地下水预测的新方法研究
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摘要
随着人类社会的发展和生产规模的扩大,逐渐出现了世界性的水资源危机,世界各国在地下水资源的开发利用过程中也都产生了许多使人畏惧的环境问题。例如,由于地下水资源的过度开发与不合理利用,不仅加剧了供需矛盾,而且引发了一系列环境地质问题,如地面沉降、地面塌陷、海水入侵以及地裂缝、矿区地质灾害等。国际社会及世界各国逐步认识到地下水资源的合理利用及保护的重要性,地下水资源的定量评价逐渐成了水资源管理的中心课题,地下水数值模拟也伴随着人类对地下水定量评价的发展而发展起来。由于计算技术和计算方法的进步,解算数学模型已经成为地下水定量评价的主要手段。数值方法不仅可以有效地解地下水流问题,也能用于解地下水质问题以及其他模型问题,地下水数值模拟已经成为对水资源进行定量评价管理的有效工具。
尽管地下水数值计算技术和计算方法发展很快,地下水数值模拟仍然存在着大量的理论问题和实际问题需要研究解决,仍有很多局限性。例如,模拟结果的不确定性,数值计算方法的局限性,对输入数据的要求以及对模型的假设,边界条件估计问题,由于材料的不均匀性产生的非确定性因素等。由此可见,地下水模拟过程中充满了各种各样的不确定因素。在把地下水数值模型应用于地下水资源的预测和管理时,如何利用和改进现有技术,减少模型的不确定性,提高数值模拟过程的有效性和预测结果的可靠性,是该研究领域具有挑战性的问题之一。
从对水资源利用管理方面看,考虑到流场变化(例如,新建抽水井,降雨入渗率发生变化,新修水库等)对系统的影响,与绝对环境变化相比,人们也许更关心其在地下水系统中引起的变化量。如果用传统的地下水模拟过程来预报预测局部水文水利条件变化对地下水系统产生的影响,首先要建立模型,接着进行模型校正以满足已有的初始条件,运行模型得到应力变化前的地下水水位分布,流场改变以后,重新运行模型,得到改变后的地下水水位分布值,最后通过计算流场改变前后水位分布的差值得到其变化量。由于建立地下水模型需要大量的现场资料作为输入数据和进行模型校正,现场资料的缺乏造成模型校正的困难,测量误差和自然环境变化引起的现场数据的不确定性转化成参数估计的不确定性,进一步影响到模型的预测结果的精确性。
某些情况下,变化值只与少数参数有关。如果我们能建立一种数学模型,直接模拟变化值,则可以减少对现场资料的需求,减少传统模拟方法中许多不确定因素的影响,提高模拟过程的效率和精确度。例如,如果只关心抽取地下水对环境的影响,可以假设河流水位变化以及降雨入渗率变化为零,模型边界可以远离影响区域并且设为无水流交换边界。在这种情况下,仅仅需要知道渗透系数及地表水体的渗漏系数等少数几类水文地质参数即可建立模型和进行模型校正。由于变化值是从零开始计算,因此模拟时的初始条件值可以设为零。由于减少对数据的需求及将边界设在远离影响区域之外,可以减少由于数据和边界条件引起的不确定性。
基于以上的考虑,本文在传统地下水数值模拟的基础上,引进摄动理论,提出了一种预测局部水文水利条件变化引起地下水环境变化的新的方法。该新方法可以在拥有较少现场资料的情况下,对局部水文水利条件变化引起的地下水水位和流量的变化直接进行的预测,减少模拟过程及模型校正过程中对输入数据的需求,简化边界条件。同时引入反复迭代方法求解控制方程,与现有计算程序结合实现数值计算。本文分别推导出承压水和非承压水的水位及流量变化控制方程,并指出对于承压水可以用与传统方法类似的数值解法来求解水位变化控制方程,得到的水位的变化值与用传统方法得到的水位差值是等价的。对于非承压水的水位变化控制方程,用传统数值方法求解可以得到近似值,但其误差根据相对变化值的增大而增加。经过迭代后,可以明显改善计算结果,减少预测误差。
全文共分七章。
第一章绪论部分。通过大量文献检索和调查研究,阐明了地下水资源的定量评价在水资源利用管理方面的重要性,总结了地下水数值模拟的发展、现状及存在的问题,指出已有的数值模拟方法存在的局限性,对大量原始数据的需求及各种不确定因素引起的预测结果的不确定性等。针对目前人们关心的局部水文水利条件变化引起的地下水环境的变化问题,目前存在的研究结果不足以有效快速地对其进行预报预测。
第二章地下水控制方程及其数值解法的研究。本章首先对传统的地下水控制方程及其数值解法进行了介绍。在大多数地下水数值模型中采用有限差分法求解地下水控制偏微方程,国外开发了许多功能多样的地下水系统数值模拟软件,以其模块化、可视化、交互性、求解方法多样化等特点得到广泛应用。在传统地下水控制方程的基础上,考虑到局部水文水利条件变化引起水文地质参数及地下水水位发生变化的情况,通过对方程中的变量进行分解,推导出含有初始项和摄动项的控制方程,消去初始项后,得到水位变化控制方程,该水位变化控制方程的变量直接是地下水水位的变化值。
承压水水位变化控制方程的形式与传统承压水水位控制方程形式相似,可以用与传统方法类似的数值解法及计算软件来进行求解。对于非承压水的水位变化控制方程,其方程的形式与传统承压水水位控制方程不同,用传统数值解法只能得到近似值,其误差根据水头相对变化量的增大而增加。对于局部水文水利条件变化引起的地下水水位相对变化量较小的地区,可以用传统数值方法和现有数值模拟软件近似计算。对于地下水水位相对变化量较大地区,为了减少预测误差,本研究引入迭代方法,推导出迭代求解的水位变化控制方程及其相关的有限差分表达式,并用MATLAB编写出相关有限差分方程求解的计算程序,将其与现有计算方法和计算软件结合起来可以求得非承压水水位变化控制方程的精确解。
第三章流量变化控制方程。与水位变化控制方程推导的方法一样,在传统流量控制方程的基础上,通过变量分解,分别推导出承压水和非承压水流量变化控制方程,该流量变化控制方程的变量直接是地下水流量的变化值。对于承压水的流量变化控制方程,其方程形式与传统流量控制方程形式相似,可以用与传统方法类似的数值解法及计算软件来进行求解。对于非承压水的流量变化控制方程,其方程的形式与传统控制方程不同,本文给出了相应的有限差分求解公式。
第四章水位变化控制方程及其迭代求解方法的验证。本章借助具有解析解的简单的一维地下水模型来对水位变化控制方程及迭代求解方法进行验证。对于承压水,首先求出一维承压水含水层在局部应力改变后其地下水水位分布的解析解,然后推导出一维承压水水位变化控制方程的解,将两种解对照可以看出,解析解和水位变化控制方程的解是一样的。
对于非承压水含水层,推导出解析解和水位变化控制方程的初始数值解及引入迭代方法后的不同迭代步的数值解。将各种求解结果进行比较分析可以看出,水位变化控制方程的初始解有别于解析解,但迭代后的解收敛于解析解。这一验证结果进一步说明了本文推导的水位变化控制方程的正确性,引入的迭代求解方法是可行的。
本文通过对非承压水在不同原因引起的局部水文水利条件变化引起的地下水水位变化的计算分析,定量地给出地下水水位相对变化值与迭代步骤地关系。分析结果表明,如果相对水头变化值较小,解水位变化控制方程时较少的迭代步即可收敛,反之亦然。本文给出迭代步与相对水头变化值的关系公式,对今后该方法的实际应用具有指导意义。
第五章新方法的应用。针对近年来人们对局部水文水利条件变化对地下水环境的影响越来越关注的情况,本文对新建抽水井,降雨入渗率发生变化,河流或湖泊水位发生变化三种典型的局部水文水利条件变化情况下地下水水位变化问题进行预测分析。
首先建立传统意义的地下水模型,用有限差分方法,借助国际上通用的地下水数值模拟软件MODFLOW,分别求出流场变化前后的地下水水位分布,并计算其变化量。然后应用新的方法,重新建立水位变化模型,结合自编的求解迭代方程的有限差分计算程序和现有的MODFLOW计算软件,直接求出各种情况下地下水水位的变化值。将新方法求解结果与传统方法求解结果进行比较分析,进一步展示了新方法在模拟过程中具有概念模型清晰,建模过程简单,原始数据需求较少,边界条件容易确定,模拟结果可靠的优越性。
第六章用统计的方法分析模型校正过程中新方法与传统方法的不同。为了使模型校正过程具有可信性,本章根据经验建立了一个相对复杂的地下水概念模型,划分了密集的离散网格,用有限差分的数值解法求解出局部水文水利条件变化引起的地下水水位的变化量的分布,并以此作为校正目标。分别用传统方法和新方法建立校正模型,对模型进行校正,并对校正结果进行统计分析。由校正过程可以看出,传统模型的校正涉及的参数较多,参数调整和组合过程相对复杂。新方法的校正过程涉及的参数明显减少,参数调整及组合过程变得明显简单。从校正后模型的预测结果可以看出,由于传统模型的校正得到的参数组合较多,模型具有很大的不确定性,因此其预测结果也具有很大的不确定性,预测值的变化范围较大。而新方法建立的模型校正参数组合数量较少,既参数的不确定性减少,其预测值的变化范围就远远少于传统方法得到的预测范围的分布,减少了预测结果的不确定性。
第七章对本研究成果做出总结。本文提出的新的地下水模拟方法,可以在拥有较少现场资料的情况下,对局部水文水利条件变化引起的地下水水位及流量的变化直接进行的预测。与传统数值模拟方法比较,新方法明显减少模拟过程及模型校正过程中对输入数据的需求,边界条件容易确定,预测结果的不确定性降低等。
由于新方法概念清晰,迭代方法简单有效,今后很容易将其与传统的数值模拟方法结合,通过对传统数值模拟软件进行改进,直接将新方法融合到传统的数值模拟软件中去,可以快速有效地对局部水文水利条件变化引起的地下水水位及流量的变化值进行预测。
新方法不仅可以直接用来预测由局部水文水利条件变化引起的地下水水位等的变化值,而且将所得到的变化值与区域地下水水位分布结合起来还可以得到流场变化后的区域地下水水位等的分布。借此模拟方法可以避免直接应用复杂的传统模拟方法来进行预报和预测,该研究成果应用到实践中去,可以大大提高人们对地下水环境的预报预测能力,此项研究及其成果将对水资源管理产生重大影响。
本文的主要创新点:
1在传统地下水数值模拟方法的基础上,引入摄动理论,推导出直接反应局部水文水利条件变化引起地下水水位变化及流量变化控制方程。该新方法的提出,避免了传统方法在建模及模型校正过程中对大量输入数据的需求,简化边界条件,降低输入数据的不确定性引起的预测结果的不确定性。
2将迭代方法应用到对非承压水水位变化控制方程的求解中,推导出迭代求解公式及其有限差分求解公式,编写有关有限差分求解的计算程序,并与现有计算软件相结合来实现计算求解。
3对本文提出的新方法进行验证。验证结果表明本文推导的水位变化控制方程是正确的,引入迭代求解方法是可行的。推导出不同情况下达到迭代收敛所需要的迭代步与相对水头变化量之间的关系公式,为把该方法应用到实践中去提供具体指导。
Due to the development of human society and expansion of the production scales, the global water resources crisis began to surface. During the process of utilizing and exploitation of the ground water in the world, dreadful environmental problems prevail. For example, because of the over exploitation and inappropriate utilization of the ground water, not only has it intensified the demand and supply imbalance, it has also set off a series of geological problems, such as ground-surface settlement and collapse, seawater invasion, ground fissure, mining area geological catastrophic events etc.
The importance of the appropriate utilization and protection of the ground water has gradually become known to the world community. The quantitative assessment of groundwater resources has become the center topic in water resource management. The groundwater numerical simulation started to evolve after the development of the quantitative assessment of groundwater resources. Through the improvement of the calculation methods and technologies, numerical simulation has become the mainstream method in quantitative assessment. The numerical simulation not only can simulate the ground water flow problems, it can also be used to simulate the water quality problems or problems occur in other models. The ground water numerical simulation has turned into an effective tool for ground water quantitative assessment management.
Although there is a vast development of the calculation methods and technologies of ground-water numerical simulation, it has the following limitations in its theories and practicalities and needs to be further researched and resolved. For examples: the uncertainty of the simulation results, the limitation of the numeric calculations, problems incurred in the input data requirements, the assumptions to the models, and the assessment of the boundary condition, as well as some other uncertainties caused by the non-heterogeneous of aquifer. Therefore, the groundwater modeling process is still permeated with all kinds of uncertain factors. It is one of the biggest challenges in this research field to fully utilize and improve the technologies on hand to eliminate the modeling uncertainty, promote the validity of the numerical simulation process and the reliability of the forecasted results.
Looking from the aspect of the water resource management and considering the stress changes (such as, new wells, changes in rainfall filtration rate, new reservoirs), people are probably more concerned with the net change incurred in the groundwater system. If we are using the traditional groundwater modeling process to predict the influences to the groundwater system caused by the local stress changes, we need to build a model first, followed by modifying the model to satisfy the existing conditions, manipulating the model to obtain the distribution of the groundwater water table before the stress changes, reworking the model to get the distribution after the changes, eventually calculating the difference between the two distribution values to obtain the net change. Constructing the groundwater modeling requires inputting tremendous amount of field data to modify the model. The difficulties in model calibration caused by the lack of field information, the measurement errors, and natural environmental variables all affect the uncertainty of the field data which in turn causes the parameter uncertainty and the inaccuracy of the modeling predictions.
Under some circumstances, the net change only depends on a few parameters. If we can build a mathematical modeling to simulate the net change directly, we can reduce the need for field data, and the ill effects of the uncertain factors found in the traditional modeling in order to raise the efficiency and accuracy of the modeling process. For example: If we are only concerned with the environment impact caused by a groundwater pumping stress, we can regard river stage and recharge changes equal to zero. The boundary could be set far from the impact area and set to no flow. The only parameters we need to know then are the aquifer hydraulic conductivities and leakance of surface water features. Since we calculate the change from zero, the preliminary conditions of the modeling can be set to zero. By reducing data requirements and moving the boundaries away from the impact area of local stresses, the uncertainties related to the data and boundary conditions can be reduced.
According to what have been discussed, this study, based on the traditional groundwater numerical modeling, has developed a new groundwater modeling to make direct predictions on the groundwater system changes caused by the local stress fluctuations. The proposed new modeling reduces the requirement of input data in the modeling processes, simplifies the boundary conditions, at the same time, and introduces the iteration method in solving the governing equation. The change governing equations related to confined and unconfined aquifer have been developed. It was verified that the simulation process for confined aquifer is as the same as those of traditional numerical simulation process. Otherwise, it is only approximation for unconfined aquifer. The predicted error will be increased as long as the increase of the absolute value of net change. This study introduced an iteration scheme which can improve the perturbation results.
There are seven chapters in this paper.
Chapter One is the introduction. Referencing through a large quantity of documents, literatures, surveys and researches, it illuminates the importance of the groundwater resources quantitative assessment in water resources management and utilization. It summarizes the development of the groundwater numerical modeling, its present function and exiting issues. It points out the limitations, the requirement for larger amount of raw data and the outcome uncertainty caused by the different variables in the existing numerical modeling. The available research results are inadequate in predicting warnings for the highly concerned issue of groundwater net change caused by local stress fluctuations.
Chapter Two is a study of the governing equations and numerical simulations. This chapter gives an introduction to governing equation used in traditional modeling and numerical solutions. In most groundwater models, these equations are solved numerically using finite difference schemes. There are an array of multifunctional software developed abroad for groundwater modeling and are widely adopted for their modularization, visualization, interactivities and variety in simulation method.
Considering the geological parameters and head changes caused by local stress fluctuations, we developed the groundwater head change governing equation for the fluctuating component of variables by decomposing the variables in the governing equation into an initial term and a perturbation term. The format of groundwater head change governing equation for confined aquifer is similar with that of traditional governing equation. Theoretically, therefore, we can use the same numerical schemes to directly model the hydraulic head perturbation in confined aquifers.
The perturbation governing equation for unconfined aquifer is different from traditional governing equation. Theoretically, if the change in head is small compared to the saturated thickness, we can use the same numerical method as for traditional groundwater modeling to estimate hydraulic head change. However, if the change in head is large compared to the saturated thickness, the numerical schemes employed in traditional groundwater equations cannot be directly used.
For the more complicated case of unconfined aquifers, we introduced an iteration scheme which improved the perturbation results. We compiled finite difference scheme programs for iteration scheme in MATLAB which can be combined with those exiting software, such as MODFLOW and IGW, in solving the unconfined aquifer perturbation equation.
Chapter Three: Considering the geological parameters and head changes caused by local stress fluctuations, we developed the flux change governing equation by decomposing the variables in the governing equation into an initial term and a perturbation term. The format of change governing equation for confined aquifer is similar with that of traditional governing equation. Theoretically, therefore, we can use the same numerical schemes to directly model the flux perturbation in confined aquifers.
The flux change governing equation for unconfined aquifer is different from traditional governing equation. Theoretically, if the change in head is small compared to the saturated thickness, we can use the same numerical method as for traditional groundwater modeling to estimate flux change. However, if the change in head is large compared to the saturated thickness, the numerical schemes employed in traditional groundwater equations cannot be directly used. We derive finite difference scheme for solving flux change governing equation in unconfined aquifer.
Chapter Four: We use a 1D groundwater model for which an analytical solution existed to verify the results obtained from perturbation equations and to verify the improvements through the iteration scheme. The verification results show that the perturbation solution is exactly as the same as the analytical solution in confined aquifer. In unconfined aquifer, the iteration solutions converged to the analytical solution. We also found out that when the relative change in head is small, it takes lesser iterations to converge to the analytical solution and vice versa.
Chapter Five: New method application. The aim of new method is to directly calculate the change caused by local stress fluctuations without having to separately model pre-and post-stress conditions. To test this methodology, we created a one-layer groundwater model for an unconfined aquifer. We simulated a typical scenario in which more groundwater pumping is required for a certain new development in an area; the development also creates impervious surfaces reducing recharge rates; and, the water levels drop in the surface reservoirs due to droughts/climate change etc. We first ascertained the net changes in aquifer levels using the traditional modeling approach, i.e., separately modeling pre- and post-conditions and then calculating the net change by subtracting one from the other. After ascertaining the change using traditional modeling approach we set up a 'perturbation model' to directly calculate the 'change'. Theoretically, the model should give us the same change as the traditional models as long as change in head is small relative to saturated thickness of the aquifer. However, that not being the case in our example model, the two solutions did not match. The difference can be minimized by iterating it several times using the iteration scheme. As expected, the results of 'modeling change only' improved with every iteration step until the solution converged to the traditional modeling solution.
We have already seen that the new method solution converges to the analytical one for a 1D situation. It is difficult to ascertain an analytical solution for a relatively more complicated (2D or 3D) groundwater situation where different kinds of stresses change simultaneously. However, using a traditional modeling approach with very fine model discrete grid we obtained a solution close to the analytical one and demonstrated that 'modeling the change only' can converge to this solution after a few iterations.
Chapter Six: Using statistics to analyze the differences between the new method and the traditional method in model calibration.
In order to build up the credibility of the model calibration, we have created a relatively complex groundwater conceptual model based on experiences and set a relative large density of discrete grid. We simulated the net change by employing finite difference numerical simulation method in traditional way and using the results as calibration targets. Reset the model with less density of discrete grid for calibration. We then calibrated the model respectively in traditional method and new method. From the calibration results we can see that applications of groundwater models do require extensive field information for input data and for calibration. The uncertainties in data due to measurement errors and natural variability translate into the uncertainty of estimated parameters which further translates into uncertainty in model predictions. The groundwater modeling process is marred by uncertainties all along. Our new approach is more efficient in terms of time and resources required to build the model and also reduce uncertainties related to data, parameter estimation and boundary conditions leading to reduced uncertainty in model predictions.
Chapter Seven: Conclusions. This chapter points out the distinct concept of the new method and the simplicity and efficiency of the iteration scheme .By programming the new method into the traditional numerical modeling software, we can obtain fast and efficient predictions on the groundwater head net change caused by local stress fluctuations.
Perturbation method not only can be used to predict the groundwater head net change caused by local stress fluctuations, it can also associate the obtained net change to the groundwater level distribution to compute the system groundwater head distribution after the stress fluctuations. By using this method, we avoid the complexity of the traditional modeling in making predictions. When we put the results into practical use, it will increase our ability to make such predictions. This research and its payoffs will bring great contributions to the water resources management.
The innovations of this study are as follows:
1. Based on the traditional groundwater numerical modeling, this study introduces the perturbation theory to derive the change governing equation to directly reflect the groundwater head or flux net change caused by local stress fluctuations. This new approach is more efficient in terms of time and resources required to build the model and also reduce uncertainties related to data, parameter estimation and boundary conditions leading to reduced uncertainty in model predictions.
2. This study introduces an iteration scheme which improves the simulation results and compiles finite difference scheme programs which can be combined with those existing software in solving the unconfined aquifer head change governing equation.
3. Making validation using analysis solution and the predictions from the outcomes, quantifying the iteration steps requested by convergence, providing guidance in making practical applications.
尽管地下水数值计算技术和计算方法发展很快,地下水数值模拟仍然存在着大量的理论问题和实际问题需要研究解决,仍有很多局限性。例如,模拟结果的不确定性,数值计算方法的局限性,对输入数据的要求以及对模型的假设,边界条件估计问题,由于材料的不均匀性产生的非确定性因素等。由此可见,地下水模拟过程中充满了各种各样的不确定因素。在把地下水数值模型应用于地下水资源的预测和管理时,如何利用和改进现有技术,减少模型的不确定性,提高数值模拟过程的有效性和预测结果的可靠性,是该研究领域具有挑战性的问题之一。
从对水资源利用管理方面看,考虑到流场变化(例如,新建抽水井,降雨入渗率发生变化,新修水库等)对系统的影响,与绝对环境变化相比,人们也许更关心其在地下水系统中引起的变化量。如果用传统的地下水模拟过程来预报预测局部水文水利条件变化对地下水系统产生的影响,首先要建立模型,接着进行模型校正以满足已有的初始条件,运行模型得到应力变化前的地下水水位分布,流场改变以后,重新运行模型,得到改变后的地下水水位分布值,最后通过计算流场改变前后水位分布的差值得到其变化量。由于建立地下水模型需要大量的现场资料作为输入数据和进行模型校正,现场资料的缺乏造成模型校正的困难,测量误差和自然环境变化引起的现场数据的不确定性转化成参数估计的不确定性,进一步影响到模型的预测结果的精确性。
某些情况下,变化值只与少数参数有关。如果我们能建立一种数学模型,直接模拟变化值,则可以减少对现场资料的需求,减少传统模拟方法中许多不确定因素的影响,提高模拟过程的效率和精确度。例如,如果只关心抽取地下水对环境的影响,可以假设河流水位变化以及降雨入渗率变化为零,模型边界可以远离影响区域并且设为无水流交换边界。在这种情况下,仅仅需要知道渗透系数及地表水体的渗漏系数等少数几类水文地质参数即可建立模型和进行模型校正。由于变化值是从零开始计算,因此模拟时的初始条件值可以设为零。由于减少对数据的需求及将边界设在远离影响区域之外,可以减少由于数据和边界条件引起的不确定性。
基于以上的考虑,本文在传统地下水数值模拟的基础上,引进摄动理论,提出了一种预测局部水文水利条件变化引起地下水环境变化的新的方法。该新方法可以在拥有较少现场资料的情况下,对局部水文水利条件变化引起的地下水水位和流量的变化直接进行的预测,减少模拟过程及模型校正过程中对输入数据的需求,简化边界条件。同时引入反复迭代方法求解控制方程,与现有计算程序结合实现数值计算。本文分别推导出承压水和非承压水的水位及流量变化控制方程,并指出对于承压水可以用与传统方法类似的数值解法来求解水位变化控制方程,得到的水位的变化值与用传统方法得到的水位差值是等价的。对于非承压水的水位变化控制方程,用传统数值方法求解可以得到近似值,但其误差根据相对变化值的增大而增加。经过迭代后,可以明显改善计算结果,减少预测误差。
全文共分七章。
第一章绪论部分。通过大量文献检索和调查研究,阐明了地下水资源的定量评价在水资源利用管理方面的重要性,总结了地下水数值模拟的发展、现状及存在的问题,指出已有的数值模拟方法存在的局限性,对大量原始数据的需求及各种不确定因素引起的预测结果的不确定性等。针对目前人们关心的局部水文水利条件变化引起的地下水环境的变化问题,目前存在的研究结果不足以有效快速地对其进行预报预测。
第二章地下水控制方程及其数值解法的研究。本章首先对传统的地下水控制方程及其数值解法进行了介绍。在大多数地下水数值模型中采用有限差分法求解地下水控制偏微方程,国外开发了许多功能多样的地下水系统数值模拟软件,以其模块化、可视化、交互性、求解方法多样化等特点得到广泛应用。在传统地下水控制方程的基础上,考虑到局部水文水利条件变化引起水文地质参数及地下水水位发生变化的情况,通过对方程中的变量进行分解,推导出含有初始项和摄动项的控制方程,消去初始项后,得到水位变化控制方程,该水位变化控制方程的变量直接是地下水水位的变化值。
承压水水位变化控制方程的形式与传统承压水水位控制方程形式相似,可以用与传统方法类似的数值解法及计算软件来进行求解。对于非承压水的水位变化控制方程,其方程的形式与传统承压水水位控制方程不同,用传统数值解法只能得到近似值,其误差根据水头相对变化量的增大而增加。对于局部水文水利条件变化引起的地下水水位相对变化量较小的地区,可以用传统数值方法和现有数值模拟软件近似计算。对于地下水水位相对变化量较大地区,为了减少预测误差,本研究引入迭代方法,推导出迭代求解的水位变化控制方程及其相关的有限差分表达式,并用MATLAB编写出相关有限差分方程求解的计算程序,将其与现有计算方法和计算软件结合起来可以求得非承压水水位变化控制方程的精确解。
第三章流量变化控制方程。与水位变化控制方程推导的方法一样,在传统流量控制方程的基础上,通过变量分解,分别推导出承压水和非承压水流量变化控制方程,该流量变化控制方程的变量直接是地下水流量的变化值。对于承压水的流量变化控制方程,其方程形式与传统流量控制方程形式相似,可以用与传统方法类似的数值解法及计算软件来进行求解。对于非承压水的流量变化控制方程,其方程的形式与传统控制方程不同,本文给出了相应的有限差分求解公式。
第四章水位变化控制方程及其迭代求解方法的验证。本章借助具有解析解的简单的一维地下水模型来对水位变化控制方程及迭代求解方法进行验证。对于承压水,首先求出一维承压水含水层在局部应力改变后其地下水水位分布的解析解,然后推导出一维承压水水位变化控制方程的解,将两种解对照可以看出,解析解和水位变化控制方程的解是一样的。
对于非承压水含水层,推导出解析解和水位变化控制方程的初始数值解及引入迭代方法后的不同迭代步的数值解。将各种求解结果进行比较分析可以看出,水位变化控制方程的初始解有别于解析解,但迭代后的解收敛于解析解。这一验证结果进一步说明了本文推导的水位变化控制方程的正确性,引入的迭代求解方法是可行的。
本文通过对非承压水在不同原因引起的局部水文水利条件变化引起的地下水水位变化的计算分析,定量地给出地下水水位相对变化值与迭代步骤地关系。分析结果表明,如果相对水头变化值较小,解水位变化控制方程时较少的迭代步即可收敛,反之亦然。本文给出迭代步与相对水头变化值的关系公式,对今后该方法的实际应用具有指导意义。
第五章新方法的应用。针对近年来人们对局部水文水利条件变化对地下水环境的影响越来越关注的情况,本文对新建抽水井,降雨入渗率发生变化,河流或湖泊水位发生变化三种典型的局部水文水利条件变化情况下地下水水位变化问题进行预测分析。
首先建立传统意义的地下水模型,用有限差分方法,借助国际上通用的地下水数值模拟软件MODFLOW,分别求出流场变化前后的地下水水位分布,并计算其变化量。然后应用新的方法,重新建立水位变化模型,结合自编的求解迭代方程的有限差分计算程序和现有的MODFLOW计算软件,直接求出各种情况下地下水水位的变化值。将新方法求解结果与传统方法求解结果进行比较分析,进一步展示了新方法在模拟过程中具有概念模型清晰,建模过程简单,原始数据需求较少,边界条件容易确定,模拟结果可靠的优越性。
第六章用统计的方法分析模型校正过程中新方法与传统方法的不同。为了使模型校正过程具有可信性,本章根据经验建立了一个相对复杂的地下水概念模型,划分了密集的离散网格,用有限差分的数值解法求解出局部水文水利条件变化引起的地下水水位的变化量的分布,并以此作为校正目标。分别用传统方法和新方法建立校正模型,对模型进行校正,并对校正结果进行统计分析。由校正过程可以看出,传统模型的校正涉及的参数较多,参数调整和组合过程相对复杂。新方法的校正过程涉及的参数明显减少,参数调整及组合过程变得明显简单。从校正后模型的预测结果可以看出,由于传统模型的校正得到的参数组合较多,模型具有很大的不确定性,因此其预测结果也具有很大的不确定性,预测值的变化范围较大。而新方法建立的模型校正参数组合数量较少,既参数的不确定性减少,其预测值的变化范围就远远少于传统方法得到的预测范围的分布,减少了预测结果的不确定性。
第七章对本研究成果做出总结。本文提出的新的地下水模拟方法,可以在拥有较少现场资料的情况下,对局部水文水利条件变化引起的地下水水位及流量的变化直接进行的预测。与传统数值模拟方法比较,新方法明显减少模拟过程及模型校正过程中对输入数据的需求,边界条件容易确定,预测结果的不确定性降低等。
由于新方法概念清晰,迭代方法简单有效,今后很容易将其与传统的数值模拟方法结合,通过对传统数值模拟软件进行改进,直接将新方法融合到传统的数值模拟软件中去,可以快速有效地对局部水文水利条件变化引起的地下水水位及流量的变化值进行预测。
新方法不仅可以直接用来预测由局部水文水利条件变化引起的地下水水位等的变化值,而且将所得到的变化值与区域地下水水位分布结合起来还可以得到流场变化后的区域地下水水位等的分布。借此模拟方法可以避免直接应用复杂的传统模拟方法来进行预报和预测,该研究成果应用到实践中去,可以大大提高人们对地下水环境的预报预测能力,此项研究及其成果将对水资源管理产生重大影响。
本文的主要创新点:
1在传统地下水数值模拟方法的基础上,引入摄动理论,推导出直接反应局部水文水利条件变化引起地下水水位变化及流量变化控制方程。该新方法的提出,避免了传统方法在建模及模型校正过程中对大量输入数据的需求,简化边界条件,降低输入数据的不确定性引起的预测结果的不确定性。
2将迭代方法应用到对非承压水水位变化控制方程的求解中,推导出迭代求解公式及其有限差分求解公式,编写有关有限差分求解的计算程序,并与现有计算软件相结合来实现计算求解。
3对本文提出的新方法进行验证。验证结果表明本文推导的水位变化控制方程是正确的,引入迭代求解方法是可行的。推导出不同情况下达到迭代收敛所需要的迭代步与相对水头变化量之间的关系公式,为把该方法应用到实践中去提供具体指导。
Due to the development of human society and expansion of the production scales, the global water resources crisis began to surface. During the process of utilizing and exploitation of the ground water in the world, dreadful environmental problems prevail. For example, because of the over exploitation and inappropriate utilization of the ground water, not only has it intensified the demand and supply imbalance, it has also set off a series of geological problems, such as ground-surface settlement and collapse, seawater invasion, ground fissure, mining area geological catastrophic events etc.
The importance of the appropriate utilization and protection of the ground water has gradually become known to the world community. The quantitative assessment of groundwater resources has become the center topic in water resource management. The groundwater numerical simulation started to evolve after the development of the quantitative assessment of groundwater resources. Through the improvement of the calculation methods and technologies, numerical simulation has become the mainstream method in quantitative assessment. The numerical simulation not only can simulate the ground water flow problems, it can also be used to simulate the water quality problems or problems occur in other models. The ground water numerical simulation has turned into an effective tool for ground water quantitative assessment management.
Although there is a vast development of the calculation methods and technologies of ground-water numerical simulation, it has the following limitations in its theories and practicalities and needs to be further researched and resolved. For examples: the uncertainty of the simulation results, the limitation of the numeric calculations, problems incurred in the input data requirements, the assumptions to the models, and the assessment of the boundary condition, as well as some other uncertainties caused by the non-heterogeneous of aquifer. Therefore, the groundwater modeling process is still permeated with all kinds of uncertain factors. It is one of the biggest challenges in this research field to fully utilize and improve the technologies on hand to eliminate the modeling uncertainty, promote the validity of the numerical simulation process and the reliability of the forecasted results.
Looking from the aspect of the water resource management and considering the stress changes (such as, new wells, changes in rainfall filtration rate, new reservoirs), people are probably more concerned with the net change incurred in the groundwater system. If we are using the traditional groundwater modeling process to predict the influences to the groundwater system caused by the local stress changes, we need to build a model first, followed by modifying the model to satisfy the existing conditions, manipulating the model to obtain the distribution of the groundwater water table before the stress changes, reworking the model to get the distribution after the changes, eventually calculating the difference between the two distribution values to obtain the net change. Constructing the groundwater modeling requires inputting tremendous amount of field data to modify the model. The difficulties in model calibration caused by the lack of field information, the measurement errors, and natural environmental variables all affect the uncertainty of the field data which in turn causes the parameter uncertainty and the inaccuracy of the modeling predictions.
Under some circumstances, the net change only depends on a few parameters. If we can build a mathematical modeling to simulate the net change directly, we can reduce the need for field data, and the ill effects of the uncertain factors found in the traditional modeling in order to raise the efficiency and accuracy of the modeling process. For example: If we are only concerned with the environment impact caused by a groundwater pumping stress, we can regard river stage and recharge changes equal to zero. The boundary could be set far from the impact area and set to no flow. The only parameters we need to know then are the aquifer hydraulic conductivities and leakance of surface water features. Since we calculate the change from zero, the preliminary conditions of the modeling can be set to zero. By reducing data requirements and moving the boundaries away from the impact area of local stresses, the uncertainties related to the data and boundary conditions can be reduced.
According to what have been discussed, this study, based on the traditional groundwater numerical modeling, has developed a new groundwater modeling to make direct predictions on the groundwater system changes caused by the local stress fluctuations. The proposed new modeling reduces the requirement of input data in the modeling processes, simplifies the boundary conditions, at the same time, and introduces the iteration method in solving the governing equation. The change governing equations related to confined and unconfined aquifer have been developed. It was verified that the simulation process for confined aquifer is as the same as those of traditional numerical simulation process. Otherwise, it is only approximation for unconfined aquifer. The predicted error will be increased as long as the increase of the absolute value of net change. This study introduced an iteration scheme which can improve the perturbation results.
There are seven chapters in this paper.
Chapter One is the introduction. Referencing through a large quantity of documents, literatures, surveys and researches, it illuminates the importance of the groundwater resources quantitative assessment in water resources management and utilization. It summarizes the development of the groundwater numerical modeling, its present function and exiting issues. It points out the limitations, the requirement for larger amount of raw data and the outcome uncertainty caused by the different variables in the existing numerical modeling. The available research results are inadequate in predicting warnings for the highly concerned issue of groundwater net change caused by local stress fluctuations.
Chapter Two is a study of the governing equations and numerical simulations. This chapter gives an introduction to governing equation used in traditional modeling and numerical solutions. In most groundwater models, these equations are solved numerically using finite difference schemes. There are an array of multifunctional software developed abroad for groundwater modeling and are widely adopted for their modularization, visualization, interactivities and variety in simulation method.
Considering the geological parameters and head changes caused by local stress fluctuations, we developed the groundwater head change governing equation for the fluctuating component of variables by decomposing the variables in the governing equation into an initial term and a perturbation term. The format of groundwater head change governing equation for confined aquifer is similar with that of traditional governing equation. Theoretically, therefore, we can use the same numerical schemes to directly model the hydraulic head perturbation in confined aquifers.
The perturbation governing equation for unconfined aquifer is different from traditional governing equation. Theoretically, if the change in head is small compared to the saturated thickness, we can use the same numerical method as for traditional groundwater modeling to estimate hydraulic head change. However, if the change in head is large compared to the saturated thickness, the numerical schemes employed in traditional groundwater equations cannot be directly used.
For the more complicated case of unconfined aquifers, we introduced an iteration scheme which improved the perturbation results. We compiled finite difference scheme programs for iteration scheme in MATLAB which can be combined with those exiting software, such as MODFLOW and IGW, in solving the unconfined aquifer perturbation equation.
Chapter Three: Considering the geological parameters and head changes caused by local stress fluctuations, we developed the flux change governing equation by decomposing the variables in the governing equation into an initial term and a perturbation term. The format of change governing equation for confined aquifer is similar with that of traditional governing equation. Theoretically, therefore, we can use the same numerical schemes to directly model the flux perturbation in confined aquifers.
The flux change governing equation for unconfined aquifer is different from traditional governing equation. Theoretically, if the change in head is small compared to the saturated thickness, we can use the same numerical method as for traditional groundwater modeling to estimate flux change. However, if the change in head is large compared to the saturated thickness, the numerical schemes employed in traditional groundwater equations cannot be directly used. We derive finite difference scheme for solving flux change governing equation in unconfined aquifer.
Chapter Four: We use a 1D groundwater model for which an analytical solution existed to verify the results obtained from perturbation equations and to verify the improvements through the iteration scheme. The verification results show that the perturbation solution is exactly as the same as the analytical solution in confined aquifer. In unconfined aquifer, the iteration solutions converged to the analytical solution. We also found out that when the relative change in head is small, it takes lesser iterations to converge to the analytical solution and vice versa.
Chapter Five: New method application. The aim of new method is to directly calculate the change caused by local stress fluctuations without having to separately model pre-and post-stress conditions. To test this methodology, we created a one-layer groundwater model for an unconfined aquifer. We simulated a typical scenario in which more groundwater pumping is required for a certain new development in an area; the development also creates impervious surfaces reducing recharge rates; and, the water levels drop in the surface reservoirs due to droughts/climate change etc. We first ascertained the net changes in aquifer levels using the traditional modeling approach, i.e., separately modeling pre- and post-conditions and then calculating the net change by subtracting one from the other. After ascertaining the change using traditional modeling approach we set up a 'perturbation model' to directly calculate the 'change'. Theoretically, the model should give us the same change as the traditional models as long as change in head is small relative to saturated thickness of the aquifer. However, that not being the case in our example model, the two solutions did not match. The difference can be minimized by iterating it several times using the iteration scheme. As expected, the results of 'modeling change only' improved with every iteration step until the solution converged to the traditional modeling solution.
We have already seen that the new method solution converges to the analytical one for a 1D situation. It is difficult to ascertain an analytical solution for a relatively more complicated (2D or 3D) groundwater situation where different kinds of stresses change simultaneously. However, using a traditional modeling approach with very fine model discrete grid we obtained a solution close to the analytical one and demonstrated that 'modeling the change only' can converge to this solution after a few iterations.
Chapter Six: Using statistics to analyze the differences between the new method and the traditional method in model calibration.
In order to build up the credibility of the model calibration, we have created a relatively complex groundwater conceptual model based on experiences and set a relative large density of discrete grid. We simulated the net change by employing finite difference numerical simulation method in traditional way and using the results as calibration targets. Reset the model with less density of discrete grid for calibration. We then calibrated the model respectively in traditional method and new method. From the calibration results we can see that applications of groundwater models do require extensive field information for input data and for calibration. The uncertainties in data due to measurement errors and natural variability translate into the uncertainty of estimated parameters which further translates into uncertainty in model predictions. The groundwater modeling process is marred by uncertainties all along. Our new approach is more efficient in terms of time and resources required to build the model and also reduce uncertainties related to data, parameter estimation and boundary conditions leading to reduced uncertainty in model predictions.
Chapter Seven: Conclusions. This chapter points out the distinct concept of the new method and the simplicity and efficiency of the iteration scheme .By programming the new method into the traditional numerical modeling software, we can obtain fast and efficient predictions on the groundwater head net change caused by local stress fluctuations.
Perturbation method not only can be used to predict the groundwater head net change caused by local stress fluctuations, it can also associate the obtained net change to the groundwater level distribution to compute the system groundwater head distribution after the stress fluctuations. By using this method, we avoid the complexity of the traditional modeling in making predictions. When we put the results into practical use, it will increase our ability to make such predictions. This research and its payoffs will bring great contributions to the water resources management.
The innovations of this study are as follows:
1. Based on the traditional groundwater numerical modeling, this study introduces the perturbation theory to derive the change governing equation to directly reflect the groundwater head or flux net change caused by local stress fluctuations. This new approach is more efficient in terms of time and resources required to build the model and also reduce uncertainties related to data, parameter estimation and boundary conditions leading to reduced uncertainty in model predictions.
2. This study introduces an iteration scheme which improves the simulation results and compiles finite difference scheme programs which can be combined with those existing software in solving the unconfined aquifer head change governing equation.
3. Making validation using analysis solution and the predictions from the outcomes, quantifying the iteration steps requested by convergence, providing guidance in making practical applications.
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[41]李树忱.地下结构三维有限元仿真分析.山东科技大学硕士论文,2001.
[42]陈传淼.有限元法的超收敛性[J].数学进展,1985,14(1):39-51.
[43]李大潜.有限元素法续讲[M].北京:科学出版社,1979:23-45.
[44]吕和祥,蒋和洋.非线性有限元[M].北京:化学工业出版社,1992:55-84.
[45]Kitahara M.Boundary integral equation methods in eigenvalue problems of elastic dynamics and thin Plates[M].New York:Elsevier Science Publishing Company Inc,1985.
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[52]姚磊华.地下水水流模型的摄动待定系数随机有限元法[J].水利学报,1999,7:60-64
[53]Freeze,R.A.A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media[J].Water Resources Research,1975,11:725-741.
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