Surrogate modeling for unsaturated infiltration via the physics and equality-constrained artificial neural networks

Machine learning(ML)provides a new surrogate method for investigating groundwater flow dynamics in unsaturated soils.Traditional pure data-driven methods(e.g.deep neural network,DNN)can provide rapid predictions,but they do require sufficient on-site data for accurate training,and lack interpretability to the physical processes within the data.In this paper,we provide a physics and equalityconstrained artificial neural network(PECANN),to derive unsaturated infiltration solutions with a small amount of initial and boundary data.PECANN takes the physics-informed neural network(PINN)as a foundation,encodes the unsaturated infiltration physical laws(i.e.Richards equation,RE)into the loss function,and uses the augmented Lagrangian method to constrain the learning process of the solutions of RE by adding stronger penalty for the initial and boundary conditions.Four unsaturated infiltration cases are designed to test the training performance of PECANN,i.e.one-dimensional(1D)steady-state unsaturated infiltration,1D transient-state infiltration,two-dimensional(2D)transient-state infiltration,and 1D coupled unsaturated infiltration and deformation.The predicted results of PECANN are compared with the finite difference solutions or analytical solutions.The results indicate that PECANN can accurately capture the variations of pressure head during the unsaturated infiltration,and present higher precision and robustness than DNN and PINN.It is also revealed that PECANN can achieve the same accuracy as the finite difference method with fewer initial and boundary training data.Additionally,we investigate the effect of the hyperparameters of PECANN on solving RE problem.PECANN provides an effective tool for simulating unsaturated infiltration.

GROWTH AND YIELD MODELS FOR DAHURIAN LARCH PLANTATIONS

Growth and yield models were developed for individual tress and stands based on 336 temporary plots with 405 stem analysis trees of dahurian larch ( Larix gmelinii( Rupr. )Rupr.) plantations throughout Daxing'anling mountains. Several equations were selected using nonlinear regression analysis. Results showed that the Richards equation was the best model for estimating tree height, stand mean height and stand dominant height from age; The Power equation was the best model for prediction tree volume from DBH and tree height; The logarithmic stand volume equation was good for predicting stand volume from age, mean height, basal area and other stand variables. These models can be used to construct the volume table, the site index table and other forestry tables for dahurian larch plantations.

An Efficient Multiple-Dimensional Finite Element Solution for Water Flow in Variably Saturated Soils

Multiple-dimensional water flow in variably saturated soils plays an important role in ecological systems such as irrigation and water uptake by plant roots; its quantitative description is usually based on the Richards＇ equation. Because of the nonlinearity of the Richards＇ equation and the complexity of natural soils, most practical simulations rely on numerical solutions with the nonlinearity solved by iterations. The commonly used iterations for solving the nonlinearity are Picard and Newton methods with the former converging at first-order rate and the later at second-order rate. A recent theoretical analysis by the authors, however, revealed that for solving the diffusive flow, the classical Picard method is actually a chord-Newton method, converging at a rate faster than first order; its linear convergence rate is due to the treatment of the gravity term. To improve computational efficiency, a similar chord-Newton method as for solving the diffusive term was proposed to solve the gravity term. Testing examples for one-dimensional flow showed significant improvement. The core of this method is to produce a diagonally dominant matrix in the linear system so as to improve the iteration-toiteration stability and hence the convergence. In this paper, we develop a similar method for multiple-dimensional flow and compare its performance with the classical Picard and Newton methods for water flow in soils characterised by a wide range of van Genuchten parameters.

Effects of Soil Hydraulic Properties on Soil Moisture Estimation

Accurate quantification of soil moisture is essential to understand the land surface processes.Soil hydraulic properties influence water transport in soil and thus affect the estimation of soil moisture.However,some soil hydraulic properties are only observable at a few field sites.In this study,the effects of soil hydraulic properties on soil moisture estimation are investigated by using the one-dimensional(1-D)Richards equation at ELBARA,which is part of the Maqu monitoring network over the Tibetan Plateau(TP),China.Soil moisture assimilation experiments are then conducted with the unscented weighted ensemble Kalman filter(UWEnKF).The results show that the soil hydraulic properties significantly affect soil moisture simulation.Saturated soil hydraulic conductivity(Ksat)is optimized based on its observations in each soil layer with a genetic algorithm(GA,a widely used optimization method in hydrology),and the 1-D Richards equation performs well using the optimized values.If the range of Ksat for a complete soil profile is known for a particular soil texture(rather than for arbitrary layers within the horizon),optimized Ksat for each soil layer can be obtained by increasing the number of generations in GA,although this increases the computational cost of optimization.UWEnKF performs well with optimized Ksat,and improves the accuracy of soil moisture simulation more than that with calculated Ksat.Sometimes,better soil moisture estimation can be obtained by using optimized saturated volumetric soil moisture content Ksat.In summary,an accurate soil profile can be obtained by using soil moisture assimilation with optimized soil hydraulic properties.

Stable expression and control parameters in the numerical simulation of unsaturated flow

The Richards’equation describes the flow phenomenon in unsaturated porous media and is essential to hydrology and environmental science.This study evaluated the numerical stability of two different forms of the Richards’equation.Sensitivity analyses were performed to investigate the control parameters of the equation.The results show that the h-form Richards’equation has better applicability for calculating variable saturation flows than theθ-form Richards’equation.For the h-form Richards’equation,the hydraulic conductivity of the soil in the low-suction range and the specific moisture capacity in the high-suction range primarily influenced the solution.In addition,sensitivity analyses indicated that the saturated hydraulic conductivity,initial condition,and air-entry pressure have a higher sensitivity to the simulation results than the saturated water content,rainfall intensity,and decline rate of hydraulic conductivity.Moreover,their correctness needs to be guaranteed first in numerical simulations.The research findings can provide a helpful reference for improving the reliability of numerical simulations of unsaturated flows.

An h-Adaptive Finite Element Solution of the Relaxation Non-Equilibrium Model for Gravity-Driven Fingers

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media.In this paper,a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon.Towards the efficiency and accuracy of the numerical simulations,a predictor-corrector process is used for resolving the nonlinearity of the equation,and an h-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region,in which a heuristic a posteriori error indicator is designed for the purpose.In numerical simulations,a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method.The effectiveness of the h-adaptive method is also successfully demonstrated by numerical experiments.Finally the mechanism on generating fingers is discussed by numerically studying several examples.