地下水系統孔隙介質尺度效應於異質性地層參數化方法之研究

Methodology for Characterization of Scale Effects in Heterogeneous Subsurface flow system

吳呈懋(Cheng-Mau Wu)

指導教授 ： 李天浩 ； 徐年盛

國立臺灣大學/工學院/土木工程學研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.03077

摘要

由於地下含水層參數有與生俱來的異質性，含水層特性隨空間變異很大，藉由含水層有效參數可用來描述含水層水流傳輸的巨觀行為，有效參數是將異質性孔隙介質為視等價均質介質時，所得到通過孔隙介質的通量會有大約相同的結果。本研究首先探討地下水系統空間異質性孔隙介質之有效參數的推估方法，其次是以小波理論為分析工具建立一套能估計地下水模式之特性尺度和其相關物理參數的特徵化方法。研究中除了的對小波理論基本特性進行瞭解外，並初步應用在異質性孔隙介質尺度效應的特徵化 (或參數化) 上。
本文探討的內容主要分兩大部分：(1)空間異質性孔隙介質有效參數的推估方法：主要探討內容包括重新思考有效參數的定義及其觀念，探討傳統抽水試驗推估異質性含水層有效參數的一些問題，其次提出估計異質性地下水系統有效參數兩個合理方法：洩降-距離曲線分析和洩降分佈的空間矩分析，並以數值實例解釋其合理性，文中藉由分析數值模擬的結果探討說明該問題。研究結果在空間異質性孔隙介質有效參數的推估方法方面，顯示利用傳統抽水試驗分析來估計有效參數時，從長時間的抽水試驗分析所得的流通係數代表在抽水井有效半徑內所有流通係數的某種平均值；此平均值與抽水井及觀測井附近的地質有很大的關係，而且可能受到有效半徑內地質異質性的影響，而儲蓄係數的估計值變異很大與抽水井與觀測井間地質的儲蓄係數特性有關，特別是觀測井的位置。 (2)小波理論在異質性孔隙介質尺度效應的特徵化的應用，探討內容包括： 將小波多尺度分析應用於異質性地下水系統正解問題的參數均一化，將參數數量作最有效率的減少並使水頭計算的誤差維持在一合理範圍內。其次探討把參數局部的微小擾動量除去時，狀態變數所依循守恆的原則。最後建立一套小波多尺度分析參數逆推演算法，此方法的邏輯是利用小波多尺度分析的概念從低解析度(粗)到高解析度(細)來決定參數結構及數量，符合物理現象的原則並能以合理的參數結構和參數值來描述參數場。
小波理論在異質性孔隙介質尺度效應的特徵化的初步應用結果方面，首先小波多尺度分析與地下水參數檢定問題結合，原創一套小波多尺度解析逆推演算法。其次將小波多尺度解析逆推演算法中的AIC統計檢定法推廣至水頭變化與參數有相關性時，藉由參數的協變異矩陣來推求水頭的協變異矩陣，進而計算具有水頭協變異矩陣的損失函數，並結合點量測參數值推求區塊平均參數值的觀念使實際量測時的資料(一般包括點量測參數資料及區塊量測參數資料)，和數值計算時所採用的參數資料(區塊參數)可以妥適的整合。

關鍵字

異質性 ； 參數化 ； 尺度效應 ； 有效參數 ； 洩降-距離曲線分析 ； 空間矩分析 ； 小波多尺度分析 ； 參數均一化 ； 小波多尺度分析參數逆推演算法 ； AIC統計檢定法

並列摘要

Spatial heterogeneity is ubiquitous in nature. Variability of parameter in subsurface is extensive. Researchers have devoted decades to seek a proper solution to represent the effective (or equivalent) parameters of heterogeneous media. The effective parameters are obtained by conceptualizing the heterogeneous soil formation as an equivalent homogeneous medium that will discharge approximately the same flux as the ensemble flux of the heterogeneous formations. This study explores solutions for the estimation of effective parameters in heterogeneous media first. Additionally, using wavelets as an analyzing tool, a solution for characterizing scale effects in heterogeneous subsurface porous media is proposed. The article is composed of two main parts:
(1) Estimations of effective parameter in heterogeneous media: Firstly, we clarify the concept of the effective parameter and address the problems of parameter estimation by traditional pumping test. Secondly, we present two estimation approaches (i.e., distance-drawdown and spatial moment analyses) for Seff and Teff, which are consistent with Theis' homogeneous aquifer assumption.
Results of estimation of effective parameters in heterogeneous show that: i) Seff and Teff values evolve with time, as well as the principal directions of the transmissivity; ii) Seff approaches the arithmetical mean of the field; iii) Teff converges to its geometric mean at large time for the Gaussian random field we generated; and iv) the averages of local T and S values within the cone of depression at early times differ from the Teff and Seff values. Both the averages and effective parameters, however, agree at large times, indicative of the existence of an REV in our domain if the pumping time is sufficiently long and there are no other effects (such as boundaries). At early time, estimated and values change with time, deviating significantly from the geometric means of the fields. The values stabilize rather quickly at the value dominated by the geology between the pumping and the observation well. At late times, values of approach but do not equal the geometric mean, and are influenced by the location, size, and degree of heterogeneity as the cone of depression evolves.
(2) Investigation and application of wavelet theory for characterization of scale effects in heterogeneous porous media. Three themes are explored: Firstly, by analogy to Fourier transform, we use a wavelet kernel function to derive the theoretical relationship between the wavelet spectrum of piezometric head and hydraulic conductivity in a one-dimensional heterogeneous system. Secondly, a homogenization process with wavelet multiresolution aspects is to determine the representative equivalent homogenized signal and to remove high frequency fluctuations under a specific threshold. Thirdly, we integrate wavelet multiresolution analysis with inverse problem to identify parameters with scale effect in heterogeneous media. An innovative parameter identification scheme is developed namely MRAIA (MultiResolution Analysis Inverse Algorithm), which combines wavelet multiresolution analysis (MRA), Gauss-Newton minimization scheme, and statistical test (AIC). Identify the structure and value of parameter properly and simultaneously.
The preliminary results show that wavelet based approaches have excellent achievement for homogenization and parameter identification. The potential exists for using wavelet analysis as the multiscale spatial heterogeneity analysis method for characterizing detailed geologic structures.

並列關鍵字

heterogeneity ； characterization ； scale effects ； effective parameter ； distance-drawdown analyses ； spatial moment analyses ； wavelet multiresolution analysis ； parameter homogenization ； multiresolution analysis inverse algorithm ； AIC test

參考文獻

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