• 沒有找到結果。

典型抽水所引致三維壓密沉陷解析

N/A
N/A
Protected

Academic year: 2022

Share "典型抽水所引致三維壓密沉陷解析 "

Copied!
260
0
0

加載中.... (立即查看全文)

全文

(1)

行政院國家科學委員會專題研究計畫 期末報告

典型抽水所引致三維壓密沉陷解析

計 畫 類 別 : 個別型

計 畫 編 號 : NSC 100-2221-E-216-025-

執 行 期 間 : 100 年 08 月 01 日至 101 年 10 月 31 日 執 行 單 位 : 中華大學土木工程學系

計 畫 主 持 人 : 呂志宗

計畫參與人員: 碩士班研究生-兼任助理人員:謝適任

公 開 資 訊 : 本計畫可公開查詢

中 華 民 國 102 年 01 月 31 日

(2)

中 文 摘 要 : 本研究旨在探討週期性抽水、非點狀抽水及瞬時抽水等所引 致之地表位移、地層有效應力變化及地層超額孔隙水壓等,

數學模式中,擬引用 Biot 之三維壓密理論,將地層模擬為均 質等向或橫向等向性之多孔隙的彈性半無限域,本研究擬採 用符號運算軟體 Mathematica 與積分轉換方法,推導出各種 典型抽水速率所引致的暫態暨長期之地層反應的閉合解。各 項研究成果均將進行參數影響分析,並將所獲得之研究成果 製作成容易應用之工程圖表,以利於其在工程上之應用。

中文關鍵詞: 抽水,孔彈性力學,積分轉換,半無限域,閉合解。

英 文 摘 要 : This study presents the ground surface displacements, effective stresses and excess pore water pressures induced by periodic groundwater withdrawal, non-point groundwater withdrawal, and transient groundwater withdrawal, etc., in a homogeneous isotropic/cross- anisotropic poroelastic half space. The formulation of the mathematical model is based on Biot's three- dimensional consolidation theory of porous media.

Using symbolic computation with Mathematica and

integral transforms, the closed-form solutions of the transient and long-term responses of the stratum subjected to some typical pumping rates are derived.

The consolidation affected by the critical consolidation parameters are illustrated and

discussed. Appropriate figures are constructed for the engineering applications.

英文關鍵詞: Groundwater Withdrawal, Poroelasticity, Integral Transform, Half Space, Closed-form Solution.

(3)

行政院國家科學委員會補助專題研究計畫 □期中進度報告

□期末報告

典型抽水所引致三維壓密沉陷解析

計畫類別:□個別型計畫 □整合型計畫 計畫編號:NSC100-2221-E-216-025-

執行期間:100 年 8 月 1 日至 101 年 10 月 31 日 執行機構及系所:中華大學土木工程學系

計畫主持人:呂志宗 共同主持人:

計畫參與人員:謝適任

本計畫除繳交成果報告外,另含下列出國報告,共 1 份:

□移地研究心得報告

□出席國際學術會議心得報告

□國際合作研究計畫國外研究報告

處理方式:除列管計畫及下列情形者外,得立即公開查詢

□涉及專利或其他智慧財產權,□一年□二年後可公開查詢 中 華 民 國 102 年 1 月 31 日

(4)

I

摘 要

本研究旨在探討週期性抽水、非點狀抽水及瞬時抽水等所引致之地表位移、地層有效應力變化及 地層超額孔隙水壓等,數學模式中,擬引用 Biot 之三維壓密理論,將地層模擬為均質等向或橫向等向 性之多孔隙的彈性半無限域,本研究擬採用符號運算軟體 Mathematica 與積分轉換方法,推導出各種典 型抽水速率所引致的暫態暨長期之地層反應的閉合解。各項研究成果均將進行參數影響分析,並將所 獲得之研究成果製作成容易應用之工程圖表,以利於其在工程上之應用。

關鍵詞:抽水,孔彈性力學,積分轉換,半無限域,閉合解。

(5)

II

ABSTRACT

This study presents the ground surface displacements, effective stresses and excess pore water pressures induced by periodic groundwater withdrawal, non-point groundwater withdrawal, and transient groundwater withdrawal, etc., in a homogeneous isotropic/cross-anisotropic poroelastic half space. The formulation of the mathematical model is based on Biot’s three-dimensional consolidation theory of porous media. Using symbolic computation with Mathematica and integral transforms, the closed-form solutions of the transient and long-term responses of the stratum subjected to some typical pumping rates are derived. The consolidation affected by the critical consolidation parameters are illustrated and discussed. Appropriate figures are constructed for the engineering applications.

Keywords: Groundwater Withdrawal, Poroelasticity, Integral Transform, Half Space, Closed-form Solution.

(6)

III

目 錄

摘要 ... I ABSTRACT ... II 目錄 ... III

一、前言 ... 1

二、研究目的 ... 2

三、文獻探討 ... 2

四、研究方法 ... 3

五、結果與討論 ... 4

參考文獻 ... 5 附錄 1 研究成果投稿後已被接受之 EI 等級期刊論文一篇 ... 附 1 附錄 2 研究成果投稿後依審查意見修訂中之 EI/SCI 等級期刊論文一篇 ... 附 30 附錄 3 研究成果投稿後已發表於 EI 等級之國際會議論文(第一篇) ... 附 61 附錄 4 研究成果投稿後已發表於 EI 等級之國際會議論文(第二篇) ... 附 74 附錄 5 出席國際會議並擔任會議主持人之心得報告 ... 附 86 附錄 6 初步整理後擬進行投稿之其他相關研究成果(1) ... 附 95 附錄 7 初步整理後擬進行投稿之其他相關研究成果(2) ... 附 104 附錄 8 初步整理後擬進行投稿之其他相關研究成果(3) ... 附 111 附錄 9 參與本計畫案之兼任研究助理謝適任同學的碩士論文:單井抽水所引致軸對稱

彈性沉陷之研究 ... 附 119

(7)

1

一、前言

歷年來超抽地下水所引起的地層下陷問題一直廣受國人重視,審視民國 101 年以來與該主題相關 之研究計畫案發現,除本計畫案以外,至少仍有 16 個相關之研究計畫案仍獲得各界之經費支持[1-16],

其中有 12 個研究計畫案[1-12]是由國科會予以經費支持、3 個研究計畫案[13-15]是由水利署給予經費 補助、另有 1 個研究計畫案[16]是由農委會給予經費補助。這些計畫案[1-16]與本計畫案[17]所欲探討 之研究主題,均是當今各界相當關注之抽水所引致的地層下陷之相關議題。

如圖 1 所示,計畫申請人自服務於教職以來,即兢兢業業從事於研究工作,並擬訂適合個人之研 究主題,曾多次獲得國科會之經費補助,對累積個人之研究潛能,提昇個人之研究水平,助益甚大。

今本研究計畫案能順利執行完畢,最重要的就是能獲得國科會之經費補助,謹致個人萬分之謝忱。

圖 1 計畫主持人歷年來之研究主題規劃說明 本計畫案之各項研究成果如附錄 1~9 所示,共計包括:

1. 已發表於 EI 等級期刊論文一篇[18],如附件 1 所示。

2. 投稿後依審查意見修訂中之 EI/SCI 等級期刊論文一篇[19],如附件 2 所示。

3. 已發表於 EI 等級之國際會議論文兩篇[20,21],如附件 3 與附件 4 所示。

4. 研究成果經初步整理後擬繼續進行投稿之論文三篇[22-24],如附件 6 至附件 8 所示。

5. 出席國際會議並擔任會議主持人,如附件 5 所示。

6. 人才培育:指導參與本計畫案之兼任研究助理謝適任同學完成其碩士論文「單井抽水所引致軸對稱

(8)

2 彈性沉陷之研究」[25],如附件 9 所示。

本研究已研討出瞬時抽水[22]、週期性抽水[24]、點狀抽水[18,22,25]、圓形平面抽水[21]及線 狀抽水[23,25]等所引致之地表沉陷、水平位移及地層超額孔隙水壓等。由於抽水問題與熱壓密問題具 有類比關係,故亦將部分研究成果應用於熱作用源所引致的熱壓密問題之解析[18-20]。

二、研究目的

水為維繫所有生態體系運作之所需,地球上的水分布於各個角落,包括海洋、冰川、河川、湖泊、

地層及大氣中。台灣雖然平均降雨量約為世界平均降雨量之 2.6 倍,但由於降雨多集中在 5~10 月之豐 水期,故每人所分配之平均降雨量約只有世界平均值的 1/6,因此抽用地下水為解決用水問題的辦法之 一,但由於大量使用並且管制成效不彰,導致地層下陷問題日益嚴重。當地下水因人為使用而被抽取 出 來 時 , 原 先 由 固 體 土 壤 與 孔 隙 水 共 同 承 擔 之 荷 重 , 會 逐 漸 移 轉 至 固 體 土 壤 上 , 故 造 成 壓 密 (Consolidation)現象,所造成的地表下陷情況稱為壓密沉陷(Consolidation Settlement)。地層下陷的過程 通常不易被發現,往往在排水設施功能降低、地下管線受損及土地淹水時才被發現,所以有人稱其為 沉默的土地危機。台灣因工程上或是經濟上的因素過度抽取地下水且未考慮適當的補注機制,已造成 不少縣市深「陷」危機,不只造成國土的流失,且對民生安全的影響亦甚鉅。

地層下陷為目前世界各國普遍遭遇的問題,其成因包括自然因素與人為因素兩方面,而過量抽取 地下水乃是最嚴重的人為影響因素。地下水並非不可抽用,在台灣地下水也是解決部分季節缺水危機 的辦法之一,但如何適量的取用,才不會造成無法挽回的後果是相當重要的。台灣西部沿海由於養殖 業的發展,過量抽取地下水,已造成相當嚴重的地層下陷,致使地下水層的水位面下降、海水入侵地 下水層、地下水層被汙染、土壤鹽化、排水不易等惡果。由於地層下陷為不可逆之現象,故應以嚴肅 態度面對。

為解決抽水所引致的地層下陷問題,除依賴適當的教育宣導與法規的訂定,改變國人用水習慣與 觀念外,亦應從工程上之學理分析層面切入,進行相關之學理分析,藉以瞭解抽水所引致的地層下陷 機制與沉陷結果,其關鍵課題包括抽水所引致的地層位移變化量、及地層孔隙水壓變化量等的探討,

本研究擬分別探討瞬時抽水[22]、週期性抽水[24]、點狀抽水[18,22,25]、圓形平面抽水[21]及線狀抽 水[23,25]等所引致之三維壓密沉陷,並繪製相關之應用圖表,使有助於瞭解此一問題的關鍵影響因素 及其影響結果,研究成果希冀能提供相關主管機關做為訂立相關規範之參考依據。

三、文獻探討

除本國外,抽水所引致的地層下陷亦是世界各國在經濟發展過程中常見的問題[26-35],這些國家 不僅包括中國大陸[26-28]、墨西哥[29,30]、泰國[31,32]、希臘[33,34],連已開發國家如美國[35]、

日本、義大利等等國家亦然,因這一類的問題,常是世界各國於經濟發展過程中,於面對水資源不足 的課題時,必須使用地下水為替代性水資源所導致的問題。

探討地層因抽水所引致之力學與滲流行為變化問題時,常引用多孔介質壓密理論(Consolidation

(9)

3 Theory)建立問題之基本方程式。Terzaghi[36]首先提出單向度之壓密理論,其係以有效應力的觀念,

說明土壤的單向度壓密過程。在陸續的研究中,Biot[37]所建立之耦合三維壓密模式,一般公認較為嚴 謹合理,而廣受重視,此一理論模式,Biot[38]曾將其推廣至異向性介質情況。基於 Biot[37,38]壓密 模式,Rice 與 Cleary[39]曾改寫 Biot 之耦合三維壓密模式,因其基本參數較易由試驗獲得,故所建立 之基本方程式較常被引用。Booker 與 Carter[40-43]曾解析單點抽水所引致的半無限域壓密沉陷問題,

數學模式中地層係模擬為均質等向性之線彈性飽和多孔介質,地下水的滲流則模擬為等向性[40]或橫斷 面等向性[41-43]模式;Tarn 與 Lu[44]則進一步探討地層力學與滲流性質均模擬為橫斷面等向性情況 下,抽水引致的半無限域地層下陷問題,其中地表面係模擬為透水暨不透水兩種情況;Barends[45]、

Worsak 與 Chau[46]等人亦曾解析半無限域中之單點抽水所引致的地層下陷問題。直到如今,這一類問 題仍廣受各界的重視[47-57]。

根據各界對抽水所引致地層下陷問題之學理分析發現,抽水時除會引起地層下陷外,亦會引起地 層之水平位移,國內外亦已有許多相關之文獻[47-52]的監測結果印證此一結論。然而,地層水平位移 常被忽略卻是事實,根據計畫主持人的研究發現[53-56],當地表模擬為透水情況時,抽水所引致之最 大地表水平位移,約為地表最大沉陷量之 30%,顯然這是地表水平位移不宜忽略的重要學理依據;以 學理分析為基礎之相關研究成果,亦已有許多文獻[57-60]證明抽水所引致之地表水平位移相當顯著。

本計畫之研究探討內容,亦包括抽水所引致之地表水平位移量之估算。

Tenney 等人[61]、Tenney 與 Lastoskie[62]曾探討地下水整治時,瞬時抽水問題之數學模式與其解 析,但僅探討其滲流行為,並未進一步研討出所引致的壓密沉陷量。Renner 與 Messar[63]則研究週期 性抽水試驗,對地下水位變化的影響;Propst[64]、Townley[65]、Gendelman 等人[66]、及 Zhao 等人[67]

亦曾探討類似的問題。關於含水層的模擬方式,可包括自由含水層、限制含水層及半無限域含水層等 模擬方式。有關線狀抽水的模擬方式,亦為近年來各界之研究重點。這一類問題,計畫主持人均已順 利以 Biot 三維壓密沉陷理論為基礎,推導出問題的閉合解[17-25]。

一般而言,以數學模擬(Mathematical Modelling)方式探討抽水所引起的地層下陷問題前,基本 而重要的課題是需能事先研討出地層受單點抽水影響時,其所引起的地層力學與滲流行為變化之解析 解,因為這類的解相當關鍵,故亦稱為基本解(Fundamental Solution)。只要這類問題的解可以取得,

則任意形態之抽水的影響,只要對所研討出之解作適當之積分,即可研討出各類相關問題之解。計畫 主持人以往已研討出一系列問題之基本解,本計畫係將問題延伸至瞬時抽水[22]、線狀抽水[18,22,

25]

、週期性抽水[24]和圓形平面取水[21]等所引起的三維壓密沉陷問題之閉合解的解析,並已順利研討 出各類問題的閉合解。本計畫於執行過程中,計畫主持人是將 65%左右的時間、人力與資源等投入於 本計畫案中,其餘的時間、人力與資源等則專注於:(1)研究成果的投稿。(2)留意世界各國關於這一類 問題的最新研究趨勢與成果。(3)人才的培育等。

四、研究方法

本研究係採用數學模擬(Mathematical Modelling)的方式進行相關之研究,並推導出問題之理論解析 解(Theoretical Analytical Solution),使用理論解析方式推估合理之壓密沉陷量。數學模式中乃將含水層 模擬為一半無限域(Half Space),含水層力學行為與滲流性質是考慮為均質(Homogeneous)等向性 (Isotropy)或橫向等向性(Cross-anisotropy),並考慮含水層為完全飽和狀態,且適於引用線彈性飽和多孔

(10)

4 介質彈性力學理論建立數學模式。基於此,研討出地層受抽水作用影響時之穩態(Steady State)或暫態 (Transient)壓密沉陷解析解,其中抽水強度與抽水速率考慮為定值,此外地表邊界則模擬為完全透水與 完全不透水兩種情況加以探討。

本研究基於 Biot 三維壓密理論,並根據以往所推導出之點抽水所導致地層下陷問題的基本解 (Fundament Solution),利用積分轉換方法(Integral Transform)與符號運算軟體 Mathematica 輔助積分之運 算,用以討論因瞬時抽水[22]、線狀抽水[18,22,25]、週期性抽水[24]和圓形平面取水[21]等所引起的 壓密沉陷問題,所研討出之解包括含水層的垂直位移、水平位移與超額孔隙水壓變化等,並繪製出相 關之工程應用圖表。各項研究結果均可使用簡單之數學符號加以表達,故稱之為閉合解(Closed-form Solution),此亦可為邊界元素法等數值模擬方法建立研究基礎。

本文之研究成果可應用於以下所述工程實務情況:(1)若能取得含水層的基本水文地質參數資料,

即可進行簡易之沉陷量的估算,因本文是採用線彈性理論進行數學模擬,故通常沉陷量估算結果會較 大,與工程上常希望進行保守評估相符。(2)若能確實考慮以下各種尺度因素的影響,包括含水層厚度、

抽水深度、抽水量、抽水型態、抽水所引致之沉陷影響範圍等,則各項研究成果均可提供工程界於進 行沉陷量估算時之參考。

五、結果與討論

本研究旨在引用數學模擬方式,探討抽水所引致的地表邊界沉陷量、水平位移量與超額孔隙水壓 等,係採用 Biot 三維壓密理論建立基本方程式,將抽水行為模擬為點抽水、瞬時抽水[22]、線狀抽水[18,

22,25]

、週期性抽水[24]和圓形平面取水[21]等情況,並將地表邊界模擬為透水和不透水兩種滲流邊界

條件。研討過程中有引用點抽水問題之基本解,再對其進行線積分或面積分的運算,推導出各類抽水 所引致之壓密沉陷行為。經仔細研究與討論後,獲得以下結論:

1. 含水層柏松比對抽水所引致壓密行為有重要的影響:當含水層之柏松比變大時,地層較容易產生 變形,因此會反映出較大之地表沉陷。

2. 模擬線狀抽水行為時,就取水長度與井深的比值之影響而言:當取水長度與井深的比值增加時,

地表沉陷量也呈增加的現象,這是因為取水長度增加時,代表抽水量也會增加,故壓密沉陷量也 跟著增加。

3. 各種抽水行為之模擬均呈地表滲流邊界條件有重要的影響:地表模擬為不透水情況下所引致的地 表沉陷會較大,這是因為當地表面模擬為不透水邊界時,抽水所引起之負的超額孔隙水壓無法消 散,含水層之有效應力因而升高,壓密沉陷之效應變大,故所導致的地表沉陷量會變大。

4. 抽水所引起的地表水平位移量會在水井邊逐漸升高後逐步降低:這是因為單井抽水所引起的地表 水平位移是一軸對稱問題,故地表面在對稱點上之水平位移量應為零;另外,含水層遠處受抽水 擾動的影響很小,故地表遠處之水平位移量亦很小,因此抽水所引起的地表水平位移量會在水井

邊逐漸升高後逐步降低。另外,點抽水所引致的地表最大水平位移是落在 位置

上,其中符號 r 表徑向距離座標變數,h 是抽水深度,

則是黃金數,



 1.618。

基於以上研究成果,擬對未來提供一些研究方向與建議:

1. 後續之研究可考慮進行視窗程式設計,讓使用者僅需輸入簡單之參數即可快速求得三維壓密沉陷 之各項結果,以利研究成果之工程應用與推廣。

(11)

5 2. 可引用多孔介質彈力理論與熱彈力理論之類比關係,將相關研究成果推廣至熱彈性力學問題的應

用與解析。

參考文獻

1. 葉弘德,2012/8/1~2013/7/31,「具不規則邊界且為非均質非等向水層的解析解」,國科會補助專題 研究計畫,NSC99-2221-E009-062-MY3。

2. 譚義績,2012/8/1~2013/7/31,「地層下陷分析監測、防護管理與防治策略研擬-總計畫暨子計畫:利 用 類 神 經 網 路 解 析 地 下 水 位 變 化 與 地 層 下 陷 參 數 關 聯 性 (I) 」, 國 科 會 補 助 專 題 研 究 計 畫 , NSC101-2625-M002-008。

3. 羅偉誠,2012/8/1~2013/7/31,「地層下陷分析監測、防護管理與防治策略研擬-子計畫:地層下陷區 域下陷量及海水入侵預測趨勢之研究(I)」,國科會補助專題研究計畫,NSC101-2625-M006-007。

4. 張誠信、陳世楷,2012/8/1~2013/7/31,「整合水質、水量及水文地質空間變異資訊評估地下水使用 在水資源永續經營之角色」,國科會補助專題研究計畫,NSC100-2410-H424-017-MY2。

5. 張良正,2012/8/1~2013/7/31,「因應環境與社會變遷之穩定供水與減災總合政策研究-子計畫:區域 地 下 水 補 注 機 制 探 討 及 管 制 與 運 用 策 略 研 究 (I) 」, 國 科 會 補 助 專 題 研 究 計 畫 , NSC101-2625-M009-003。

6. 徐國錦,2012/8/1~2013/7/31,「貝氏統計法於地下水文參數推估之研究」,國科會補助專題研究計 畫,NSC101-2221-E006-194-MY3。

7. 黃志彬、袁如馨、楊磊,,2012/8/1~2013/7/31「台灣南部地區水產養殖業水資源永續發展對策 - 綠 色水產養殖池及水循環回收系統之研發」,國科會補助專題研究計畫,NSC101-2119-M009-004。

8. 楊紹洋,2012/8/1~2013/7/31,「滲漏含水層考慮彎曲效應之地下水流半解析解」,國科會補助專題 研究計畫,NSC101-2221-E238-009。

9. 黃安斌、馮道偉、張文忠、何彥德、馮正一、蔡東霖、張胤隆,2012/8/1~2013/7/31,「雲林地區地 層下陷特性調查監測與模擬」,國科會補助專題研究計畫,NSC101-2119-M009-003。

10. 倪春發、張中白、董家鈞,2012/8/1~2013/7/31,「地層下陷多尺度時空觀測資訊整合、含水層沈陷 參數推估及長期行為預測」,國科會補助專題研究計畫,NSC101-2116-M008-004。

11. 賴進松,2012/8/1~2013/7/31,「地層下陷分析監測、防護管理與防治策略研擬-子計畫:地層嚴重下 陷地區防洪排水系統改善方案研擬(I)」,國科會補助專題研究計畫,NSC101-2625-M002-016。

12. 謝嘉聲,2012/8/1~2013/7/31,「整合多源衛星影像雷達干涉技術偵測屏東地區地表變形之研究」,

國科會補助專題研究計畫,NSC101-2116-M151-001。

13. 蘇惠珍、連惠邦,2012/2/1~2012/12/31,「沿海低地環境改善之研究-以嘉義沿海魚塭區為例」,經濟 部水利署補助專題研究計畫,MOEAWRA1010282。

14. 張良正、鄭蔚辰,2012/2/1~2012/12/31,「應用資料同化方法推估區域地下水利用之研究」,經濟部

(12)

6 水利署補助專題研究計畫,MOEAWRA1010209。

15. 李振誥、徐國錦、丁崇峯,2012/3/1~2012/12/31,「鳥嘴潭人工湖設置對彰化地區地層下陷防制之 研究(1/2)」,經濟部水利署補助專題研究計畫,MOEAWRA1010216。

16. 許榮庭,2012/4/1~2012/12/31,「水產養殖經營管理研究-雲林沿海地區養殖產業現況及用水分析與 養殖用水規劃」,行政院農業委員會補助專題研究計畫,101 農科-11.3.1-漁-F7。

17. 呂志宗,2011/8/1~2012/10/31,「典型抽水所引致三維壓密沉陷解析」,國科會補助專題研究計畫,

NSC100-2221-E-216-025。

18. Lu, John C.-C. and Feng-Tsai Lin, 2013, “Golden Ratio in the Green’s Functions of Poromechanics and Thermomechanics,” International Journal of Modelling and Simulation, ISSN: 0228-6203, Accepted for

Publication. (This work is supported by the National Science Council through grant

NSC81-0410-E-216-503 & NSC100-2221-E-216-025.) (EI)

19. Lu, John C.-C., Meng-Qi Chen and Feng-Tsai Lin, 2013, “Point Heat Source Induced Thermoelastic Responses of the Cross-anisotropic Strata,” Submitted on October 25, 2011 to International Journal for

Numerical and Analytical Methods in Geomechanics, ISSN: 0363-9061, Under Revision. (This work is

supported by the National Science Council through grant NSC100-2221-E-216-025.) (EI, SCI)

20. Lu, John C.-C. (Session Chair) and Feng-Tsai Lin, 2012/6/25~27, “Modelling of a Buried Deep Horizontal Line Heat Source in a Cross-Anisotropic Thermoelastic Medium,” Proceedings of the 20th

IASTED International Conference on Applied Simulation and Modelling, CD ISBN: 978-0-88986-925-7,

Napoli, Italy, pp. 150-157. (This work is supported by the National Science Council through grants NSC100-2221-E-216-025.) (EI)

21. Lu, John C.-C. (Session Chair) and Feng-Tsai Lin, 2012/6/25~27, “Modelling of Consolidation Settlement Due to a Circularly Symmetric Fluid Sink,” Proceedings of the 20th

IASTED International Conference on Applied Simulation and Modelling, CD ISBN: 978-0-88986-925-7, Napoli, Italy, pp.

107-113. (This work is supported by the National Science Council through grants NSC100-2221-E-216-025.) (EI)

22. Lu, John C.-C. and Feng-Tsai Lin, 2013, “Closed-form Solutions of the Axisymmetric Elastic Consolidation Settlement Due to an Impulsive Point sink,” Ready to Submit for Reviewing. (This work is supported by the National Science Council through grants NSC100-2221-E-216-025.)

23. Lu, John C.-C. and Feng-Tsai Lin, 2013, “Consolidation Settlement of a Poroelastic Half Space Subjected to a Line Sink,” Ready to Submit for Reviewing. (This work is supported by the National Science Council through grants NSC100-2221-E-216-025.)

24. Lu, John C.-C. and Feng-Tsai Lin, 2013, “Elastic Subsidence Subjected to Periodic Pumping,” Ready to

Submit for Reviewing. (This work is supported by the National Science Council through grants

NSC100-2221-E-216-025.)

25. 謝適任,2013,「單井抽水所引致軸對稱彈性沉陷之研究」,碩士論文,中華大學土木工程學系,共 104 頁。

(13)

7 26. Hu, R.L., Z.Q. Yue, L.C. Wang, and S.J. Wang, 2004, “Review on Current Status and Challenging Issues

of Land Subsidence in China,” Engineering Geology, Vol. 76, pp. 65-77.

27. 馬志強,2007,「漳州某基坑抽水引起地面沉降的初探」,科技信息,第 24 卷,第 104-105 頁。

28. 潘國營、鐘福平、姜衍祥、林雲,2006,「應用灰色關聯分析法識別導致地面沉降的抽水層位」,河 南理工大學學報,第 25 卷,第 1 期,第 18-21 頁。

29. Ovando-Shelley, E., A. Ossa, and M.P. Romo, 2007, “The Sinking of Mexico City: Its Effects on Soil Properties and Seismic Response,” Soil Dynamics and Earthquake Engineering, Vol. 27, pp. 333-343.

30. Gonzalez-Moran, T., R. Rodriguez, and S.A. Cortes, 1999, “The Basin of Mexico and its Metropolitan Area: Water Abstraction and Related Environmental Problems,” Journal of South American Earth

Sciences, Vol. 12, pp. 607-613.

31. Phien-wej, N., P.H. Giao, and P. Nutalaya, 2006, “Land Subsidence in Bangkok, Thailand,” Engineering

Geology, Vol. 82, pp. 187-201.

32. Phien-wej, N., P.H. Giao, and P. Nutalaya, 1998, “Field Experiment of Artificial Recharge Through a Well with Reference to Land Subsidence Control,” Engineering Geology, Vol. 50, pp. 187-201.

33. Stiros, S.C., 2001, “Subsidence of the Thessaloniki (Northern Greece) Coastal Plain, 1960-1999,”

Engineering Geology, Vol. 61, pp. 243-256.

34. Psimoulis, P., M. Ghilardi, E. Fouache, and S. Stiros, 2007, “Subsidence and Evolution of the Thessaloniki Plain, Greece, Based on Historical Leveling and GPS Data,” Engineering Geology, Vol. 90, pp. 55-70.

35. Poland, J.F., 1984, Guidebook to Studies of Land Subsidence Due to Ground-water Withdrawal, The United Nations Educational Scientific and Cultural Organization, Unesco, Paris.

36. Terzaghi, K., 1943, Theoretical Soil Mechanics, John Wiley & Sons, New York, N.Y., pp. 256-296.

37. Biot, M.A., 1941, “General Theory of Three-Dimensional Consolidation,” J. Appl. Phys., Vol. 12, No. 2, pp. 155-164.

38. Biot, M.A., 1955, “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid,” J. Appl. Phys., Vol. 26, No. 2, pp. 182-185.

39. Rice, J.R. and M.P. Cleary, 1976, “Some Basic Stress Diffusion Solutions for Fluid-Saturated Elastic Porous Media with Compressible Constituents,” Review Geophys. Space Phys., Vol. 14, No. 2, pp.

227-241.

40. Booker, J.R. and J.P. Carter, 1986a, “Analysis of a Point Sink Embedded in a Porous Elastic Half Space,”

Int. J. Numer. Anal. Methods Geomech., Vol. 10, No. 2, pp. 137-150.

41. Booker, J.R. and J.P. Carter, 1986b, “Long Term Subsidence Due to Fluid Extraction from a Saturated, Anisotropic, Elastic Soil Mass,” Q. J. Mech. Appl. Math., Vol. 39, Pt. 1, pp. 85-97.

42. Booker, J.R. and J.P. Carter, 1987a, “Elastic Consolidation Around a Point Sink Embedded in a

(14)

8 Half-Space with Anisotropic Permeability,” Int. J. Numer. Anal. Methods Geomech., Vol. 11, No. 1, pp.

61-77.

43. Booker, J.R. and J.P. Carter, 1987b, “Withdrawal of a Compressible Pore Fluid from a Point Sink in an Isotropic Elastic Half Space with Anisotropic Permeability,” Int. J. Solids Struct., Vol. 23, No. 3, pp.

369-385.

44. Tarn, J.-Q. and C.-C. Lu, 1991, “Analysis of Subsidence Due to a Point Sink in an Anisotropic Porous Elastic Half Space,” Int. J. Numer. Anal. Methods Geomech., Vol. 15, No. 8, pp. 573-592. (EI, SCI) 45. Barends, F.B.J., 1981, “Landsubsidence Due to a Well in an Elastic Saturated Subsoil,” in A. Verruijt and

F.B.J. Barends(ed.), Flow and Transport in Porous Media, Proceedings of Eurmech 143/Delft/2-4 September 1981, A.A. Balkema/Rotterdam, pp. 11-18.

46. Worsak, K.N. and K.T. Chau, 1990, “Point Sink Fundamental Solutions for Subsidence Prediction,” J.

Engng. Mech., ASCE, Vol. 116, No. 5, pp. 1176-1182.

47. Burbey, T.J., S.M. Warner, G. Blewitt, J.W. Bell, and E. Hill, 2006, “Three-dimensional Deformation and Strain Induced by Municipal Pumping, Part 1: Analysis of Field Data,” Journal of Hydrology, Vol. 319, No. 1-4, pp. 123-142.

48. Burbey, T.J., 2006, “Three-dimensional Deformation and Strain Induced by Municipal Pumping, Part 2:

Numerical Analysis,” Journal of Hydrology, Vol. 330, pp. 422-434.

49. Kim, J.M. and R.R. Parizek, 1999, “Three-dimensional Finite Element Modelling for Consolidation Due to Groundwater Withdrawal in a Desaturating Anisotropic Aquifer System,” Int. J. Numer. Anal. Methods

Geomech., Vol. 23, No. 6, pp. 549-571.

50. Bawden, G.W., W. Thatcher, R.S. Stein, K.W. Hudnut, and G. Peltzer, 2001, “Tectonic Contraction Across Los Angeles After Removal of Groundwater Pumping Effects,” Nature, Vol. 412, No. 6849, pp. 812-815.

51. Hou, C.S., J.C. Hu, L.C. Shen, J.S. Wang, C.L. Chen, T.C. Lai, C. Huang, Y.R. Yang, R.F. Chen, Y.G.

Chen, and J. Angelier, 2005, “Estimation of Subsidence Using GPS Measurements, and Related Hazard:

the Pingtung Plain, Southwestern Taiwan,” Comptes Rendus Geoscience, Vol. 337, No. 13, pp. 1184-1193.

52. Gambolati, G., M. Putti, and P. Teatini, 1996, “Coupled and Uncoupled Poroelastic Solutions to Land Subsidence Due to Groundwater Withdrawal,” Proceedings of Engineering Mechanics, ASCE, Vol. 1, pp.

483-486.

53. Lin, Feng-Tsai and John C.-C. Lu, 2010/4, “Golden Ratio in the Point Sink Induced Consolidation Settlement of a Poroelastic Half Space,” Modelling, Simulation and Optimization, Shkelzen Cakaj (ed.),

ISBN: 978-953-307-056-8, I-Tech Education and Publishing, Vienna, Austria, www.intechweb.org, pp.

25-40.

54. Lu, John C.-C. and Feng-Tsai Lin, 2011/7/4~6, “Elastic Consolidation Settlement Due to Periodic Pumping,” Proceedings of the 3rd

IASTED International Conference on Environmental Management and

Engineering, CD ISBN: 978-0-88986-886-1, Calgary, Alberta, Canada, pp. 53-59. (EI)

(15)

9 55. Lu, John C.-C. and F.-T. Lin, 2006, “The Transient Ground Surface Displacements Due to a Point

Sink/Heat Source in an Elastic Half-Space,” Geotechnical Special Publication No. 148, GeoShanghai

International Conference 2006, ASCE, Shanghai, China, pp. 210-218. (EI)

56. Lu, John C.-C. and Feng-Tsai Lin, 2011/7/4~6, “Consolidation Settlement Due to a Point Sink with Compressible Constituents,” Proceedings of the 22nd

IASTED International Conference on Modelling and Simulation, CD ISBN: 978-0-88986-887-8, Calgary, Alberta, Canada, pp. 275-281. (EI)

57. Yeh, H.-D., R.-H. Lu, and G.-T. Yeh, 1996, “Finite Element Modelling for Land Displacements Due to Pumping,” Int. J. Numer. Anal. Methods Geomech., Vol. 20, pp. 79-99.

58. Bear, J. and M.Y. Corapcioglu, 1981, “Mathematical Model for Regional Land Subsidence Due to Pumping, 2. Integrated Aquifer Subsidence Equations for Vertical and Horizontal Displacements,” Water

Resour. Res., Vol. 17, pp. 947-958.

59. Bear, J. and M.Y. Corapcioglu, 1984, Fundamentals of Transport Phenomena in Porous Media, NATO ASI Series, Martinus Nijhoff Published.

60. 曾鈞敏,「地下水超抽引致地層下陷之三維解析研究」,博士論文,國立臺灣大學土木工程研究所 (2009) 。

61. Tenney, C.M., C.M. Lastoskie, and M.J. Dybas, 2004, “A Reactor Model for Pulsed Pumping Groundwater Remediation,” Water Research, Vol. 38, pp. 3869-3880.

62. Tenney, C.M. and C.M. Lastoskie, 2007, “Pulsed Pumping Process Optimization Using a Potential Flow Model,” Journal of Contaminant Hydrology, Vol. 93, pp. 111-121.

63. Renner, J. and M. Messar, 2006, “Periodic Pumping Tests,” Geophysical Journal International, Vol. 167, No. 1, pp. 479-493.

64. Propst, G., 2006, “Pumping Effects in Models of Periodically Forced Flow Configurations,” Physica D, Vol. 217, pp. 193-201.

65. Townley, L.R., 1995, “The Response of Aquifers to Periodic Forcing,” Advances in Water Resources, Vol.

18, No. 3, pp. 125-146.

66. Gendelman, O.V., E. Gourdon, and C.H. Lamarque, 2006, “Quasiperiodic Energy Pumping in Coupled Oscillators under Periodic Forcing,” Journal of Sound and Vibration, Vol. 294, pp. 651-662.

67. Zhao, C.-Y. and W.-H. Tan, 2009, “Einstein–Podolsky–Rosen Entanglement in Time-dependent Periodic Pumping Non-degenerate Optical Parametric Amplifier,” Chinese Phys. B, Vol. 18, pp. 4143-4153.

(16)

附錄 1

研究成果投稿後已被接 受之 EI 等級期刊論文 一篇

Lu, John C.-C. and Feng-Tsai Lin, 2013,

“Golden Ratio in the Green’s Functions of Poromechanics and Thermomechanics,”

International Journal of Modelling and Simulation, ISSN: 0228-6203, Accepted for Publication. (This work is supported by the National Science Council through grant

NSC81-0410-E-216-503 &

NSC100-2221-E-216-025.) (EI)

附 1

(17)

論文接受函

附 2

(18)

1

John Lu

寄件者: journals@actapress.com

寄件日期: 2012年10月16日星期二 下午 8:41

收件者: cclu@chu.edu.tw

主旨: Paper Status - Paper 205-5645 (Accepted for Publication)

標幟狀態: 已標幟

Re: Paper Number 205‐5645  Dear Dr. John Lu, 

 

I am pleased to inform you that your paper, entitled GOLDEN RATIO IN THE GREEN?S FUNCTIONS OF  POROMECHANICS AND THERMOMECHANICS, has been accepted for publication in the journal 

International Journal of Modelling and Simulation. This paper has passed peer‐review and final approval by  the Editor‐in‐Chief. Please access our online system to view the reviewers' comments; try to include these  suggestions in the final paper. 

 

In order to begin processing your paper, we require several items: 

1. The final paper and accompanying files (email it to journals@actapress.com by Jan‐16‐2013)   

  A double‐spaced MS Word document of your final paper. If your paper is in Latex (.tex) format, please also  submit your final paper as a PDF file. Please visit http://www.actapress.com/submissioninfo.aspx for the  final paper formatting guide. 

  An electronic copy of the figures (.eps, .ps, .jpg, etc.). Please ensure that the filenames of the figures are  labelled according to the figure number (e.g., for Figure 1 it is "figure 1.eps"). 

  A brief biography for each author, and if possible, a photo of each author. Photos must be of high  resolution (300 dpi), black and white, and no less than 3.5 cm in width. 

 

***NOTE: For papers exceeding 8 (EIGHT) printed pages (single‐spaced, 12‐pt font size) inclusive of  illustrations, there is a charge of $100.00 (US currency) per additional page.*** 

 

To access the review assessments of your paper, please visit 

http://www.actapress.com/review/UI/AuthorViewResults.aspx?pn=205‐5645 

 

Username: cclu@chu.edu.tw  Password: ilovejesus 

 

Thank you very much, and we look forward to publishing your paper. 

     

Sincerely,   

ACTA Press / IASTED 

Building B6, Suite #101, 2509 Dieppe Avenue SW, Calgary, AB T3E 7J9 Canada  phone: (403) 288‐1195 

fax: (403) 247‐6851 

e‐mail: journals@actapress.com  Website: http://www.actapress.com/ 

附 3

(19)

2

Peer‐review portal: http://www.actapress.com/review/   

     

附 4

(20)

論文全文

附 5

(21)

Department of Civil Engineering, Chung Hua University, Taiwan, R.O.C.; email: cclu@chu.edu.tw

 Department of Naval Architecture, National Kaohsiung Marine University, Taiwan, R.O.C.; email: ftlin@mail.nkmu.edu.tw

GOLDEN RATIO IN THE GREEN’S FUNCTIONS OF POROMECHANICS AND THERMOMECHANICS

John C.-C. Lu

*

and Feng-Tsai Lin

**

Abstract

This paper presents the transient responses of a point fluid sink or a point heat source in the strata. Green’s functions of the elastic displacements and excess pore fluid pressure or temperature increment of strata are derived by using Laplace-Hankel integral transforms. The strata are modeled as a poroelastic or thermoelastic half space in the mathematical modelling. Poroelasticity and thermoelasticity are applied on the formulation of basic governing equations, and analogy is drawn between poroelasticity and thermoelasticity. Attention is focused on the golden ratio which appears in the magnitude of maximum ground surface horizontal displacement and corresponding vertical displacement of the half space Green’s functions. The study concludes that golden ratio exists in these phenomena, and the horizontal displacement should be properly considered in the prediction of displacements induced by groundwater withdrawal or buried heat source.

Key Words

Golden ratio, Green’s function, point fluid sink, point heat source, poromechanics, thermomechanics

1. Introduction

The golden ratio

[1, 2] is an irrational algebraic number 1.6180339887498948482… which is well known in mathematics, science, biology, art, architecture, nature and beyond [3]. It is interesting to

附 6

(22)

2

discover that the golden ratio exists in the point fluid sink and point heat source induced elastic displacements of a homogeneous isotropic half space. Examples of the golden ratio in engineering include the shear flow in porous half space [4], classical mechanics of coupled-oscillator problem [5], and magnetic compound [6], etc. This study is focused on the transient responses of a point fluid sink or a point heat source in the strata. The derived closed-form solutions are defined as Green’s functions of poromechanics and thermomechanics.

The three-dimensional consolidation theory introduced by Biot [7, 8] is generally regarded as the fundamental theory for modelling land subsidence. The approach is followed by Rice and Cleary [9] who provided an elegant formulation of Biot’s theory by using easily identifiable quantities and material constants. Bear and Corapcioglu [10, 11] presented the modified Biot’s equations when the pore fluid is treated as compressible and the solid skeleton is assumed as incompressible. Based on Biot’s theory modified by Bear and Corapcioglu [10, 11], Booker and Carter [12-15], Tarn and Lu [16] presented solutions of subsidence by a point fluid sink embedded in the saturated elastic half space at a constant rate.

Chen [17, 18], Kanok-Nukulchai and Chau [19] presented analytic solutions for the steady-state responses of displacements and stresses in a half space subject to a fluid point sink. Lu and Lin [20, 21] displayed transient displacements of the pervious half space due to steady pumping rate [20] and impulsive pumping [21]. The results presented by Hou et al. [22] shown that ground horizontal displacement occurred during groundwater withdrawal from an aquifer.

Nuclear wastes are usually deposited at a great depth, such as 200 to 700 meters below ground surface to be isolated from the living environment of human beings. Hueckel and Peano [23] indicated that European guidelines require that temperature increments in the soil close to the heat source should not exceed 80C while the temperature increments at the ground surface is limited to less than 1C. It suggested that linear theory was adequate for a repository design based on technical conservatism [23].

附 7

(23)

3

Given these modest temperature increments, Hollister, Anderson and Health [24] observed that any significant non-linear behaviour and/or plastic deformation of the soil would be confined to a relatively small volume of soil around the waste canister itself. In this case, a linear model can provide reasonable approximation to the assessment of a proposed design [25]. The responses of the strata were satisfactorily modeled by assuming it as a thermoelastic continuum [26]. Booker and Savvidou [26, 27], Savvidou and Booker [28] derived an extended Biot’s theory including the thermal effects and presented solutions of thermo-consolidation around the spherical and point heat sources. The analogy between thermoelasticity and poroelasticity was drawn by Lu and Lin [20], Norris [29], Manolis and Beskos [30], Cheng et al. [31], etc.

Based on the axially symmetric poromechanics and thermomechanics, the Green’s functions of the transient elastic deformations in half spaces due to a point fluid sink and a point heat source are presented in this paper. The transient closed-form solutions are derived through Laplace-Hankel integral transforms.

The homogeneous isotropic stratum is modeled as either poroelastic or thermoelastic half space in the mathematical model. The golden ratio, known as

 1.618, appears in the maximum ground surface horizontal displacement and corresponding vertical displacement. The study concludes that golden ratio exists in these phenomena, and the horizontal displacement should be properly considered in the prediction of displacements induced by groundwater withdrawal or buried heat source.

2. The Golden Ratio

The golden ratio

, approximately 1.6180339887498948482…, is an irrational mathematical constant.

The symbol

is also known as golden section, golden mean, divine proportion, divine section, golden proportion, golden cut, golden number, etc. The golden ratio

can be derived from a geometrical line segment and ratio as shown in Figure 1, where the ratio of the full length 1 to the length of x is equal to the

附 8

(24)

4 ratio of longer section x to shorter section 1 x  :

1 1

x xx

 (1)

Assuming x  1

, hence,

satisfies

2

1 0

  

(2)

The golden ratio is the positive solution of equation (2) as shown below:

(1 5) 2

  (3)

Figure 1. Dividing a segment into the golden ratio.

Figure 2. The golden rectangle.

Figure 2 displayed another geometric description of golden ratio through the golden rectangle. Giving a rectangle with sides’ ratio a:b, the removing of square section leaves remaining rectangle with the same ratio as original rectangle, i.e.,

b a

a bb

 (4)

Thus, this solution is the golden ratio

:

1 5

2 a

  b (5)

The golden ratio is a remarkable number that arises in various areas of mathematics, nature, and arts,

附 9

(25)

5

etc. There are many interesting mathematical properties of

. For example,

can be expressed as a continuous fraction with the single number 1 [1]:

1 1 1 1

1 1 1 1

1

 

 

(6)

Also, the golden ratio

can be expressed as a continuous square root of the number 1:

1 1 1 1

     (7)

However, the most interesting is that

is within Fibonacci series [1, 2]. The Fibonacci series is a set of numbers that begins with two 1s, and each following term is the sum of the prior two terms, i.e., 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ... . The relationship between two successive numbers of Fibonacci series tends to approach

.

Based on the theory of poromechanics and thermomechanics, the strata are modeled as homogeneous isotropic half spaces. This paper presents the Green’s functions of the transient and long-term ground surface elastic deformations of the strata due to a point fluid sink or a point heat source. It is interesting to find that the golden ratio

appears in this study for the maximum ground surface horizontal displacement and corresponding vertical displacement.

3. Poroelastic Modelling 3.1 Governing Equations

The formulation of Biot’s equations follows that of Rice and Cleary [9] with easily identifiable quantities and material constants. Four basic material constants are selected in the constitutive equations including the shear modulus G , the drained Poisson’s ratio

, the undrained Poisson’s ratio

u

and Skempton’s

附 10

(26)

6

pore pressure coefficient B [32]. The physical ranges of B and

u

are obviously 0   and B 1

1

0  

 u

 [9], respectively. For the situation of incompressible constituents, the poroelastic

2

coefficients B  and 1

u

 . According to Rice and Cleary [9], the reformulated constitutive relations

12

can be expressed as [33]:

 

  

2 3

2 1 2 1 2 1

u

ij ij ij ij

u

G G p

B

  

   

  

   

   (8)

 

    

  

2 2

2 1 2 1 2 1

3 1 2 9 1 2

u u

u u u

GB GB

p

    

   

  

  

   (9)

in which

ij

, p and

ij

are the total stress components, excess pore fluid pressure and solid strain components of the poroelastic media, respectively. The fluid pressure p is positive for compression. The parameter

is the variation of fluid content per unit reference volume. The volumetric strain of the skeletal material is denoted by

and

 

11

22

33

. The symbol

ij

is the Kronecker delta. The inversions of equations (8) and (9) are shown as the form:

 

  

1 3

2 1 2 1 1

u

ij ij kk ij ij

u

G GB p

  

    

  

  

          (8*)

  

  

 

  

2 2

9 1 2 3

1 2 1

2 1 2 1

u u u

u u

p B

GB

    

 

 

 

  

 

 

  (9*)

The solid strain components

ij

and displacement components u are governed by the linear kinematic

i

equation:

, ,

1

ij

2 u

i j

u

j i

  (10)

The total stress components

ij

must satisfy the equilibrium equation:

,

0

ij j

b

i

  (11)

where b denotes the body force components. The mass balance for the fluid phase is denoted by:

i

附 11

(27)

7

,

v

i i

t

 

  

 (12)

in which v is the specific discharge velocity components, and

i

is the rate of injected fluid source into the saturated porous aquifer per unit volume. Assuming that the pore fluid flow is governed by Darcy’s law, we have

,

i i

f

v k p

 

(13)

in which k denotes the permeability of the porous media and

f

is the unit weight of pore fluid.

The governing equations (8) to (13) are combined to yield the field equations for solutions of the boundary value problems. Substituting (8) and (10) into (11), (9*) and (13) into (12), then the equilibrium equation (11) and mass balance equation (12) are expressed in terms of displacement components u and

i

excess pore fluid pressure p as below:

2

0

i

1 2

i

i i

G p

G u b

x x

  

 

    

   (14a)

  

  

2

2 2

9 1 2

2 1 2 1

u u

f u

k p

p GB t t

     

  

   

    

 

  (14b)

where

is known as Biot’s coefficient of effective stress which is defined as

 

1 2 3

u

 1

u

B

  

 

 

  (15)

The above mathematical model is known as coupled model of poroelasticity where the flow field is dependent on the displacement field. The coupling term   in equation (14b) is neglected in this

t study.

附 12

(28)

8

Figure 3. Mechanics of poroelastic point fluid sink problem.

Figure 3 presents a point fluid sink buried in a saturated porous half space at a depth h. The constant sink strength is denoted as Q at the location   0, h . Applying the equilibrium equations to axisymmetric poromechanics problem with a vertical axis of symmetry and neglecting the effects of body forces b , then

i

equation (14a) is transformed to equations (16a) and (16b). Moreover, assuming the flow field is independent of the displacement field, thus the mass balance equation (14b) is expressed as (16c).

Therefore, the uncoupled governing equations in axially symmetric coordinates   r z are derived in , terms of displacements u i

i

  r z , and excess pore fluid pressure p as following:

2

2

0

1 2

r r

u

G p

G u G

r r r

 

 

    

   (16a)

2

0

z

1 2

G p

G u

z z

  

 

   

   (16b)

  

        

2

2 2

9 1 2

2 0

2 1 2 1

u u

f u

k p Q

p r z h u t

t r

GB

  

 

   

  

     

   (16c)

where

2 2

2

2 2

1

r r r z

  

   

   and u

r

u

r

u

z

r r z

 

  ;

  x and u t   are the Dirac delta function and Heaviside unit step function, respectively. Equations (16a) to (16c) are the uncoupled basic field equations with a point fluid sink at constant rate, in which the fluid and solid are treated as compressible

附 13

(29)

9 constituents.

3.2 Boundary Conditions and Initial Conditions

The half space ground surface is treated as pervious traction-free boundary for all times t  The 0.

mathematical statements of the ground surface boundary z  in axisymmetric coordinates 0   r z are: ,

, 0,0

rz

r t

 ,

zz

r , 0, t   , and 0 p r, 0, t 0 (17a)

The displacements and excess pore fluid pressure at the remote boundary due to the effect of a point fluid sink must be nil at any time. These conditions are written as

     

 

lim

r

, , ,

z

, , , , , 0, 0, 0

z

u r z t u r z t p r z t



 (17b)

Assuming no initial displacements and seepage of the strata, the initial conditions at time t  0

of the mathematical model due to a point fluid sink are treated as

, , 00

u r z

r

 , u

z

r z , , 0

 and 0 p r z, , 0

0 (18)

The mathematical model in this study is based on the governing equations (16a)-(16c), the corresponding boundary conditions (17a)-(17b) and initial conditions (18).

4. Thermoelastic Modelling 4.1 Governing Equations

The constitutive behavior of the isotropic body with a point heat source buried in an isotropic thermoelastic half space at depth h as shown in Figure 4 is expressed as

 

2 1

2 2

1 2 1 2

s

ij ij ij ij

G G

G

  

   

 

   

  (19)

 

0

2 1

1 2

G

s

c

s T

  

  

 (20)

Here,

ij

and

ij

are the thermal stress components and strain components of the thermoelastic medium,

附 14

(30)

10

respectively. The symbol

denotes the volumetric strain of the thermoelastic medium and

11 22 33

 

 

. The temperature increment

is measured from the reference state. The entropy s is the function for internal state of the thermoelastic system. The average temperature in the natural state corresponding with

ij

 is denoted by 0 T . The material constants

0

, G ,

s

and c

are the Poisson’s ratio, shear modulus, linear thermal expansion coefficient and specific heat at constant strain of the thermoelastic medium, respectively. The coefficient c

c , where the constants

and c define the density and specific heat of the thermoelastic medium, respectively.

The conservation of energy in the form of the entropy flow is as:

0 i i,

T s q W

t

  

 (21)

where q is the heat flux, and W is the quantity of heat generated in a unit volume and unit time. The

i

thermal flow is assumed to follow the Fourier law for heat conduction. In the case of an isotropic body, the Fourier heat conduction law has the form

,

i t i

q  



(22)

in which

t

is the coefficient of heat conduction.

The thermal stresses

ij

should satisfy the equilibrium relations in equation (11). Substituting the linear kinematic equation (10) and constitutive equation (19) into the equilibrium equation (11), and the entropy equation (20) and Fourier heat conduction law (22) into the conservation of energy (21), respectively. Then the equations (11) and (21) are expressed in terms of thermal displacements u and

i

temperature increment

of the thermoelastic medium as follows:

 

2

2 1

1 2 1 2 0

s i

i i

G G

G u

x x

   

 

  

   

    (23a)

附 15

(31)

11

 

0

2

2 1

1 2

s t

G T

c W

t t

   

  

  

    

   (23b)

Figure 4 presents a point heat source buried in a thermoelastic half space at a depth h. The constant heat generating rate is denoted as H at the location   0, h in axisymmetric coordinates system   r z . ,

The equilibrium equations are applied to axisymmetric thermomechanics problem with a vertical axis of symmetry and neglecting the effects of body forces b . Moreover, assuming the thermal flow field is

i

independent from the displacement field in the conservation of energy. Therefore, the uncoupled governing equations in axially symmetric coordinates   r z , are derived in terms of thermal displacements u i

i

  r z ,  and temperature increment

as following:

 

2

2

2 1

1 2 1 2 0

r s r

u G

G u G G

r r r

   

 

  

    

    (24a)

 

2

2 1

1 2 1 2 0

s z

G G

G u

z z

   

 

  

   

    (24b)

     

2

0

t

2

c H r z h u t

t r

   

      

 (24c)

Equations (24a) to (24c) constitute the fundamental equations of transient responses for a thermoelastic medium subjected to a point heat source.

Figure 4. Mechanics of thermoelastic point heat source problem.

附 16

(32)

12 4.2 Boundary Conditions and Initial Conditions

The half space ground surface, z = 0, is considered as traction-free, and it remains the same temperature at all times t  . Therefore, the boundary conditions on surface z = 0 in axisymmetric coordinates 0   r z ,

are given by

, 0,0

rz

r t

 ,

zz

r , 0, t   , and 0

r , 0, t 0 (25a)

The remote boundary conditions due to the effect of a point heat source must be nil at any time as below:

     

 

lim

r

, , ,

z

, , , , , 0, 0, 0

z

u r z t u r z t

r z t



 (25b)

Assuming there are no initial change of thermal displacements and temperature increment for the thermoelastic medium, the initial conditions at time t  0

due to a point heat source are treated as

, , 00

u

r

r z

 , u

z

r z , , 0

  , and 0

r z , , 0

0 (26)

From these basic governing equations, the corresponding quantities of poroelasticity and thermoelasticity are shown in Table 1.

Table 1

Analogy of Poroelastic and Thermoelastic Quantities

Poromechanics Thermomechanics

p

,

u i

i

r z u i

i

  r z ,

 

1 2 3

u

 1

u

B

 

 

  21

1 2 G

 s

  

  

2

2

9 1 2

2 1 2 1

u u

GB

u

  

 

 

 

0

c T

w

k

T

0t

0

W T

附 17

參考文獻

相關文件

This research is conducted with the method of action research, which is not only observes the changes of students’ creativity, but also studies the role of instructor, the

To ensure the Xianbei and Han people would live together peacefully, Emperor Xiaowen (reigned 471-499) not only moved the capital from Pingcheng to Luoyang, but also carried out

The compilers of the biographies of monks not only wrote about the crucial life experiences of these eminent monks, but also revealed wonderful affi nities that brought them

It is interesting that almost every numbers share a same value in terms of the geometric mean of the coefficients of the continued fraction expansion, and that K 0 itself is

Fayun’s annotation is according to Kumarajiva’s original translation, not only sentences by the strict branch demonstrates it to the Lotus Sutra, also aware of

When the fan has not been opened, type B(without fan、Heat Sink), D(without fan、without heat Sink) streamline with experiment value are identical.. Besides, consider whether

and Feng-Tsai Lin, “Analysis of the Transient Ground Surface Displacements Subject to a Point Sink in a Poroelastic Half Space,” Chung Hua Journal of Science and Engineering,

In our experiment, knowledge structures can not only effectively present representative topics, related techniques and issues, but also help understand the research