描述受壓、自由或滲漏含水層中由定水頭、定流量或微水試驗引起的水力水頭分布之半解析解

Abstract

本研究發展一個適用於三種水層試驗(定水頭試驗、定流量試驗及微水試驗)，且在有限井半徑與井部分貫穿條件下，可描述受壓含水層、自由含水層及滲漏受壓含水層的水力水頭分佈之數學模式，並利用拉普拉斯轉換及分離變數法求出其半解析解，再利用Stehfest法作數值逆轉，可求得時間域的結果。此三種試驗的井質量平衡邊界，會有不同的表示方式：在定流量試驗下之抽水量為定值，在微水試驗下僅於井內瞬間注入或取出一小體積的水，在定水頭試驗下井水位維持定值。本模式依水層不同而上邊界條件有異：受壓含水層的上邊界為不透水層，自由含水層的上邊界為自由液面，而滲漏受壓含水層的上邊界考慮上方滯水層之滲漏。所求得的解預測結果顯示，在相同假設條件下，各試驗在井部分貫穿於受壓、自由含水層之解，會與前人之解吻合。預測結果也顯示，在微水試驗井完全或部分貫穿於自由含水層下，忽略自由液面的重力排水會低估井內的水位。此外，在試驗井部分貫穿滲漏受壓含水層下，作定水頭試驗時，水層之水頭變化隨觀測距離、滯水層的滲漏係數及垂直水力傳導係數的減少而降低，且隨井的貫穿率的增加而降低；當作定流量試驗時，水層之洩降變化隨觀測距離、貫穿率、滯水層的滲漏係數及垂直水力傳導係數的增加而降低。

This study develops a general mathematical model for analyzing the hydraulic head distribution in confined, unconfined or leaky confined aquifers with a finite radius and partial penetration well during different types of aquifer tests (i.e., constant-head test, constant-flux test, and slug test). The semi-analytical solution is derived using the methods of Laplace transform and separation of variables, and the time-domain results are obtained by Stehfest method. The conservation of mass flux across the well screen is applied to describe the groundwater flow in these three aquifer tests. The pumping rate is maintained constant in the constant-flux test. In the slug test, a small volume of water is quickly added or removed from the well. In a constant-head test is performed to keep a constant well water level. A general upper boundary condition of the model is applied for three types of aquifers. For a confined aquifer, the top boundary of the aquifer is impermeable; for an unconfined aquifer, the free surface equation is used; for leaky confined aquifer, an equation describing the leakage rate from the aquitard is applied. The proposed solution agrees well with the solutions for confined and unconfined aquifers presented in previous studies. The result predicted from the present solution demonstrates that the well water level is underestimated when neglecting the delayed gravity response for slug test at a fully or partially penetrating well conducted in unconfined aquifers. For the constant head test at a partially penetrating well in the leaky confined aquifer, the hydraulic head decreases with the distance of observation well, the leakance and the vertical hydraulic conductivity of aquitard, and increases with decreasing the penetration ratio. Furthermore, the drawdown decreases with increasing the distance of observation well, the penetration ratio, the leakance and the vertical hydraulic conductivity of aquitard during the constant flux test.