Investigation of a stochastic solute transport model and development of inverse methods in hydrogeology

Abstract

This thesis encompasses two aspects of stochastic solute transport modelling in porous media. One part is dedicated to investigating a stochastic transport model. In the second part, two types of inverse methods are developed to solve the inverse problem in stochastic contaminant transport modelling of saturated porous media in heterogeneous aquifers.

In the first part, a one-dimensional stochastic solute transport model (SSTM) is investigated to identify its behaviour under different heterogeneous conditions in porous media. This stochastic computational model has been developed without using Fickian assumptions. The model consists of two main parameters, correlation length and variance of solute velocity, which is assumed as a fundamental stochastic variable. The velocity is represented by the mean velocity described by Darcy's law, with added white noise, which is correlated in space and time. As the statistical nature of the SSTM changes with the parameters, the computational solutions of the model are explored in relation to these parameters. The variance is found to be the dominant parameter, however, there is a correlation between both parameters. Moreover, these parameters influence the stochasticity of the flow in a complex manner. We hypothesise that the variance is inversely proportional to the pore size and the correlation length represents the geometry of the flow. The computational exploration results from different scales show that the hypotheses are reasonable. The results further reveal that the SSTM is capable of simulating the scale effect of solute dispersion and, to some extent, they agree with the past literature.

In second part, we develop two different inverse methods to estimate the hydrogeologic parameters. In the first inverse method, the maximum likelihood approach is used to estimate the parameters of a stochastic partial differential equation that describes the dynamics of a heterogeneous groundwater flow. The applications are within the context of one and two-dimensional solute transport models. Initially, the method is used to estimate an unknown single parameter of a system. It is then extended to determine two parameters simultaneously. The results show that this inverse method is reasonably reliable in the presence of noise. However, the investigation of parameter estimates also shows an inverse relationship to the noise level of the system. The main advantage of this estimation method is its direct dependence on field observations of the state variables of natural systems in the presence of uncertainty.

In the second inverse method, the inverse problem in groundwater modelling is solved using a hybrid Artificial Neural Networks (ANN) approach. Initially, a Multi Layer Perceptron (MLP) network is developed and it is found that the network produces better results when the target range of the parameters is smaller. Therefore, a Self-Organising Map (SOM) is used to identify the objective subrange of the parameters and then the MLP model is employed to obtain final estimates. The data for the ANN is obtained from a numerical model that simulates the solute transport in a saturated groundwater flow. The forward problem of the numerical model is solved to generate solute concentration data for a range of parameters. These input data are fed into a MLP network to train the network, along with the corresponding parameter values. A sufficiently trained ANN model is used to estimate hydraulic conductivity (single parameter) and hydraulic conductivity and longitudinal dispersion coefficient (two parameters). This approach is first tested on synthetic data to identify its feasibility and robustness. Then, an experimental dataset obtained from the artificial aquifer at Lincoln University, New Zealand is used to validate the method. It is found that ANN produce accurate estimates in the presence of uncertainty.

The hybrid ANN inverse method is then applied to determine the parameters of the SSTM. Results reveal that the ANN model predictions are satisfactory and the average absolute error of the estimates for a highly random data range is approximately 5.5% and outcomes are better for other ranges. Finally, the method is applied to the artificial aquifer to attain its parameters. Results are validated using a more conventional curve fitting technique. This validation shows that the estimates of ANN are dependable. However, ANN are able to produce accurate results only if the pattern of the dataset used to estimate parameters are similar to that of the training data. Therefore, it is important to adequately simulate the aquifer system of interest with a large enough training dataset. Nevertheless, due to the stochastic nature of the real world heterogeneous aquifers, it is not a trivial undertaking to identify the behaviour of the aquifers. Thus, it is found that prior information about the system is of utmost importance to obtaining reasonably accurate estimates. Additionally, as ANN's extrapolation capabilities beyond its calibration range are not reliable, it is necessary to set a calibration range sufficient to meet the limits of actual data.