Generally speaking, there are two main methods to study natural or social phenomena: mechanism analysis and statistical analysis. The former uses classical mathematical tools to analyze the causal relationship of phenomena; The latter takes stochastic mathematics as a tool, seeks statistical laws through a large number of observation data, and finally describes it with some mathematical relationship or mathematical model. Among them, it is very important to establish a mathematical model to simulate and predict the phenomenon. In hydrogeology, from the formula established by Qiu Buyi and Tess to the three-dimensional flow model simulating the quantity and quality of groundwater at present, it shows that mathematical model method is a very important technical method in hydrogeology. Especially in recent years, with the high development of digital computer and the continuous improvement of numerical analysis technology, mathematical models have been widely used in hydrogeology (Lin et al., 1995).

I. Mathematical model

The so-called mathematical model is a mathematical structure obtained by making some necessary descriptions and assumptions about specific objects in the real world for specific purposes and then using appropriate mathematical tools. It can not only explain the actual form of specific phenomena, but also predict the future situation of the object, and also provide the optimal decision or control to deal with the object. Mathematical model of groundwater is the result of generalization, transformation and induction of hydrogeological conditions and water resources information by using mathematical language and tools. After deductive reasoning, the mathematical model gives mathematical analysis, prediction, decision-making or control, and then returns to practical application after explanation. Finally, after practice, if the result is correct or basically correct, it can be used to guide practice; Otherwise, we should reconsider the process of induction and modify the mathematical model, as shown in figure 15- 1.

Figure 15- 1 Relationship between Hydrogeological Problems and Mathematical Models

From the current practical application, mathematical model of groundwater can be divided into three categories, namely analytical model, numerical model and multivariate statistical model. The analytical model consists of various analytical solutions of differential equations describing groundwater flow, such as Theis formula and Qiubuyi formula. The analytical model is only suitable for groundwater flow problems with relatively uniform aquifer, simple geometry, small scope and simple source and sink terms. When establishing groundwater analysis model, a "model aquifer" with straight boundary, effective width, thickness and length is usually used to simulate the groundwater conditions in the study area. The solution of the model is obtained by using the ideal aquifer with average hydraulic properties and according to the mirror image theory and certain groundwater flow equation.

For complex conditions that are not suitable for analytical model, numerical model can be used to establish corresponding partial differential equations and get numerical solutions. In order to establish a numerical model, it is necessary to discretize the aquifer system with continuous parameters into several subdivision units and discretize the time variables equally. Then a set of linear algebraic equations is formed by using finite difference principle, finite element principle or boundary element principle. Then, with the help of digital computer, this group of linear algebraic equations is solved numerically. According to the different principles of establishing equations, different numerical models such as finite difference method, finite element method and boundary element method can be produced.

Because groundwater system is a multivariable system, some multivariate statistical models can also be used to solve groundwater flow problems. The multivariate statistical analysis method is used to process all kinds of hydrogeological observation data, evaluate some characteristics or laws of groundwater, and predict and explore the distribution and variation laws of chemical components of groundwater. , you can get some quantitative information. For example, multiple regression analysis can quantitatively establish the mathematical relationship expression between one variable and another variable or several variables in groundwater system, so as to study the restrictive relationship and correlation between variables, and make evaluation and prediction. For another example, the factor analysis model or correspondence analysis model is to reduce some complex factors in the groundwater system to a few comprehensive factors through some internal relations, and then analyze the distribution and genetic relationship between groundwater samples and variables to obtain regular information. With the development of science and technology, some new groundwater multivariate analysis models have appeared in recent years, such as time series model and grey system model. They all play an active role in the process of groundwater management.

Second, the establishment and application of mathematical model of groundwater

There is no certain pattern in the steps of establishing a mathematical model, but there are generally the following processes.

First of all, we should understand and master the hydrogeological conditions, various phenomena, information and statistical data, and make clear the purpose of establishing the model and the practical problems to be solved; Then the specific hydrogeological conditions are generalized and the hydrogeological conceptual model is established. This process is the key to establishing the model, and different generalizations will lead to different models. If the generalization is unreasonable or too simple, it will lead to model failure or partial failure; If the summary is too detailed, it may be difficult or even impossible to continue the next step if you try to take all the factors of complex practical phenomena into account. Therefore, at this stage, the modeler is required to have rich hydrogeological theory and practical experience, so as to distinguish the main factors from the secondary factors of the problem and try to homogenize and linearize the problem.

After the hydrogeological conceptual model is established, the relationship between various quantities (constants and variables) is established by using appropriate mathematical tools, such as using partial differential equations to describe the movement of groundwater. This is the second step in building the model. This work often requires extensive mathematical knowledge, such as calculus, differential equations, linear algebra, probability statistics and planning theory.

The third step is model solving and parameter identification. Before applying the model, the established model should be verified. This is also very important for the success or failure of the model. In the study of water resources, the reliability and credibility of groundwater model must be simulated and verified by using the historical data of groundwater before using groundwater model for evaluation and prediction.

Because the response of groundwater system is generated by pulse excitation outside the system, for groundwater quantity model, the response is groundwater level, and the pulse is groundwater recharge or exploitation. Therefore, the response of the system to historical pulses is also reflected in the historical water level data of the system. If the groundwater model can well simulate the prototype of the groundwater system, then the model should be able to reproduce the historical groundwater level and its changes, which is the basic starting point of the model verification idea.

For the verification of groundwater model, according to the field and indoor test results and regional hydrogeological investigation data, the upper and lower limit range values of a series of hydrogeological parameters are given, and the response of the system to external pulses with time is determined by using the initial values of a group of optimal parameters of the system. The result of this response is the calculated value of the system state variable, which can be expressed as the change of groundwater level or salt concentration in water. Then, the calculated value is compared with the known historical data of the system. If the data sorting and modeling work is more accurate and complete, the model will get better fitting results in the first operation. However, there are some differences between the model and the entity, and the model coefficients (such as water storage coefficient, hydraulic conductivity coefficient, infiltration rate, dispersion and dispersion coefficient, etc.) need to be adjusted reasonably. ), and recalculated by computer, and then compared the calculated value with historical data. Within the limits of parameters, this adjustment and fitting process is often repeated until the calculated results are well fitted with historical data. The "adaptation" here generally has two meanings: one refers to the adaptation between observation wells; Secondly, the overall flow field of the system is well fitted. Practice has proved that it is incorrect to overemphasize the final "fitting" of the model and ignore the inspection of the distortion of the hydrogeological conceptual model. It is very important to remember Chamberlain's warning in this regard. He said: "The rigor of mathematical analysis gives people a deep impression and gives people a precise and meticulous feeling, but this should not make us turn a blind eye to the defects of the premise that restricts the whole process. The meticulous mathematical process based on unreliable premise is probably more hidden and dangerous than any other deception. "

Once the groundwater model is corrected and verified, it can be used for evaluation and prediction. By studying the response law of groundwater system to various inputs, different groundwater management schemes can be evaluated reasonably and comprehensively. By coupling groundwater model and optimization model, we can make a comprehensive economic, ecological and environmental evaluation of each groundwater management scheme. Therefore, using the model technology, we can not only choose the most technical and economic management scheme, but also meet various constraints of the system.

In hydrogeology, mathematical model technology plays a very important role, and there are many kinds of mathematical models used, such as analytical model and numerical model mentioned earlier in this book, in addition to models established by stochastic mathematical theory and optimization theory. Because there are many kinds of models, here are only a few model methods.