Interaction of Stream and Sloping Aquifer Receiving Constant Recharge

An analytical solution and finite-element numerical solution of a linearized and nonlinear Boussinesq equation, respectively, were obtained to describe water table variation in a semi-infinite sloping∕horizontal aquifer caused by the sudden rise or fall of the water level in the adjoining stream. Transient water table profiles in recharging and discharging aquifers having 0, 5, and 10% slopes and receiving zero or constant replenishment from the land surface were computed for t = 1 and 5 days by employing analytical and finite-element numerical solutions. The effect of linearization of the nonlinear governing equation, recharge, and slope of the impermeable barrier on water table variation in a semi-infinite flow region was illustrated with the help of a numerical example. Results suggest that linearization of the nonlinear equation has only a marginal impact on the predicted water table heights (with or without considering constant replenishment). The relative errors between the analytical and finite-element numerical solution varied in the range of ?0.39 to 1.59%. An increase in slope of the impermeable barrier causes an increase in the water table height at all the horizontal locations, except at the boundaries for the recharging case and a decrease for the discharging case.     

Falling Water Tables in Horizontal∕Sloping Aquifer

To describe falling water tables between two drains lying on a horizontal∕sloping impermeable barrier, analytical solutions of the Boussinesq equation linearized by Baumann's and Werner's methods and numerical solutions of the nonlinear form of the Boussinesq equation using finite-difference and finite-element methods were obtained. A hybrid finite analytic method, in which the nonlinear Boussinesq equation was locally linearized and solved analytically after approximating the unsteady term by a simple finite-difference formula to approximately preserve the overall nonlinear effect by the assembly of locally analytic solutions, was also used to obtain a solution of the Boussinesq equation. Midpoints of falling water tables between two drains in a horizontal∕sloping aquifer as obtained from various solutions were compared with already existing experimental values. Euclidean L2 and Tchebycheff L∞ norms were used to rank the performance of various solutions with respect to experimental data. It was observed that the performance of the hybrid finite analytic solution is the best, followed by finite element, finite difference, analytical with Werner's linearization method, and analytical with Baumann's linearization method, respectively.     

Water Table Fluctuation in the Presence of a Time-Varying Exponential Recharge and Depth-Dependent ET in a Two-Dimensional Aquifer System with an Inclined Base

An analytical solution is presented for water table fluctuation between ditch drains in presence of exponential recharge and depth-dependent evapotranspiration (ET) from groundwater table in a two-dimensional gently sloping aquifer. The groundwater head above the drain is small compared to the saturated thickness of the aquifer. A sound mathematical transformation is devised to transform the two-dimensional groundwater flow equation into a simple form, which makes possible to obtain an analytical solution. The transient midpoint water table variations from the proposed solution compare well with the already existing solutions for horizontal aquifer. A numerical example is used to illustrate the combined effect of depth-dependent ET coupled with a time-varying exponential recharge on the water table fluctuation. The inclusion of a depth-dependent ET in the solution results in water table decline at a faster rate as compared to the case when ET is not considered. With an increase in slope of the aquifer base, water table profiles become asymmetric and the water table divide shifts towards the lower drain. The height of the water table profiles increases on moving away from the boundary of the aquifer and the highest level of the ground water table is obtained in the central portion of the aquifer basin due to the presence of drainage ditches on the aquifer boundary. When the effect of ET is incorporated in combination with recharge, the analytical solution results in accurate and reliable estimates of water table fluctuations under situations subjected to a number of controlling factors. This study will be useful for alleviation of drainage problems of the aquifers receiving surface recharge and surrounded by streams.     

Analytical Solution for Transient Water Table Heights and Outflows from Inclined Ditch-Drained Terrains

This paper presents two analytical solutions of the linearized Boussinesq equation for an inclined aquifer, drained by ditches, subjected to a constant recharge rate. These solutions are based on different initial conditions. First, the transient solution is obtained for an initially fully saturated aquifer. Then, an analytical expression is derived for the steady state solution by allowing time to approach infinity. As this solution represents the groundwater table shape more realistically, this water table profile is used as an initial condition in the derivation of the second analytical solution for the groundwater table height, and the in- and outflow into the ditches. The solutions allow the calculation of the transient behavior of the groundwater table, and its ouflow, due to changing percolation rates or water level heights in both ditches.     

Water Table Profiles and Discharges for an Inclined Ditch-Drained Aquifer under Temporally Variable Recharge

This paper presents the solution of the linearized Boussinesq equation for an inclined, ditch-drained aquifer, with a temporally varying recharge rate. Water-table profiles and flow rates into the ditches are calculated. As an initial condition the steady-state profile for a constant recharge rate is used, and the linearized Boussinesq equation is solved for a different recharge rate. Then, at a specified time, the transient water table profile is used as initial condition for the Boussinesq equation with a new recharge rate. The transient solution at a new specified time is then used as the initial condition for the Boussinesq equation with a different recharge rate, and so on. Using the Darcy equation, analytical expressions for the flow rates into the ditches can be obtained. The solution allows the calculation of the transient behavior of the groundwater table and its flow rates due to temporally variable recharge rates.     

Drainage of Sloping Lands with Variable Recharge: Analytical Formulas and Model Development

Semianalytical transient equations for shallow subsurface transverse drainage systems installed in sloping lands are developed. They provide a general relationship between drain flow rates, water table elevations, and recharge rates. This relationship demonstrates that, depending on the recharge intensity, several drain flow rates can be observed at a given water table elevation. The recharge contribution is shown to depend on a water table shape factor and to decrease when the water table is low or the slope is steep. For very steep slopes, the recharge intensity no longer influences the drain flow rate. These equations can be used to confirm previous results obtained in steady-state conditions and to determine precisely under which conditions slope needs to be considered in drainage design. They have been incorporated into the field drainage model SIDRA, which simulates hourly values of water table elevations and drain flow rates. The model predictions are compared with the predictions of a steady-state equation and a numerical model, which solves the Boussinesq equation (SLOP model).     

Water Table Fluctuation between Drains in the Presence of Exponential Recharge and Depth-Dependent Evapotranspiration

A linearized form of the Boussinesq equation was solved analytically to predict the water table fluctuation in subsurface drained farmland in the presence of recharge and evapotranspiration (ET). The recharge was assumed to be variable with time and the ET considered decreasing linearly with a decrease in the water table height above the drains. The proposed analytical solution was verified for special cases with the existing solutions. There was a close match between the solutions. Applications of the solution in prediction of the water table height in a drainage system are illustrated with the help of physical examples.     

Transient Drainage to Partially Penetrating Drains in Sloping Aquifers

The nonlinear Boussinesq unsteady-state differential equation used for evaluating drainage of sloping lands with drains lying at a distance above the impermeable layer was solved. A combination of explicit and implicit difference methods was used to obtain a finite-difference solution for a linearized system of equations of Graute-Nicolson type on two time levels, ensuring the stability of the solution. The maximum height of the water tables was obtained as a function of time for different slopes varying from 0 to 70%. Model results were compared with the available experimental solutions of Luthin and Guitjens and Chauhan et al. as well as the numerical solution of Moody and were found to be in reasonable agreement.     

Flow Depletion Induced by Pumping Well from Stream Perpendicularly Intersecting Impermeable/Recharge Boundary

Analytical solutions for rate and volume of flow depletion induced by pumping a well from a stream that intersects an impermeable or a recharge boundary at right angles are derived using the basic flow depletion factor defined earlier by the author. A new concept of directly obtaining stream flow depletion using the method of images is proposed. The solutions are derived for five different management cases of a stream and boundary intersecting at right-angles, assuming the aquifer to be confined with semi-infinite areal extent. A computationally simple function is proposed for accurately approximating the error function. The existing analytical solution in the case of a right-angle bend of stream given by Hantush was obtained for unconfined aquifers using a linearization of the governing partial differential equation. The solution for this case obtained using the proposed method for confined aquifer is the same as obtained by Hantush for unconfined aquifers, which shows that the linearization adopted by Hantush does not actually solve this problem for unconfined aquifers.     

Analytical Ground-Water Flow Solutions for Channel-Aquifer Interaction

A general solution scheme for determining ground-water levels for channel∕group-water systems with recharge is developed and verified. The analytical solution uses the Laplace transform method to solve a linearized form of the Boussinesq equation. Unlike other solutions, this scheme allows for both boundaries and sources∕sinks to vary as a function of time and space. To verify the analytical scheme, three one-dimensional case studies of flow between two line sources in an unconfined aquifer were explored through a base run and a set of sensitivity analyses. These runs involved comparisons to MODFLOW and changes in the boundary conditions and dimensions. As noted, the flow equations were linearized about a point called the representative flow depth. A value of havg, defined as the average water depth between the initial and steady flow conditions, was used as the representative flow depth. Results of the proposed method matched very well with MODFLOW solutions for all times and locations using an optimal linearization point. In addition, using havg improved the solutions compared to those obtained previously.     

Drainage of Sloping Lands with Variable Recharge Model Validation and Applications

Semianalytical transient equations describing the behavior of water tables in subsurface drained soils when drains rest on a sloping impervious barrier have been derived previously and represent the bases of the SIDRA model. To validate (SIDRA), water table elevations and drainflow rates have been monitored for six years in the French Alps on fields with a slope of 8%. The predicted drain flow rates and water table shapes compare reasonably well with data of a drainage experiment site but the improvement provided by taking the slope into consideration was limited. Running the model with different slopes confirmed that high water table elevations and peak flows were not significantly changed with a slope of 8%. As could be predicted from the analysis in steady state, low water table elevations were the most affected by slope. With the soil parameters of the field experiment and from an analysis in tail recession conditions, it was shown that there is a clear threshold of 12% slope below which the slope has no significant influence and can be neglected in drainage design.     

Dupuit-Forchheimer Analyses of Steady-State Water-Table Heights due to Accretion in Drained Lands Overlying Undulating Sloping Impermeable Beds

The Dupuit-Forchheimer approximation is used in an investigation of steady-state water-table heights due to accretion in ditch-drained lands resting on an undulating impermeable bed that slopes away from a peak midway between drainage ditches toward a lower level at the drains. Analytical expressions are obtained for the water-table profiles assuming both horizontal flow and flow parallel to the impermeable base. These are compared with numerical results obtained for the Laplace solution of the flow problem that show the equipotentials to be better approximated as being normal to the base than vertical. There is good agreement for large slopes between the water-table heights obtained assuming one-dimensional flow parallel to the sloping base and the two-dimensional numerical results. Poorer agreement is obtained as the slope becomes less with results approaching those given by assuming horizontal flow which always results in underestimates. At small accretion rates agreement is obtained with both Dupuit-Forchheimer analyses and the Laplace solution. The maximum height of the water table above the base decreases and is closer to the drain as the slope increases.     

Tide-Induced Water Table Fluctuations in Coastal Aquifers Bounded by Rhythmic Shorelines

Previous studies on tidal water table dynamics in unconfined coastal aquifers have focused on the inland propagation of oceanic tides in the cross-shore direction based on the assumption of a straight coastline. Here, two-dimensional analytical solutions are derived to study the effects of rhythmic coastlines on tidal water table fluctuations. The computational results demonstrate that the alongshore variations of the coastline can affect the water table behavior significantly, especially in areas near the centers of the headland and embayment. With the coastline shape effects ignored, traditional analytical solutions may lead to large errors in predicting coastal water table fluctuations or in estimating the aquifer’s properties based on these signals. The conditions under which the coastline shape needs to be considered are derived from the new analytical solution.     

Theory of Seepage into an Auger Hole in a Confined Aquifer

A steady-state theory is presented for predicting flow into an auger hole partially penetrating a homogeneous and anisotropic confined aquifer that is underlain by an impermeable layer. The developed equations can be directly applied (i.e., without resorting to a coordinate transformation) to translate the rate of rise of the water in a pumped auger hole into directional conductivities of soil. The study shows that the conductivity values calculated by neglecting the confining pressure of an artesian aquifer (i.e., by applying the existing unconfined auger-hole seepage theories to experimental auger data obtained from a confined aquifer) may lead to serious error; hence, the confining head of an aquifer must be considered while the conductivity values are computed. Further, the distance of the outer layer also plays an important role in determining the flow to an auger hole penetrating a confined aquifer, and this parameter must therefore be included in the theoretical analysis of the problem. The validity of the proposed theory is checked by comparing a few results obtained from the theory with corresponding results obtained from numerical and analytical works. The developed theory is an addition to already existing auger-hole seepage theories for water-table aquifers; together with the available theories, the proposed solution is expected to cover the most commonly encountered auger hole experimental flow situations in the field.     

Vapor Flow to Trench in Leaky Aquifer

Soil-vapor extraction has become the most common innovative technology for treating subsurface soils contaminated with volatile and semivolatile organic compounds. This popularity is due partly to the low cost of vapor extraction and partly to the fact that mitigation is completed in situ. Previous applications of this technology have generally considered flow to either vertical or horizontal wells. However, vapor flow to a trench offers the advantages of a more uniform velocity field and lower construction costs at sites with shallow water tables. Therefore, an analytical solution is obtained for steady flow to a trench. The trench is assumed to partially penetrate an anisotropic aquifer and to have a finite horizontal length. The bottom aquifer boundary is assumed to be an impermeable water table, and the top boundary is a semipermeable aquitard. A comparison is made with field measurements to illustrate the application of the solution and to give confidence in its use.     

Study of ionic currents across a model membrane channel using Brownian dynamics

Brownian dynamics simulations have been carried out to study ionic currents flowing across a model membrane channel under various conditions. The model channel we use has a cylindrical transmembrane segment that is joined to a catenary vestibule at each side. Two cylindrical reservoirs connected to the channel contain a fixed number of sodium and chloride ions. Under a driving force of 100 mV, the channel is virtually impermeable to sodium ions, owing to the repulsive dielectric force presented to ions by the vestibular wall. When two rings of dipoles, with their negative poles facing the pore lumen, are placed just above and below the constricted channel segment, sodium ions cross the channel. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength does not increase the channel conductance further. When only those ions that acquire a kinetic energy large enough to surmount a barrier are allowed to enter the narrow transmembrane segment, the channel conductance decreases monotonically with the barrier height. This barrier represents those interactions between an ion, water molecules, and the protein wall in the transmembrane segment that are not treated explicitly in the simulation. The conductance obtained from simulations closely matches that obtained from ACh channels when a step potential barrier of 2-3 kTr is placed at the channel neck. The current-voltage relationship obtained with symmetrical solutions is ohmic in the absence of a barrier. The current-voltage curve becomes nonlinear when the 3 kTr barrier is in place. With asymmetrical solutions, the relationship approximates the Goldman equation, with the reversal potential close to that predicted by the Nernst equation. The conductance first increases linearly with concentration and then begins to rise at a slower rate with higher ionic concentration. We discuss the implications of these findings for the transport of ions across the membrane and the structure of ion channels.     

Simultaneous Prediction of Saturated Hydraulic Conductivity and Drainable Porosity Using the Inverse Problem Technique

The saturated hydraulic conductivity K and the effective porosity f are two important input parameters needed for lateral drain spacing design, as well as some other applications. The technical and economic justification, of most drainage projects, is mainly connected to these two parameters. The current design procedure is based upon calculation of the lateral spacing, using some average values of K and f within the drainage area. The objectives of this study were to introduce a new method for simultaneous estimation of K and f parameters using the inverse problem technique, and to evaluate five different unsteady drainage analytical models of the Boussinesq equation, suggested by different researchers for simultaneous prediction of the parameters. Consequently, five different analytical models for predicting water table profiles were solved, using the inverse problem technique. Each model was then evaluated. A physical drainage model of 2.2?m length, 0.3?m width, and 0.5?m height was established in the laboratory and carefully packed with a sandy loam soil. A perforated drainage pipe of 4.5?cm in diameter was installed at the bottom end of the model. Many piezometers were inserted in the soil for spatial and temporal water table monitoring. Different data sets from the experiments and literature were used for model calibration. The newly proposed approach that is based upon measuring water table profiles, at different times, was then evaluated with both constant and variable f. The predicted values of the proposed approach indicated reasonable agreement with the measured data. With variable effective porosity, the method was even more accurate to predict the water table profiles. Using the inverse problem technique, all the analytical models provided good agreement with the measured data. Among these, however, the Topp and Moody model predicted more accurate results than other models.     

Agroecological Impacts from Salinization and Waterlogging in an Irrigated River Valley

Extensive field data and calibrated flow and salt-transport models characterize the spatial and temporal patterns of salinity and waterlogging in an irrigated western river valley. Over three irrigation seasons, average seasonal aquifer recharge from irrigated fields in a 50,600?ha study area ranges from 0.59?to?0.99?m, including contribution from precipitation. The salinity of irrigation water varies from 618?to?1,090?mg/L. The water table is shallow, with 16 to 33% of irrigated land underlaid by an average water table less than 2?m deep. Average water table salinity ranges from 2,680?to?3,015?mg/L, and average soil salinity from 2,490?to?3,860?mg/L. Crop yield reductions from salinity and waterlogging range from 0 to 89% on fields, with regional averages ranging from 11 to 19%. Annual salt loading to the river from subsurface return flows, generated in large part by dissolution from irrigation recharge, averages about 533?kg/irrigated?ha?per?km. Upflux from shallow water tables under fallow ground contributes to about 65?million?m3 (52,600?acre-ft) per year of nonbeneficial consumption. Beyond problem identification, the developed database and models provide a basis for effectively addressing these problems through a systematic and comparative assessment of alternative solutions.     

Analysis of Seepage from Polygon Channels

An exact analytical solution for the quantity of seepage from a trapezoidal channel underlain by a drainage layer at a shallow depth has been obtained using an inverse hodograph and a Schwarz-Christoffel transformation. The symmetry about the vertical axis has been utilized in obtaining the solution for half of the seepage domain only. The solution also includes relations for variation in seepage velocity along the channel perimeter and a set of parametric equations for the location of phreatic line. From this generalized case, particular solutions have also been deduced for rectangular and triangular channels with a drainage layer at finite depth and trapezoidal, rectangular, and triangular channels with a drainage layer and water table at infinite depth. Moreover, the analysis includes solutions for a slit, which is also a special case of polygon channels, for both cases of the drainage layer. These solutions are useful in quantifying seepage loss and/or artificial recharge of groundwater through polygon channels.