目前時間分數階傳輸模式只被用來分析與地質材料不起吸脫附作用的非反應性示蹤劑實驗，因此本研究的目的是發展可用來分析與地質材料起吸脫附作用的反應性示蹤劑實驗的時間分數階傳輸模式。假設溶質與地質材料有非均衡吸脫附作用(nonequilibrium sorption)和溶質衰減，本模式修改Schumer et al.[2003]的冪律記憶函數(power-law memory function)來處理時間分數階的微分，但是加上時間尺度有效率係數F[Tγ-1]，用以使模式裡每項因次相同；γ為時間分數階階數，0<γ<1。此時間分數階傳輸模式對時間(包括分數階)作拉普拉斯逆轉換法求得數值解，並且使用有限差分隱示法求解作驗證，成功地分析Starr et al.[1985]的一組非反應性(溴離子)實驗得到γ=0.59，F=1.09[d-0.41]與四組反應性 (鍶-85)實驗得到γ=0.9，F值隨著有效縱向流速的變小而減少。反應性實驗γ值大於非反應性實驗γ值說明非均衡吸脫附作用抓住示蹤劑造成溶質質量損失，且不同類型的示蹤劑會有不同γ值。 For non-Fickian transport problems, the current fractional-in-time models can only deal with nonreactive tracers that do not chemically reacted with geological materials. This study develops a fractional-in-time model suitable for reactive tracers, assuming a non-equilibrium sorption process and biological/radioactive decay. The model development adopts the same modeling approach as given by Schumer et al. [2003], while allowing the power-law memory function involved to be modified by including a time-scaling effective rate coefficient, F in [Tγ-1], where 0<γ<1 is the fractional order of the time derivative in the governing equation. This modification renders a uniform dimensionality for the resultant fractional-in-time governing equation. Solutions to the model developed are determined using the Laplace transform with respect to time and a numerical Laplace inversion technique. The solutions so obtained are verified using a finite-difference model based on Grünwald series for the fractional-in-time term. There are four sets of reactive 85Sr tracer test data from a sand box consisted of a sand layer sandwiched between two silt layers. They cannot be analyzed by the conventional advective-dispersion equation models, but can be successfully analyzed using the fractional-in-time model developed herein. For the bromide breakthrough, it is found thatγ=0.59 and F=1.09 [d-0.41]. For the 85Sr breakthroughs, it is found that γ=0.9 which is larger than that of the bromide because of extra solute mass losses due to non-equilibrium sorption and radioactive decay, and F changes from 15.2~17.5 [d-0.1] under the influence of the flow velocity of the tracer tests. The fractional-in-time model developed appears to be useful for non-Fickian solute transport for both nonreactive and reactive tracers.
