含井膚層水層在有限邊界條件下井緣流量解與洩降解之研究

Abstract

工程上，定水頭試驗及定流量試驗的數據通常被用來推估含水層的參數。定水頭試驗是藉由注水或抽水產生固定水頭，進而量測井緣流量，通常是應用在低透水性的水層；定流量試驗則是藉由固定抽水量，在觀測井量測洩降的分佈值，此試驗適合用於透水性高的水層。在過去的文獻中，已有單層或含井膚層水層的解析解，得知當時間越久或觀測井距離試驗井越遠的情況下，井膚層的效應很小可忽略不計。但是，當外邊界為有限值的情況下，此問題對於井緣流量解和洩降解的影響，較少被討論。另一方面，由於此問題相關的解析解，形式複雜且不易計算數值，因此，近似解是值得探討的議題，過去的文獻顯示，利用拉普拉斯域變數與時間域變數成反比的關係，可自拉普拉斯解求得近似解。本論文目的為討論有限邊界的定流量試驗及定水頭試驗問題，利用拉普拉斯轉換，分別求得洩降及井緣流量的半解析解，再利用Crump數值逆轉方法，求得時間域的數值。此外，本文探討有限邊界對於井緣流量解及洩降解的影響，也考慮於遠處邊界為無限或有限值的情況下，分別推導得含井膚層水層在長時間條件下的洩降及井緣流量近似解。所得結果顯示，當遠處邊界為無限時，井緣流量解、洩降解及此兩者的近似解經過簡化，可分別得到先前單層含水層的解。

The constant-head test and constant-flux test are commonly employed for estimating the aquifer parameters in engineering practice. The constant-head test injects or pumps water with a variable flow rate for maintaining a constant hydraulic head in a low-permeability aquifer while the constant-flux test keeps a constant flow rate to record the drawdown distribution from the observation well of a high-permeability aquifer. The solutions for the wellbore flow rate and drawdown at a well with a finite radius in an infinite confined aquifer with or without a skin zone have been reported in the groundwater literature. The effects of well radius and skin zone are negligible if the test period is very long and/or the distance between the observation well and test well is large. However, little attention has been paid to the effect of a finite boundary on the flow-rate and drawdown solutions in the groundwater community. The main objectives of this thesis are first to develop new semi-analytical solutions for exploring the effect of finite boundary on the wellbore flow-rate and drawdown solutions in a confined aquifer where a finite skin zone is present. These solutions are then calculated by the modified Crump algorithm. The Laplace-domain solution can reduce to the existing infinite-domain solution in some special cases. In addition, an approximate solution for small- or large-time condition is useful if the analytical solution is very complicated and not easy to evaluate accurately. The second objective of this thesis is to derive approximate solutions with considering the effect of skin zone in a finite or infinite confined aquifer based on the relationship between the Laplace variable and time. An approximate solution for an infinite confined aquifer with a skin zone can reduce to the solution without a skin zone if the skin is absent. The large-time solution is equal to the steady-state solution for a finite confined aquifer with a skin zone. In addition, this solution can reduce to Thiem’s equation if the skin zone is absent.