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1
Bauget, F., and M. Fourar. "Non-Fickian dispersion in a single fracture." Journal of Contaminant Hydrology 100, no. 3-4 (September 2008): 137–48. http://dx.doi.org/10.1016/j.jconhyd.2008.06.005.
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Maryshev, Boris, Maminirina Joelson, Dimitri Lyubimov, Tatiana Lyubimova, and Marie-Christine Néel. "Non Fickian flux for advection–dispersion with immobile periods." Journal of Physics A: Mathematical and Theoretical 42, no. 11 (February 18, 2009): 115001. http://dx.doi.org/10.1088/1751-8113/42/11/115001.
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Pannone, Marilena. "An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions." Water 9, no. 10 (September 30, 2017): 751. http://dx.doi.org/10.3390/w9100751.
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Kwaw, Albert Kwame, Zhi Dou, Jinguo Wang, Yuting Zhang, Xueyi Zhang, Wenyuan Zhu, and Portia Annabelle Opoku. "Influence of Clay on Solute Transport in Saturated Homogeneous Mixed Media." Geofluids 2021 (August 25, 2021): 1–14. http://dx.doi.org/10.1155/2021/1207971.
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In this study, four homogeneous porous media (HPM1-HPM4), consisting of distinct proportions of sand-sized and clay-sized solid beads, were prepared and used as single fracture infills. Flow and nonreactive solute transport experiments in HPM1-HPM4 under three flow rates were conducted, and the measured breakthrough curves (BTCs) were quantified using conventional advection-dispersion equation (ADE), mobile-immobile model (MIM), and continuous time random walk (CTRW) model with truncated power law transition time distribution. The measured BTCs showed stronger non-Fickian behaviour in HPM2-HPM4 (which had clay) than in HPM1 (which had no clay), implying that clay enhanced the non-Fickian transport. As the fraction of clay increased, the global error of ADE fits also increased, affirming the inefficiency of ADE in capturing the clay-induced non-Fickian behaviour. MIM and CTRW performed better in capturing the non-Fickian behaviour. Nonetheless, CTRW’s performance was robust. 12.5% and 25% of clay in HPM2 and HPM3, respectively, decreased the flowing fluid region and increased the solute exchange rate between the flowing and stagnant fluid regions in MIM. For CTRW, the power law exponent ( β CTRW ) values were 1.96, 1.75, and 1.63 in HPM1-HPM3, respectively, implying enhanced non-Fickian behaviour. However, for HPM4, whose clay fraction was 50%, the β CTRW value was 1.87, implying a deviation in the trend of non-Fickian enhancement with increasing clay fraction. This deviation indicated that non-Fickian behaviour enhancement depended on the fraction of clay present. Moreover, increasing flow rate enhanced the non-Fickian transport based on β CTRW .
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Levy, Melissa, and Brian Berkowitz. "Measurement and analysis of non-Fickian dispersion in heterogeneous porous media." Journal of Contaminant Hydrology 64, no. 3-4 (July 2003): 203–26. http://dx.doi.org/10.1016/s0169-7722(02)00204-8.
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Zoua, Shimin, Jianghai Ma, and Antonis D. Koussis. "Analytical solutions to non-Fickian subsurface dispersion in uniform groundwater flow." Journal of Hydrology 179, no. 1-4 (May 1996): 237–58. http://dx.doi.org/10.1016/0022-1694(95)02830-7.
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Bottacin-Busolin, Andrea. "Non-Fickian dispersion in open-channel flow over a porous bed." Water Resources Research 53, no. 8 (August 2017): 7426–56. http://dx.doi.org/10.1002/2016wr020348.
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Neuman, Shlomo P. "A quasilinear theory of non-Fickian and Fickian subsurface dispersion: 1. Theoretical analysis with application to isotropic media." Water Resources Research 26, no. 5 (1990): 887–902. http://dx.doi.org/10.1029/89wr03602.
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Shin, Jaehyun, Dongsop Rhee, and Inhwan Park. "Applications of Two-Dimensional Spatial Routing Procedure for Estimating Dispersion Coefficients in Open Channel Flows." Water 13, no. 10 (May 17, 2021): 1394. http://dx.doi.org/10.3390/w13101394.
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In this study, the performance of two routing procedures were evaluated to estimate the two-dimensional dispersion coefficients. The two-dimensional Stream-Tube Routing Procedure (2D ST-RP) has been widely used to obtain the dispersion coefficients from measured concentration-time curves under the frozen cloud assumption. Meanwhile, the Spatial Routing Procedure (2D S-RP) employs the spatial distributions of concentration to estimate the dispersion coefficients. The performance of the two routing methods were evaluated in aspect of the validity of the frozen cloud assumption and the applicability in the non-Fickian mixing. From the estimation results of dispersion coefficients, the results by the 2D ST-RP included errors due to skewed concentration-time curves which were created by violating the frozen cloud assumption. On the other hand, the 2D S-RP provides accurate dispersion coefficients in the same condition. The estimated results of dispersion coefficients in the meandering channel show that both the 2D ST-RP and the 2D S-RP contained errors due to the non-Fickian mixing properties of the test case. Even with the discrepancies, the 2D S-RP presented more appropriate spatial variabilities along the meander cycle than the results by the 2D ST-RP.
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Cherubini, C., C. I. Giasi, and N. Pastore. "Evidence of non-Darcy flow and non-Fickian transport in fractured media at laboratory scale." Hydrology and Earth System Sciences 17, no. 7 (July 9, 2013): 2599–611. http://dx.doi.org/10.5194/hess-17-2599-2013.
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Abstract. During a risk assessment procedure as well as when dealing with cleanup and monitoring strategies, accurate predictions of solute propagation in fractured rocks are of particular importance when assessing exposure pathways through which contaminants reach receptors. Experimental data obtained under controlled conditions such as in a laboratory allow to increase the understanding of the fundamental physics of fluid flow and solute transport in fractures. In this study, laboratory hydraulic and tracer tests have been carried out on an artificially created fractured rock sample. The tests regard the analysis of the hydraulic loss and the measurement of breakthrough curves for saline tracer pulse inside a rock sample of parallelepiped shape (0.60 × 0.40 × 0.08 m). The convolution theory has been applied in order to remove the effect of the acquisition apparatus on tracer experiments. The experimental results have shown evidence of a non-Darcy relationship between flow rate and hydraulic loss that is best described by Forchheimer's law. Furthermore, in the flow experiments both inertial and viscous flow terms are not negligible. The observed experimental breakthrough curves of solute transport have been modeled by the classical one-dimensional analytical solution for the advection–dispersion equation (ADE) and the single rate mobile–immobile model (MIM). The former model does not properly fit the first arrival and the tail while the latter, which recognizes the existence of mobile and immobile domains for transport, provides a very decent fit. The carried out experiments show that there exists a pronounced mobile–immobile zone interaction that cannot be neglected and that leads to a non-equilibrium behavior of solute transport. The existence of a non-Darcian flow regime has showed to influence the velocity field in that it gives rise to a delay in solute migration with respect to the predicted value assuming linear flow. Furthermore, the presence of inertial effects enhance non-equilibrium behavior. Instead, the presence of a transitional flow regime seems not to exert influence on the behavior of dispersion. The linear-type relationship found between velocity and dispersion demonstrates that for the range of imposed flow rates and for the selected path the geometrical dispersion dominates the mixing processes along the fracture network.
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Zhang, Haoming, Jiazhong Qian, Yong Liu, Yi Xu, Hongguang Sun, Xuequn Chen, and Fulin Li. "Applicability of the Time Fractional Derivative Model on Fickian and Non-Fickian Transport in the Single Fracture: An Experimental Investigation." Geofluids 2022 (July 7, 2022): 1–14. http://dx.doi.org/10.1155/2022/4426630.
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The understanding of solute transport in rock fractures is of great importance in many engineering fields. In this study, two groups of experiments on artificial and natural single fractures with different fracture apertures and roughness were conducted to investigate the changes of solute transport regimes. The time fractional advection–dispersion equation (tFADE) as a promising model was applied to describe the anomalous transport. The performance of the classical advection-dispersion equation (ADE) and tFADE was compared according to the fitting precision of breakthrough curves (BTCs). The responses of the fitting parameters in the tFADE to the experimental conditions were also discussed. The results indicated that the non-Fickian transport more likely occurred in the short distance transport, and the larger Peclet number ( Pe ) led to the increase of the exponent of the power-law function in the phase of concentration decline. The tFADE was superior to ADE in capturing the non-Fickian transport especially the tailing behavior. The fractional order of time α in the tFADE was the key parameter to describe the anomalous transport process, and its responding mechanisms of were revealed: the best-fit α decreased with the increase of flow velocity and the decrease of the fracture aperture. The roughness of the single fracture which leads to a complex flow field had a significant effect on the best-fit α . The findings of this study can help for better understanding the effectiveness and physical significance of the tFADE.
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Neuman, Shlomo P., and You-Kuan Zhang. "A quasi-linear theory of non-Fickian and Fickian subsurface dispersion: 1. Theoretical analysis with application to isotropic media." Water Resources Research 26, no. 5 (May 1990): 887–902. http://dx.doi.org/10.1029/wr026i005p00887.
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Moradi, G., and B. Mehdinejadiani. "Modelling solute transport in homogeneous and heterogeneous porous media using spatial fractional advection-dispersion equation." Soil and Water Research 13, No. 1 (January 24, 2018): 18–28. http://dx.doi.org/10.17221/245/2016-swr.
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This paper compared the abilities of advection-dispersion equation (ADE) and spatial fractional advection-dispersion equation (sFADE) to describe the migration of a non-reactive contaminant in homogeneous and heterogeneous soils. To this end, laboratory tests were conducted in a sandbox sizing 2.5 × 0.1 × 0.6 m (length × width × height). After performing a parametric sensitivity analysis, parameters of sFADE and ADE were individually estimated using the inverse problem method at each distance. The dependency of estimated parameters on distance was examined. The estimated parameters at 30 cm were used to predict breakthrough curves (BTCs) at subsequent distances. The results of sensitivity analysis indicated that average pore-water velocity and dispersion coefficient were, respectively, the most and least sensitive parameters in both mathematical models. The values of fractional differentiation orders (α) for sFADE were smaller than 2 in both soils. The scale-dependency of the dispersion coefficients of ADE and sFADE was observed in both soils. However, the application of sFADE to describe solute transport reduced the scale effect on the dispersion coefficient, especially in the heterogeneous soil. For the homogeneous soil, the predicting results of ADE and sFADE were nearly similar, while for the heterogeneous soil, the predicting results of sFADE were more satisfactory in comparison with those of ADE, especially when the transport distance increased. Compared to ADE, the sFADE simulated somewhat better the tailing parts of BTCs and showed the earlier arrival of tracer. Overall, the solute transport, especially in the heterogeneous soil, was non-Fickian and the sFADE somewhat better described non-Fickian transport.
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Uddin, Md Zahir, Jakir Ahmed Chowdhury, Ikramul Hasan, and Md Selim Reza. "Enhancement of Dissolution Profile of Poorly Water Soluble Loratadine by Solid Dispersion Technique." Dhaka University Journal of Pharmaceutical Sciences 15, no. 2 (January 2, 2017): 195–201. http://dx.doi.org/10.3329/dujps.v15i2.30937.
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This study was carried out to improve the dissolution properties of loratadine by solid dispersion technique. A series of solid dispersions of loratadine in PVP K-30 (1:1, 1:3 and 1:5) were prepared by kneading technique. The prepared solid dispersions were characterized by various physicochemical properties (fourier transform infrared spectroscopy, X-ray diffraction and scanning electron microscopy) and the dissolution characteristics were compared with loratadine and the physical mixtures of loratadine. It was revealed from the physicochemical analyses that there was a good compatibility between drug and carrier. On the other hand, the drug release from the prepared binary solid dispersions was significantly enhanced in comparison to both drug alone and the physical mixtures. Finally solid dispersion of loratadine: PVP K-30 prepared as 1:5 ratio was found to be described by non-Fickian release mechanism and was selected as the best formulation in this study.Dhaka Univ. J. Pharm. Sci. 15(2): 195-201, 2016 (December)
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Zhang, You-Kuan, and Shlomo P. Neuman. "A quasi-linear theory of non-Fickian and Fickian subsurface dispersion: 2. Application to anisotropic media and the Borden site." Water Resources Research 26, no. 5 (May 1990): 903–13. http://dx.doi.org/10.1029/wr026i005p00903.
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Zaheer, Muhammad, Hadayat Ullah, Saad Ahmed Mashwani, Ehsan ul Haq, Syed Husnain Ali Shah, and Fawaz Manzoor. "SOLUTE TRANSPORT MODELLING IN LOW-PERMEABILITY HOMOGENEOUS AND SATURATED SOIL MEDIA." Rudarsko-geološko-naftni zbornik 36, no. 2 (2021): 25–32. http://dx.doi.org/10.17794/rgn.2021.2.3.
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Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.
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Liu, Meng, Dong Chen, Hong-Guang Sun, and Feng Zhang. "Vertical Distribution of Suspended Sediment Concentration in the Unsaturated Jingjiang Reach, Yangtze River, China." Fractal and Fractional 7, no. 6 (June 2, 2023): 456. http://dx.doi.org/10.3390/fractalfract7060456.
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The Rouse formula and its variants have been widely used to describe the vertical distribution of the sediment concentration in sediment-laden flows in equilibrium. Han’s formula extends the Rouse formula to non-equilibrium regimes, where the diffusive flux is still assumed to be Fickian. The turbulent flow and suspension regimes downstream of a mega-reservoir, e.g., the Three Gorges Reservoir, usually exhibit fractal and unsaturated properties, respectively. To characterize the non-Fickian dynamics of suspended sediment and the non-equilibrium regime in natural dammed rivers, this study proposes a new formula for the concentration profile of unsaturated sediment based on the Hausdorff fractal derivative advection–dispersion equation. In addition, we find that the order of the Hausdorff fractal derivative is related to the sizes of the sediment and the degrees of non-equilibrium. Compared to Rouse and Han’s formulae, the new formula performs better in describing the sediment concentration profiles in the Jingjiang Reach, approximately 100 km below the Three Gorges Dam.
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Kim, Byunguk, Siyoon Kwon, and Il Won Seo. "An Explicit Solution for Characterizing Non-Fickian Solute Transport in Natural Streams." Water 15, no. 9 (April 27, 2023): 1702. http://dx.doi.org/10.3390/w15091702.
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One-dimensional solute transport modeling is fundamental to enhance understanding of river mixing mechanisms, and is useful in predicting solute concentration variation and fate in rivers. Motivated by the need of more adaptive and efficient model, an exact and efficient solution for simulating breakthrough curves that vary with non-Fickian transport in natural streams was presented, which was based on an existing implicit advection-dispersion equation that incorporates the storage effect. The solution for the Gaussian approximation with a shape-free boundary condition was derived using a routing procedure, and the storage effect was incorporated using a stochastic concept with a memory function. The proposed solution was validated by comparison with analytical and numerical solutions, and the results were efficient and exact. Its performance in simulating non-Fickian transport in streams was validated using field tracer data, and good agreement was achieved with 0.990 of R2. Despite the accurate reproduction of the overall breakthrough curves, considerable errors in their late-time behaviors were found depending upon the memory function formulae. One of the key results was that the proper formula for the memory function is inconsistent according to the data and optimal parameters. Therefore, to gain a deeper understanding of non-Fickian transport in natural streams, identifying the true memory function from the tracer data is required.
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Zheng, Lizhi, Lichun Wang, and Scott C. James. "When can the local advection–dispersion equation simulate non-Fickian transport through rough fractures?" Stochastic Environmental Research and Risk Assessment 33, no. 3 (March 2019): 931–38. http://dx.doi.org/10.1007/s00477-019-01661-7.
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Ross, Katharina, Falk Heße, Jude L. Musuuza, and Sabine Attinger. "Ensemble and effective dispersion in three-dimensional isotropic fractal media." Stochastic Environmental Research and Risk Assessment 33, no. 11-12 (November 2, 2019): 2089–107. http://dx.doi.org/10.1007/s00477-019-01739-2.
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Abstract We determine the time-dependent behavior of the dispersion coefficient for transport in formations with isotropic log-conductivity fields showing fractal behavior. We consider two different dispersion coefficients for point-like injection: (1) the ensemble dispersion coefficients, defined as half the rate of change of the second central moments of the ensemble-averaged concentration distribution and (2) the effective dispersion, which is half the rate of change of the expected second central moments. Our results show, that the two longitudinal macrodispersion coefficients steadily grow with time and remain different at all times in a fully fractal regime, indicating that no Fickian transport regime is ever reached. The resulting effective longitudinal transport model is consequently a fractional advection–dispersion equation. In the semi-fractal regime, a Gaussian transport regime is reached eventually. However, compared to the case of a classic non-fractal regime, the transient non-Gaussian regime lasts much longer. In the transverse direction, the two dispersion coefficients approach the same large-time limit also in fractal media highlighting the fundamental difference between longitudinal and transverse dispersion.
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Jha, Raman K., Steven L. Bryant, and Larry W. Lake. "Effect of Diffusion on Dispersion." SPE Journal 16, no. 01 (October 4, 2010): 65–77. http://dx.doi.org/10.2118/115961-pa.
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Summary It is known that dispersion in porous media results from an interaction between convective spreading and diffusion. However, the nature and implications of these interactions are not well understood. Dispersion coefficients obtained from averaged cup-mixing concentration histories have contributions of convective spreading and diffusion lumped together. We decouple the contributions of convective spreading and diffusion in core-scale dispersion and systematically investigate interaction between the two in detail. We explain phenomena giving rise to important experimental observations such as Fickian behavior of core-scale dispersion and power-law dependence of dispersion coefficient on Péclet number. We track movement of a swarm of solute particles through a physically representative network model. A physically representative network model preserves the geometry and topology of the pore space and spatial correlation in flow properties. We developed deterministic rules to trace paths of solute particles through the network. These rules yield flow streamlines through the network comparable to those obtained from a full solution of Stokes’ equation. Paths of all solute particles are deterministically known in the absence of diffusion. Thus, we can explicitly investigate purely convective spreading by tracking the movement of solute particles on these streamlines. Then, we superimpose diffusion and study dispersion in terms of interaction between convective spreading and diffusion for a wide range of Péclet numbers. This approach invokes no arbitrary parameters, enabling a rigorous validation of the physical origin of core-scale dispersion. In this way, we obtain an unequivocal, quantitative assessment of the roles of convective spreading and diffusion in hydrodynamic dispersion in flow through porous media. Convective spreading has two components: stream splitting and velocity gradient in pore throats in the direction transverse to flow. We show that, if plug flow occurs in the pore throats (accounting only for stream splitting), all solute particles can encounter a wide range of independent velocities because of velocity differences between pore throats and randomness of pore structure. Consequently, plug flow leads to a purely convective spreading that is asymptotically Fickian. Diffusion superimposed on plug flow acts independently of convective spreading (in this case, only stream splitting), and, consequently, dispersion is simply the sum of convective spreading and diffusion. In plug flow, hydrodynamic dispersion varies linearly with the pore-scale Péclet number when diffusion is small in magnitude compared to convective spreading. For a more realistic parabolic velocity profile in pore throats, particles near the solid surface of the medium do not have independent velocities. Now, purely convective spreading (caused by a combination of stream splitting and variation in flow velocity in the transverse direction) is non-Fickian. When diffusion is nonzero, solute particles in the low-velocity region near the solid surface can move into the main flow stream. They subsequently undergo a wide range of independent velocities because of stream splitting, and, consequently, dispersion becomes asymptotically Fickian. In this case, dispersion is a result of an interaction between convection and diffusion. This interaction results in a weak nonlinear dependence of dispersion on Péclet number. The dispersion coefficients predicted by particle tracking through the network are in excellent agreement with the literature experimental data for a broad range of Péclet numbers. Thus, the essential phenomena giving rise to hydrodynamic dispersion observed in porous media are (1) stream splitting of the solute front at every pore, causing independence of particle velocities purely by convection; (2) velocity gradient in pore throats in the direction transverse to flow; and (3) diffusion. Taylor's dispersion in a capillary tube accounts only for the second and third of these phenomena, yielding a quadratic dependence of dispersion on Péclet number. Plug flow in the bonds of a physically representative network accounts only for the first and third phenomena, resulting in a linear dependence of dispersion on Péclet number. When all the three phenomena are accounted for, we can explain effectively the weak nonlinear dependence of dispersion on Péclet number.
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Park, Inhwan, and Il Won Seo. "Modeling non-Fickian pollutant mixing in open channel flows using two-dimensional particle dispersion model." Advances in Water Resources 111 (January 2018): 105–20. http://dx.doi.org/10.1016/j.advwatres.2017.10.035.
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Davis, P. M., T. C. Atkinson, and T. M. L. Wigley. "Longitudinal dispersion in natural channels: 2. The roles of shear flow dispersion and dead zones in the River Severn, U.K." Hydrology and Earth System Sciences 4, no. 3 (September 30, 2000): 355–71. http://dx.doi.org/10.5194/hess-4-355-2000.
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Abstract. The classical one-dimensional advection-diffusion equation (ADE) gives an inadequate description of tracer cloud evolution in the River Severn, U.K. A solute transport model incorporating the effects of tracer storage in dead zones is presented in which the channel is conceived as being divided into two parallel regions. The bulk flow region occurs in the central part. Its longitudinal dispersive properties are described by the ADE. Adjacent to this, an additional cross-sectional area is defined in which tracer can be stored temporarily in regions of slowly moving water called dead zones. Exchange between the two regions follows a first order rate equation. Applying the model to the River Severn shows that a dispersing cloud’s evolution occurs in two distinct stages with a rapid transitional phase. Initially, shear-dispersion is dominant while the tracer particles mix fully over the bulk flow. Once this has occurred, dead zone storage accounts well for the non-Fickian evolution of the cloud. After the transitional phase the dead zone storage mechanism clearly dominates over shear-dispersion. Overall, the combined shear flow dispersion – dead zone model (D-DZM) provides a much better, physically consistent description of the tracer cloud’s evolution than the simple classical ADE approach can do alone. Keywords: Channels; dispersion; dead zones; tracers; River Severn
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Bolster, D., and M. Dentz. "Anomalous dispersion in chemically heterogeneous media induced by long-range disorder correlation." Journal of Fluid Mechanics 695 (February 13, 2012): 366–89. http://dx.doi.org/10.1017/jfm.2012.25.
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AbstractWe study transport in an idealized porous medium characterized by a spatially varying retardation factor, which models linear instantaneous chemical adsorption of a solute. Using a stochastic modelling approach, we study the impact of disorder correlation on the large-scale dispersion behaviour. We consider short, long-range and intermediate-range disorder correlations, and demonstrate that (truncated) power-law correlation causes anomalous dispersion, even in the presence of weak heterogeneity. We identify different preasymptotic and asymptotic regimes of anomalous dispersion that shed new light on the disorder and local-scale transport mechanisms leading to non-Fickian behaviour. The analytical results are complemented by numerical random walk particle tracking simulations, which are found to be in good agreement with the derived dispersion behaviour. We conclude the paper by deriving an effective transport equation for this system, which can be shown to be tied to the family of continuous-time random walk models.
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Cherubini, C., C. I. Giasi, and N. Pastore. "Evidence of non-Darcy flow and non-Fickian transport in fractured media at laboratory scale." Hydrology and Earth System Sciences Discussions 10, no. 1 (January 10, 2013): 221–54. http://dx.doi.org/10.5194/hessd-10-221-2013.
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Abstract. Accurate predictions of solute propagation in fractured rocks are of particular importance when assessing exposure pathways through which contaminants reach receptors during a risk assessment procedure, as well as when dealing with cleanup and monitoring strategies. The difficulty in modeling fractured media leads to the application of simplified analytical solutions that fail to reproduce flow and transport patterns in such complex geological formations. A way for understanding and quantifying the migration of contaminants in groundwater systems is that of analyzing tracer transport. Experimental data obtained under controlled conditions such as in a laboratory allow to increase the understanding of the fundamental physics of fluid flow and solute transport in fractures. In this study laboratory hydraulic and tracer tests have been carried out on an artificially created fractured rock sample. The tests regard the analysis of the hydraulic loss and the measurement of breakthrough curves for saline tracer pulse inside a rock sample of parallelepiped (0.60 × 0.40 × 0.8 m) shape. The effect of the experimental apparatus on flow and transport tests has been estimated. In particular the convolution theory has been applied in order to remove the effect of acquisition apparatus on tracer experiment. The experimental results have shown evidence of a non-Darcy relationship between flow rate and hydraulic loss that is best described by Forchheimer's law. The observed experimental breakthrough curves of solute transport have been modeled by the classical one-dimensional analytical solution for advection–dispersion equation (ADE) and the single rate mobile–immobile model (MIM). The former model does not fit properly the first arrival and the tail while the latter provides a very decent fit.
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Yan, Xiaosan, Jiazhong Qian, Lei Ma, Mu Wang, and Aofeng Hu. "Non-Fickian Solute Transport in a Single Fracture of Marble Parallel Plate." Geofluids 2018 (June 13, 2018): 1–9. http://dx.doi.org/10.1155/2018/7418140.
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Accurate prediction of solute transport processes in a fracture aquifer is an important task not only for proper management of the groundwater but also for pollution control. A key issue of this task is how to accurately obtain the experimental data and to analyze the solute transport in fracture in subsurface hydrology, which would greatly help us to understand the releasing mechanism and transport of the solute in a fracture. In this study, a fracture experiment is conducted in a laboratory based on previous studies. The fracture used with a length of 60 cm and a width of 10 cm is sealed with glass glue to avoid leakage of tracer due to uneven fracture walls. The sodium chloride (NaCl) solute is injected from the left of the fracture. And an electrical conductivity monitoring system is installed on the right of the fracture. Then breakthrough curves (BTCs) of solute transport are fitted using the classical advection-dispersion equation (ADE) and the truncation power-law function (TPL) model in the package of the continuous time random walk (CTRW). The results show that the flow satisfies non-Darcian law in the experimental conditions, which can be better fitted using the Forchheimer equation and Izbash equation. The solute transport presents non-Fickian phenomena and shows a long tailing. The fitting results of the TPL model are far better than ADE in fitting the long tailing at three different flow velocities. Furthermore, electrical conductivity monitoring method not only is effective but also has an advantage of no disturbance to water and concentration fields in a fracture.
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Zhang, Yong, Graham E. Fogg, HongGuang Sun, Donald M. Reeves, Roseanna M. Neupauer, and Wei Wei. "Adjoint subordination to calculate backward travel time probability of pollutants in water with various velocity resolutions." Hydrology and Earth System Sciences 28, no. 1 (January 11, 2024): 179–203. http://dx.doi.org/10.5194/hess-28-179-2024.
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Abstract. Backward probabilities, such as the backward travel time probability density function for pollutants in natural aquifers/rivers, have been used by hydrologists for decades in water quality applications. Calculating these backward probabilities, however, is challenging due to non-Fickian pollutant transport dynamics and velocity resolution variability at study sites. To address these issues, we built an adjoint model by deriving a backward-in-time fractional-derivative transport equation subordinated to regional flow, developed a Lagrangian solver, and applied the model/solver to trace pollutant transport in diverse flow systems. The adjoint model subordinates to a reversed regional flow field, transforms forward-in-time boundaries into either absorbing or reflective boundaries, and reverses the tempered stable density to define backward mechanical dispersion. The corresponding Lagrangian solver efficiently projects backward super-diffusive mechanical dispersion along streamlines. Field applications demonstrate the adjoint subordination model's success with respect to recovering release history, groundwater age, and pollutant source locations for various flow systems. These include systems with upscaled constant velocity, nonuniform divergent flow fields, or fine-resolution velocities in a nonstationary, regional-scale aquifer, where non-Fickian transport significantly affects pollutant dynamics and backward probabilities. Caution is needed when identifying the phase-sensitive (aqueous vs. absorbed) pollutant source in natural media. The study also explores possible extensions of the adjoint subordination model for quantifying backward probabilities of pollutants in more complex media, such as discrete fracture networks.
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Hou, Yusong, Jianguo Jiang, and J. Wu. "The Form of Waiting Time Distributions of Continuous Time Random Walk in Dead-End Pores." Geofluids 2018 (2018): 1–6. http://dx.doi.org/10.1155/2018/8329406.
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Anomalous dispersion of solute in porous media can be explained by the power-law distribution of waiting time of solute particles. In this paper, we simulate the diffusion of nonreactive tracer in dead-end pores to explore the waiting time distributions. The distributions of waiting time in different dead-end pores show similar power-law decline at early time and transit to an exponential decline in the end. The transition time between these two decline modes increases with the lengths of dead-end pores. It is well known that power-law distributions of waiting time may lead to anomalous (non-Fickian) dispersion. Therefore, anomalous dispersion is highly dependent on the sizes of immobile zones. According to the power-law decline, we can directly get the power index from the structure of dead-end pores, which can be used to judge the anomalous degree of solute transport in advance.
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Hu, Yingtao, Wenjie Xu, Liangtong Zhan, Zuyang Ye, and Yunmin Chen. "Non-Fickian Solute Transport in Rough-Walled Fractures: The Effect of Contact Area." Water 12, no. 7 (July 18, 2020): 2049. http://dx.doi.org/10.3390/w12072049.
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The influence of contact area, caused by normal deformation, on fluid flow and solute transport through three-dimensional (3D) rock fractures is investigated. Fracture surfaces with different Hurst exponents (H) were generated numerically using the modified successive random addition (SRA) method. By applying deformations normal to the fracture surface (Δu), a series of fracture models with different aperture distributions and contact area ratios (c) were simulated. The results show that the contact area between the two fracture surfaces increases and more void spaces are reduced as deformation (Δu) increases. The streamlines in the rough-walled fractures show that the contact areas result in preferential flow paths and fingering type transport. The non-Fickian characteristics of the “early arrival” and “long tail” in all of the breakthrough curves (BTCs) for fractures with different deformation (Δu) and Hurst parameters (H) were determined. The solute concentration distribution index (CDI), which quantifies the uniformity of the concentration distribution within the fracture, decreases exponential as deformation (Δu) and/or contact area ratios (c) increase, indicating that increased contact area can result in a larger delay rate of mass exchange between the immobile zone around the contact areas and the main flow channel, thus, resulting in a longer time for the solute to fill the entire fracture. The BTCs were analyzed using the continuous time random walk (CTRW) inverse model. The inverse modeling results show that the dispersion exponent β decreases from 1.92 to 0.81 as c increases and H decreases, suggesting that the increase in contact area and fracture surfaces enhance the magnitude of the non-Fickian transport.
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Guo, Zhilin, Rui Ma, Yong Zhang, and Chunmiao Zheng. "Contaminant transport in heterogeneous aquifers: A critical review of mechanisms and numerical methods of non-Fickian dispersion." Science China Earth Sciences 64, no. 8 (July 7, 2021): 1224–41. http://dx.doi.org/10.1007/s11430-020-9755-y.
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Villamil, Yeison, J. A. Vidal Vargas, and Osvair V. Trevisan. "APPLYING THE CONTINUOUS-TIME RANDOM WALK MODEL TO NON-FICKIAN DISPERSION IN MISCIBLE DISPLACEMENT THROUGH CARBONATE ROCK." Journal of Porous Media 22, no. 1 (2019): 87–105. http://dx.doi.org/10.1615/jpormedia.2019020224.
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Chen, Zhou, Jin Guo Wang, Wen Zhang Zhang, and Jia Hui Shi. "Experimental Study and Models Comparison for Solute Transport through Riparian Zones." Advanced Materials Research 1073-1076 (December 2014): 1604–8. http://dx.doi.org/10.4028/www.scientific.net/amr.1073-1076.1604.
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Solute transport through riparian zone was studied experimentally and numerically with the consideration of silt layer. The silt layer had markable change on flow field and lead to a significant variation of the breakthrough curves (BTCs). BTCs of solute tracer tests show non-Fickian features as early arrival of peak value and long tailings. BTCs were fitted by advection dispersion equation (ADE), mobile and immobile model (MIM) and the continuous time random walk (CTRW) models. MIM and CTRW can fit BTCs better than ADE and MIM fit better on the capture of the peak value and CTRW fit better in description of the long tailing.
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Suresh Kumar, G., M. Sekhar, and D. Misra. "Time dependent dispersivity behavior of non-reactive solutes in a system of parallel fractures." Hydrology and Earth System Sciences Discussions 3, no. 3 (June 7, 2006): 895–923. http://dx.doi.org/10.5194/hessd-3-895-2006.
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Abstract. In order to obtain meaningful predictions of contaminant transport, an accurate way of quantifying dispersivity needs to be developed. Results from the theoretical studies suggest that dispersion and the associated dispersivity is non-fickian near the source of contaminant and it grows with travel time and distance. In most tests of a limited duration it is quite probable that the asymptotic regime is not reached, and a proper interpretation of the test should be based on the time-dependent results due to the difficulty associated with the expensive experimental setups added to the marked scarcity of field data. An attempt has been made using spatial moment analysis to evaluate the time dependent dispersivity for a system of parallel fractures with matrix diffusion. The study is limited to non-reactive solutes, having a constant continuous source. An empirical relation to evaluate the dispersivity was developed by us based on the sensitivity analysis, when distinct parallel fractures have constant aperture width and is found to be functions of matrix porosity, matrix diffusion coefficient and injected fracture velocity at pre-asymptotic stage. The system becomes more complex when the aperture widths of the distinct parallel fractures are varied, as it appears that the initial development period of non-fickian behavior may be long due to the continuous lateral mixing of the solute body. It is found that dispersivity at pre-asymptotic regime increases with the coefficient of variation for distinct parallel fractures with varying aperture widths.
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Abou-Saleh, Khalil, Jalal Dweik, Youssef Haidar, and Abbas Ghaddar. "Solving Diffusion Time in Heterogeneous Microscale Rock Matrix by 3D Computations: Non-Fickian Dispersion Observed in Porous Media." Journal of Geoscience and Environment Protection 07, no. 12 (2019): 42–52. http://dx.doi.org/10.4236/gep.2019.712003.
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Zhang, You-Kuan. "A quasilinear theory of non-Fickian and subsurface dispersion: 2. Application to anisotropic media and the Borden site." Water Resources Research 26, no. 5 (1990): 903–13. http://dx.doi.org/10.1029/89wr03641.
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Salandin, Paolo, and Andrea Rinaldo. "The influence of the form of the log-conductivity covariance on non-Fickian dispersion in random permeability fields." International Journal for Numerical Methods in Engineering 27, no. 1 (July 1989): 185–93. http://dx.doi.org/10.1002/nme.1620270115.
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Park, Inhwan, Jaehyun Shin, Hoje Seong, and Dong Sop Rhee. "Comparisons of Two Types of Particle Tracking Models Including the Effects of Vertical Velocity Shear." Water 12, no. 12 (December 16, 2020): 3535. http://dx.doi.org/10.3390/w12123535.
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In this study, two types of particle tracking models were presented to investigate the applicability in the two-dimensional solute mixing simulations. The conventional particle tracking model, denoted as PTM, was developed based on Fick’s law, which adopted the dispersion coefficient to calculate the random displacements. The other model is the particle dispersion model (PDM), which computes the shear dispersion process by dividing into two computation procedures as the shear translation and the vertical mixing. The PTM and the PDM included the effects of vertical profiles of velocity in the computation of dispersion coefficients and the shear translation step, respectively. The main difference between the two models is whether the shear dispersion process is reproduced using Fick’s law or the direct computation method. These differences were clearly revealed by comparing with the analytic solution of the advection-dispersion equation. The concentration curve resulting from the PTM shows the Gaussian curves, which were well-fitted with the analytic solution in both initial and Taylor periods. Meanwhile, the PDM presented skewed curves in the initial period and gradually turned to the symmetric shape in the Taylor period. The inherent differences of the two particle tracking models were scrutinized against the two-dimensional tracer test results, which show the non-Fickian mixing properties. The comparisons of concentration–time curves reveal that the PDM reproduced a more accurate shape of the curves than the results by the PTM by demonstrating skewed concentration curves.
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Frank, Sascha, Thomas Heinze, and Stefan Wohnlich. "Comparison of Surface Roughness and Transport Processes of Sawed, Split and Natural Sandstone Fractures." Water 12, no. 9 (September 10, 2020): 2530. http://dx.doi.org/10.3390/w12092530.
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In single fractures, dispersion is often linked to the roughness of the fracture surfaces and the resulting local aperture distribution. To experimentally investigate the effects of diverse fracture types and surface morphologies in sandstones, three fractures were considered: those generated by sawing and splitting, and a natural sedimentary fracture. The fracture surface morphologies were digitally analyzed and the hydraulic and transport parameters of the fractures were determined from Darcy and the tracer tests using a fit of a continuous time random walk (CTRW) and a classical advection–dispersion equation (ADE). While the sawed specimen with the smoothest surface had the smallest dispersivity, the natural fracture has the largest dispersivity due to strong anisotropy and non-matching fracture surfaces, although its surface roughness is comparable to the split specimen. The parameterization of the CTRW and of the ADE agree well for β > 4 of the truncated power law. For smaller values of β, non-Fickian transport processes are dominant. Channeling effects are observable in the tracer breakthrough curves. The transport behavior in the fractures is controlled by multiple constraints such as several surface roughness parameters and the equivalent hydraulic aperture.
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Salomoni, Valentina Anna Lia, and Nico De Marchi. "Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations." Fractal and Fractional 6, no. 1 (December 31, 2021): 21. http://dx.doi.org/10.3390/fractalfract6010021.
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Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.
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Rao, Monica RP, and Gajanan Paul. "Vaginal delivery of clotrimazole by mucoadhesion for treatment of candidiasis." Journal of Drug Delivery and Therapeutics 11, no. 6 (November 15, 2021): 6–14. http://dx.doi.org/10.22270/jddt.v11i6.5116.
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The aim of this study was to prepare mucoadhesive vaginal tablets of clotrimazole for treatment of vaginal candidiasis. A combination of mucoadhesive polymers like HPMC K100M, Sodium CMC, and Eudragit L100 were used in different ratios prepare solid dispersions to enhance its solubility. Tablets were prepared by the wet granulation method. Solid dispersions were evaluated for saturation solubility. All tablet batches were evaluated for weight variation, hardness, friability, drug content, swelling index, in vitro drug release study, ex vivo diffusion and mucoadhesive strength. FTIR spectra showed there was no interaction between the drug and the excipients. HPMC K100 M increased the solubility of clotrimazole in simulated vaginal fluid at pH 4.5. Eudragit L100 was shown to increase the swelling and mucoadhesive strength of the tablet, i.e., 3.03 and 3.29 g. The in vitro and ex vivo release of all 9 batches showed between 61 to 78% release in 8h. Ex vivo diffusion studies using sheep vaginal membrane showed 43 to 59% in 6h in simulated vaginal fluid. The release and flux were nearly comparable to marketed tablet i.e., Candid-V6. The drug release of all batches followed the Korsmeyer-Peppas kinetic model, and the mechanism was found to be non‐Fickian/anomalous.
Keywords: Clotrimazole, solid dispersion, HPMC K100M, Eudragit L100, Sodium CMC.
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Pang, Guofei, Wen Chen, and Kam Yim Sze. "A Comparative Study of Finite Element and Finite Difference Methods for Two-Dimensional Space-Fractional Advection-Dispersion Equation." Advances in Applied Mathematics and Mechanics 8, no. 1 (December 21, 2015): 166–86. http://dx.doi.org/10.4208/aamm.2014.m693.
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AbstractThe paper makes a comparative study of the finite element method (FEM) and the finite difference method (FDM) for two-dimensional fractional advection-dispersion equation (FADE) which has recently been considered a promising tool in modeling non-Fickian solute transport in groundwater. Due to the non-local property of integro-differential operator of the space-fractional derivative, numerical solution of FADE is very challenging and little has been reported in literature, especially for high-dimensional case. In order to effectively apply the FEM and the FDM to the FADE on a rectangular domain, a backward-distance algorithm is presented to extend the triangular elements to generic polygon elements in the finite element analysis, and a variable-step vector Grünwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.
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Changdeo, Jagdale Swati, Musale Vinod, Kuchekar Bhanudas Shankar, and Chabukswar Anuruddha Rajaram. "Physicochemical characterization and solubility enhancement studies of allopurinol solid dispersions." Brazilian Journal of Pharmaceutical Sciences 47, no. 3 (September 2011): 513–23. http://dx.doi.org/10.1590/s1984-82502011000300009.
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Allopurinol is a commonly used drug in the treatment of chronic gout or hyperuricaemia associated with treatment of diuretic conditions. One of the major problems with the drug is that it is practically insoluble in water, which results in poor bioavailability after oral administration. In the present study, solid dispersions of allopurinol were prepared by solvent evaporation, kneading method, co-precipitation method, co-grinding method and closed melting methods to increase its water solubility. Hydrophilic carriers such as polyvinylpyrrolidone, polyethylene glycol 6000 were used in the ratio of 1:1, 1:2 and 1:4 (drug to carrier ratio). The aqueous solubility of allopurinol was favored by the presence of both polymers. These new formulations were characterized in the liquid state by phase solubility studies and in the solid state by differential scanning calorimetry, powder X-ray diffraction, UV and Fourier Transform Infrared spectroscopy. Solid state characterizations indicated that allopurinol was present as an amorphous material and entrapped in polymer matrix. In contrast to the very slow dissolution rate of pure allopurinol, the dispersion of the drug in the polymers considerably enhanced the dissolution rate. Solid dispersion prepared with polyvinylpyrrolidone showed highest improvement in wettability and dissolution rate of allopurinol. Mathematical modeling of in vitro dissolution data indicated the best fitting with Korsemeyer-Peppas model and the drug release kinetics primarily as Non-Fickian diffusion. Therefore, the present study showed that polyvinylpyrrolidone and polyethylene glycol 6000 have a significant solubilizing effect on allopurinol.
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Ullah, Md Bashir, Md Rezaul Karim, Md Shamsul Alam, Md Rajib Hassan, Mohiuddin Ahmed Bhuiyan, and Md Sohel Rana. "Formulation and In vitro Evaluation of Unfolding Type Expandable Gastroretentive Film of Enalapril Maleate." Bangladesh Pharmaceutical Journal 20, no. 2 (August 14, 2018): 148–56. http://dx.doi.org/10.3329/bpj.v20i2.37868.
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The present work was based on the development and characterization of unfolding type gastroretentive dosage form appropriate for controlled release of enalapril maleate. Drug loaded films were prepared by solid dispersion technique using methocel K15 and eudragit RSPO and eudragit RLPO as polymers and polyethylene glycol 400 (PEG 400) as the plasticizer. The film folded in a capsule shell was shown to unfold in the gastric juice and provide drug release up to 12 h in the acidic medium. Formulations provided satisfactory unfolding characteristics allowing expansion to remain in the stomach. Formulation containing above 60% content of eudragit RSPO and eudgrait RLPO combination of total polymer content provided satisfactory film integrity over 12 hours. The result revealed that formulation F1 showed a minimum percentage of drug release of 63.41% followed by formulation F2, F3, F4 and F5 with 66.76%, 80.21%, 83.26% and 86.92% release in 8 hour respectively. Formulation with high proportion of eudragit RLPO and RSPO combination in total polymer content was found to be slow in drug release and lower the release from the polymeric film over time. As the concentration of HPMC K 15 in total polymer content increased the release rate of enalapril maleate as well as % release from the polymeric film also increased over time. Most of the formulation followed Higuchi release kinetics followed by Korsmeyer release kinetics. The drug release mechanism from the film follows Fickian and Non Fickian release kinetics. The films were evaluated for mechanical properties, in vitro drug release and unfolding behavior based on the mechanical shape memory of polymers. Absence of drug polymer interaction and uniform drug dispersion in the polymeric layers was revealed by DSC, FT-IR and SEM studies.Bangladesh Pharmaceutical Journal 20(2): 148-154, 2017
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Cherubini, C., C. I. Giasi, and N. Pastore. "On the reliability of analytical models to predict solute transport in a fracture network." Hydrology and Earth System Sciences 18, no. 6 (June 24, 2014): 2359–74. http://dx.doi.org/10.5194/hess-18-2359-2014.
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Abstract. In hydrogeology, the application of reliable tracer transport model approaches is a key issue to derive the hydrodynamic properties of aquifers. Laboratory- and field-scale tracer dispersion breakthrough curves (BTC) in fractured media are notorious for exhibiting early time arrivals and late time tailing that are not captured by the classical advection–dispersion equation (ADE). These "non-Fickian" features are proven to be better explained by a mobile–immobile (MIM) approach. In this conceptualization the fractured rock system is schematized as a continuous medium in which the liquid phase is separated into flowing and stagnant regions. The present study compares the performances and reliabilities of the classical MIM and the explicit network model (ENM), taking expressly into account the network geometry for describing tracer transport behavior in a fractured sample at bench scale. Though ENM shows better fitting results than MIM, the latter remains still valid as it proves to describe the observed curves quite well. The results show that the presence of nonlinear flow plays an important role in the behavior of solute transport. First, the distribution of solute according to different pathways is not constant, but it is related to the flow rate. Second, nonlinear flow influences advection in that it leads to a delay in solute transport respect to the linear flow assumption. However, nonlinear flow is not shown to be related with dispersion. The experimental results show that in the study case the geometrical dispersion dominates the Taylor dispersion. However, the interpretation with the ENM shows a weak transitional regime from geometrical dispersion to Taylor dispersion for high flow rates. Incorporating the description of the flow paths in the analytical modeling has proven to better fit the curves and to give a more robust interpretation of the solute transport.
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Cherubini, C., C. I. Giasi, and N. Pastore. "On the reliability of analytical models to predict solute transport in a fracture network." Hydrology and Earth System Sciences Discussions 10, no. 12 (December 6, 2013): 14905–48. http://dx.doi.org/10.5194/hessd-10-14905-2013.
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Abstract. In hydrogeology, the application of reliable tracer transport model approaches is a key issue to derive the hydrodynamic properties of aquifers. Laboratory and field-scale tracer dispersion breakthrough curves (BTC) in fractured media are notorious for exhibiting early time arrivals and late-time tailing that are not captured by the classical advection–dispersion equation (ADE). These "non-Fickian" features are proved to be better explained by a mobile–immobile (MIM) approach. In this conceptualization the fractured rock system is schematized as a continuous medium in which the liquid phase is separated into flowing and stagnant regions. The present study compares the performances and reliabilities of classical Mobile–Immobile Model (MIM) and the Explicit Network Model (ENM) that takes expressly into account the network geometry for describing tracer transport behavior in a fractured sample at bench scale. Though ENM shows better fitting results than MIM, the latter remains still valid as it proves to describe the observed curves quite well. The results show that the presence of nonlinear flow plays an important role in the behaviour of solute transport. Firstly the distribution of solute according to different pathways is not constant but it is related to the flow rate. Secondly nonlinear flow influences advection, in that it leads to a delay in solute transport respect to the linear flow assumption. Whereas nonlinear flow does not show to be related with dispersion. However the interpretation with the ENM model shows a weak transitional regime from geometrical dispersion to Taylor dispersion for high flow rates. The experimental results show that in the study case the geometrical dispersion dominates the Taylor dispersion. Incorporating the description of the flowpaths in the analytical modeling has proved to better fit the curves and to give a more robust interpretation of the solute transport.
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Mathias, S. A., A. P. Butler, T. C. Atkinson, S. Kachi, and R. S. Ward. "A parameter identifiability study of two chalk tracer tests." Hydrology and Earth System Sciences Discussions 3, no. 4 (August 29, 2006): 2437–71. http://dx.doi.org/10.5194/hessd-3-2437-2006.
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Abstract. As with most fractured rock formations, Chalk is highly heterogeneous. Therefore, meaningful estimates of model parameters must be obtained at a scale comparable with the process of concern. These are frequently obtained by calibrating an appropriate model to observed concentration-time data from radially convergent tracer tests (RCTT). Arguably, an appropriate model should consider radially convergent dispersion (RCD) and Fickian matrix diffusion. Such a model requires the estimation of at least four parameters. A question arises as to whether or not this level of model complexity is supported by the information contained within the calibration data. Generally modellers have not answered this question due to the calibration techniques employed. A dual-porosity model with RCD was calibrated to two tracer test datasets from different UK Chalk aquifers. A multivariate sensitivity analysis, which assumed only a priori upper and lower bounds for each model parameter, was undertaken. Rather than looking at measures of uncertainty, the shape of the multivariate objective function surface was used to determine whether a parameter was identifiable. Non-identifiable parameters were then removed and the procedure was repeated until all remaining parameters were identifiable. It was found that the single fracture model (SFM) (which ignores mechanical dispersion) obtained the best mass recovery, excellent model performance and best parameter identifiability in both the tests studied. However, there was no objective evidence suggesting that mechanical dispersion was negligible. Moreover, the SFM (with just two parameters) was found to be good at approximating the Single Fracture Dispersion Model SFDM (with three parameters) when different, and potentially erroneous parameters, were used. Overall, this study emphasises the importance of adequate temporal sampling of breakthrough curve data prior to peak concentrations, to ensure adequate characterisation of mechanical dispersion processes, and continued monitoring afterwards, to ensure adequate characterisation of fracture spacing (where possible), when parameterising dual-porosity solute transport models.
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de Dreuzy, Jean-Raynald, and Jesus Carrera. "On the validity of effective formulations for transport through heterogeneous porous media." Hydrology and Earth System Sciences 20, no. 4 (April 5, 2016): 1319–30. http://dx.doi.org/10.5194/hess-20-1319-2016.
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Abstract. Geological heterogeneity enhances spreading of solutes and causes transport to be anomalous (i.e., non-Fickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through heterogeneous porous media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the multi-rate mass transfer (MRMT) model to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, in general non-dispersive mixing cannot.
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de Dreuzy, J. R., and J. Carrera. "On the validity of effective formulations for transport through heterogeneous porous media." Hydrology and Earth System Sciences Discussions 12, no. 11 (November 25, 2015): 12281–310. http://dx.doi.org/10.5194/hessd-12-12281-2015.
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Abstract. Geological heterogeneity enhances spreading of solutes, and causes transport to be anomalous (i.e., non-Fickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through Heterogeneous Porous Media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the Multi-Rate Mass Transfer (MRMT) to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, non-dispersive mixing cannot.
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Sohail, Kashif, Ikram Ullah Khan, Yasser Shahzad, Talib Hussain, and Nazar Muhammad Ranjha. "pH-sensitive polyvinylpyrrolidone-acrylic acid hydrogels: Impact of material parameters on swelling and drug release." Brazilian Journal of Pharmaceutical Sciences 50, no. 1 (March 2014): 173–84. http://dx.doi.org/10.1590/s1984-82502011000100018.
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In this study, we fabricated pH-sensitive polyvinylpyrrolidone/acrylic acid (PVP/AA) hydrogels by a free-radical polymerisation method with variation in the content of monomer, polymer and cross-linking agent. Swelling was performed in USP phosphate buffer solutions of pH 1.2, 5.5, 6.5 and 7.5 with constant ionic strength. Network structure was evaluated by different parameters and FTIR confirmed the formation of cross-linked hydrogels. X-ray crystallography showed molecular dispersion of tramadol HCl. A drug release study was carried out in phosphate buffer solutions of pH 1.2, 5.5 and 7.5 for selected samples. It was observed that swelling and drug release from hydrogels can be modified by changing composition and degree of cross-linking of the hydrogels under investigation. Swelling coefficient was high at higher pH values except for the one containing high PVP content. Drug release increased by increasing the pH of the medium and AA contents in hydrogels while increasing the concentration of cross-linking agent had the opposite effect. Analysis of the drug release mechanism revealed non-Fickian transport of tramadol from the hydrogels.
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Pini, Ronny, Nicholas T. Vandehey, Jennifer Druhan, James P. O’Neil, and Sally M. Benson. "Quantifying solute spreading and mixing in reservoir rocks using 3-D PET imaging." Journal of Fluid Mechanics 796 (May 10, 2016): 558–87. http://dx.doi.org/10.1017/jfm.2016.262.
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We report results of an experimental investigation into the effects of small-scale (mm–cm) heterogeneities on solute spreading and mixing in a Berea sandstone core. Pulse-tracer tests have been carried out in the Péclet number regime $Pe=6{-}40$ and are supplemented by a unique combination of two imaging techniques. X-ray computed tomography (CT) is used to quantify subcore-scale heterogeneities in terms of permeability contrasts at a spatial resolution of approximately $10~\text{mm}^{3}$, while [11C] positron emission tomography (PET) is applied to image the spatial and temporal evolution of the full tracer plume non-invasively. To account for both advective spreading and local (Fickian) mixing as driving mechanisms for solute transport, a streamtube model is applied that is based on the one-dimensional advection–dispersion equation. We refer to our modelling approach as semideterministic, because the spatial arrangement of the streamtubes and the corresponding solute travel times are known from the measured rock’s permeability map, which required only small adjustments to match the measured tracer breakthrough curve. The model reproduces the three-dimensional PET measurements accurately by capturing the larger-scale tracer plume deformation as well as subcore-scale mixing, while confirming negligible transverse dispersion over the scale of the experiment. We suggest that the obtained longitudinal dispersivity ($0.10\pm 0.02$ cm) is rock rather than sample specific, because of the ability of the model to decouple subcore-scale permeability heterogeneity effects from those of local dispersion. As such, the approach presented here proves to be very valuable, if not necessary, in the context of reservoir core analyses, because rock samples can rarely be regarded as ‘uniformly heterogeneous’.