Numerical Modeling of Groundwater Flow in Steady State Using Isogeometric Method

Abstract
Solving the governing equations of a system is the most important issues that is always discussed in science and engineering fields. Since there are few equations that have analytical solution, many numerical methods have been proposed for solving the equations that have no analytical solutions. Numerical methods are developed by the advent of computers. Today, with using computers and these methods together, complicated equations in diverse areas can be resolved. Several numerical methods such as finite element method (FEM), finite difference method (FDM) and meshless (MFree) method have been suggested for solving partial differential equations. In this study, Isogeometric analysis method is engaged as a numerical method. Isogeometric analysis was developed by Hughes in 2005 in order to eliminate the gap between the world of finite element analysis and computer modeling. This method uses the same basis functions, in the process of modelling. Isogeometric method provides the possibility of simulation in irregular and complex geometry domains and also removes errors due to the multiple elements. Two variable NURBS basis functions are defined by B-spline basis functions. B-spline basis functions are calculated by the Cox–de Boor recursion. In this study, Birjand aquifer is modeled in two dimensions by the Isogeometric analysis using four-point Gauss integration method. The mentioned aquifer is defined by 1274 points and 836 control elements. After creating the geometry of the aquifer by control points and knot vector, NURBS basis functions and their derivatives were calculated. Then, with using input information, such as hydraulic conductivity coefficients, boundary conditions, precipitation rates and the sources and sinks, water table is computed. In order to allocate hydraulic conductivity coefficients of the aquifer, the domain is divided by the GIS software to multiple homogeneous Thiessen. According to the location of NURBS elements in the aquifer, a value has been assigned to NURBS elements. In Birjand aquifer there are boundary conditions with constant head. For enforcing the boundary conditions, 83 points of control points were defined as fixed head. There are 190 wells in the Birjand aquifer, the Extracted water from the wells were used as the discharge rate in the model. Also, 15 percent of the amount of rainfall was considered as the recharge rate in 2011-2012 period, the value of recharge rate is 0.0000727 m/day based on rain gauges. In order to ensure the accuracy of modeling the results of Isogeometric method is compared with finite difference method solutions and observation data, the relative mean error of Isogeometric method is 0.000256. With using Isogeometric method the consumed time for running the MatLab code is around 400 seconds. In order to evaluate the model, three criteria is calculated. Mean error (ME), mean absolute error (MAE) and root mean square error (RMSE) whose values are 0.09, 0.34, and 0.459 respectively. The values of the error and computation time has shown the power of this method in modeling of groundwater flow. Finally, Birjand aquifer groundwater balance was calculated using the input values, extracted water and water storage in plain. By studying the model balance and actual balance of aquifer and comparing them with each other, it is determined that the change in the volume of the aquifer in the time period considered is close to that of the aquifer, which indicates the accuracy of the model.

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Subjects

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Modares Civil Engineering journal
Volume 18, Issue 3
September and October 2018
Pages 195-206