本研究發展了時間分數階移流延散模式(FADE)之半解析解，模擬單一裂隙(single fracture)於徑向發散流場中，溶質的非費克傳輸(non-Fickian transport)，以應用於現地的示蹤劑試驗資料分析。所建立的三個時間分數階模式考慮冪律記憶函數(power law memory function)，其分數階階數或稱為時間尺度指數(time scale index)0<μ<1。第一個模式，FADE_ADV_NF，僅考慮移流，且考慮裂隙中因非費克擴散進入母岩層的濃度損失(non-Fickian matrix diffusion)，需加上另一個時間分數階微分項，其時間尺度指數0<ω<0.5。兩個時間尺度指數μ和ω較小者，將控制濃度長時間的冪律拖尾，但無其他獨立的方法判定何者較小，將產生參數非惟一性的問題，故此模式對於分析的幫助不大。第二個模式，FADE_ADV，僅考慮移流，但不考慮母岩層的非費克擴散。第三個模式，FADE_DISP，考慮移流和延散效應，但不考慮母岩層的非費克擴散。當靠近注入井時，小時間可忽略延散影響；而隨著徑向距離越遠，延散之影響越大。裂隙介質中移動相和非移動相(mobile/immobile，MIM)的多重質量轉移率機制(multiple rate mass transfer)，將使溶質傳輸時產生遲滯的效果。本研究中修正分數階容積係數(fractional capacity coefficient)ψ＝F(θim/θm)，其中F為「時間尺度有效轉換率係數」(time-scaling effective rate coefficient)，因次為[Tμ-1]，目的是為了使裂隙中非移動相和移動相之有效孔隙率比值θim/θm為無因次。裂隙岩層示蹤劑試驗資料分析結果顯示，忽略延散影響時可得到較佳的分析結果。在同一裂隙岩層中的兩次試驗具有相同的μ，故可做為異質性程度的特徵參數；分數階容積係數ψ具有尺度效應，與試驗時間呈正比。

The three time-fractional advection dispersion equation (FADE) models are developed for the single fractured formations. The semianalytical solutions are derived for analyzing field tracer test data alike the non-Fickian transport on the divergent flow fields. In the three models, the fractional-in-time derivative within the time scale index 0<μ<1 originates from the considerations of the multiple rate mass transfer between the mobile and immobile phases(MIM) in the fractured media that can be formulated using the power law memory function. The first model, FADE_ADV_NF, assumes the pure advection in the fractured media while allowing non-Fickian matrix diffusion, which adds to the governing equation another fractional-order term within the time scale index 0<ω<0.5. It has been found that the additional term contributes little to the problem of interest because smaller μ or ω is determined from the power law tail, but there is no independent method to determine which is smaller. The second model, FADE_ADV, assumes pure advection in the fractured media without non-Fickian matrix diffusion. The third model, FADE_DISP, assumes both advection and dispersion in the fracture without non-Fickian matrix diffusion. In the vicinity of the injection well where the groundwater velocity is relative large, dispersion can be neglected at the early time. As the radial distance increases, the groundwater velocity decreases and dispersion effects are not negligible. The multiple rate mass transfer between the MIM tends to delay the advancement of solutes. In the current study, the fractional capacity coefficient ψ has been modified by including a time-scaling effective rate coefficient, F in [Tμ-1] where 0<μ<1 is the time scale index to make the porosity ratio between the MIM within the fractures to be dimensionless. From the analysis of the field tracer test data, while neglecting dispersion, model is much better fit to data. The same time scale index μ for two experiments in the same fracture aquifer denotes that it can be the characteristic of the heterogeneity. Fractional capacity coefficient ψ is proportional to the experiment durations and its scale dependent property presents here.

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