分析斜斷層帶非受壓含水層中抽水引起的地下水流考慮非線性自由液面方程式

Abstract

非線性自由液面方程式(nonlinear free-surface equation, NFSE)描述非受壓含水層的地下水位變動情形，基於兩個假設可以簡化成線性的自由液面方程式(linear free-surface equation, LFSE)，1. 忽略水力梯度平方項 2. 地下水位變化遠小於初始水層厚度。本研究應用疊加原理來處理NFSE，針對斜斷層帶非受壓含水層中的抽水問題，討論NFSE和LFSE預測的地下水位差異。考慮一個矩形的含水層區域，露頭的水位變化滿足NFSE，其他三個邊界滿足不透水邊界條件，虛井沿著水層邊界佈設。水層的洩降可表示為一個一階級數，級數的每一項是Theis[1935]的洩降解析解，反映抽水井和虛井的影響。將本研究的解代入NFSE和不透水邊界條件可獲得一組聯立方程式，各虛井的體積流率為聯立方程式的根。研究結果顯示LFSE的適用性取決於無因次化參數 q=Q/(4π Tw_1 sinθ)的大小，其中Q為井的抽水速率，T為水層導水係數，w_1是沿著斜層方向抽水井到露頭的距離，θ為斜層與地面的夾角。當 q  10-2 時，LFSE和NFSE皆能預測相同的地下水位變化。當 q > 10-2 時，相較於NFSE，LFSE在預測的地下水位位置會有顯著的誤差。 關鍵字：自由液面方程式，二次項水力梯度，虛假井，疊加原理，傾斜斷層水層

The nonlinear free-surface equation (NFSE) has been simplified to a linear one, called linear free-surface equation, LFSE, based on two assumptions that (1) squared hydraulic gradients are ignorable and (2) water table movement is much smaller than the initial aquifer thickness. This study relaxes these two assumptions by applying the superposition principle to deal with the NFSE for the problem of flow toward a well in a sloping fault-zone unconfined aquifer. We consider two-dimensional flow in rectangular aquifer domain with one side, moving boundary satisfying the NFSE, at the outcrop and the other three sides, no-flow strata, under the ground. Image wells surrounds the aquifer boundary. The aquifer drawdown subject to the NFSE equals the sum of the Theis [1935] solution for the extraction well and a simple series for the image wells. The volumetric rates of the image wells are the solution of a system of equations arisen from substituting the present solution into the NFSE. Research results indicate that the validity of the LFSE depends on the magnitude of dimensionless parameter q=Q/(4πTw_1 sin⁡θ) where Q is the volumetric rate of the extraction well, T is the aquifer transmissivity, w_1 is a distance between the extraction well and outcrop in the sloping direction, and θ is the inclination of the aquifer. When q <= 0.01, conventional analytical solutions based on the LFSE can give good predictions. When q > 0.01, the LFSE causes significant error in predicting water table location compared with that based on the NFSE. Keywords: free-surface equation, squared hydraulic gradient, image well, superposition principle, sloping fault-zone aquifer