徑向雙層受壓含水層定水頭部分貫穿的水力水頭近似解

Abstract

定水頭試驗在部分貫穿井下的水井水力模式是混合邊界的問題，其解析解相當複雜，且數值計算不容易。因此，本研究目的在推導一個新的近似解，用來描述有限受壓含水層在定水頭部分貫穿井條件下的暫態水力水頭。本研究參考電化學領域中所提出與時間相關的擴散層近似方法，以獲得此近似解。此近似解在部分貫穿井的條件下，由一修正型Bessel函數的組合，並且於井完全貫穿含水層時可以簡化為較簡單的自然對數函數的解。當含水層沒有膚層並且為完全貫穿井時，此解可進一步簡化為 Perrochet (2005)的解。在部分貫穿井下的水力水頭近似解，將與Yang and Yeh (2005)的半解析解做比較；而在完全貫穿井下的水力水頭近似解，將與Yang and Yeh (2006)的解析解做比較。比較結果顯示利用近似解計算水力水頭，在時間較小的情況會有差異，然而，上述計算膚層和含水層水力水頭的差值，皆在小數後3位以下。顯然，此近似解的數值計算很簡單，且精度不差，有工程應用的價值。

A mathematical model describing the transient hydraulic head distribution induced by a constant-head pumping/injection in a radial two-zone confined aquifer at a partially penetrating well is a mixed-type boundary value problem. The analytical solution of the model is very complicated and difficult to evaluate accurately. This study aims at developing a new approximate solution describing the transient hydraulic head distribution for a constant-head test (CHT) at a partially penetrating well in aquifers of infinite extent. This approximate solution is acquired based on a time-dependent diffusion layer approximation proposed in the field of electrochemistry. The approximate solution is in terms of modified Bessel functions for aquifers with a partially penetrating well and can reduce to a simpler form in terms of natural logarithmic function for the case of well full penetration. Moreover, it can further reduce to Perrochet’s solution (2005) if the aquifer is without the skin zone and at a fully penetrating well. The predicted transient hydraulic heads from the present approximate solutions are compared with those estimated by Yang and Yeh’s Laplace-domain solution (2005) for partial penetration case and Yang and Yeh’s time-domain solution (2006) for full penetration case. The results show that the differences between the present approximate solution and Yang and Yeh’s solutions (2005, 2006) are large when the test time is small. The differences are however less than 10-3. This newly developed approximate solution has advantages of easy computing and good accuracy from practical viewpoint and thus is a handy tool to evaluate temporal and spatial hydraulic head distributions for the CHT.