Evaluation of Hopscotch Method for Transient Ground-Water Flow

The hopscotch finite-difference technique is shown to be a fast and accurate way to simulate transient, saturated, ground-water flow in relatively typical but heterogeneous 2D and 3D domains. The odd-even hopscotch (OEH) and line hopscotch methods are reviewed, and their implementation for saturated ground-water flow is presented. The OEH scheme, which is a second-order accurate explicit process, is efficient, requiring only six floating point operations per mesh node and time step, and is unconditionally stable (for saturated ground-water flow). Numerical experiments on typical 2D meshes (2,500 nodes) with synthetic, randomly heterogeneous hydraulic conductivity, suggest that the OEH process is approximately 1.5 times faster than the alternating direction implicit method and 3–4 times faster than the Crank-Nicolson implicit method using preconditioned conjugate gradient iteration. Similar experiments on medium-sized 3D meshes (87,500 nodes) suggest that the OEH process is between 7 and 10 times faster than the Crank-Nicolson preconditioned conjugate gradient method. Although the numerical results presented illustrate only typical test problem performance, they nevertheless clearly indicate promise for using OEH to simulate transient ground-water flow in 2D and, especially, 3D heterogeneous domains requiring fine spatial meshes.     

Flow past Bench-Scale Vertical Ground-Water Cutoff Walls

An experimental study was conducted to evaluate factors affecting flow rates past soil-bentonite (SB), geomembrane (GM), and composite geomembrane-soil (CGS) vertical cutoff walls. Intact walls and walls containing defects (inadequate keys, windows, and poor joint seals) were studied. For intact cutoff walls, CGS walls had the lowest flow rate, followed by SB and GM walls. CGS walls typically had flow rates as much as 100 times lower than comparable GM walls. Flow rates increased by a factor of 2–160 when the GM walls contained defective joints or the SB walls contained permeable windows. For all wall types, an effective key was required to achieve low flow rate past the wall. For GM walls, under seepage was not effectively controlled when the GM was placed in direct contact with the aquitard. Flow rates for GM walls placed in direct contact with the aquitard were nearly equal to the aquifer flow rate without a wall. Better control of underseepage was possible with SB and CGS walls placed in direct contact with the aquitard because the soft bentonitic backfill conformed to the surface of the aquitard. Hydraulic conductivity of unsealed joints in GM walls was estimated from flow rates past GM walls where a portion of the joint was unsealed. These hydraulic conductivities ranged between 1.8 × 10?4 and 5.6 × 10?3 cm∕s.     

Analytical Solutions for Estimating Effective Discharge

The concept of a dominant discharge that governs the cross-sectional and planform characteristics of a channel is used widely in stream restoration practice. This dominant discharge can be based on field measurements, a chosen recurrence interval of flooding or related to the effective discharge. The effective discharge is defined as the flow which transports the most sediment over the period of record. Significant differences in the relation between the computed effective discharge and bankfull discharge have been documented in the literature. Herein, different methods of estimating the effective discharge are described and analytical solutions derived. These analytical solutions are easy to apply and can be used to predict the effective discharge under existing conditions and the historic flow records. These solutions can be used to explain some of the discrepancies between different methods of estimating the effective discharge. In addition, the equations can be used to predict future trends in the effective discharge should the hydrologic, sediment transport or channel characteristics change in the future. The analysis procedures are illustrated with examples at two different scales: The Red River in Idaho and the Russian River in Northern California.     

Analytical Solutions for Stability of Slurry Trench

Coulomb-type force equilibrium analyses are presented for general two- and three-dimensional stability of a slurry-supported trench. Analytical solutions are derived for the factor of safety and critical failure plane angle in each case for drained effective stress and undrained total stress conditions. The solutions can accommodate variable trench depth, trench length, slurry depth, groundwater table elevation, surcharge loading, tension crack depth, and level of fluid in the tension cracks. Drained analyses can account for c′??′ soil strength and the effect of soil suctions above the groundwater table using the total cohesion method. Undrained analyses can account for undrained shear strength that varies linearly with depth. The solutions reduce to previously published expressions for simplified cases. An example problem is provided to illustrate variations of the factor of safety and critical failure plane angle with the length of a three-dimensional slurry trench. Finally, the method shows excellent agreement with the results of a full-scale field experiment of the failure of a diaphragm wall slurry trench constructed in silty sand.     

Kinematic and Diffusion Waves: Analytical and Numerical Solutions to Overland and Channel Flow

Flood wave propagation is the unifying concept in representing open channel and overland flow. Therefore, understanding flood wave routing theory and solving the governing equations accurately is an important issue in hydrology and hydraulics. In an attempt to contribute to the understanding of this subject, in this study: (1) an analytical solution is derived for diffusion waves with constant wave celerity and hydraulic diffusivity applied to overland flow problems; and (2) an algorithm is developed using the MacCormack explicit finite difference method to solve the kinematic and diffusion wave governing equations for both overland and open channel flow. The MacCormack method is particularly well suited to approximate nonlinear differential equations. The analytical solutions provide the practicing engineer with computational speed in obtaining results for overland flow problems, and a means to check the validity of the numerical models. On the other hand, for larger scale catchment-stream problems, the verified numerical methods provide efficient and accurate algorithms to obtain solutions. Both the analytical approaches and the MacCormack algorithm are used to solve the same synthetic examples. Comparison of results shows that the numerical and analytical solutions are in close agreement. Furthermore, the MacCormack algorithm is applied to a real catchment: a segment of the Duke University West Campus storm water drainage system. In order to check the accuracy of the results obtained by the MacCormack method, the results are compared to predictions of the Environmental Protection Agency storm water management model (SWMM) as calibrated with measured rainfall and surface runoff flow data. The results obtained from SWMM are in good agreement with the results obtained from applying the MacCormack algorithm.     

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.     

Simulation of Ground-Water Flow in Steep Basin with Shallow Surface Soil

A coupled ground-water∕channel flow distributed model has been developed for continuous simulation in a 123-km2 basin. The aim was to analyze the streamflow generation processes in natural vegetated environments. Finite-difference schemes have been used to solve conservation equations of the 2D saturated subsurface flow and the 1D kinematic surface flow. Because of the high hydraulic conductivity of the surface soil, only the saturation excess mechanism of runoff production has been considered. Parameter sensitivity analysis showed the overriding influence of soil storage capacity and conductivity. A grid discretization >100 m produces a hydraulic conductivity greater than physically meaningful, which considerably increases as the space-grid step increases. Results indicate that the model can satisfactorily simulate the water-flow behavior of the catchment after fitting the three parameters of surface hydraulic conductivity, effective porosity, and evapotranspiration losses. These are done after calculating the conductivity as a function of the height of the water table. The simulation efficiency has varied from 87% in the first 5-year calibration period to 85.8% in the subsequent 5-year validation period.     

Analytical Solutions for Contaminant Diffusion in Double-Layered Porous Media

Analytical solutions for conservative solute diffusion in one-dimensional double-layered porous media are presented in this paper. These solutions are applicable to various combinations of fixed solute concentration and zero-flux boundary conditions (BC) applied at each end of a finite one-dimensional domain and can consider arbitrary initial solute concentration distributions throughout the media. Several analytical solutions based on several initial and BCs are presented based on typical contaminant transport problems found in geoenvironmental engineering including (1) leachate diffusion in a compacted clay liner (CCL) and an underlying stratum; (2) contaminant removal from soil layers; and (3) contaminant diffusion in a capping layer and underlying contaminated sediments. The analytical solutions are verified against numerical solutions from a finite-element method based model. Problems related to leachate transport in a CCL and an underlying stratum of a landfill and contaminant transport through a capping layer over contaminated sediments are then investigated, and the suitable definition of the average degree of diffusion is considered.     

Analytical Solution for Circular Gates as Flow Metering Structures

Water measurement in irrigation canals is frequently hindered by low head availability and high capital investment costs associated with construction of compatible hydraulic structures. Often irrigation systems have circular sliding gates in place used as diversion and flow control structures. The Fresno Irrigation District investigated the feasibility of using such circular gates (Armco Model 101) as flow metering stations in the 1920s. This early work demonstrated that circular gates could be used simultaneously for both flow control and as flow measurement structures. The original work is compiled by USBR as 10,500 data points and is presented in tabular fashion for gate diameters varying from 20.3?to?121.9?cm (8–48?in.). An analytical equation of the form Q = CD(y,D)yD, [where CD(y,D) is a discharge function which depends on the gate displacement y and the nominal gate diameter D, g represents the gravitational acceleration, and H is the hydraulic headloss through the crescent-shaped orifice] accurately predicts most tabulated values. Equations are provided to compute the discharge function for nominal gate diameters varying from 20.3?to?121.9?cm (8–48?in.) for gate displacements between 5.1?cm (2?in.) and fully open conditions. The precision of the proposed algorithms are excellent (predicted values are within ±5% of the corresponding reported values 95% of the time) for gates greater than 30.5?cm (12?in.).     

Approximate Solutions for Forchheimer Flow to a Well

An exact solution for transient Forchheimer flow to a well does not currently exist. However, this paper presents a set of approximate solutions, which can be used as a framework for verifying future numerical models that incorporate Forchheimer flow to wells. These include: a large time approximation derived using the method of matched asymptotic expansion; a Laplace transform approximation of the well-bore response, designed to work well when there is significant well-bore storage and flow is very turbulent; and a simple heuristic function for when flow is very turbulent and the well radius can be assumed infinitesimally small. All the approximations are compared to equivalent finite-difference solutions.     

Analytical Solutions of Linear Finite-Strain One-Dimensional Consolidation

Numerical solutions of linear finite-strain one-dimensional consolidation of both initially unconsolidated and initially fully consolidated soil layers with both one-way and two-way drainage have been available for some time. However, no solutions of the limiting cases for initially unconsolidated soil have been available. Nor, except for the limiting cases corresponding to small-strain consolidation of initially fully consolidated layers, has the accuracy of these solutions and the solution charts based on them been evaluated. Analytical counterparts of the earlier numerical solutions and solution charts, analytical solutions of the limiting cases for initially unconsolidated soil, and analytical solutions of the small-strain Terzaghi equation expressed in material coordinates are presented in this paper. The analytical solutions clarify aspects of the numerical solutions, improve marginally the accuracy of the solution charts, and enable the latter to be easily replicated and extended. They may also be applied to the validation of numerical solutions of nonlinear finite-strain consolidation.     

Considerations for Monitoring Permeable Ground-Water Treatment Walls

Permeable ground-water treatment walls (PTWs) have been implemented as a means by which innovative ground-water treatment technologies can be applied in-situ. Though not widely addressed in the technical literature, the ground-water monitoring program for a PTW at a commercial site should consider several factors including: (1) design elements of the PTW; (2) the remediation process to be implemented through the PTW; (3) the distribution of contaminants in the affected aquifer; (4) ground-water sampling methods; and (5) regulatory issues. Also, the compliance monitoring well network within the PTW and sampling plan should be designed to assure that (1) design ground-water residence time goals within the PTW are achieved prior to sampling; (2) the uniformity of ground-water flow through the PTW is accounted for; and (3) ground-water samples are collected using techniques (e.g., micropurging) that reduce the potential for collecting nontreated ground water from down- or upgradient of the PTW. A case study illustrates the concepts used to develop a ground-water monitoring program for a PTW that was accepted by regulatory agencies for a commercial site.     

Analytical Construction of Transient Flow Nets in Homogeneous and Isotropic Flow Medium

The purpose of this study is to construct time-dependent flow nets, also called transient flow nets in homogenous and isotropic flow medium. Transient flow nets under hydraulic structures are developed in response to reservoir head fluctuations. An analytical solution for a transient flow net has not been reported in the literature. Time-dependent flow net equations are limited in engineering applications to simple boundary conditions. The geometry of transient flow nets does not change with time, as only the numerical values assigned to equipotential lines and flow lines change with time.     

Analytical Solutions to General Orthotropic Plates under Patch Loading

Orthotropic plates are widely used in bridge deck systems. However, these are not commonly treated as such within design specifications, and semianalytical solutions are not presently available for all deck types. This paper develops deflection equations for infinitely wide and simply supported thin plates considering each of the three cases of orthotropy: (1)?relatively torsionally stiff, flexurally soft; (2)?uniformly thick plate; and (3)?torsionally soft, flexurally stiff; subjected to arbitrary patch loading. These are common boundary and loading conditions encountered for bridge deck applications. The reported analytical solutions enable rapid evaluation of multiple moving patch loads to determine maximum design load effects and permit validation of numerical and finite-element methods. Application of the solutions will produce guidelines that can prescribe design demands and establish practical design simplifications for treatment of different bridge deck and slab systems in a uniform and consistent manner.     

Analytical Solutions for Open-Channel Temperature Response to Unsteady Thermal Discharge and Boundary Heating

Analytical solutions are derived for a one-dimensional model of the bulk temperature response of open-channel flow with unsteady and nonuniform heating at the upstream end, the water surface, and the riverbed. The solutions are explicit formulas comprised of transient terms, which play dominant roles in the upstream region, and equilibrium terms, which determine the temperature far downstream. The applicability of the solutions to practical problems is illustrated for two cases: (1) a stream bounded at its upstream end by a dam and with a midreach inflow; and (2) Boulder Creek, Colo., which is impacted by effluent released from a wastewater treatment plant. The model prediction is in reasonable agreement with gauged data.     

Method of Fundamental Solutions for Three-Dimensional Stokes Flow in Exterior Field

The main purpose of the present paper is to provide practical and numerical implementations of the method of fundamental solutions for three-dimensional exterior Stokes problems with quiet far-field condition and discuss the issues therein. The solutions of the steady Stokes problems are obtained by utilizing the boundary collocation method as well as the expansion of Stokeslets, which are the fundamental solutions of the steady Stokes equations. To validate the proposed model, numerical results of a lid-driven cavity flow, uniform flow passing a sphere, and a rotating dumbbell-shaped body show good agreement with the numerical and analytical solutions available in the literature. Also, a hypothetical problem with both vorticity and velocity boundary conditions is solved and compared with the analytical solution. The proposed model is then properly exploited to obtain the flow results of uniform flow passing a pair of vertical spheres in tandem and uniform flow passing a pair of horizontal spheres in tandem. Furthermore, the accuracy of the present numerical scheme is addressed and the detail flow characteristics, such as pressure distribution, streamline contour, velocity field, and vorticity fields are sketched.     

Analytical Solution for Fluid-Structure Interaction in Liquid-Filled Pipes Subjected to Impact-Induced Water Hammer

In this paper, an analytical solution for the four-equation model describing fluid-structure interaction in liquid-filled pipes subjected to impact-induced water hammer is presented. The analytical solution is derived for the case of both Poisson coupling and junction coupling. The results obtained from the analytical solution are in good agreement with experimental measurements and with numerical solutions determined by the method of characteristics. Taking advantage of the analytical solution, some features of coupled wave propagation are presented and discussed.     

Infiltration Considerations for Ground-Water Recharge with Waste Effluent

Water reuse and ground-water recharge can be used to meet the growing demands for water, particularly in arid regions. Ground-water recharge using fresh water or treated wastewater is most often accomplished by infiltration from surface basins. The water percolates through the unsaturated soil region to an underlying aquifer for storage and future use. In the case of wastewater, additional treatment occurs as the effluent flows through the soil. The system hydraulics of recharge basins have been examined through a combination of field and laboratory investigations. These studies indicate that infiltration rates and soil aquifer treatment of wastewater are influenced by soil type and soil profile characteristics, surface clogging material, pond depth, and wetting∕drying cycle times. The surface-clogging layer was found to be susceptible to consolidation and to associated reduction in hydraulic conductivity under seepage forces.