示蹤劑試驗於徑向收斂受壓水層之解析模式:考慮膚層與垂直延散效應

Abstract

過去文獻常忽略鑿井過程形成的膚層，因膚層的水文地質特性與地層特性不同，若忽略膚層的存在，可能會錯估示蹤劑試驗的濃度分佈與逆推水層的水文地質參數值；再者，若注入井只部分貫穿水層而抽水井完全貫穿水層，則溶質(示蹤劑)在水層中會有垂直延散(vertical dispersion)的問題。據我們所知，過去與示蹤劑抽出試驗的研究，常忽略膚層及垂直延散對溶質傳輸的影響，有鑒於此，本研究發展一個溶質傳輸的數學模式，此模式將前述兩個效應納入考慮，以描述徑向雙層的受壓水層連續注入示蹤劑後，在觀測點觀測其溶質濃度隨時間的變化。此模式的內邊界為井緣，設定為定流通量條件(Neumann boundary condition)；外側邊界則設於注入井處，井開篩處設為第三類邊界條件(Robin boundary condition)，此外，抽與注兩口井皆考慮井蓄水效應。應用拉普拉斯轉換與有限傅立葉餘弦轉換(finite Fourier cosine transform)，可求得模式的半解析解；接著利用數值逆轉方法，可用來推算任意觀測點溶質濃度的時空分佈，可探討膚層與垂直延散對濃度分佈的影響。

Solute transport from an injection well in aquifers to a pumping well can be simplified to radial convergent transport. Existing models made two assumptions: (1) absence of skin zone due to well installation and (2) exclusion of vertical dispersion from the partially-penetrating injection well, which may cause errors in predicting solute concentration and aquifer parameters. This study relaxes these two assumptions to develop a new mathematical model for radially convergent transport in a two-zone aquifer system. The rim of the pumping well is under the Neumann condition with the wellbore storage effect. The screen location of the injection well is treated as the Robin boundary while the unscreened part is considered as the no-flow boundary. The semi-analytical solution of the model is obtained by the methods of Laplace transform and finite Fourier cosine transform. The Crump algorithm for the inverse Laplace transform is applied to estimate solute concentration in time domain. The present solution is used to explore quantitative conditions that a model can predict the same concentration as the present model despite making the two assumptions. The research results are applicable in the preliminary planning stage of aquifer remediation.