定水頭部分貫穿井之水井水力學模式為混合邊界值問題，因為在井的邊界上同時存在兩種邊界：在井篩上為定水頭邊界，井篩之外為不透水邊界。一般而言，混合邊界值問題無法直接由積分轉換求解。然而，本研究使用拉普拉斯轉換及有限傅利葉餘弦轉換求得定水頭部分貫穿井於受壓含水層之半解析解。在此方法中，將井篩不均勻地分割成許多小段，每一小段上皆維持相同的定水頭，但有不同的流通量。然後，將井篩上的定水頭邊界置換成對應的流通量邊界，即將混合邊界轉換成均勻邊界，利用積分轉換便可求解。最後，由井篩上的定水頭可求得井篩上的流通量變化，獲得混合邊界值問題的半解析解。由此半解析解可獲得當含水層厚度為無限大時的暫態解及穩態解。當井篩底部（或頂部）至含水層底部（或頂部）的距離大於100倍的井篩長度時，含水層厚度可視為無限大，地下水流可到達穩定狀態，抽水井流量是從無限遠的含水層底部所供應的。若含水層厚度為有限時，地下水流為暫態，抽水井的流量隨時間增加而衰減。部份貫穿效應在定水頭井周圍最大，隨著水平距離增加而減小，在1至2倍含水層厚度除以含水層異向性之平方根的距離上消失，視井篩位置而定。水平方向之水力導數及儲水係數可由全程貫穿井的比洩降獲得。當井篩由含水層頂部貫穿時，垂直方向之水力導數可由抽水井的比洩降求得。 An analytical approach using integral transform techniques is developed to deal with a well hydraulics model involving a mixed boundary of a flowing partially penetrating well, where constant drawdown is stipulated along the well screen and no-flux condition along the remaining unscreened part. The aquifer is confined and of finite thickness. First, the mixed boundary is changed into a homogeneous Neumann boundary by discretizing the well screen into a finite number of segments, each of which at constant drawdown is subject to unknown a priori well bore flux. Then, the Laplace and the finite Fourier transforms are used to solve this modified model. Finally, the prescribed constant drawdown condition is reinstated to uniquely determine the well bore flux function, and to restore the relation between the solution and the original model. The transient and the steady-state solutions for infinite aquifer thickness can be derived from the semi-analytical solution, complementing the currently available dual integral solution. If the distance from the edge of the well screen to the bottom/top of the aquifer is 100 times greater than the well screen length, aquifer thickness can be assumed infinite for times of practical significance, and groundwater flow can reach a steady-state condition, where the well will continuously supply water under a constant discharge. However, if aquifer thickness is smaller, the well discharge decreases with time. The partial penetration effect is most pronounced in the vicinity of the flowing well, decreases with increasing horizontal distance, and vanishes at distances larger than 1 to 2 times the aquifer thickness divided by the square root of aquifer anisotropy. The horizontal hydraulic conductivity and the specific storage coefficient can be determined from vertically averaged drawdown as measured by fully penetrating observation wells. The vertical hydraulic conductivity can be determined from the well discharge under two particular partial penetration conditions.
