matrix boundary are then evaluated.     

Penny-shaped fractures revisited

All methods of seismic characterization of fractured reservoirs are based on effective media theories that relate geometrical and material properties of fractures and surrounding rock to the effective stiffnesses. In exploration seismology, the first-order theory of Hudson is the most popular. It describes the effective model caused by the presence of a single set of thin, aligned vertical fractures in otherwise isotropic rock. This model is known to be transversely isotropic with a horizontal symmetry axis (HTI). Following the theory, one can invert the effective anisotropy for the crack density and type of fluid infill of fractures, the quantities of great importance for reservoir appraisal and management.Here I compute effective media numerically using the finite element method. I deliberately construct models that contain a single set of vertical, ellipsoidal, non-intersecting and non-interconnected fractures to check validity of the first-order Hudson’s theory and establish the limits of its applicability. Contrary to conventional wisdom that Hudson’s results are accurate up to crack density e ≈ 0.1, I show that they consistently overestimate the magnitudes of all effective anisotropic coefficients ε^(V), δ^(V), and γ^(V). Accuracy of theoretically derived anisotropy depends on the type of fluid infill and typically deteriorates as e grows. While the theory gives | ε^(V)|, |δ^(V)|, |γ^(V)| and close to the upper bound of the corresponding numerically obtained values for randomly distributed liquid-filled fractures, theoretical predictions of ε^(V), δ^(V) are not supported by numerical computations when the cracks are dry. This happens primarily because the first-order Hudson’s theory makes no attempt to account for fracture interaction which contributes to the final result much stronger for gas- than for liquid-filled cracks. I find that Mori-Tanaka’s theory is superior to Hudson’s for all examined crack densities and both types of fluid infill.The paper was presented at the 11th International Workshop on Seismic Anisotropy (11IWSA) held in St. John’s, Canada in 2004.     

An experimental investigation of vertical vibration of model footings on sand

Many free and forced vertical vibrations tests were conducted on surface and embedded models for footings on dry and moist poorly graded sand that has the following properties: D₁₀ = 0·21 mm, D₃₀ = 0·305 mm, D₆₀ = 0·423 mm, C_u = 2, C_c = 1·05, G_s = 2·66, γ_max = 1·74 cm⁻³ and γ_min = 1·4 g cm⁻³. The tests were conducted at relative density Dr = 82%. The effect of mass, area, geometry, embedment, saturation, load amplitude and frequency were studied. For this purpose square, rectangular and circular models of concrete footings were chosen. Swieleh sand was chosen as the foundation soil. Results have been obtained for models having different mass, same base shape and area; models of different base area and equal base shape and mass; and models of different base shape geometry and about equal masses and base areas.Forced vertical vibration tests results showed an increase in natural frequency and a reduction in amplitude with the increase in embedment depth, degree of saturation and footing base area. Increasing the mass of model footing resulted in a decrease in the natural frequency while the dynamic response increased. Also, results showed that the circular model footing gives low values of dynamic response in comparison to other models.Free vertical vibration test results showed an increase in damping ratio with increase in the base area of the model footing, depth of embedment and saturation of sand. On the other hand, the results showed a decrease in damping ratio with increase in the footing mass. Circular footing gives the highest value of damping ratio among other footings.Results showed that the best method for evaluating the dynamic stiffness for vertical vibration is using Dobry and Gazetas¹ (J. Geotech. Engng, ASCE, 1986, 112, 109–133) equations together with the formula proposed by Ronald and Bojan² (J. Geotech. Engng, ASCE, 1995, 121, 274–286) for evaluating the equivalent shear modulus.     

Effective macroscopic transport parameters between parallel plates with constant concentration boundaries

A macroscopic transport model is developed, following the Taylor shear dispersion analysis procedure, for a 2D laminar shear flow between parallel plates possessing a constant specified concentration. This idealized geometry models flow with contaminant dissolution at pore-scale in a contaminant source zone and flow in a rock fracture with dissolving walls. We upscale a macroscopic transient transport model with effective transport coefficients of mean velocity, macroscopic dispersion, and first-order mass transfer rate. To validate the macroscopic model the mean concentration, covariance, and wall concentration gradient are compared to the results of numerical simulations of the advection–diffusion equation and the Graetz solution. Results indicate that in the presence of local-scale variations and constant concentration boundaries, the upscaled mean velocity and macrodispersion coefficient differ from those of the Taylor–Aris dispersion, and the mass transfer flux described by the first-order mass transfer model is larger than the diffusive mass flux from the constant wall. In addition, the upscaled first-order mass transfer coefficient in the macroscopic model depends only on the plate gap and diffusion coefficient. Therefore, the upscaled first-order mass transfer coefficient is independent of the mean velocity and travel distance, leading to a constant pore-scale Sherwood number of 12. By contrast, the effective Sherwood number determined by the diffusive mass flux is a function of the Peclet number for small Peclet number, and approaches a constant of 10.3 for large Peclet number.     

Footings under seismic loading: Analysis and design issues with emphasis on bridge foundations

The paper provides state-of-the-art information on the following aspects of seismic analysis and design of spread footings supporting bridge piers: (1) obtaining the dynamic stiffness (“springs” and “dashpots”) of the foundation; (2) computing the kinematic response; (3) determining the conditions under which foundation–soil compliance must be incorporated in dynamic structural analysis; (4) assessing the importance of properly modeling the effect of embedment; (5) elucidating the conditions under which the effect of radiation damping is significant; (6) comparing the relative importance between kinematic and inertial response. The paper compiles an extensive set of graphs and tables for stiffness and damping in all modes of vibration (swaying, rocking, torsion), for a variety of soil conditions and foundation geometries. Simplified expressions for computing kinematic response (both in translation and rotation) are provided. Special issues such as presence of rock at shallow depths, the contribution of foundation sidewalls, soil inhomogeneity and inelasticity, are also discussed. The paper concludes with parametric studies on the seismic response of bridge bents on embedded footings in layered soil. Results are presented (in frequency and time domains) for accelerations and displacements of bridge and footing, while potential errors from some frequently employed simplifications are illustrated.     

Response spectral amplification ratios from 1- and 2-dimensional nonlinear soil site models

In order to determine the effect of geometry on the ground response of 2-dimensional (2-D) basins filled with soils that can develop nonlinear response, we use three basin models with width/depth ratios 3, 6 and 10. The three basins are subjected to a suite of rock site records with various magnitudes and source distances. We compute response spectral amplification ratios at four locations on the surface of the 2-D basins, and determine the average variation of the amplification ratios with respect to excitation spectra, for peak ground acceleration (PGA) and 3 spectral periods of 0.2, 0.5, 1 s. Similarly, we compute the average response spectral amplification ratios for two 1-dimensional (1-D) nonlinear models, one having the soil profile at the basin centre and the other having a soil profile at half the depth of the basin. From the relationship between the average amplification ratios and excitation spectra, we determine the cross-over point in terms of excitation spectral values that separate the amplification range from the deamplification range. Our results show that the cross-over point varies significantly from one location to another on the ground surface and from one basin to another, in a range of 0.3–1.1g for PGA. The effects of basin geometry are very strong at weak and moderate excitation, but decrease with increasing excitation spectra in a significant portion around the basin centre. Our results provide some justification for using 1-D models for 2-D basins with a width/depth ratio ?6 if the soil site is subjected to strong ground shaking.     

计算部分熔融岩石电导率方法的综述

部分熔融岩石的电导率是由固体岩石的低电导率和硅酸盐熔体的高电导率组成，因此熔体的几何形态和空间分布对部分熔融岩石的电导率将产生重大影响.本文介绍了熔体在岩石中的分布特征和它们的几何分布形态，同时，评述了各种数学模型在模拟计算部分熔融岩石电导率中的优缺点.     

Analytical and Numerical Modeling of Solute Intrusion,Recovery, and Rebound in Fractured Bedrock

Contaminated groundwater in fractured bedrock can expose ecosystems to undesired levels of risk for extended periods due to prolonged back-diffusion from rock matrix to permeable fractures. Therefore, it is key to characterize the diffusive mass loading (intrusion) of contaminants into the rock matrix for successful management of contaminated bedrock sites. Even the most detailed site characterization techniques often fail to delineate contamination in rock matrix. This study presents a set of analytical solutions to estimate diffusive mass intrusion into matrix blocks, it is recovered by pumping and concentration rebound when pumping ceases. The analytical models were validated by comparing the results with (1) numerical model results using the same model parameters and (2) observed chloride mass recovery, rebound concentration, and concentration in pumped groundwater at a highly fractured bedrock site in Alberta, Canada. It is also demonstrated that the analytical solutions can be used to estimate the total mass stored in the fractured bedrock prior to any remediation thereby providing insights into site contamination history. The predictive results of the analytical models clearly show that successful remediation by pumping depends largely on diffusive intrusion period. The results of initial mass from the analytical model was used to successfully calibrate a three-dimensional discrete fracture network numerical model further highlighting the utility of the simple analytical solutions in supplementing the more detailed site numerical modeling. Overall, the study shows the utility of simple analytical methods to support long-term management of a contaminated fractured bedrock site including site investigations and complex numerical modeling.     

Fractal dimension of fault systems in Japan: Fractal structure in rock fracture geometry at various scales

Based on fault maps, whether or not the fracture geometry of rocks is self-similar, was examined by using a box-counting algorithm. The statistical self-similarity (fractal structure) of the fault fracture systems holds well at the scale of about 2 to 20 km. The fractal dimension in Japan varied from 1.05 to 1.60. The fractal dimension is about 1.5–1.6 at the central part of the Japan Arc, and decreases with distance from the center. At a smaller scale, the fractal structure also holds well in the rock fracture geometry. The fractal dimension of the North Izu Peninsula fault system (branching faults) is 1.49 at the scale of 0.625 to 10 km, the fractal dimension of rock fracture geometry at the scale order of 10^–1 to 10^–2 meters is about 1.49–1.61. The upper limit of the fractal dimension of rock fracture geometry is about 1.6, judging from the estimation of fractal dimension on actual fracture geometry of rocks. This value may impose a restraint on modeling of faulting and the fracture process of rocks.     

Conceptual model of fracture-limited sea cliff erosion: Erosion of the seaward tilted flyschs of Socoa,Basque Country,France

Sea cliff morphology and erosion rates are modulated by several factors, including rock control that reflects both lithology and rock structure. Erosion is anticipated to preferentially exploit ‘fractures’, broadly meant as any discontinuity in an otherwise continuous medium, where the rock mass is weakest. Unpicking the direct control of such fractures on the spatial and temporal pattern of erosion remains, however, challenging. To analyse how such fractures control erosion, we monitored the evolution of a 400 m-long stretch of highly structured sedimentary cliffs in Socoa, Basque Country, France. The rock is known as the Socoa flysch formation. This formation combines decimetre-thick turbidites composed of repeat triplets of medium to strong calcareous sandstone, laminated siltstones and argillaceous marls. The sequence plunges at 45° into the sea with a shore-parallel strike. The cliffs are cross-cut by two normal and reverse fault families, with 10–100 m alongshore spacing, with primary and secondary strata-bound fractures perpendicular to the bedding, which combined delimit the cliff rock mass into discrete blocks that are exploited by the erosion process. Erosion, and sometimes plucking, of such beds and blocks on the cliff face was monitored using ground-based structure-from-motion (SfM) photogrammetry, over the course of 5.7 years between 2011 and 2017. To compare with longer time change, cliff-top retreat rate was assessed using SfM-orthorectified archive aerial photographs spanning 1954–2008. We show that the 13,250 m² cliff face released 4500 blocks exceeding 1.45 × 10⁻³ m³, removing a total volume of 170 m³. This equates to an average cliff erosion rate of 3.4 mm/year, which is slightly slower than the 54-year-long local cliff-top retreat (10.8 ± 1.8 mm/year). The vertical distribution of erosion reflects the height of sea water inundation, where the maximum erosion intensity occurs ca. 2 m above high spring-tide water level. Alongshore, the distribution of rockfall scars is concentrated along bed edges bounding cross-cutting faults; the extent of block detachment is controlled by secondary tectonic joints, which may extend through several beds locally sharing similar mechanical strength; and rockfall depth is always a multiple of bed thickness. Over the longer term, we explain block detachment and resultant cliff collapse as a cycle. Erosion nucleates on readily exploitable fractures but elsewhere, the sea only meets defect-free medium-strong to strong rock slabs offering few morphological features for exploitation. Structurally delimited blocks are quarried, and with sufficient time, carve semi-elliptic scars reaching progressively deeper strata to be eroded. Lateral propagation of erosion is directed along mechanical weaknesses in the bedding, and large episodic collapses affect the overhanging slabs via sliding on the weak marl beds. Collapse geometry is confined to one or several triplets of turbidite beds, but never reaches deeper into the cliff than the eroded depth at the foot. We contend that this fracture-limited model of sea-cliff erosion, inferred from the Socoa site dynamics and its peculiar sets of fractures, applies more broadly to other fractured cliff contexts, albeit with site-specific geometries. The initiation of erosion, the propagation of incremental block release and the ultimate full failure of the cliff, have each been shown to be fundamentally directly controlled by structure, which remains a vital control in understanding how cliffed coasts have changed in the past and will change in the future.     

Correlations between electrical and elastic properties of solid–liquid composites with interfacial energy-controlled equilibrium microstructures

Electrical conductivity and seismic velocity are studied for plausible pore geometries in the Earth's interior for reliable quantitative analysis of experimental data such as seismic tomography and magnetotelluric explorations. Electrical conductivity of a two-phase system with equilibrium, interfacial energy-controlled phase geometry is calculated for the dihedral angles θ = 40°–100° that are typical for rock–aqueous fluid and θ = 20°–60° for rock–melt systems of lower crust and upper mantle for the case of tetrakaidecahedral grains. Electrical conductivity vs. seismic velocity correlations are acquired by combining of the simulated electrical conductivities with the seismic velocity calculated with the help of equilibrium geometry model Takei [Takei, Y., Effect of pore geometry on V_P/V_S: From equilibrium geometry to crack. J. Geophys. Res. 107 (2002): 10.1029/2001JB000522.] for the same pore geometries. The results show that electrical conductivity gradually decreases reaching zero when seismic velocities reach seismic velocities of intact rock for rock–melt systems, while for rock–aqueous fluid systems with θ ≥ 60° conductivity drops to zero at velocities up to 10% smaller. This can explain the seeming discrepancy of the low seismic velocity region, attributed to the high fluid fraction, and the low electrical conductivity of the same region, which is sometimes faced at collocated electromagnetic and seismic experiments.     

Isotope geochemistry of minerals and fluids from Newberry volcano, Oregon

Isotopic compositions were determined for hydrothermal quartz, calcite, and siderite from core samples of the Newberry 2 drill hole, Oregon. The δ¹⁵O values for these minerals decrease with increasing temperatures. The values indicate that these hydrothermal minerals precipitated in isotopic equilibrium with water currently present in the reservoirs. The δ¹⁸O values of quartz and calcite from the andesite and basalt flows (700–932 m) have isotopic values which require that the equilibrated water δ¹⁸O values increase slightly (− 11.3 to −9.2‰) with increasing measured temperatures (150–265°C). The lithic tuffs and brecciated lava flows (300–700 m) contain widespread siderite. Calculated oxygen isotopic compositions of waters in equilibrium with siderite generally increase with increasing temperatures (76–100°C). The δ¹⁸O values of siderite probably result from precipitation in water produced by mixing various amounts of the deep hydrothermal water (− 10.5 ‰) with meteoric water (− 15.5 ‰) recharged within the caldera. The δ¹³C values of calcite and siderite decrease with increasing temperatures and show that these minerals precipitated in isotopic equilibrium with CO₂ of about −8 ‰.The δ¹⁸O values of weakly altered (<5% alteration of plagioclase) whole-rock samples decrease with increasing temperatures above 100°C, indicating that exchange between water and rock is kinetically controlled. The water/rock mass ratios decrease with decreasing temperatures. The δ¹⁸O values of rocks from the bottom of Newberry 2 show about 40% isotopic exchange with the reservoir water.The calculated δ¹⁸O and δD values of bottom hole water determined from the fluid produced during the 20 hour flow test are −10.2 and −109‰, respectively. The δD value of the hydrothermal water indicates recharge from outside the caldera.     

Can Karst Conduit Models Be Calibrated? A Dual Approach Using Dye Tracing and Temperature

Modeling flow and transport using both temperature and dye tracing provides constraints that can improve understanding of karst networks. A laminar flow and transport model using the finite element subsurface flow model simulated the conduit connection between a sinking stream and spring in central Pennsylvania to evaluate how conduit morphology might affect dye transport. Single and overly tortuous conduit models resulted in high concentrations as dye flowed back into the conduit from the matrix after dye injections ceased. A forked conduit model diverted flow from the main conduit, reducing falling limb dye concentration. Latin hypercube sampling was performed to evaluate the sensitivity of 52 parameter combinations (conduit hydraulic conductivity, conduit cross-sectional area, matrix transmissivity, matrix porosity, and dispersivity) for four conduit geometry scenarios. Sensitivity of arrival time for 50% of the dye indicated no parameter combinations which simulate falling limb dye concentrations for tortuous geometries, confirming the importance of the forked geometry regardless of other parameters. Temperature data from high-resolution loggers were then incorporated into the forked conduit model to reproduce seasonal spring temperature using variable sink inflow. Unlike the dye trace models, the thermal models were sensitive to other model parameters, such as conduit cross-sectional area and matrix transmissivity. These results showed this dual approach (dye and temperature) to karst network modeling is useful for (1) exploring the role of conduit and matrix interaction for contaminant storage, (2) constraining karst conduit geometries, which are often poorly understood, and (3) quantifying the effect of seasonal trends on karst aquifers.     

Fluids in the lithosphere, 1. Experimentally-determined wetting characteristics of CO2H2O fluids and their implications for fluid transport, host-rock physical properties, and fluid inclusion formation

The equilibrium distribution of CO₂H₂O fluids in synthetic rock samples (principally dunite and quartzite) has been characterized by measurements of the dihedral wetting angle, θ, resulting from 5-day annealing periods at 950–1150°C and 1 GPa. For fluids in equilibrium with polycrystalline quartz, θ varies systematically from 57° for pure H₂O to 90° at X_CO2 0.9. Similarly, for San Carlos olivine, θ varies from 65° for pure H₂O to 90° at X_CO2 0.9. The addition of solutes (NaCl, KCl, CaF₂, Na₂CO₃) to H₂O causes a major decrease in θ in the quartz/fluid system (to values as low as 40°), but has no effect on fluid wetting in dunite. Reconnaissance experiments on other mono- and polymineralic aggregates indicate universally high wetting angles (θ 60°) in upper mantle assemblages and for CO₂ in felsic compositions. For diopside + H₂O, θ 80°, with large variation due to crystalline anisotropy. In no case does θ approach 0°, the condition necessary for fluid to be present along all grain boundaries.Because a value of θ greater than 60° precludes the existence of an interconnected fluid phase in a rock, our results have important implications not only for fluid transport but also for the physical properties of the bulk fluid/rock system. Any static fluid present in the upper mantle must exist as isolated pores located primarily at grain corners, and transport can occur only by hydrofracture. In the continental crust, aqueous fluids (especially saline ones) are likely to form an interconnected network along grain edges, thus contributing to high electrical conductivity and allowing the possibility of fluid transport by porous flow or surface energy-driven infiltration.     

The effects of rock fragments on hydrologic and hydraulic responses along a slope

A model describing the three‐dimensional matrix flow along a slope with rock fragments or impermeable blocks was developed. The model was combined with modified Picard's iteration to ensure mass conservation in the unsaturated flow. We found that rock fragments obstruct water flow along the slope. The groundwater table must be raised to provide a sufficient pore water pressure gradient to facilitate water flow, but higher pore water pressure may induce slope failure. We also conducted a bench‐scale laboratory flume experiment to examine the effects of impermeable blocks on downstream seepage flow. In addition, a numerical experiment was conducted to examine how different arrangements of impermeable blocks affect downstream seepage flow and pore water pressure. This research demonstrated that the hydraulic phenomena were affected when impermeable blocks were present, and pore water pressure increased as the position of impermeable blocks was lowered. Copyright © 2006 John Wiley & Sons, Ltd.     

PART I: Theoretical Site Response Estimation for Microzoning Purposes

— We estimate the theoretical site response along seven cross sections located in the city of Thessaloniki (Greece). For this purpose the 2-D structural models used are based on the known geometry and the dynamic soil properties derived from borehole measurements and other geophysical techniques. Several double-couple sources have been employed to generate the seismic wavefield, and a hybrid method that combines the modal summation with finite differences, has been deployed to produce synthetic accelerograms to a maximum frequency of 6 Hz for all components of motion. The ratios between the response spectra of signals derived for the 2-D local model and the corresponding spectra of signals derived for the 1-D bedrock reference model at the same site, allow us to estimate the site response due to lateral heterogeneities. We interpret the results in terms of both geological and geometrical features of the models and of the characteristics of the wave propagation. The cases discussed confirm that the geometry and depth of the rock basement, along with the impedance contrast, are responsible for ground amplification phenomena such as edge effects and generation and entrapment of local surface waves. Our analysis also confirms that the peak ground acceleration is not well correlated with damage and that a substantially better estimator for possible damage is the spectral amplification.     

Wave propagation in a seven-story reinforced concrete building: I. Theoretical models

For transient, high frequency, and pulse like excitation of structures in the near field of strong earthquakes, the classical design approach based on relative response spectrum and mode superposition may not be conservative. For such excitations, it is more natural to use wave propagation methods. In this paper (Part I), we review several two-dimensional wave propagation models of buildings and show results for theoretical dispersion curves computed for these models. We also estimate the parameters of these models that would correspond to a seven-story reinforced concrete building in Van Nuys, California. Ambient vibration tests data for this building imply vertical shear wave velocity β_z=112 m/s and anisotropy factor β_x/β_z=0.55 for NS vibrations, and β_z=88 m/s and β_x/β_z=1 for EW vibrations. The velocity of shear waves propagating through the slabs is estimated to be about 2000 m/s. In the companion paper (Part II), we estimate phase velocities of vertically and horizontally propagating waves between seven pairs of recording points in the building using recorded response to four earthquakes.     

Oxygen and hydrogen isotopes in the volcano-sedimentary complex of Huelva (Iberian Pyrite Belt): example of water circulation through a volcano-sedimentary sequence

The minerals of basic and acidic rocks from the volcano-sedimentary sequence in the Huelva area, Spain, Iberian Pyrite Belt, display an extendedδ¹⁸O enrichment. Quartzδ¹⁸O values from quartz-keratophyres vary from +10.5 to +17.0 and feldsparδ¹⁸O values from +14.4 to +16.0. For the spilite or spilitized doleritesδ¹⁸O values vary from +9.9 to +13.4 for feldspar, from +6.4 to +9.8 for chlorite, from +3.7 to +4.3 for ilmenite and from +13.6 to +14.0 for quartz, but pyroxene exhibits magmatic values, from +5.3 to +6.1 with an exception at +7.5. The chloriteδD values vary from −34 to −43‰.This is attributed to hydrothermal alteration with seawater enriched inδ¹⁸O by circulation through sediments.The temperatures of interaction determined from isotopic fractionations between minerals range from 400° to 520°C.CalculatedδD andδ¹⁸O values for water in equilibrium with the minerals at isotopic temperatures range from −16 to +5 and from +8.3 to +12.8, respectively.A model of circulation of seawater through a pile of sedimentary rocks and then through basaltic rocks is proposed to explain the high¹⁸O compositions of the rocks from the Huelva District. Water/rock mass ratios calculated from this model range between 0.3 and 0.7 for the determined range of temperatures.     

A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches

The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, _N=αcotβ (in which β is the beach slope, α is the amplitude parameter and is the shallow water parameter) and are limited to tan⁻¹(α)βπ/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as and α increase, and reaches 7% of the linear solution.     

A new lumped-parameter model for flow in unsaturated dual-porosity media

A new lumped-parameter approach to simulating unsaturated flow processes in dual-porosity media such as fractured rocks or aggregated soils is presented. Fluid flow between the fracture network and the matrix blocks is described by a non-linear equation that relates the imbibition rate of the local difference in liquid-phase pressure between the fractures and the matrix blocks. Unlike a Warren-Root-type equation, this equation is accurate in both the early and late time regimes. The fracture/matrix interflow equation has been incorporated into an existing unsaturated flow simulator, to serve as a source/sink term for fracture gridblocks. Flow processes are then simulated using only fracture gridblocks in the computational grid. This new lumped-parameter approach has been tested on two problems involving transient flow in fractured/porous media, and compared with simulations performed using explicit discretisation of the matrix blocks. The new procedure seems to accurately simulate flow processes in unsaturated fractured rocks, and typically requires an order of magnitude less computational time than do simulations using fully-discretised matrix blocks.