Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains

Generalized analytical solutions for groundwater head in horizontal aquifers in the presence of parallel subsurface drains are obtained considering a transient rate of recharge as a power series (polynomial) function and depth-dependent rate of evapotranspiration. A function, new to analytical drainage studies, is proposed for correctly representing the depth-dependent rate of evapotranspiration. The solutions are obtained considering the practical situation of drains placed at a shallow depth in a considerable depth of aquifer. Two conditions of large and small saturated thicknesses in comparison to the changes in groundwater head are considered. A mathematical criterion is proposed to distinguish between large and small saturated thicknesses.     

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.     

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.     

Comparison of Models for Computing Drainage Discharge

The WAVE model describes the transport and transformations of matter and energy in the soil, crop, and vadose environment. A lateral field drainage subprogram was added to the WAVE model to simulate lateral subsurface drainage flow. The subsurface drainage is considered as the drainage provided by evenly spaced parallel drains with a free outlet: drain tubing or ditch. The rate of subsurface water movement into drain tubes or ditches depends on the hydraulic conductivity of the soil, drain or ditch spacing, hydraulic head in the drains, profile depth, and water table elevation. Hooghoudt's steady-state equation was selected for incorporation in the WAVE model. The subsurface drainage subprogram was calibrated and validated by comparison with the SWAP model (The Netherlands) and DRAINMOD (the United States) and partially by using 7 years of drain outflow data from an experimental field under fallow and cropped conditions. The comparative study revealed that the three models performed equally well and that the models were reliable and accurate tools for predicting the drainage flux as a function of rainfall-evapotranspiration and local conditions. The WAVE model, in comparison to the SWAP and DRAINMOD model, provided as good a prediction of the lateral subsurface drainage flow to drains. The statistical analysis between each model and observed data revealed that the three models were able to predict with sufficient accuracy the observed drainage discharge. The DRAINMOD model, however, has the advantage of giving a more accurate estimate of the discharge, resulting in a more precise modeling. The models were consistent in predicting water table levels, but they could not be verified against field data because of a lack of suitable measurements.     

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.     

Functional Relationship to Describe Drains with Entrance Resistance

Numerical flow models usually represent drains as a system dependent boundary condition. If soil is saturated, drains act as the Dirichlet boundary condition with pressure head set equal to zero, and if soil is unsaturated, drains act as the Neumann boundary condition with flow set equal to zero. The underlying assumption is that drains exhibit ideal behavior. In reality, however, this is generally not so, and the flow encounters additional resistances due to pipe slotting and clogging of the envelope material around the drains. To account for the resulting resistance, a Hooghoudt-type boundary condition was developed that prescribes drain flow in relation to the groundwater level at a reference position. The measured drain discharge in an old drainage system was compared with calculated discharge assuming an ideal drain. It was found that the ideal drain assumption led to large errors in simulated discharge. With a correctly formulated and calibrated Hooghoudt boundary condition, however, more accurate drain discharges were obtained.     

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.     

Applying MODFLOW Model for Drainage Problem Solution: A Case Study from Jahir Irrigated Fields, Israel

High water table and soil salinization processes are common in irrigated fields in Israel. Subsurface drainage systems are a common technique to solve soil salinity problems. Subsurface drainage models can contribute to the efficiency of the drainage system as it can assist in the selection of the proper drainage system and its proper placement in the field. In this study we used the MODFLOW groundwater flow model to simulate groundwater levels in Jahir irrigated fields, the Jordan Valley, Israel. Using a three-layer groundwater flow model, the most efficient drainage system was found to be a combination of deep drains with relief wells and a pump placed in the area with soil salinity problem and upward hydraulic pressure. It was found that simulated drainage system can yield nearly 200,000?m3 of water per year. Given certain information, a spatially distributed groundwater flow model such as MODFLOW can provide more reliable information than different analytical solutions for planning of an effective subsurface drainage system.     

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).     

Design of Minimum Seepage-Loss Nonpolygonal Canal Sections with Drainage Layer at Shallow Depth

Analytical solutions exist for the seepage discharge from polygonal and nonpolygonal canal sections underlain by a drainage layer at a hydraulically infinite depth. These solutions lead to underestimation of the seepage discharge if a drainage layer occurs at a shallow depth. This paper presents solutions for seepage discharge from circular and exponential sections overlying a shallow drainage layer. The discharge has been calculated by a finite-difference-based numerical solution of the differential governing the seepage flow. The phreatic boundaries of the flow domain were described in terms of two parameters that were estimated by a minimization process. Such seepage computations were performed for a large number of independent dimensionless variables of the section geometry. Subjecting the computed seepage to regression analyses, explicit equations for seepage discharge loss have been obtained. Using these seepage loss equations, the design variables for minimum seepage loss have been obtained. The use of the design equations has been illustrated by design examples.     

Analytical Solutions for Shallow Tunnels in Saturated Ground

Estimates of ground deformations and liner stresses in a tunnel are usually obtained from empirical correlations or from past experience on similar tunnels. These correlations account for only a few of the significant factors, and extrapolation to other cases is questionable because similitude conditions are not generally fulfilled. In this paper, complete analytical solutions for a shallow tunnel in saturated ground are obtained. Two different drainage conditions have been considered: full drainage at the ground-liner interface, and no drainage. The solutions cover different construction processes and soil conditions: (1) dry ground; (2) saturated ground with and without air pressure; (3) with and without a gap between the ground and the liner; and (4) applicability for short term analysis (i.e., undrained excavation and liner installation) and for long term analysis. Since the ground and the liner are assumed to behave elastically, the solutions obtained are restricted to cases where ground deformations are small, such as stiff clays and rocks, or when the excavation method prevents large deformations of the ground.     

Ramp Kernels for Aquifer Responses to Arbitrary Stream Stage

Analytical expressions for ramp kernels (new kernels) for an improved convolution for obtaining aquifer responses, viz, groundwater head, rate, and cumulative volume of groundwater flow, to an arbitrary stage, are obtained. The use of the ramp kernels gives accurate aquifer responses and is superior to the conventional convolution in which numerical integration or pulse kernels are used. The extent of improvement in the results with the use of the ramp kernels is discussed and quantified for three examples, where the results are compared to analytical solutions. For the comparisons, the analytical solutions for linear and sinusoidal stream stages are derived. The use of the ramp kernels reproduces accurately the analytical solutions. The concept of ramp kernels can also be used for obtaining an accurate solution of convolution integrals observed in other fields.     

Hydraulic Considerations in Geosynthetic and Aggregate Subsurface Drains

The hydraulic design of certain types of subsurface drains has recently been put on a more rational footing, and deficiencies in earlier design methods have been demonstrated. However, significant limitations remain in hydraulic design methods for geosynthetic and aggregate subsurface drains. It is important to decouple the groundwater hydrology from the internal hydraulics of the drain, and properly design subsurface drains for open-channel rather than pressurized conditions. Present design methods can inadvertently result in pressurized flow. The assumption of uniform flow (Manning’s equation alone) is also improperly made in some present design methods. Consequences can include unintended pressurized flow and attendant nonuniformity of inflow on the one hand and uneconomical design on the other. Current standard guidelines provide relatively little guidance for the design of geosynthetic and aggregate drains. A current ASTM standard, commonly referenced by geosynthetic manufacturers, has significant limitations. Deficiencies and qualifications are identified for present design methods. Guidance is given for the improved design of geosynthetic and aggregate subsurface drains based on sound hydraulic principles.     

Mathematical Formulation and Validation of a Mixed Finite Element–Finite Difference Model for Simulating Phreatic Surfaces

The phreatic surface in an unconfined aquifer exists as a movable interface between the saturated and unsaturated zones. The movement of the phreatic surface depends on recharge, hydraulic conductivity, porosity, and horizontal and vertical flows. The location of the phreatic surface helps define the variably saturated flow domain in the subsurface. The variably saturated flow process in the subsurface is described by a parabolic partial differential equation. In this equation, the hydraulic conductivity and soil moisture capacity are used as the subsurface characteristics. The location of the phreatic surface is governed by a first-order partial differential equation. The governing parabolic partial differential equation is solved using a variational finite element formulation. The first order phreatic surface equation is then solved by loosely coupling with the governing parabolic partial differential equation describing the variably saturated flow. In the present study, a two-dimensional space is used to investigate the movement of the phreatic surface in a variably saturated unconfined flow domain. Based on the time-varying solutions of hydraulic heads, the location of the phreatic surface is simulated in a finite two-dimensional space.     

Quantifying Rainfall and Flooding Impacts on Groundwater Levels in Irrigation Areas: GIS Approach

Landscapes continuously irrigated without proper drainage for a long period of time frequently experience a rise in water-table levels. Waterlogging and salinization of irrigated areas are immediate impacts of this situation in arid areas, especially when groundwater salinity is high. Flooding and heavy rainfall further recharge groundwater and accelerate these impacts. An understanding of regional groundwater dynamics is required to implement land and water management strategies. The purpose of this study is to quantify the impact of flood and rain events on spatial scales using a geographic information system (GIS). This paper presents a case study of shallow water-table levels and salinity problems in the Wakool irrigation district located in the Murray irrigation area with groundwater average electrical conductivity greater than 25,000?μS/cm. This area has experienced several large flood events during the past several decades. Piezometric data are interpolated to generate a water-table surface for each event by applying the Kriging method of spatial interpolation using the linear variogram model. Spatial and temporal analysis of major flood events over the last four decades is conducted using calculated water-table surfaces to quantify the change in groundwater storage and shallow water-table levels. The drainage impact of a subsurface drainage scheme partially covering the area has also been quantified in this paper. The results show that flooding and local rainfall have a significant impact on shallow groundwater. The study also found that postflood climatic conditions (evaporation and rainfall) play a significant role in the groundwater dynamics of the area. The spatial net average groundwater recharge during the flooding events ranges from 0.19 to 0.52?ML/ha. The GIS-based techniques described in this paper can be used for net recharge estimation in semiarid regions where it is important to quantify net recharge impacts of regional flooding and local rainfall. The spatial visualization of the net recharge in a GIS environment can help prioritize management actions by local communities.     

Analytical Model of Surge Flow in Nonprismatic Permeable Channels and Its Application in Arid Regions

A surge running down a dry wadi bed as a consequence of a controlled water release from a reservoir—e.g., for artificial groundwater recharge—represents a free boundary problem. After some time, when aiming for groundwater recharge, the infiltration equals inflow and thus forms a kind of “standing” wave. The numerical solution of such phenomena generally involves considerable problems. For avoiding the numerical inconvenience resulting from the complex interacting surface/subsurface flow, we present an analytical solution of the slightly modified zero-inertia (ZI) equations. The development introduces a momentum-representative cross section for portraying the transient development of momentum and refers to a channel with constant slope, irregular geometry, and a permeable channel bed with significant infiltration. Due to the structure of the solution, any arbitrary infiltration model can be used for quantifying the infiltration losses. For both synthetic prismatic and nonprismatic test channels, the robust and easy-to-use analytical ZI model shows an excellent match with the results of a comparative numerical simulation. Finally, the ZI model is employed for simulating a surge flow downstream of the Wadi Ahin groundwater recharge dam (Oman), in order to perform a scenario for artificial groundwater recharge in a natural wadi channel reach. This realistic application illustrates the potential of the new approach by even computing an almost standing wave and shows its applicability for an accurate and robust evaluation of release strategies.     

2. Numerical Simulations of Water Flow and Solute Transport Applied to Acid Sulfate Soils

Field investigations of Rassam et al. in 2001 have highlighted the effects of infiltration, drainage, and evapotranspiration on the dynamics of water flow and solute transport in acid sulfate (AS) soils. In this work, HYDRUS-2D is adopted as the modeling tool to elucidate the trends observed in that field experiment. Hypothetical simulations have shown that the relative contribution of drains to lowering the water table is significant only when closely spaced drains are installed in coarse textured soils, evapotranspiration being the main driving force in all other cases. AS soils reaction products that are close to a drain are readily transportable during infiltration and early drainage, but those produced farther away from it near the midpoint between drains are only slowly transported during a prolonged drainage process. Simulating the field trial of Rassam et al. has shown that drain depth and evapotranspiration significantly affect solute fluxes exported to the ecosystem. Managing AS soils should target minimal drain depth and density. Partial or full lining of the drains should be considered as a management option for ameliorating the environmental hazards of AS soils.     

Effect of Periodic and Continuous Irrigation on Water Transport through Porous Media

In this paper, we compare the effect of two modes of irrigation on water transport through an unsaturated porous medium. The two modes studied are (1) periodic and (2) continuous irrigation. The former process consists of two stages, i.e., imbibition and drainage. The entire volume of water is assumed to be added during the imbibition stage at a constant rate. The time during which imbibition occurs is taken as a fraction of the total time period. In the drainage stage, no fresh water is added and the water inside the soil redistributes itself and drains from the soil. In the continuous mode, water is added at a constant rate during the whole time period. To ensure a fair comparison, the rate of water addition in the continuous mode is kept the same as the average rate of water addition over a time period in the periodic mode that includes the drainage and imbibition steps. The recharge is calculated as the volume of water drained from the bottom of the soil during a time period. The transport of water in the unsaturated zone is studied in the presence of water uptake by plant roots. The two modes of operation were simulated using a mass conservative algorithm based on a modification of Picard’s iterative scheme. Predictions in the periodic mode were performed using direct simulation and the results obtained were compared with an algorithm based on a shooting method. The performances of these two modes have been evaluated by calculating the recharge amount. When the sink term due to the plant roots was included, it was found that the recharge is significantly higher for the case of periodic operation. A physical explanation for the results obtained is proposed. The effect of hysteresis in the water retention curves was simulated using an empirical method. We have found that the total recharge amount in the periodic operation calculated using a mean nonhysteretic curve is very close to that obtained when we include the effect of hysteresis.     

Exact Equations for Critical Depth in a Trapezoidal Canal

Critical depth is an important parameter in the analysis of varied flow in canals and natural streams. For triangular, rectangular, and parabolic channel sections it is possible to express critical depth analytically. However, for many practical sections, including the trapezoidal section, the governing equations are implicit in the critical depth. For these sections the critical depth is presently obtained either by trial and error procedure, or by using empirical equations based on curve fitting. In this Technical Note exact analytical solutions of critical depth for the trapezoidal open channel section have been obtained in the form of fast converging infinite series.