Stochastic estimation of states in unconfined aquifers subject to artificial recharge

Abstract

An extended Kalman filter model for characterizing minimum variance estimates of the piezometric heads and coefficients defining an unconfined aquifer subject to artificial recharge is developed. The system evolution model employs Hantush's (1967) model. Sensitivity analyses are used to test the estimation capability of the technique. The ability of the extended Kalman filter to use all available information from both the system model and measurements of the state to provide approximate minimum variance estimates of the state and confidence limits is shown. The extended Kalman filter version of Hantush's model is applied to a record data length of one year obtained at the study area located in Norwood, Ontario. INTRODUCTIONIn many regions, increases in demand for water have been met by groundwater extractions resulting in the 'mining' of the water. Concurrently, the quality of the surface water body receiving the increased effluent has deteriorated. To halt or even reverse these trends, artificial groundwater recharge is receiving increased attention. In some situations the water source for use in the recharge is secondary sewage effluent '[e.g., Bouwer, 1976]. The basis of the recharge concept is that the vadose zone and the aquifers purify the water chemically and biologically, thereby, allowing its reuse.Groundwater systems are unique •in water resources managemerit because they can only be observed and controlled indirectly, using measurements obtained from the surface. Many deterministic mathematical models to infer the groundwater system response and parameters have been developed, all of them essentially being based on hydrologic and geologic information such as the soil characteristics, infiltration, and pumpage rates. The parameter estimates for the majority of the models are obtained either directly in the field or from disturbed samples in a laboratory. These model parameters are then assumed to be constant and perfectly known. However, these parameters exhibit large temporal and spatial fluctuations due to the heterogeneity and anisotropy of the aquifer, the stochastic meteorological conditions, and other changing environmental factors. In addition, due to the measurement errors there is always some uncertainty implicit in assignment of parameter magnitudes at any point in time or space. As well, any model represents only an abstraction of the real system. As a consequence the modeling of a stochastic process by a deterministic model may be subject to significant errors. Therefore the development of methodologies to interpret the observable information and thereby provide more accurate estimates of the responses of the system is of increasing interest. Only recently have the uncertainties of hydrologic system parameters been incorporated into a groundwater modeling technique. Wilson et al. [1978] utilized a state estimation technique, the extended Kalman filter, which combines an uncertain system model and noisy measurements of system Paper number 2W0709. 0043-1497/82/002 W-0709505.00 behavior to dete...

Kalman filtering in groundwater flow modelling: problems and prospects

“…The presence of measurement nonlinearity in the observation equations was overcome by performing local iterations within the algorithm while retaining the recursive structure of the ®lter. Schmidtke et al (1982) applied an extended Kalman ®lter (EKF) for joint on-line state-parameter estimation in an uncon®ned aquifer subject to arti®cial recharge, with equations of state based on the model of Hantush (1967). The major weakness here, however, is that although analytic and semi-analytic groundwater ow models are well suited for use with the EKF, the¯ow problems must have geometrically regular features, such as linear vertical boundaries, horizontally layered media, and equally spaced fractures.…”

Simultaneous Estimation of Transmissivity Values and Zonation

“…Some success has been achieved in using the KF to improve current state estimates of head [Schmidtke et al, 1982;Zhou et al, 1991;Graham and Tankersley, 1993;Hantush and Mariño, 1994] and contaminant concentrations [Jinno et al, 1989;Yu et al, 1989;McLaughlin, 1989, 1991;Zou and Parr, 1995]. The filter continues to show promise in the optimal design of monitoring networks Andricevic, 1993] and in determining optimal pumping rates [Andricevic and Kitanidis, 1990;Lee and Kitanidis, 1991;Andricevic, 1993].…”

“…This is generally done by predetermining the parameter structure in one of three ways: (1) the transmissivity field is considered to be homogeneous [Schmidtke et al, 1982;Van Geer and Van Der Kloet, 1985;Jinno et al, 1989;Andricevic and Kitanidis, 1990], (2) heterogeneous transmissivities values are considered distributed over all the nodes or elements in the discretized domain [Wilson et al, 1978], or (3) the transmissivity field is parameterized by a priori "zonation" of the transmissivity field into piecewise constant partitions [Jinno et al, 1989;Lee and Kitanidis, 1991] or interpolated from a few basis nodes [Yeh and Yoon, 1981; M. C. Hill et al, A controlled experiment in groundwater flow modeling, 1, Calibration using nonlinear regression to estimate parameter values, submitted to Water Resources Research, 1994, hereinafter referred to as Hill et al, submitted manuscript, 1994. Each of these three methods has its advantages and disadvantages.…”

“…Most applications in this area tend to force low dimensionality of the parameter space (through assumptions of homogeneity or a priori zonation), utilize small synthetic domains, and have a large number of high-quality observations available for conditioning. Schmidtke et al [1982] used EKF to estimate homogeneous permeability, fillable porosity, and recharge rate in a small two-dimensional transient flow domain based on filtered head measurements. Van Geer and Van Der Kloet [1985] estimated the single lumped ratio of transmissivity over storativity in a onedimensional transient flow problem by filtering perfectly known head measurements from two observation wells; they found that a suboptimal version of the filter, in which head estimates were not conditioned on measurements, introduced little error into the transmissivity estimation.…”

Analytical series expressions for Hantush's M and S functions

“…Since that time the mounding functions M* and S* and the component error and exponential integral functions, have been studied repeatedly and to high numerical accuracy by several workers in various hydrological contexts [Schmidtke et al, .1982;Allen, 1986;Kinzelbach, 1986]. The advent of advanced mathematical tools, for example, the computer package Mathematica [Wolfram, 1992] and others like it, has made such numerical evaluations easy.…”