Investigating network models as a numerical method for solving groundwater equations

Doctoral dissertation in the field of civil-water and environmental engineering

Abstract

         Network methods are among the most widely used tools for underground water modeling, which have expanded and gained a lot of acceptance during the last two decades. On the other hand, the advancement of computing power of computers and the ease of access to them have led to the rapid development of numerical methods to solve groundwater problems. In this research, with a new approach to network methods, these methods are introduced as a numerical model for simulating the movement of underground water. For this purpose, by substituting a square grid instead of the porous medium and solving the aforementioned grid, a system of algebraic equations is obtained, which solves the hydraulic head distribution inside the porous medium. Also, more accurate answers have been obtained by making corrections in the aforementioned network, including the addition of diagonal members and the use of imaginary members. Finally, with the development of this network model, a non-systematic model has been presented to solve the flow in an arbitrary network. To validate the model, various problems were solved and the answers were compared with finite difference numerical methods and finite elements in permanent and non-permanent states. The persistent problems solved in this research are: simulation of flow without the presence of springs and wells (Laplace's equation) in square, rectangular, triangular domains, 90° intersection and flow passing around the cylinder, and simulation of flow with the presence of wells (Poisson's equation) in a rectangular domain. The unsteady problems investigated also include flow simulation in one-dimensional and square domains. The results obtained from solving these problems indicate that, firstly, network methods can be developed as a numerical tool and used to model flow in porous media. Secondly, the use of diagonal and imaginary members leads to more accurate answers than finite difference and finite element methods. Another advantage of the network method is the possibility of making this model in a laboratory. In this research, a network of pipes was built in the laboratory and issues such as water movement around an impermeable rectangle, in an environment with compound boundaries, under the dam seal curtain, in an open aquifer, and in a heterogeneous aquifer were investigated by it. The results obtained from the set of these tests showed that, firstly, this laboratory device is simpler from the point of view of construction and use, compared to other devices such as sandbox, spherical balls, electric circuits and other common laboratory models, and secondly, despite its simplicity, the results obtained from it are in good agreement with the numerical solutions presented in the research.

1.       General

1-1 Introduction

The simulation of flow in porous media has attracted the attention of many researchers during the past three decades. (Vafai, 2005) Various applications of this simulation can be seen in branches such as water engineering, environmental engineering, petroleum engineering, and groundwater hydrology. (Afzali, 1387).

The underground water obtained by the pump from the underground structures is the main source of many water supply systems. The amount of water output of a spring is considered as the output of the underground water system, it can be affected to a great extent by the amount of pumping done from the same area. Water for storage can be injected into wells dug for this purpose and the underground water level can be raised using this technique. These issues are among the issues that can have an impact on groundwater management.

Actually, in the groundwater management system, qualitative and quantitative issues cannot be considered separately from each other. In many parts of the world, due to the extraction of underground water in excess of the allowed limit, the quality of these waters has been steadily declining, which has drawn the attention of both consumers and producers to this issue. Deterioration of the quality of underground water can be due to the increase in water salinity or the increase in the concentration of ions such as nitrate.

In recent years, in addition to the aforementioned general issues affecting water quality, public attention has been paid to the issue of groundwater pollution by toxic industrial wastes, leachate from waste disposal, petroleum and other toxic liquids, fertilizers, herbicides and insecticides used in agriculture, and radioactive substances buried deep in the ground.Although many of these issues happen on the surface of the earth, these pollutants join the underground water after penetrating the earth. After joining the underground water, these pollutants are transferred by the movement of underground water and reach the rivers, lakes and harvesting wells. On the other hand, the limited amount of underground water has caused the increasing importance of underground water as a source of human drinking water.

Any planning for control and cleaning operations requires estimation and estimation of the studied values. Subsequently, any operation that is used to read the studied values, needs to know how the underground water behaves. Therefore, proper management is realized when the response of the desired system to the intended activities can be known.

One of the first steps required to estimate the behavior of groundwater is to find a mathematical model, and the use of these models in turn requires the collection of information. The more accurate the collected information is, the more reliable the answers of the presented model can be, although the collection of information is always faced with errors and uncertainty. By using test systems, the cause of errors that are caused by human errors or human ignorance can be greatly reduced and eliminated. For this reason, there are many studies on issues such as taking care of the head of a well, designing drinking water supply systems, estimating the movement and transfer of pollutants in the aquifer, and so on. (Vedat, 2006) and mathematical models have been presented for these problems.

1-2 The purpose of this research

As mentioned, the first step to estimate the behavior of underground water is to obtain a mathematical model. According to Darcy's law and the establishment of mass constancy, it can be shown that the governing equation for underground water in stable conditions is Laplace's equation. Therefore, by obtaining the physical properties of the desired aquifer using experiments and finding out the governing boundary conditions of that aquifer, the complete mathematical model governing the desired problem can be obtained. Unfortunately, since the physical properties of the problems in nature are not homogeneous and the boundaries of the studied problems are geometrically irregular, it is impossible to solve these problems analytically. To solve this problem, researchers have been benefiting from numerical and laboratory methods for many years.

All numerical methods to solve Laplace's equation first decompose the domain of the desired problem, which means that they either divide it into several nodes or into several elements, and then find algebraic relationships between those nodes or different elements using mathematical methods. In other words, after analyzing the domain of the problem, they convert Laplace's equation into a system of linear equations.

One of the broadest fields that is used in the investigation and study of porous media is the Pore Network Model (PNM) method (Joekar et al., 2012). In this method, the holes in the porous medium are simulated as a network of holes that are connected to each other by connecting channels. This method has gained a lot of acceptance during the past few decades and in recent years, due to the progress of computing systems and also the progress in producing high quality photos of the porous medium, its acceptance and use by researchers has increased. In this research, with a completely new and different attitude from what has been presented so far about the PNM method, the solution of the groundwater equation in the saturated state is done using the PNM method. In this research, instead of solving the flow equations in a porous medium, the equations of a network of pipes are solved. Since the relationship between the head of two points and the amount of flow passing between those two points in slow flow is linear, it is possible to write an equation governing the slow flow in a network of pipes as a linear system of equations and unknowns, and by solving this system, the approximate solution of the groundwater equations can be obtained. The purpose of this research can be summarized as follows: How to construct the matrix of coefficients using PNM method for First, second and third type boundaries.

The effect of factors such as inhomogeneous and inhomogeneous porous medium on the matrix obtained by PNM method.