IDENTIFICATION OF AQUIFER DISPERSIVITIES: METHODS OF ANALYSIS AND PARAMETER UNCERTAINTY1

ABSTRACT: This paper presents a method for estimating aquifer dispersivities in solute transport models. Sensitivity equations are derived for the calculation of sensitivity coefficients. A modified Gauss-Newton algorithm is used to perform the least-squares minimization. A statistical procedure is outlined to assess reliability of the estimated parameters. The solute transport model is solved by the upstream weighted, multiple cell balance method which combines the concepts of local mass balance and finite element approximations. A one-dimensional solute transport problem in a vertical column system is first used to illustrate the inverse technique. A second example considers the parameter identification problem for three-dimensional solute transport with a unidirectional steady and uniform flow field. The third example solves the parameter identification problem in a three-dimensional, stream-aquifer, solute transport system with steady state flow. Numerical experiments are conducted to study data requirements for parameter identification.     

SOLUTE TRANSPORT IN THREE‐DIMENSIONAL SATURATED POROUS MEDIA HAVING SELF‐SIMILAR HYDRAULIC CONDUCTIVITY DISTRIBUTION1

ABSTRACT: Customarily, it has been assumed that hydraulic conductivity is a stationary, homogeneous stochastic process with a finite variance for stochastic analysis of solute transport in the subsurface. That the distribution of hydraulic conductivity may have a fractal behavior with long range correlations was suggested from field data analyses. This motivates us to further investigate how the fractal behavior of permeability distribution impacts solute transport in porous media. This study provides longitudinal and transverse macrodispersivity coefficients and the variance of the solute concentration. Longitudinal and transverse macrodispersivity coefficients are found to depend strongly on the fractal dimension (D) of logarithmic hydraulic conductivity (logK). The longitudinal and transverse macrodispersivity coefficients are the highest when D = 1, and the values decrease monotonically to zero at D = 2. Both coefficients correspond to the characteristic length scale of the logK distribution, thus are scale dependent parameters. The ratio of the transverse to the longitudinal macrodispersivity coefficient is on the order of 10^‐1 to 10^‐4. Concentration variance also decreases with the fractal dimension of logK. There is no spatial spreading of solute for D = 2, and the concentration variance reaches zero for this case.     

A HYBRID OPTIMIZATION APPROACH TO THE ESTIMATION OF DISTRIBUTED PARAMETERS IN TWO-DIMENSIONAL CONFINED AQUIFERS1

ABSTRACT: In using non-linear optimization techniques for estimation of parameters in a distributed ground water model, the initial values of the parameters and prior information about them play important roles. In this paper, the genetic algorithm (GA) is combined with the truncated-Newton search technique to estimate groundwater parameters for a confined steady-state ground water model. Use of prior information about the parameters is shown to be important in estimating correct or near-correct values of parameters on a regional scale. The amount of prior information needed for an accurate solution is estimated by evaluation of the sensitivity of the performance function to the parameters. For the example presented here, it is experimentally demonstrated that only one piece of prior information of the least sensitive parameter is sufficient to arrive at the global or near-global optimum solution. For hydraulic head data with measurement errors, the error in the estimation of parameters increases as the standard deviation of the errors increases. Results from our experiments show that, in general, the accuracy of the estimated parameters depends on the level of noise in the hydraulic head data and the initial values used in the truncated-Newton search technique.     

Enhancing Hyrdological Simulation Program – FORTRAN Model Channel Hydraulic Representation1

Abstract: The hydrological simulation program – FORTRAN (HSPF) is a comprehensive watershed model that employs depth‐area‐volume‐flow relationships known as the hydraulic function table (FTABLE) to represent the hydraulic characteristics of stream channel cross‐sections and reservoirs. An accurate FTABLE determination for a stream cross‐section site requires an accurate determination of mean flow depth, mean flow width, roughness coefficient, longitudinal bed slope, and length of stream reach. A method that uses regional regression equations to estimate mean flow depth, mean flow width, and roughness coefficient is presented herein. FTABLES generated by the proposed method (Alternative Method) and FTABLES generated by Better Assessment Science Integrating Point and Nonpoint Sources (BASINS) were compared. As a result, the Alternative Method was judged to be an enhancement over the BASINS method. First, the Alternative Method employs a spatially variable roughness coefficient, whereas BASINS employs an arbitrarily selected spatially uniform roughness coefficient. Second, the Alternative Method uses mean flow width and mean flow depth estimated from regional regression equations whereas BASINS uses mean flow width and depth extracted from the National Hydrography Dataset (NHD). Third, the Alternative Method offers an option to use separate roughness coefficients for the in‐channel and floodplain sections of compound channels. Fourth, the Alternative Method has higher resolution in the sense that area, volume, and flow data are calculated at smaller depth intervals than the BASINS method. To test whether the Alternative Method enhances channel hydraulic representation over the BASINS method, comparisons of observed and simulated streamflow, flow velocity, and suspended sediment were made for four test watersheds. These comparisons revealed that the method used to estimate the FTABLE has little influence on hydrologic calibration, but greatly influences hydraulic and suspended sediment calibration. The hydrologic calibration results showed that observed versus simulated daily streamflow comparisons had Nash‐Sutcliffe efficiencies ranging from 0.50 to 0.61 and monthly comparisons had efficiencies ranging from 0.61 to 0.84. Comparisons of observed and simulated suspended sediments concentrations had model efficiencies ranging from 0.48 to 0.56 for the daily, and 0.28 to 0.70 for the monthly comparisons. The overall results of the hydrological, hydraulic, and suspended sediment concentration comparisons show that the Alternative Method yielded a relatively more accurate FTABLE than the BASINS method. This study concludes that hydraulic calibration enhances suspended sediment simulation performance, but even greater improvement in suspended sediment calibration can be achieved when hydrological simulation performance is improved. Any improvements in hydrological simulation performance are subject to improvements in the temporal and spatial representation of the precipitation data.     

NONLINEAR MODELING AND PREDICTION OF A RIVER FLOW SYSTEM1

ABSTRACT: Model estimation and prediction of a river flow system are investigated using nonlinear system identification techniques. We demonstrate how the dynamics of the system, rainfall, and river flow can be modeled using NARMAX (Nonlinear Autoregressive Moving Average with eXogenuous input) models. The parameters of the model are estimated using an orthogonal least squares algorithm with intelligent structure detection. The identification of the nonlinear model is described to represent the relationship between local rainfall and river flow at Enoree station (inputs) and river flow at Whitmire (output) for a river flow system in South Carolina.     

Estimating Lag to Peak between Rainfall and Peak Streamflow with a Mixed‐Effects Model

We test the use of a mixed‐effects model for estimating lag to peak for small basins in Maine (drainage areas from 0.8 to 78 km²). Lag to peak is defined as the time between the center of volume of the excess rainfall during a storm event and the resulting peak streamflow. A mixed‐effects model allows for multiple observations at sites without violating model assumptions inherent in traditional ordinary least squares models, which assume each observation is independent. The mixed model includes basin drainage area and maximum 15‐min rainfall depth for individual storms as explanatory features. Based on a remove‐one‐site cross‐validation analysis, the prediction errors of this model ranged from ?42% to +73%. The mixed model substantially outperformed three published models for lag to peak and one published model for centroid lag for estimating lag to peak for small basins in Maine. Lag to peak estimates are a key input to rainfall–runoff models used to design hydraulic infrastructure. The improved accuracy and consistency with model assumptions indicates that mixed models may provide increased data utilization that could enhance models and estimates of lag to peak in other regions.     

Does the Temporal Resolution of Precipitation Input Influence the Simulated Hydrological Components Employing the SWAT Model?

This study aimed to evaluate the influence of sub‐daily precipitation time steps on model performance and hydrological components by applying the Green and Ampt infiltration method using the Soil and Water Assessment Tool (SWAT). Precipitation was measured at a resolution of 0.1 mm and aggregated to 5‐, 15‐, 30‐, and 60‐min time steps. Daily discharge data over a 10‐year period were used to calibrate and validate the model. Following a global sensitivity analysis, relevant parameters were optimized through an automatic calibration procedure using SWAT‐CUP for each time step. Daily performance statistics were almost equal among all four time steps (NSE ≈ 0.47). Discharge mainly consisted of groundwater flow (55%) and tile flow (42%), in reasonable proportions for the investigated catchment. In conclusion, model outputs were almost identical, showing simulations responded nearly independently of the chosen precipitation time step. This held true for (1) the selection of sensitive parameters, (2) performance statistics, (3) the shape of the hydrographs, and (4) flow components. However, a scenario analysis revealed that the precipitation time step becomes important when saturated hydraulic conductivities are low and curve numbers are high. The study suggests that there is no need in using precipitation time steps <1 h for lowland catchments dominated by soils with a low surface runoff potential if daily flow values are being considered. Editor's note : This paper is part of the featured series on SWAT Applications for Emerging Hydrologic and Water Quality Challenges. See the February 2017 issue for the introduction and background to the series.     

MODELING GROUND WATER FLOW USING FLUX BOUNDARY CONDITIONS1

ABSTRACT: Determination of the boundary conditions for modeling ground water flow is a critical point especially in regional models. Normally the regional models require model areas that are greater than the given area of interest. This work focuses on the prediction of hydraulic heads in regional models using flux boundary conditions. The model uses flux boundary conditions that were estimated using a radial flow analog and Darcy's law. The regional model that is presented uses no parameter identification (inverse estimation) procedures. In the present work, the Houston area was used. The simulation of the hydrological conditions of the Chicot and Evangeline Aquifers that underlie the Houston area were made using the available information about the geological profile in the Houston region and the current information about the existing production wells. The regional model works as a forward problem. The system parameters such as hydraulic conductivity, specific storage, and hydrological stresses were specified, and the model predicts the hydraulic head. Actual data from piezometers operated by the U.S. Geological Survey (USGS) in many places throughout Houston were used as initial conditions. Some piezometric head data were generated using the regional variable theory called kriging to supply head estimates in areas where data were unavailable. The Modular Three Dimensional Finite Difference Groundwater Flow Model developed by the USGS was used to predict the hydraulic heads. The predicted ground water heads are compared to the actual data. The results show that the model performs well for locations where data were available.     

MODEL DEVELOPMENT FOR CONJUNCTIVE USE STUDY OF THE SAN JACINTO BASIN,CALIFORNIA1

ABSTRACT: A two-layered confined-unconfined numerical model for flow and mass transport is developed for the San Jacinto Basin. The model structure is determined by the geological structure of the Basin and model parameters are calibrated using 20 years of historical records. The total number of historical head observations used for the flow model calibration is 1,117 and the total number of the estimated parameters is 91. The two-layered transport model is also calibrated using historical water quality records. Sensitivity analysis of the flow model shows that only 68 parameters (out of a total of 91) are relatively sensitive and reliable. However, the unreliable parameters (23 of them) are found to be insensitive and thus not significant to the prediction and management of conjunctive use of surface water and ground water. The developed flow model has been used to study the two proposed artificial recharge scenarios for the San Jacinto Basin. We have found that during a relatively dry condition, an artificial recharge rate of 80 acre-ft/day can be achieved during the recharge period October through January. However, for a relatively wet condition, only 80 percent of the proposed rate can be effectively stored in the Basin during these months.     

HYSTAR Sediment Model: Distributed Two‐Dimensional Simulation of Watershed Erosion and Sediment Transport Using Time‐Area Routing

An erosion and sediment transport component incorporated in the HYdrology Simulation using Time‐ARea method (HYSTAR) upland watershed model provides grid‐based prediction of erosion, transport and deposition of sediment in a dynamic, continuous, and fully distributed framework. The model represents the spatiotemporally varied flow in sediment transport simulation by coupling the time‐area routing method and sediment transport capacity approach within a grid‐based spatial data model. This avoids the common, and simplistic, approach of using the Universal Soil Loss Equation (USLE) to estimate erosion rates with a delivery ratio to relate gross soil erosion to sediment yield of a watershed, while enabling us to simulate two‐dimensional sediment transport processes without the complexity of numerical solution of the partial differential governing equations. In using the time‐area method for routing sediment, the model offers a novel alternative to watershed‐scale sediment transport simulation that provides detailed spatial representation. In predicting four‐year sediment hydrographs of a watershed in Virginia, the model provided good performance with R² of 0.82 and 0.78 and relative error of ?35% and 11% using the Yalin and Yang's sediment transport capacity equations, respectively. Prediction of spatiotemporal variation in sediment transport processes was evaluated using maps of sediment transport rates, concentrations, and erosion and deposition mass, which compare well with expected behavior of flow hydraulics and sediment transport processes.     

Seawater Intrusion and Sustainable Yield of Basal Aquifers1

Liu, Clark C.K. and John J. Dai, 2012. Seawater Intrusion and Sustainable Yield of Basal Aquifers. Journal of the American Water Resources Association (JAWRA) 48(5): 861‐870. DOI: 10.1111/j.1752‐1688.2012.00659.x Abstract: Basal aquifers, in which freshwater floats on top of saltwater, are the major freshwater supply for the Hawaiian Islands, as well as many other coastal regions around the world. Under unexploited or natural conditions, freshwater and the underlying seawater are separated by a relatively sharp interface located below mean sea level at a depth of about 40 times the hydraulic head. With forced draft, the hydraulic head of a basal aquifer would decline and the sharp interface would move up. It is a serious problem of seawater intrusion as huge amounts of freshwater storage is replaced by saltwater. Also, with forced draft, the sharp interface is replaced by a transition zone in which the salinity increases downward from freshwater to saltwater. As pumping continues, the transition zone expands. The desirable source‐water salinity in Hawaii is about 2% of the seawater salinity. Therefore, the transition zone expansion is another serious problem of seawater intrusion. In this study, a robust analytical groundwater flow and salinity transport model (RAM2) was developed. RAM2 has a simple mathematical structure and its model parameters can be determined satisfactorily with the available field monitoring data. The usefulness of RAM2 as a viable management tool for coastal ground water management is demonstrated by applying it to determine the sustainable yield of the Pearl Harbor aquifer, a principal water supply source in Hawaii.     

Karst Hydrological Processes and Grey System Model1

Hao, Yonghong, Jiaojuan Zhao, Huamin Li, Bibo Cao, Zhongtang Li, and Tian‐Chyi J. Yeh, 2012. Karst Hydrological Processes and Grey System Model. Journal of the American Water Resources Association (JAWRA) 48(4): 656‐666. DOI: 10.1111/j.1752‐1688.2012.00640.x Abstract: The karst hydrological processes are the response of karst groundwater system to precipitation. This study provided a concept model of karst hydrological processes. The hydraulic response time of spring discharge to precipitation includes the time that precipitation penetrates through the vadose zone, and the subsequent groundwater pressure wave propagates to a spring outlet. Due to heterogeneities in karst aquifers, the hydraulic response time is different in different areas. By using grey system theory, we proposed a karst hydrological model that offers a calculation of hydraulic response time, and a response model of spring discharge to precipitation. Then, we applied the models to the Liulin Springs Basin, China. In the south part of the Liulin Springs Basin, where large fields of carbonate rocks outcrop with intensive karstification, the hydraulic response time is one year. In the north, where the Ordovician karst aquifer is covered by Quaternary loess sediments, the response time is seven years. The grey system GM(1,3) response model of spring discharge to precipitation was applied in consideration of the hydraulic response time. The model calibration showed that the average error was 6.55%, and validation showed that the average error was 12.19%.     

MANAGEMENT OF FLOOD CONTROL SUMPS AND POLLUTANT TRANSPORT1

ABSTRACT: Levee sump systems are used by many riverine communities for temporary storage of urban wet weather flows. The hydrologic performance and transport of stormwater pollutants in sump systems, however, have not been systematically studied. The objective of this paper is to present a case study to demonstrate development and application of a procedure for assessing the hydraulic performance of flood control sumps in an urban watershed. Two sumps of highly variable physical and hydraulic characteristics were selected for analysis. A hydrologic modeling package was used to estimate the flow hydrograph for each outfall as part of the flow balance for the sump. To validate these results, a water balance was used to estimate the total runoff using sump operational data. The hydrologic model calculations provide a satisfactory estimate of the total runoff and its time‐distribution to the sump. The model was then used to estimate pollutant loads to the sump and to the river. Although flow of stormwater through a sump system is regulated solely by flood‐control requirements, these sumps may function as sedimentation basins that provide purification of stormwater. A sample calculation of removals of several conventional pollutants in the target sumps using a mass balance approach is presented.     

ANALYSIS OF PUMPING TESTS: SIGNIFICANCE OF WELL DIAMETER,PARTIAL PENETRATION,AND NOISE1

ABSTRACT: The nonlinear least squares (NLS) method was applied to pumping and recovery aquifer test data in confined and unconfined aquifers with finite diameter and partially penetrating pumping wells, and with partially penetrating piezometers or observation wells. It was demonstrated that noiseless and moderately noisy drawdown data from observation points located less than two saturated thicknesses of the aquifer from the pumping well produced an exact or acceptable set of parameters when the diameter of the pumping well was included in the analysis. The accuracy of the estimated parameters, particularly that of specific storage, decreased with increases in the noise level in the observed drawdown data. With consideration of the well radii, the noiseless drawdown data from the pumping well in an unconfined aquifer produced good estimates of horizontal and vertical hydraulic conductivities and specific yield, but the estimated specific storage was unacceptable. When noisy data from the pumping well were used, an acceptable set of parameters was not obtained. Further experiments with noisy drawdown data in an unconfined aquifer revealed that when the well diameter was included in the analysis, hydraulic conductivity, specific yield and vertical hydraulic conductivity may be estimated rather effectively from piezometers located over a range of distances from the pumping well. Estimation of specific storage became less reliable for piezometers located at distances greater than the initial saturated thickness of the aquifer. Application of the NLS to field pumping and recovery data from a confined aquifer showed that the estimated parameters from the two tests were in good agreement only when the well diameter was included in the analysis. Without consideration of well radii, the estimated values of hydraulic conductivity from the pumping and recovery tests were off by a factor of four.     

Revisiting a statistical shortcoming when fitting the Langmuir model to sorption data

The Langmuir model is commonly used for describing the sorption behavior of reactive solutes to surfaces and is often fit to sorption data using nonlinear least squares regression. An important assumption of least squares regression is that the predictor variable is error free. In the case of sorption data, this assumption is not valid, and therefore the potential for parameter bias exists. Although alternative regression methods exist that either explicitly account for error in the predictor variable (Model II regression) or minimize the error in the predictor variable, these methods are not commonly used. Therefore, this paper more fully explores the differences in fitted parameters and model fits between these different data fitting methods by fitting P sorption data collected on 26 different soil samples using three different regression methods. For a majority of soils tested in this study, the differences in model fits between the three regression methods were not statistically significant. Statistical differences were observed in over a third of the soils, however, suggesting that errors in the predictor variable may be large enough to produce biased parameter estimates. These results suggest that multiple regression methods should be used when fitting the Langmuir model to sorption data to better assess the potential impact of error on model fits.     

Developing risk models of Cryptosporidium transport in soils from vegetated, tilted soilbox experiments

Transport of Cryptosporidium parvum through macroporous soils is poorly understood yet critical for assessing the risk of groundwater contamination. We developed a conceptual model of the physics of flow and transport in packed, tilted, and vegetated soilboxes during and immediately after a simulated rainfall event and applied it to 54 experiments implemented with different soils, slopes, and rainfall rates. Using a parsimonious inverse modeling procedure, we show that a significant amount of subsurface outflow from the soilboxes is due to macropore flow. The effective hydraulic properties of the macropore space were obtained by calibration of a simple two-domain flow and transport model that accounts for coupled flow in the matrix and in the macropores of the soils. Using linear mixed-effects analysis, macropore hydraulic properties and oocyst attenuation were shown to be associated with soil bulk density and rainfall rate. Macropore flow was shown to be responsible for bromide and C. parvum transport through the soil into the underlying pore space observed during the 4-h experiments. We confirmed this finding by conducting a pair of saturated soil column studies under homogeneously repacked conditions with no macropores in which no C. parvum transport was observed in the effluent. The linear mixed-effects and logistic regression models developed from the soilbox experiments provide a basis for estimating macropore hydraulic properties and the risk of C. parvum transport through shallow soils from bulk density, precipitation, and total shallow subsurface flow rate. The risk assessment is consistent with the reported occurrence of oocysts in springs or groundwater from fractured or karstic rocks protected only by shallow overlying soils.     

Empirical Estimation of Stream Discharge Using Channel Geometry in Low‐Gradient,Sand‐Bed Streams of the Southeastern Plains

Manning's equation is used widely to predict stream discharge (Q) from hydraulic variables when logistics constrain empirical measurements of in‐bank flow events. Uncertainty in Manning's roughness (n_M) is the major source of error in natural channels, and sand‐bed streams pose difficulties because flow resistance is affected by flow‐dependent bed configuration. Our study was designed to develop and validate models for estimating Q from channel geometry easily derived from cross‐sectional surveys and available GIS data. A database was compiled consisting of 484 Q measurements from 75 sand‐bed streams in Alabama, Georgia, South Carolina, North Carolina (Southeastern Plains), and Florida (Southern Coastal Plain), with six New Zealand streams included to develop statistical models to predict Q from hydraulic variables. Model error characteristics were estimated with leave‐one‐site‐out jackknifing. Independent data of 317 Q measurements from 55 Southeastern Plains streams indicated the model (Q = A_cR_H^0.6906S^0.1216; where A_c is the channel area, R_H is the hydraulic radius, and S is the bed slope) best predicted Q, based on Akaike's information criterion and root mean square error. Models also were developed from smaller Q range subsets to explore if subsets increased predictive ability, but error fit statistics suggested that these were not reasonable alternatives to the above equation. Thus, we recommend the above equation for predicting in‐bank Q of unbraided, sandy streams of the Southeastern Plains.     

OPTIMAL IDENTIFICATION OF MUSKINGUM ROUTING COEFFICIENTS1

Traditionally, identification of the Muskingum routing coefficients has been based on observations of the linearity of a loop formed by graphically plotting a forward and a reverse path. This graphical procedure is time-consuming and may not minimize the error of estimation. A procedure was developed to improve the drawbacks of the graphical method. This procedure calls for (a) the use of least square regression on the forward and reverse paths to determine their respective slopes, and (b) the use of statistical t-test to evaluate the hypothesis that these two slopes are equal. The computational procedure is repeated, using incremental values of the flow weighting coefficient, x. A graph of the computed t-value versus x can be constructed. The optimal value of x, as read from the graph, occurs at the minimum computed t-value. The procedure has been demonstrated superior to the graphical method for three illustrative examples, resulting in a reduction of the error squares by factors ranging from 5 to 6.     

DYNAMIC MODELING OF STREAM QUALITY BY INVARIANT IMBEDDING1

ABSTRACT. The estimator equations obtained using invariant imbedding is used to estimate the parameters in river or stream pollution. By using these equations, the parameters can be estimated directly from differential equations representing the pollution model and from measured noisy data such as BOD and DO. Another advantage of this approach is that a sequential estimation scheme is obtained. By using this sequential scheme, only current data are needed to estimate current or future values of the unknown parameters. Consequently, a large amount of computer time and computer memory can be saved. Furthermore, not only the parameters but also the concentrations of pollutants can be estimated. Thus, it also forms an effective forecasting technique. The classical least squares criterion is used in the estimation. Several examples are solved to illustrate the technique. (KEY WORDS: dynamic modeling; water pollution; invariant imbedding; forecasting; least squares criterion; estimation)