# Comparison of Time Nonlocal Transport Models for Characterizing Non-Fickian Transport: From Mathematical Interpretation to Laboratory Application

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## Abstract

**:**

## 1. Introduction

## 2. Review and Evaluation of Time Nonlocal Transport Models

#### 2.1. Multi-Rate Mass Transfer Model

_{m}and C

_{im,j}[ML

^{−3}] represent the aqueous concentrations in the well-mixed mobile zone and the j-th well-mixed immobile zone, respectively; β

_{j}[dimensionless] is the capacity coefficient usually defined as the ratio of porosities of the j-th immobile and the mobile phases; v [LT

^{−1}] is the velocity vector; D [L

^{2}T

^{−1}] is the dispersion coefficient tensor; n [dimensionless] is the number of distinct immobile phases; and α

_{j}[T

^{−1}] is the first-order mass transfer rate (also called the rate coefficient) associated with the j-th immobile zone. When n = 1, Equation (1) reduces to the single-rate mass transfer model.

^{−1}] is a memory function defined by the weighted sum of the exponential decay from individual immobile zones [16]:

#### 2.2. Tempered Time Fractional Advection–Dispersion Equation Model

_{im}denotes the overall chemical concentration in all immobile domains.

^{−1}] is the truncation parameter in time. This modification leads to the tt-fADE:

_{m}(x, t = 0) denotes the initial source located only in the mobile phase. At a time

#### 2.3. Continuous Time Random Walk Framework

_{1}[T] denotes a “typical median transition time” for particles. Inserting (16) into (15), and taking the Laplace and Fourier inverse transform, the following well-known CTRW framework was obtained [21]:

_{ψ}denotes the average velocity, and D

_{ψ}is the dispersion coefficient. When deriving the CTRW (17) from the master Equation (15), the jump size density $\tilde{f}(k)$ in (15) needs to be expanded as

_{ψ}and dispersion coefficient D

_{ψ}in (17):

_{1}is the mean waiting time:

_{2}= t

_{2}/t

_{1}.) keeps the integral of ϕ(t) to be 1. For simplicity and direct comparison between models, we constrain the exponent ξ to be 0 < ξ < 1 in this study. The Laplace transform of (22) is (see Equation (17) in [32])

_{M}[dimensionless] and θ

_{I}[dimensionless] are the porosity in the mobile and (total) immobile domains, respectively. Inserting (16) and (23) into (24), we obtain:

_{1}τ

_{2}:

_{2}, the non-Fickian transport transitions to Fickian diffusion. The cutoff time scale t

_{2}in (22), as explained by [32], “corresponds to the largest heterogeneity length scale”. In another numerical study by Willmann et al. [33], t

_{2}was also called “the late cutoff time”. The above analysis shows that t

_{2}is functionally equivalent to the inverse of the truncation parameter λ used in the tt-fADE model (10) and (11).

_{1}<< t << t

_{2}, one can obtain the memory function g by solving (25) numerically (e.g., using the numerical inverse Laplace transform). This was done by [32], who found that the transition probability scales as a power-law function

_{2}) are related to the parameters in the tt-fADE model (10) and (11) (γ and λ), except for t

_{1}in (17), which may be estimated by the mean diffusive time. Parameters predicted by one model (such as the tt-fADE) may also help to improve the estimated parameters of the other (i.e., the CTRW framework).

## 3. Applications: Capturing Non-Fickian Transport in Multidimensional Porous Media

## 4. Model Fit and Comparison

#### 4.1. The ADE Model with Equilibrium Sorption

^{2}] is the cross-section area; and R [dimensionless] is the retardation coefficient. From a mathematical perspective, R acts as a rescaling factor in time. Hence, we reduce the three parameters v, D, and R in the ADE model to two parameters v* (=v/R) and D* (=D/R). The fitted values for parameters v* and D* are listed in Table 1. These two parameters were calibrated manually based on visual inspection.

#### 4.2. The MRMT Model

- (1)
- MRMT model 1 (single mass-transfer rate): the immobile zones can be simplified by a single, homogeneous first-order mass-transfer rate (which is also the single-rate double-porosity model);
- (2)
- MRMT model 2 (single diffusion rate): the immobile zones have a single diffusion rate for all layers;
- (3)
- MRMT model 3 (two mass-transfer rates): the immobile zones have two sets of rate coefficients;
- (4)
- MRMT model 4 (multiple mass-transfer rates): the immobile zones have a power-law distribution of (first-order mass-transfer) rate coefficients.

#### 4.2.1. MRMT Model 1 with a Single Mass-Transfer Rate

^{−3}T

^{−1}] is a transient term accounting for rate-limited mass transfer between the mobile and immobile domains, and Da is the apparent diffusion coefficient. Here n denotes the dimensionality of the problem, and n = 1, 2, 3 denotes diffusion into layers, cylinders, and spheres, respectively. For a single rate first-order mass-transfer approximation, T(x,t) can be expressed as [35]:

_{L}does not change with the travel distance (Table 2), since the plume expansion with time is captured by the mass transfer term in the model. This is different from the standard ADE model, where the dispersivity must increase with the travel distance to capture scale-dependent dispersion.

#### 4.2.2. MRMT Model 2 with a Single Diffusion Rate

#### 4.2.3. MRMT Model 3 with Two Mass-Exchange Rates

_{1}and α

_{2}) and two capacity coefficients (β

_{1}and β

_{2}) corresponding to the first and the second immobile domains, respectively. The best-fit parameters are listed in Table 4. The dispersivity α

_{L}remains constant, since solute plume expansion (due likely to solute retention) is mainly captured by mass exchange between mobile and the two immobile zones. Or in other words, the two pairs of parameters, α

_{j}(j = 1, 2) and β

_{j}(j = 1, 2), control the mass exchange. Their values fluctuate (without predictable trends) with the travel distance and flow rate in the laboratory experiments, although α

_{2}decreases with increasing α

_{1}for the same flow rate. There is no efficient way to directly measure the rate coefficient or capacity coefficient, which creates a challenge for relating MRMT model parameters to observable physical phenomena, and for providing information to MRMT models based on ancillary observations in heterogeneous media.

#### 4.2.4. MRMT Model 4 with Power-Law Distributed Rate Coefficients

_{min}[T

^{−}^{1}] denotes the minimum rate coefficient, α

_{max}[T

^{−}^{1}] is the upper boundary of the rate coefficient, and k is the exponent.

_{L}remains stable for the same reason mentioned above for the other MRMT models. Flow velocity used in this model can be approximated by the peak velocity for the measured BTC. The total capacity coefficient (β

_{tot}) does not significantly change with the travel distance. The exponent k controls the slope of the late-time BTC in a log-log plot, and hence a larger k denotes faster decline of the late-time BTC. The best-fit k slightly increases with an increasing flow rate, due likely to the relatively faster decline of the late-time solute concentration under a larger water flow rate.

_{min}) in (37) controls the maximum waiting time for solute particles, which is functionally equivalent to the inverse of the cutoff time scale t

_{2}in the CTRW framework and the truncation parameter λ in the tt-fADE model. In general, α

_{min}increases with an increasing flow rate (Table 5). Faster flow may accelerate the mass exchange between mobile and immobile domains, generating shorter mean residence times for solute particles in the immobile domain and leading to a greater mass transfer rate. On the other hand, the maximum mass transfer rate (α

_{max}) in (37) defines the shortest waiting time (for solute particles between two displacements), whose impact on the late-time transport dynamics can be overwhelmed by the other smaller rates. Numerical results also show that α

_{max}apparently does not change with the travel distance and flow rate. Hence, α

_{max}can be kept constant for all cases (Table 5).

_{max}and smaller than α

_{min}. This subtle difference in memory functions between the tt-fADE and MRMT model 4 might be the reason for the differences observed here. In addition, the MRMT model 4 (with six model parameters) requires one more parameter than the tt-fADE model.

#### 4.3. The tt-fADE Model (10) and (11)

_{peak}, due to the fractional-order capacity coefficient β in the left-hand side of the tt-fADE Equations (10) and (11) when γ → 1:

#### 4.4. The CTRW Model

_{1}, t

_{2}, v

_{ψ}, and D

_{ψ}. The time t

_{1}expressed by formula (21) represents the approximated mean transition time. The power law behavior in the BTC begins from t

_{1}and ends at t

_{2}. Knowledge of parameters gained in the fitting exercise of the tt-fADE model in Section 4.3 improves the predictability of the CTRW model. In particular, we set the power-law exponent ξ in the CTRW model equal to the time index γ in the tt-fADE model (10) and (11), and approximate the cutoff time scale t

_{2}in the CTRW framework using the inverse of the truncation parameter λ in the tt-fADE (see Section 2.3). Here a smaller ξ represents more disorder of the host system. The remaining three parameters, including the velocity v

_{ψ}, the dispersion coefficient D

_{ψ}, and the mean waiting time t

_{1}in (17), can be fitted using the observed BTCs. The fitted parameters using the CTRW model are listed in Table 7.

## 5. Discussion

#### 5.1. Comparison of Transport Models

_{1}in the CTRW framework is related to the inverse of the mean rate coefficient in the MRMT model with power-law distributed rate coefficients. Our fitting exercise in Section 4.4 shows that the BTC is not sensitive to α

_{max}. The mean transition time t

_{1}in the CTRW framework and the maximum boundary α

_{max}used in the MRMT model might not be needed when capturing the late-time tailing of solute transport (note that the cutoff time t

_{2}in the CTRW model or the minimum rate coefficient α

_{min}in the MRMT model play a more important role than t

_{1}or α

_{max}in affecting the late-time BTC), and therefore they may be removed from the fitting parameters to simplify the model applications. Note that the tt-fADE model does not need the lower-bound of retention times. In addition, the MRMT model with power-law distributed rate coefficients tends to slightly overestimate the late-time tailing in BTCs (Figure 2 and Figures S6–S10), implying that the actual mass transfer rates may decline faster than a power-law function at late times.

_{2}in CTRW is equivalent to the inverse of the truncation parameter λ in the tt-fADE. On the other hand, the velocity and dispersion coefficient in the CTRW framework significantly differ from those in the tt-fADE model. Similarly, they are not directly related to the solution of the traditional ADE. Applications in Section 4.4 show that the average CTRW transport velocity v

_{ψ}(1.303 mm/s) is ~59% less than the average real peak velocity (3.18 mm/s) of the observed BTC and ~62% less than the average best-fit velocity (3.39 mm/s) in the tt-fADE model at the flow rate Q = 1.4 mL/s. For the field tracer tests, the CTRW transport velocity v

_{ψ}(0.3 mm/s) is two orders of magnitude larger than the real peak velocity (0.003 mm/s) of the observed BTC, which is similar to the best-fit velocity (0.004 mm/s) in the tt-fADE model. The velocity used in the tt-fADE model can be calculated by using Equation (38), instead of fitting. This procedure further eliminates the number of parameters in the tt-fADE model and makes the fitting more convenient. In addition, the dispersion coefficient in the CTRW framework D

_{ψ}cannot be kept constant under the same flow rate for different travel distances like the dispersion coefficient in the tt-fADE model for both laboratory experiments and field tracer tests. Therefore, the spatially averaged velocity defined by Formula (19) for the CTRW framework may differ from the actual pore-scale velocity. In the CTRW model, different from the traditional ADE, the advective, dispersive, and diffusive transport mechanisms are combined in the random walk formalism. The advective component and the dispersive component are calculated by spatial moments of the same joint probability density function (PDF) for particle transitions and hence cannot be disconnected [21]. In particular, according to Equation (19), velocity in the CTRW model can be estimated by determining the characteristic time and mean distance, which is the first moment of the PDF of transition displacement. It is, however, difficult to predict the effective velocity v

_{ψ}without detailed knowledge of the porous medium. It is also noteworthy that the generalized master Equation (21) does not separate the effects of the spatially varying velocity field on solute particle displacement into an advective part and a dispersive part. The concept of CTRW therefore does not build an explicit relationship between real velocity and model velocity, and the same is true for the dispersion coefficient. In addition, the power-law exponent ξ can also affect the overall magnitude of solute plume expansion, an effect that intermingles with the dispersion coefficient D

_{ψ}and the effective velocity v

_{ψ}. The above analysis is consistent with the result in [40] that many parameters, particularly v

_{ψ}and ξ, in the CTRW model are correlated with each other. Their fitting exercises showed that different parameter combinations can lead to the same mean Eulerian velocity predicted by the model, implying that the estimated model parameters were not unique.

_{peak}because the effective velocity used in the tt-fADE is adjusted by the elapsed time for solutes spent in retention, as represented by the fractional-order capacity coefficient β on the right-hand side of Equation (38). In this study, laboratory experiments with six flow rates and field tracer tests show that the best-fit velocity in the tt-fADE model is close to the measured BTC’s peak velocity, since the capacity coefficient shown in Equation (38) is relatively small. Because the transport velocity used in the tt-fADE model (10) and (11) is not significantly different from the BTC’s peak velocity, the tt-fADE model uses an independent parameter, the time index γ, to control the power-law distribution of the late-time BTC. This parameterization of the tailing is relatively simple as compared to the CTRW model, with three parameters, v

_{ψ}, D

_{ψ}, and ξ that contribute to time-nonlocal non-Fickian dispersion. We calculate the root mean square error (RMSE) for the laboratory experiments. Comparison of the RMSE (see Table 1, Table 6 and Table 7) of the two models and the ADE model demonstrates that they both perform better than the standard ADE, especially in describing tailing behavior in a heterogeneous medium. When the Formula (38) is used, the tt-fADE model is more convenient to apply in practice than the CTRW framework, since the former contains fewer parameters.

_{1}in the CTRW framework), since the slow advection can also affect the late-time BTC tail. Mass exchange can apparently affect the late-time BTC tail only when the diffusive time scale is much longer than the advective time scale, as pointed out by [16]. This may explain why t

_{1}might not be needed in the CTRW framework. Further real-world tests are needed to check the above hypotheses.

#### 5.2. Parameter Sensitivity

^{2}/s to 0.0005 cm

^{2}/s results in similar BTCs after normalization (i.e., re-scaling), implying that trapping due to the immobile zones may account at least partially for the spatial expansion of solute plumes. When D increases from 0.005 cm

^{2}/s to 0.05 cm

^{2}/s, the BTC becomes wider and its shape slightly changes (Figure 7a,b).

#### 5.3. Application to Field Transport

^{3}/h, and the injection duration was 6 h. A peristaltic pump was used to collect the samples in a chronological order, and the sample concentration was measured using a MP523-06 bromide ion concentration meter.

## 6. Conclusions

_{2}in the CTRW framework is also equivalent to the inverse of the truncation parameter λ in the tt-fADE (not shown specifically before). Hence the predictability obtained by the tt-fADE model can also improve the predictability of the CTRW framework, and vice-versa. (3) Compared to the tt-fADE model, the CTRW framework defines one additional parameter t

_{1}, which represents the mean diffusive time, corresponding to the mean of the inverse of rate coefficients in the MRMT model. Model applications, however, showed that t

_{1}in the CTRW framework is insensitive to model results, and may be neglected to alleviate model fitting burdens.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Laboratory tracer test: Comparison between the measured (symbols) and the modeled (lines) breakthrough curves (BTCs) using the advection–dispersion equation (ADE), the tempered time fractional advection–dispersion equation (tt-fADE) (red thick lines), the continuous time random walk (CTRW) (green dashed lines), and the multi-rate mass transfer (MRMT) model with the water flow rate Q = 1.4 mL/s.

**Figure 8.**Study site map showing injection well, pumping well, and continuous multichannel tubing (CMT) wells.

**Figure 9.**Field tracer test: comparison between the measured field data (symbols) and the modeled (lines) breakthrough curves using the ADE, the tt-fADE (red thick lines), the CTRW (green dashed lines), and the MRMT models.

**Table 1.**Measured and fitted parameters (using the advection–dispersion equation (ADE) model with equilibrium sorption) for Brilliant Blue FCF (BBF) transport through the “lightweight expanded clay aggregate” (Leca) beads at different flow rates and travel distances and field tests. In the legend, Q represents the flow rate; L denotes the travel distance (i.e., the length of the column); v* (=v/R) is the average flow velocity divided by the retardation coefficient; D* (=D/R) is the dispersion coefficient divided by the retardation coefficient; θ is the porosity; and RMSE stands for root mean square error between observed values and predicted values.

Experiment | Q (mL/s) | L (cm) | v* (mm/s) | D* (mm^{2}/s) | Θ | RMSE |
---|---|---|---|---|---|---|

Measured | Measured | Fitted | Fitted | Measured | Calculated | |

Lab | 0.4 | 50 | 1.20 | 5.0 | 0.39 | 2.75 |

70 | 1.12 | 7.0 | 0.39 | 1.54 | ||

100 | 1.07 | 5.3 | 0.39 | 2.30 | ||

0.6 | 50 | 1.45 | 6.8 | 0.39 | 2.16 | |

70 | 1.55 | 15 | 0.39 | 0.78 | ||

100 | 1.63 | 9.2 | 0.39 | 1.99 | ||

0.8 | 50 | 2.23 | 15 | 0.39 | 1.57 | |

70 | 2.20 | 22 | 0.39 | 0.97 | ||

100 | 2.07 | 14 | 0.39 | 1.55 | ||

1 | 50 | 2.67 | 20 | 0.39 | 1.85 | |

70 | 2.85 | 24 | 0.39 | 1.43 | ||

100 | 2.90 | 16 | 0.39 | 2.92 | ||

1.2 | 50 | 3.25 | 23 | 0.39 | 2.80 | |

70 | 3.22 | 28 | 0.39 | 1.53 | ||

100 | 3.04 | 19 | 0.39 | 2.40 | ||

1.4 | 50 | 3.48 | 40 | 0.39 | 1.09 | |

70 | 3.85 | 50 | 0.39 | 0.90 | ||

100 | 3.70 | 38 | 0.39 | 1.33 | ||

Field | 83.33 | 200 | 0.003 | 0.694 | 0.3 | n/a |

400 | 0.004 | 0.984 | 0.3 | n/a | ||

600 | 0.004 | 0.926 | 0.3 | n/a |

**Table 2.**The fitted parameters for the single mass-transfer rate mobile-immobile model (i.e., Equations (31)–(34)) at different flow rates and travel distances. In the legend, L denotes the travel distance (i.e., the length of the column); α

_{L}is the dispersivity; v is the flow velocity; β

_{tot}is the total capacity coefficient; and α stands for the mass transfer rate.

Experiment | Q (mL/s) | L (cm) | α_{L} (mm) | v (mm/s) | β_{tot} | α (s^{−1}) |
---|---|---|---|---|---|---|

Measured | Measured | Fitted | Fitted | Fitted | Fitted | |

Lab | 50 | 1.5 | 1.4 | 0.50 | 0.05 | |

0.4 | 70 | 1.5 | 1.35 | 0.50 | 0.04 | |

100 | 1.5 | 1.35 | 0.50 | 0.04 | ||

50 | 1.5 | 1.68 | 0.50 | 0.05 | ||

0.6 | 70 | 1.5 | 1.71 | 0.40 | 0.04 | |

100 | 1.5 | 1.68 | 0.31 | 0.03 | ||

50 | 1.5 | 2.38 | 0.45 | 0.06 | ||

0.8 | 70 | 1.5 | 2.32 | 0.40 | 0.04 | |

100 | 1.5 | 2.15 | 0.33 | 0.05 | ||

50 | 1.5 | 2.75 | 0.45 | 0.06 | ||

1 | 70 | 1.5 | 2.93 | 0.43 | 0.06 | |

100 | 1.5 | 2.93 | 0.35 | 0.07 | ||

50 | 1.5 | 3.15 | 0.30 | 0.06 | ||

1.2 | 70 | 1.5 | 3.15 | 0.30 | 0.06 | |

100 | 1.5 | 3.13 | 0.28 | 0.06 | ||

50 | 1.5 | 3.52 | 0.35 | 0.06 | ||

1.4 | 70 | 1.5 | 3.95 | 0.35 | 0.06 | |

100 | 1.5 | 3.98 | 0.35 | 0.06 | ||

Field | 200 | 210 | 0.004 | 0.50 | 0.14 | |

83.33 | 400 | 230 | 0.004 | 0.50 | 0.10 | |

600 | 230 | 0.004 | 0.50 | 0.03 |

**Table 3.**The fitted parameters for the single diffusion rate multi-rate mass transfer (MRMT) model at different flow rates and travel distances. In the legend, α

_{d}is the diffusion rate coefficient (α

_{d}= D

_{a}/a

^{2}, where Da is the apparent diffusion coefficient and a is the layer half-thickness).

Experiment | Q (mL/s) | L (cm) | α_{L} (mm) | v (mm/s) | β_{tot} | α_{d} (s^{−}^{1}) |
---|---|---|---|---|---|---|

Measured | Measured | Fitted | Fitted | Fitted | Fitted | |

Lab | 0.4 | 50 | 1.5 | 1.55 | 0.60 | 0.02 |

70 | 1.5 | 1.48 | 0.60 | 0.02 | ||

100 | 1.5 | 1.43 | 0.60 | 0.02 | ||

0.6 | 50 | 1.5 | 1.73 | 0.50 | 0.03 | |

70 | 1.5 | 2.22 | 0.80 | 0.02 | ||

100 | 1.5 | 2.12 | 0.61 | 0.02 | ||

0.8 | 50 | 1.5 | 2.69 | 0.61 | 0.03 | |

70 | 1.5 | 2.85 | 0.70 | 0.02 | ||

100 | 1.5 | 2.55 | 0.55 | 0.03 | ||

1 | 50 | 1.5 | 3.00 | 0.55 | 0.03 | |

70 | 1.5 | 3.30 | 0.55 | 0.02 | ||

100 | 1.5 | 3.25 | 0.48 | 0.05 | ||

1.2 | 50 | 1.5 | 3.73 | 0.48 | 0.04 | |

70 | 1.5 | 3.69 | 0.48 | 0.03 | ||

100 | 1.5 | 3.25 | 0.31 | 0.03 | ||

1.4 | 50 | 1.5 | 4.25 | 0.60 | 0.03 | |

70 | 1.5 | 4.90 | 0.60 | 0.03 | ||

100 | 1.5 | 4.10 | 0.40 | 0.02 | ||

Field | 83.33 | 200 | 210 | 0.004 | 0.50 | 0.05 |

400 | 210 | 0.004 | 0.50 | 0.04 | ||

600 | 210 | 0.005 | 0.50 | 0.03 |

**Table 4.**The fitted parameters for the two-set MRMT model at different flow rates and travel distances. In the legend, α

_{1}and β

_{1}represent the mass transfer rate and capacity coefficient of the first immobile domain, respectively; and α

_{2}and β

_{2}are the mass transfer rate and capacity coefficient of the second immobile domain, respectively.

Experiment | Q (mL/s) | L (cm) | α_{L} (mm) | v (mm/s) | α_{1} (s^{−1}) | β_{1} | α_{2} (s^{−1}) | β_{2} |
---|---|---|---|---|---|---|---|---|

Measured | Measured | Fitted | Fitted | Fitted | Fitted | Fitted | Fitted | |

Lab | 0.4 | 50 | 1.5 | 1.12 | 0.08 | 0.08 | 0.015 | 0.15 |

70 | 1.5 | 1.10 | 0.08 | 0.09 | 0.015 | 0.15 | ||

100 | 1.5 | 1.03 | 0.10 | 0.09 | 0.008 | 0.10 | ||

0.6 | 50 | 1.5 | 1.31 | 0.08 | 0.07 | 0.020 | 0.13 | |

70 | 1.5 | 1.54 | 0.15 | 0.08 | 0.020 | 0.25 | ||

100 | 1.5 | 1.55 | 0.15 | 0.08 | 0.015 | 0.15 | ||

0.8 | 50 | 1.5 | 1.96 | 0.05 | 0.15 | 0.030 | 0.08 | |

70 | 1.5 | 2.08 | 0.05 | 0.20 | 0.015 | 0.10 | ||

100 | 1.5 | 1.88 | 0.07 | 0.10 | 0.018 | 0.10 | ||

1 | 50 | 1.5 | 2.30 | 0.05 | 0.12 | 0.030 | 0.15 | |

70 | 1.5 | 2.50 | 0.07 | 0.13 | 0.025 | 0.13 | ||

100 | 1.5 | 2.57 | 0.20 | 0.11 | 0.020 | 0.11 | ||

1.2 | 50 | 1.5 | 2.85 | 0.07 | 0.12 | 0.020 | 0.13 | |

70 | 1.5 | 2.87 | 0.06 | 0.11 | 0.027 | 0.13 | ||

100 | 1.5 | 2.87 | 0.07 | 0.10 | 0.025 | 0.01 | ||

1.4 | 50 | 1.5 | 3.33 | 0.08 | 0.17 | 0.030 | 0.17 | |

70 | 1.5 | 3.80 | 0.09 | 0.17 | 0.030 | 0.17 | ||

100 | 1.5 | 3.75 | 0.08 | 0.15 | 0.030 | 0.16 | ||

Field | 83.33 | 200 | 210 | 0.004 | 0.4 | 0.2 | 0.1 | 0.2 |

400 | 210 | 0.004 | 0.4 | 0.2 | 0.1 | 0.2 | ||

600 | 210 | 0.004 | 0.4 | 0.1 | 0.1 | 0.1 |

**Table 5.**The best-fit parameters for the power-law MRMT model 4 at different flow rates and travel distances. In the legend, k stands for the slope of the late-time tail, and α

_{min}and α

_{max}stand for the lower and upper boundary of the mass transfer rates, respectively.

Experiment | Q (mL/s) | L (cm) | α_{L} (mm) | v (mm/s) | β_{tot} | k | α_{min} (s^{−}^{1}) | α_{max} (s^{−}^{1}) |
---|---|---|---|---|---|---|---|---|

Measured | Measured | Fitted | Fitted | Fitted | Fitted | Fitted | Fitted | |

Lab | 0.4 | 50 | 1.6 | 1.40 | 0.50 | 2.250 | 0.010 | 0.6 |

70 | 1.6 | 1.35 | 0.50 | 2.250 | 0.010 | 0.6 | ||

100 | 1.6 | 1.35 | 0.50 | 2.250 | 0.010 | 0.6 | ||

0.6 | 50 | 1.6 | 1.69 | 0.50 | 2.292 | 0.011 | 0.6 | |

70 | 1.6 | 1.85 | 0.60 | 2.292 | 0.012 | 0.6 | ||

100 | 1.6 | 2.10 | 0.60 | 2.292 | 0.012 | 0.6 | ||

0.8 | 50 | 1.6 | 2.83 | 0.70 | 2.290 | 0.013 | 0.6 | |

70 | 1.6 | 2.95 | 0.80 | 2.290 | 0.012 | 0.6 | ||

100 | 1.6 | 3.05 | 0.80 | 2.290 | 0.013 | 0.6 | ||

1 | 50 | 1.6 | 3.59 | 0.80 | 2.300 | 0.013 | 0.6 | |

70 | 1.6 | 4.05 | 0.85 | 2.300 | 0.013 | 0.6 | ||

100 | 1.6 | 4.16 | 0.80 | 2.300 | 0.014 | 0.6 | ||

1.2 | 50 | 1.6 | 4.63 | 0.80 | 2.335 | 0.014 | 0.6 | |

70 | 1.6 | 4.63 | 0.80 | 2.335 | 0.014 | 0.6 | ||

100 | 1.6 | 4.44 | 0.75 | 2.335 | 0.014 | 0.6 | ||

1.4 | 50 | 1.6 | 4.95 | 0.85 | 2.358 | 0.014 | 0.6 | |

70 | 1.6 | 5.48 | 0.80 | 2.358 | 0.015 | 0.6 | ||

100 | 1.6 | 5.50 | 0.80 | 2.358 | 0.015 | 0.6 | ||

Field | 83.33 | 200 | 210 | 0.004 | 0.5 | 2.25 | 0.01 | 0.6 |

400 | 210 | 0.004 | 0.5 | 2.25 | 0.01 | 0.6 | ||

600 | 210 | 0.005 | 0.5 | 2.25 | 0.01 | 0.6 |

**Table 6.**The best-fit parameters for the tempered time fractional advection–dispersion equation (tt-fADE) model (10) and (11) at different flow rates and travel distances. In the legend, α is the dispersivity; γ is the time/scale index; β is the fractional-order capacity coefficient; and λ is the truncation parameter.

Experiment | Q (mL/s) | L (cm) | v (mm/s) | D (mm^{2}/s) | α (mm) | γ | β (s^{γ}^{−1}) | λ (s^{−1}) | RMSE |
---|---|---|---|---|---|---|---|---|---|

Measured | Measured | Fitted | Fitted | α = D/v | Fitted | Fitted | Fitted | Calculated | |

Lab | 0.4 | 50 | 1.10 | 0.2 | 0.18 | 0.250 | 0.011 | 0.010 | 0.379 |

70 | 1.05 | 0.2 | 0.19 | 0.250 | 0.011 | 0.010 | 0.171 | ||

100 | 1.00 | 0.2 | 0.20 | 0.250 | 0.011 | 0.010 | 0.343 | ||

0.6 | 50 | 1.30 | 0.2 | 0.15 | 0.292 | 0.013 | 0.011 | 0.461 | |

70 | 1.40 | 0.4 | 0.29 | 0.292 | 0.013 | 0.012 | 0.296 | ||

100 | 1.49 | 0.4 | 0.27 | 0.292 | 0.013 | 0.012 | 0.489 | ||

0.8 | 50 | 1.91 | 0.5 | 0.26 | 0.290 | 0.018 | 0.013 | 0.577 | |

70 | 1.95 | 0.6 | 0.31 | 0.290 | 0.018 | 0.012 | 0.398 | ||

100 | 1.84 | 0.6 | 0.33 | 0.290 | 0.018 | 0.013 | 0.443 | ||

1 | 50 | 2.25 | 1.8 | 0.80 | 0.300 | 0.014 | 0.013 | 0.529 | |

70 | 2.40 | 1.8 | 0.75 | 0.300 | 0.014 | 0.013 | 0.218 | ||

100 | 2.40 | 1.8 | 0.75 | 0.300 | 0.014 | 0.014 | 0.326 | ||

1.2 | 50 | 2.86 | 1.0 | 0.35 | 0.335 | 0.019 | 0.014 | 0.405 | |

70 | 2.84 | 1.0 | 0.35 | 0.335 | 0.019 | 0.014 | 0.190 | ||

100 | 2.81 | 1.0 | 0.36 | 0.335 | 0.019 | 0.014 | 0.260 | ||

1.4 | 50 | 3.18 | 3.0 | 0.94 | 0.358 | 0.026 | 0.014 | 0.636 | |

70 | 3.55 | 3.0 | 0.85 | 0.358 | 0.026 | 0.015 | 0.380 | ||

100 | 3.45 | 3.0 | 0.87 | 0.358 | 0.026 | 0.015 | 0.329 | ||

Field | 83.33 | 200 | 0.004 | 0.69 | 198 | 0.358 | 0.17 | 0.01 | n/a |

400 | 0.005 | 0.69 | 148 | 0.358 | 0.17 | 0.01 | n/a | ||

600 | 0.005 | 0.69 | 148 | 0.358 | 0.17 | 0.01 | n/a |

**Table 7.**The best-fit parameters for the continuous time random walk (CTRW) model. In the legend, v

_{ψ}is the CTRW transport velocity, which can be different from the average pore velocity v; D

_{ψ}is the dispersion coefficient with the subscript ψ indicating CTRW interpretation; ξ is the power-law exponent; t

_{1}is the mean transition time; and t

_{2}is the truncation time scale. ξ is converted from γ in the tt-fADE model (10) and (11), and t

_{2}is converted from λ in the tt-fADE model.

Experiment | Q | L | v_{ψ} | D_{ψ} | ξ | log_{10}t_{1} (s) | log_{10}t_{2} (s) | RMSE |
---|---|---|---|---|---|---|---|---|

(mL/s) | (cm) | (mm/s) | (mm^{2}/s) | |||||

Measured | Measured | Fitted | Fitted | Fixed | Fitted | Fixed | Calculated | |

Lab | 0.4 | 50 | 0.465 | 0.38 | 0.250 | 1.70 | 2.00 | 0.239 |

70 | 0.637 | 0.69 | 0.250 | 1.70 | 2.00 | 0.447 | ||

100 | 0.900 | 14.00 | 0.250 | 1.70 | 2.00 | 0.862 | ||

0.6 | 50 | 0.505 | 0.50 | 0.292 | 1.70 | 1.94 | 0.932 | |

70 | 0.735 | 1.47 | 0.292 | 1.70 | 1.93 | 0.859 | ||

100 | 1.130 | 2.00 | 0.292 | 1.70 | 1.93 | 1.201 | ||

0.8 | 50 | 0.575 | 0.75 | 0.290 | 1.80 | 1.90 | 1.317 | |

70 | 0.840 | 3.43 | 0.290 | 1.80 | 1.92 | 0.491 | ||

100 | 1.150 | 4.00 | 0.290 | 1.80 | 1.90 | 0.976 | ||

1 | 50 | 0.660 | 1.00 | 0.300 | 1.80 | 1.90 | 2.039 | |

70 | 0.994 | 2.94 | 0.300 | 1.81 | 1.89 | 0.780 | ||

100 | 1.410 | 4.00 | 0.300 | 1.81 | 1.86 | 0.265 | ||

1.2 | 50 | 0.760 | 1.75 | 0.335 | 1.81 | 1.84 | 0.972 | |

70 | 1.050 | 3.43 | 0.335 | 1.81 | 1.84 | 0.457 | ||

100 | 1.500 | 4.00 | 0.335 | 1.81 | 1.84 | 0.913 | ||

1.4 | 50 | 0.820 | 2.75 | 0.358 | 1.80 | 1.84 | 1.237 | |

70 | 1.288 | 5.88 | 0.358 | 1.80 | 1.83 | 0.713 | ||

100 | 1.800 | 12.00 | 0.358 | 1.80 | 1.82 | 0.439 | ||

Field | 83.33 | 200 | 0.30 | 0.0300 | 0.358 | 1 | 2 | n/a |

400 | 0.18 | 0.0090 | 0.358 | 1 | 2 | n/a | ||

600 | 0.12 | 0.0025 | 0.358 | 1 | 2 | n/a |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lu, B.; Zhang, Y.; Zheng, C.; Green, C.T.; O’Neill, C.; Sun, H.-G.; Qian, J.
Comparison of Time Nonlocal Transport Models for Characterizing Non-Fickian Transport: From Mathematical Interpretation to Laboratory Application. *Water* **2018**, *10*, 778.
https://doi.org/10.3390/w10060778

**AMA Style**

Lu B, Zhang Y, Zheng C, Green CT, O’Neill C, Sun H-G, Qian J.
Comparison of Time Nonlocal Transport Models for Characterizing Non-Fickian Transport: From Mathematical Interpretation to Laboratory Application. *Water*. 2018; 10(6):778.
https://doi.org/10.3390/w10060778

**Chicago/Turabian Style**

Lu, Bingqing, Yong Zhang, Chunmiao Zheng, Christopher T. Green, Charles O’Neill, Hong-Guang Sun, and Jiazhong Qian.
2018. "Comparison of Time Nonlocal Transport Models for Characterizing Non-Fickian Transport: From Mathematical Interpretation to Laboratory Application" *Water* 10, no. 6: 778.
https://doi.org/10.3390/w10060778