Evaluating marginal likelihood is the most critical and computationally expensive task, when conducting Bayesian model averaging to quantify parametric and model uncertainties. The evaluation is commonly done by using Laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using Monte Carlo (MC) methods to evaluate arithmetic or harmonic mean of a joint likelihood function. This study introduces a new MC method, i.e., thermodynamic integration, which has not been attempted in environmental modeling. Instead of using samples only from prior parameter space (as in arithmetic mean evaluation) or posterior parameter space (as in harmonic mean evaluation), the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space. This is done through a path sampling that conducts Markov chain Monte Carlo simulation with different power coefficient values applied to the joint likelihood function. The thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the Laplace approximation method and three MC methods, including the nested sampling method that is recently introduced into environmental modeling. The thermodynamic integration method outperforms the other methods in terms of their accuracy, convergence, and consistency. The thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models. The application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of Bayesian model averaging. The thermodynamic integration method is mathematically rigorous, and its MC implementation is computationally general for a wide range of environmental problems.
Groundwater is often used for domestic and irrigation purposes, even in mining areas. Mine drainage, rainfall, and infiltration cause heavy metal enrichment, adversely affecting the groundwater and harming human health. In this study, water samples (October 2021) in the Suzhou southern coal mining area were analyzed for the heavy metals As, Cr, Cu, Fe, Mn, Pb, and Zn to determine potential effects of heavy metal contamination on environmental quality and human health. It was found that 22% and 31% of the sampling sites had “excellent” and “good” water quality, respectively. Excessive concentrations of Fe and Mn were detected in 47% and 72% of the samples, respectively. The non-carcinogenic health risk values of As, Cr, Cu, Fe, Mn, Pb, and Zn were below the negligible levels of health risk set by various environmental agencies. Content ranking was as follows: Fe > Mn > Cr > Cu > Pb > Zn > As, with Fe accounting for 43%. All sampling points exceeded the maximum acceptable level of Cr recommended by the agencies. Chromium, the major carcinogenic factor in the study area, contributed to 95.45% of the total health risk. Therefore, the authorities in this region must closely monitor three heavy metal elements—Fe, Mn, and Cr.
Abstract:The exploitation of shallow geothermal energy through a groundwater heat pump (GWHP) is always limited to thick and deep aquifers containing abundant water with a relatively stable temperature. Unfortunately, aquifers in hilly regions which occupy two thirds of China are usually thin and shallow. The boundary conditions in those hilly areas affect the groundwater flow that is used for geothermal energy production. To quantify the impact of boundary conditions on the shallow geothermal energy development, a shallow and thin aquifer near the Qingyi River in Anhui Province was chosen as a case study, and a three-dimensional heat-water model was developed using FEFLOW. The impact of the boundary conditions on the hydrodynamic and temperature fields of the aquifer was analyzed by using the developed model. Furthermore, the well locations of a pumping-recharging system near the river correspond to three different modes of pumping-recharging well layouts that were optimized based on the changes of pumping water temperature and the maximum drawdown. The simulation results indicated that the influence of atmospheric temperature on groundwater temperature is negligible below a depth of 11 m. When the river level is above 28 m, the optimal scheme of pumping-only was used (without considering recharging wells) with a certain distance from the river. This scheme not only operates efficiently, but also reduces the operation cost.
In a semi-infinite aquifer bounded by a channel, a transient flow model is constructed for phreatic water subjected to vertical and horizontal seepage. Based on the first linearized Boussinesq equation, the analytical solution of the model is obtained by Laplace transform. Having proven the transformation between the analytical solution and some relevant classic formulas, suitable condition for each of these formulas is demonstrated. On the base of the solution, the variation of transient flow process caused by the variables, such as vertical infiltration intensity, fluctuation range of river stage, aquifer parameters such as transmissivity and specific yield, and the distance from calculating point to channel boundary, are analyzed quantitatively one by one. Lagging effect will happen to the time, when phreatic water gets its maximum fluctuation velocity, response to the varying of the variables stated above. The condition for some variables to form equivalent lagging effect is demonstrated. Corresponding to the mathematical characteristics of the analytical solution, the physical implication and the fluctuation rule of groundwater level are discussed.
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