# Transport in Porous Media with Nonlinear Flow Condition

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## Abstract

We investigate local aspects and heterogeneities of porous medium morphology and relate them to the relevant mechanisms of momentum transfer. In the inertial flow range, there are very few experimental data that allow to recognize the effects of porous structure on the flow and transport through porous media. An experimental analysis was performed in order to understand above processes at different Reynolds numbers in randomly structured porous media. The objective of the analysis is to explore the effects of porous media particle size on inertial and viscous forces and determine range of the Reynolds numbers in which the inertial flow predominantly contributes in dispersive processes. Transport characteristics of the randomly structured porous media and the influence of inertial force on longitudinal and transverse dispersion coefficients were studied.

## Keywords

Inertial flow Stochastic finite element Uncertainty quantification Heterogeneous porous media and nonlinear flow## 1 Introduction

A hydrodynamic dispersion coefficient is the sum of a diffusion coefficient and a mechanical mixing coefficient. Slichter (1905) was the first to show interest in dispersion investigations and concluded that velocity variation is the principle cause of dispersion. The dispersion process shows a complex behaviour in porous media with uncertain flow field as it depends on the microscopic pattern of flow through porous media, and a correlation between Reynolds number and the dispersion is expected to exist.

In primary studies on dispersion (e.g. Day 1956; Scheidegger 1954), it was assumed that a perfect mixing in fluid flow through a system occurs with a constant rate, which is not always true in real natural or engineering porous media with heterogeneous microstructure. Velocity variations and pressure excitations caused by heterogeneity of porous media make the transport process within this kind of non-uniform structure anomalous and more complicated. Scheidegger (1961) proposed a general theory, which mathematically describes the dispersion mechanism of a viscous fluid moving through a porous media with random geometrical properties. Brenner (1980) used the approaches introduced by Aris (1956) to model the kinematics of flow in spatially periodic porous media based on Taylor (1953) on dispersion in cylindrical capillaries. Brenner (1990) also reviewed some subsequent developments on this Taylor–Aris model for macroscale transport processes, which is called the Taylor dispersion phenomena. This study illuminated elementary examples in macroscale modelling and physical interpretation of microscale convective–diffusive–reactive transport processes. Effects of some factors such as fluid viscosity, density, particle size distribution, shape of particles have also been investigated using packed column tests and experimental techniques of dispersion coefficients measurement (De Josselin de Jong 1958; Han et al. 1985; Delgado 2006; Çarpinlioğlu et al. 2009; Cheng 2011; Erdim et al. 2015; Allen et al. 2013; Majdalani et al. 2015; Vollmari et al. 2015; Hofmann et al. 2016; Norouzi et al. 2018).

Several theoretical and numerical techniques have been considered in order to incorporate the effects of these flow variations in the transport process modelling. For example: continuous time random walks (Montroll and Weiss 1965), integral transform methods (Cotta 1993), Taylor–Aris–Brenner moment methods (Shapiro and Brenner 1986, 1988), stochastic convective (Simmons 1982; Nezhad et al. 2011), central limit theorems (Evans and Kenney 1966) and homogenization methods (Sanchez-Palencia 1980). Most of these researches focus on understanding the dispersion process in heterogeneous porous media when the flow is within the Darcy regime (Nezhad and Javadi 2011; Nezhad et al. 2013; Dentz et al. 2004).

Fluid flow through porous media is governed by various forces that act on the fluid, including viscous forces, gravitational force, electromagnetic forces and forces due to pressure from surrounding fluid. Sum total of these forces is referred to as inertia force and according to the results of early researches on this field (i.e. Fand et al. 1987; Kececioglu and Jiang 1994; Bagcı et al. 2014), depending on the amount of contribution of inertial and viscous forces in deriving flow, four different flow regimes are generated which are distinguished by the type of relationship between fluid velocity and pressure drop though porous media. These three regimes which are defined are namely pre-Darcy, linear Darcy, nonlinear Forchheimer and turbulent flow regimes.

Inertial forces can influence the hydraulic conductivity and dispersion (Van der Merwe and Gauvin 1971). LeClair and Hamielec (1968) investigated the viscous flow at intermediate Reynolds numbers. This work employed the finite difference methods from Hamielec et al. (1967a, b) and used the results to develop the theoretical information of local velocity distributions about particles for the intermediate flow regime and discussed its application to the packed beds. Forchheimer (Forchheimer 1901 cited in Whitaker 1996) was the first to consider the nonlinear relationship between hydraulic gradient and flux at larger velocities, and Prausnitz and Wilhelm (1957) and Mickley et al. (1965) presented theoretical and experimental relationship between turbulence parameters and concentration fluctuations in packed beds. However, there are only few research investigating this correlation. Bijeljic and Blunt (2006) and Haserta et al. (2011) have investigated the contribution of inertial flow on longitudinal dispersion. It was indicated that in realistic heterogeneous porous media, the dispersion coefficient is unlikely to reach an asymptotic value and traditional advection–dispersion equation cannot model solute transport (Nezhad 2010).

Despite numerous experimental and theoretical studies on dispersion there is a lack of fundamental understanding of how complexity in flow field in high Reynolds number can influence on the dispersion and how these processes are influenced by heterogeneity of the porous media. Boundary of flow regimes would be varied among different experimental systems, because of the geometry-dependent manifestation of inertial effects, particularly from the transition to turbulence (Reddy et al. 1998). Importance of specifying a transition boundary between these regimes where fluid field gradually evolves from linear to nonlinear, and from laminar to turbulent flow are recognized when appropriate method should be used for transport modelling and calculation of dispersion coefficient.

In this paper, the effects of nonlinearity of fluid flow on dispersion are investigated using a set of tracer tests conducted on packed beds of spherical glass beads of different diameter size. The effects of fluid characteristics on the flow behaviour and transition boundary between flow regimes have not been investigated in this work and will be done in future work.

## 2 Experimental Setup

Soda lime glass spheres with diameter of 1.85 mm and 3 mm were used to make the sample columns. Both samples were compacted to have the same porosity of 0.39. The spheres are non-reactive with Rhodamine WT which was used as dye for tracer experiments. The sample columns were placed on a stainless steel mesh with a pore size smaller than the particle diameters, in order to prevent spheres from clogging the outlets, avoiding any additional delay or fiction of the fluid flow. All the tests were done under the same temperature conditions and a temperature sensor and a Rapid 318 digital multimeter were attached to the downstream column wall to monitor the temperature in the system throughout experiments. The head difference was measured at two manometer tubes which were connected to upper and lower positions of the sample column. A WATSON MARLOW 505Di peristaltic pump was used to apply a steady flow to the system with constant operation temperature. The pump has the capacity to produce discharge rates of up to 18.95 m^{3}/s, producing water flow from Darcy to non-Darcy regimes. Air bubbles were removed before doing any tests, to guarantee vacuum conditions. Only water was used as fluid in this study and effect of fluid characteristics on the flow behaviour are left to be investigated in future work.

A joint was inserted between the inflow pipes and was used to inject Rhodamine water tracing dye (Rhodamine WT) into the system with a syringe. Two Cyclops 7 Submersible Fluorometers were attached to both upstream (inlet) and downstream (outlet) columns to continuously record the cross-sectional temporal dye concentration distribution during the tests. The data measured by the fluorometers were recorded automatically and stored electronically in a Novus LogBox-AA. All sensors were calibrated prior to use and details of the calibration procedure are found in Liang (2017).

## 3 Results and Discussion

### 3.1 Flow Regime

*ρ*is water density (kg/m

^{3}),

*v*is Darcy velocity (m/s),

*μ*is dynamic viscosity [kg/(m s)] and

*L*is characteristic length (m). The

*L*is defined as the inverse of specific interfacial area, the contact area of water and solid phase in unit volume (Petrasch et al. 2008).

It is seen that under the same flow conditions a higher-pressure drop is observed for the sample with smaller particles. Although these two samples have the same porosity, the passages of flow in the \( 1.85 \times 10^{ - 3} \) m sample are narrower and much more circuitous. Therefore, the pressure gradient across the sample with the smaller particles is more sensitive to the change of the flow condition.

*R*

^{2}(coefficient of determination) so that the regression line gives a good representation of the distribution of the points on the plot. It is shown that the Reynolds number (

*R*

_{eL}) corresponding to the transition from linear to nonlinear flow is 2.48 for \( 1.85 \times 10^{ - 3} \) m sample and 3.24 for \( 3 \times 10^{ - 3} \) m sample.

### 3.2 Determination of Intrinsic Permeability and Forchheimer Coefficient

*C*

_{E}is Ergun coefficient, which depends on the geometry of pore space distribution. Forchheimer coefficient

*β*is defined as a function of the Ergun coefficient, where \( \beta = \frac{{C_{\text{E}} }}{\sqrt k } \).

*k*

_{app}is apparent permeability. It is defined as

### 3.3 Dispersion

*M*

_{w}is the accumulated mass of the tracer, and

*t*is the time.

### 3.4 Effect of Flow Regime and Particle Size on Dispersion Coefficient

*ρ*

_{β}represents the fluid density (kg/m

^{3}), 〈

*v*

_{β}〉

^{β}represents the intrinsic averaged pore water velocity (m/s),

*d*

_{p}represents the particle diameter (m),

*μ*

_{β}represents the dynamic fluid viscosity (kg m

^{−1}s

^{−1}),

*ɛ*

_{β}represents the porosity,

*D*

_{Aβ}represents the molecular diffusion coefficient for the solute (3 m

^{2}/s), and

*Sc*represents the Schmidt number.

- (1)
Molecular diffusion regime (\( Pe_{\text{p}} < 0.2 \)): the molecular diffusion predominates over mechanical dispersion;

- (2)
Transition regime (\( 0.2 < Pe_{\text{p}} < 5 \)): the molecular diffusion and mechanical dispersion have approximately the same order of magnitude;

- (3)
Major mechanical dispersion regime (\( 5 < Pe_{\text{p}} < 4 \times 10^{3} \)): the interaction between mechanical dispersion and transverse molecular diffusion causes the spreading. The relationship between the effective dispersion coefficient (

*D*) and Peclét number is expressed by the following power low equation$$ \frac{D}{{D_{AB} }} = \frac{{D_{{m,{\text{eff}}}} }}{{D_{AB} }} + \alpha_{1} Pe_{\text{p}}^{\delta } \quad 1.1 < \delta < 1.2 $$(12) - (4)
Pure mechanical dispersion regime (\( 4 \times 10^{3} < Pe_{\text{p}} < 200 \times 10^{3} \)): it is characterized by power low relation of this type:

$$ \frac{D}{{D_{AB} }} = \alpha_{2} Pe_{\text{p}}^{\delta } \quad \delta = 1 $$(13)

*Pe*

_{p}) in the first case and between \( \text{Log}\left( {\frac{D}{{D_{AB} }}} \right) \) and Log(

*Pe*

_{p}) in the second one. The results are shown in Fig. 12. For the major mechanical dispersion regime, the equation in logarithmic and non-logarithmic form results:

The *δ* value is 1.237 and it is slightly higher than indicated in the literature studies, where which *δ* is between 1.1 and 1.2.

Peclèt number corresponding to the transition from major mechanical dispersion regime to pure mechanical dispersion regime is equal to 4.78 × 10^{3}. It is determined considering a range of data so that the regression model provides a good estimate of the experimental data. The value matches with the one indicated by Wood (2007) under which the major mechanical dispersion regime takes place for a Peclét number between 0.2 and 5 × 10^{3}.

Regarding the pure mechanical dispersion regime, as shown in Fig. 12b the *R*^{2} is very low. It means that the relation (12) does not estimate correctly the experimental data in the pure mechanical regime.

*δ*is slightly lower than the one in the literature. The transition from the major mechanical dispersion regime and the pure mechanical dispersion regime occurs for a Peclét number of 5.28 × 10

^{3}, a value slightly higher than the one found for the particle of \( 1.85 \times 10^{ - 3} \) m. Instead, for the pure mechanical dispersion regime the relation found in the form of Eq. 13 is:

*δ*around 1. As shown before, the value found using experimental data is a little higher than 1 (Fig. 13).

To analyse the relation between the flow and dispersion regimes, the previous results on the flow regimes obtained by Darcian Reynolds number are combined with the ones related to the dispersion regimes calculated by Peclét number. It is done for the particles of \( 1.85 \times 10^{ - 3} \) m and \( 3 \times 10^{ - 3} \) m.

*v*) and intrinsic velocity (〈

*v*

_{β}〉

^{β}):

Dispersion and flow regime for particle size \( 1.85 \times 10^{ - 3} \) m

Dispersion and flow regime for particle size \( 3 \times 10^{ - 3} \) m

*Re*

_{d}), Peclèt number (

*Pe*

_{p}) and Particle Reynolds number (

*Re*

_{p}) for the both particles size are shown below (Table 3).

Values of Darcian Reynolds number (\( Re_{\text{d}} \)), Peclèt number (\( Pe_{\text{p}} \)) and particle Reynolds number (\( Re_{\text{p}} \)), corresponding to the transition from linear to nonlinear flow

Particle size (m) | | | |
---|---|---|---|

\( 1.85 \times 10^{ - 3} \) | 6.21 | 21,871.13 | 6.33 |

\( 3 \times 10^{ - 3} \) | 9.48 | 35,817.37 | 10.37 |

Ratio of dispersion to diffusion coefficient with respect to Péclet number is presented in Fig. 16 and are compared with some results from literature. It is proved that the tracer-determined hydrodynamic dispersion coefficient is a function of mechanical dispersion. The data show an increasing trend similar to other results from literature, but do not overlap with any other data. This is because of the dependency of dispersion coefficient to several physical properties of fluid and porous media including viscosity, density of fluid, particle size distribution and shape, fluid velocity, length of packed sample column, ratio of column diameter to particle diameter and ratio of column length to particle diameter (Delgado 2006). Any change in one of the variables would result in a different dispersion profile. For the profiles presented from literature, water was considered as the fluid of interest. The diffusion coefficient of the tracer used in this research (Rhodamine WT) is 2.9E−10 m^{2}/s as determined by Gell et al. (2001) which is different form the tracers used in the tests from the literature. Since the diffusion coefficient of Rhodamine WT is slightly larger than the other tracers, different dispersion profiles and consequently different profiles for dispersion/diffusion ratio are achieved.

## 4 Conclusion

This study focused on the impact of porous media particle size on the flow regimes and dispersion process. Water flow and dye tests through man-made cylindrical samples containing spherical glass beads were conducted to study the contribution of the flow regimes on the dispersion. Different porous media and different flow conditions were examined in this paper in order to provide a knowledge base for understanding the relationship between flow behaviour and diffusion. This understanding is essential for studying the mixing processes in porous media with particle applications in hyporheic zone modelling. Results show that by reducing particle size the magnitude of the flow velocity in that nonlinear flow occurs is reduced up to 20%.

Observation within Darcian and non-Darcian regimes showed that, as the particle size increases, the diffusion effect reduces and dispersion coefficient decreases. The travel time required for dye to move through the smaller sized particles is also less than that of the larger one. This is because the finer textured sample has a more complex pore size distribution which causes a higher dispersion coefficient value. For the same Reynolds number, the dispersion coefficient of the fine particle is larger than the coarse particle. A small change in Reynolds number for smaller size particles results in a larger change of dispersion coefficient. This indicates that fine particles are more sensitive to flow regimes.

## Notes

### Acknowledgements

Support to conduct this study was provided by the Monash Warwick Alliance Seed Fund.

## References

- Aris, R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.
**235**(1200), 67–77 (1956)CrossRefGoogle Scholar - Allen, K.G., von Backström, T.W., Kröger, D.G.: Packed bed pressure drop dependence on particle shape, size distribution, packing arrangement and roughness. Powder Technol.
**246**, 590–600 (2013)CrossRefGoogle Scholar - Barree, R., Conway, M.: Beyond beta factors: a complete model for Darcy, Forchheimer, and trans-Forchheimer flow in porous media. In: SPE Annual Technical Conference and Exhibition, Houston, Texas, USA (2004)Google Scholar
- Bagcı, O., Dukhan, N., Özdemir, M.: Flow regimes in packed beds of spheres from pre-Darcy to turbulent. Transp. Porous Media
**104**, 501–520 (2014)CrossRefGoogle Scholar - Bedmar, F., Costa, J.L., Giménez, D.: Column tracer studies in surface and subsurface horizons of two typic argiudolls. Soil Sci.
**173**(4), 237–247 (2008)CrossRefGoogle Scholar - Bijeljic, B., Blunt, M.J.: Pore-scale modeling and continuous time random walk analysis of dispersion in porous media. Water Resour. Res.
**42**, W01202 (2006). https://doi.org/10.1029/2005wr004578 CrossRefGoogle Scholar - Brenner, H.: Dispersion resulting from flow through spatially periodic porous media. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**297**(1430), 81–133 (1980)CrossRefGoogle Scholar - Brenner, H.: Macrotransport processes. Langmuir
**6**(12), 1715–1724 (1990)CrossRefGoogle Scholar - Çarpinlioğlu, M.Ö., Özahi, E., Gündoğdu, M.Y.: Determination of laminar and turbulent flow ranges through vertical packed beds in terms of particle friction factors. Adv. Powder Technol.
**20**, 515–520 (2009)CrossRefGoogle Scholar - Chandler, I.D.: Vertical Variation in Diffusion Coefficient Within Sediments. University of Warwick, Coventry (2012)Google Scholar
- Cheng, N.S.: Wall effect on pressure drop in packed beds. Powder Technol.
**210**, 261–266 (2011)CrossRefGoogle Scholar - Cotta, R.: Integral Transforms in Computational Heat and Fluid Flow. CRC Press, Boca Raton (1993)Google Scholar
- Day, P.: Dispersion of a moving salt-water boundary advancing through saturated sand. EOS Trans. AGU
**37**(5), 595–601 (1956)CrossRefGoogle Scholar - De Josselin de Jong, G.: Longitudinal and transverse diffusion in granular deposits. Eos Trans. AGU
**39**(1), 67–74 (1958)CrossRefGoogle Scholar - Delgado, J.: A critical review of dispersion in packed beds. Heat Mass Transf.
**42**(4), 279–310 (2006)CrossRefGoogle Scholar - Dentz, M., Cortis, A., Scher, H., Berkowitz, B.: Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour.
**27**, 155–173 (2004)CrossRefGoogle Scholar - Erdim, E., Akgiray, Ö., Demir, İ.: A revisit of pressure drop-flow rate correlations for packed beds of spheres. Powder Technol.
**283**, 488–504 (2015)CrossRefGoogle Scholar - Evans, E., Kenney, C.: Gaseous dispersion in packed beds at low Reynolds numbers. Trans. Inst. Chem. Eng.
**44**, T189–T197 (1966)Google Scholar - Fand, R.M., Kim, B.Y.K., Lam, A.C.C., Phan, R.T.: Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J. Fluids Eng.
**109**(3), 268–273 (1987)CrossRefGoogle Scholar - Forchheimer, P.: Wasserbewegung durch boden. Z. Ver. Deutsch. Ing.
**45**(1782), 1788 (1901)Google Scholar - Gell, C., Brockwell, D.J., Beddard, G.S., Radford, S.E., Kalverda, A.P., Smith, D.A.: Accurate use of single molecule fluorescence correlation spectroscopy to determine molecular diffusion times. Single Mol.
**2**(3), 177–181 (2001)CrossRefGoogle Scholar - Hamielec, A., Hoffman, T., Ross, L.: Numerical solution of the Navier–Stokes equation for flow past spheres: part I. Viscous flow around spheres with and without radial mass efflux. AIChE J.
**13**(2), 212–219 (1967a)CrossRefGoogle Scholar - Hamielec, A., Johnson, A., Houghton, W.: Numerical solution of the Navier–Stokes equation for flow past spheres: part II. Viscous flow around circulating spheres of low viscosity. AIChE J.
**13**(2), 220–224 (1967b)CrossRefGoogle Scholar - Han, N., Bhakta, J., Carbonell, R.: Longitudinal and lateral dispersion in packed beds: effect of column length and particle size distribution. AIChE J.
**31**(2), 277–288 (1985)CrossRefGoogle Scholar - Haserta, M., Bernsdorfa, J., Rollera, S.: Lattice Boltzmann simulation of non-Darcy flow in porous media. Procedia Comput. Sci.
**4**, 1048–1057 (2011)CrossRefGoogle Scholar - Hofmann, S., Bufe, A., Brenner, G., Turek, T.: Pressure drop study on packings of differently shaped particles in milli-structured channels. Chem. Eng. Sci.
**155**, 376–385 (2016)CrossRefGoogle Scholar - Kececioglu, I., Jiang, Y.: Flow through porous media of packed spheres saturated with water. J. Fluid Eng.
**116**, 164–170 (1994)CrossRefGoogle Scholar - LeClair, B., Hamielec, A.: Viscous flow through particle assemblages at intermediate Reynolds numbers. Steady-state solutions for flow through assemblages of spheres. Ind. Eng. Chem. Fund.
**7**(4), 542–549 (1968)CrossRefGoogle Scholar - Liang, S.: Investigation of Nonlinear Flow and Dispersion in Porous Media. MSc Thesis, University of Warwick, UK (2017)Google Scholar
- Majdalani, S., Chazarin, J.P., Delenne, C., Guinot, V.: Solute transport in periodically heterogeneous porous media: importance of observation scale and experimental sampling. J. Hydrol.
**520**, 52–60 (2015)CrossRefGoogle Scholar - Mickley, H., Smith, K., Korchak, E.: Fluid flow in packed beds. Chem. Eng. Sci.
**20**(3), 237–246 (1965)CrossRefGoogle Scholar - Montroll, E., Weiss, G.: Random walks on lattices. II. J. Math. Phys.
**6**(2), 167–181 (1965)CrossRefGoogle Scholar - Nezhad, M.M.: Stochastic Finite Element Modeling of Flow and Solute Transport in Dual Domain System. PhD thesis, University of Exeter, UK (2010)Google Scholar
- Nezhad, M.M., Javadi, A.A.: Stochastic finite-element approach to quantify and reduce uncertainty in pollutant transport modeling. J. Hazard. Toxic Radioact. Waste
**15**(3), 208–215 (2011)CrossRefGoogle Scholar - Nezhad, M.M., Javadi, A.A., Rezania, M.: Finite element modelling of contaminant transport considering effects of micro and macro heterogeneity of soil. J. Hydrol.
**404**(3–4), 332–338 (2011)CrossRefGoogle Scholar - Nezhad, M.M., Javadi, A.A., Al-Tabbaa, A., Abbasi, F.: Numerical study of soil heterogeneity effects on contaminant transport in unsaturated soil; model development and validation. Int. J. Numer. Anal. Met.
**37**(3), 278–298 (2013)CrossRefGoogle Scholar - Norouzi, A.M., Siavashi, M., Soheili, A.R., Khaliji Oskouei, M.: Experimental investigation of effects of grain size, inlet pressure and flow rate of air and argon on pressure drop through a packed bed of granular activated carbon. ICHMT
**96**, 20–26 (2018)Google Scholar - Petrasch, J., Meier, F., Friess, H., Steinfeld, A.: Tomography based determination of permeability, Dupuit-Forchheimer coefficient, and interfacial heat transfer coefficient in reticulate porous ceramics. Int. J. Heat Fluid Flow
**29**, 315–326 (2008)CrossRefGoogle Scholar - Prausnitz, J., Wilhelm, R.: Turbulent concentration fluctuations in a packed bed. Ind. Eng. Chem.
**49**(6), 978–984 (1957)CrossRefGoogle Scholar - Reddy, S., Schmid, P., Baggett, J., Henningson, D.: On the stability of streamwise streaks and transition thresholds in plane channel flows. J. Fluid Mech.
**365**, 269–303 (1998)CrossRefGoogle Scholar - Sanchez-Palencia, E.: Vibration of mixtures of solids and fluids. In: Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127, pp. 158–190. Springer, Berlin (1980)Google Scholar
- Scheidegger, A.: Statistical hydrodynamics in porous meida. J. Appl. Phys.
**25**(8), 994–1001 (1954)CrossRefGoogle Scholar - Scheidegger, A.: General theory of dispersion in porous media. J. Geophys. Res.
**66**(10), 3273–3278 (1961)CrossRefGoogle Scholar - Shapiro, M., Brenner, H.: Taylor dispersion of chemically reactive species: irreversible first-order reactions in bulk and on boundaries. Chem. Eng. Sci.
**41**(6), 1417–1433 (1986)CrossRefGoogle Scholar - Shapiro, M., Brenner, H.: Dispersion of a chemically reactive solute in a spatially periodic model of a porous medium. Chem. Eng. Sci.
**43**(3), 551–557 (1988)CrossRefGoogle Scholar - Simmons, C.S.: A stochastic-convective transport representation of dispersion in one-dimensional porous media systems. Water Resour. Res.
**18**(4), 1193–1214 (1982)CrossRefGoogle Scholar - Slichter, C.: Field Measurement of the Rate of Movement of Underground Water. Water Supply Paper. USGS Publications Warehouse, Govt. Print. Off., https://doi.org/10.3133/wsp140 (1905)
- Taylor, G.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.
**219**(1137), 186–203 (1953)CrossRefGoogle Scholar - van der Merwe, D., Gauvin, W.: Velocity and turbulence measurements of air flow through a packed bed. AIChE J.
**17**(3), 519–528 (1971)CrossRefGoogle Scholar - Vollmari, K., Oschmann, T., Wirtz, S., Kruggel-Emden, H.: Pressure drop investigations in packings of arbitrary shaped particles. Powder Technol.
**271**, 109–124 (2015)CrossRefGoogle Scholar - Whitaker, S.: The Forchheimer equation: a theoretical development. Transp. Porous Media
**25**(1), 27–61 (1996)CrossRefGoogle Scholar - Wood, B.D.: Inertial effects in dispersion in porous media. Water Resour. Res.
**43**, 12–16 (2007)CrossRefGoogle Scholar - Zeng, Z., Grigg, R.: A criterion for non-Darcy flow in porous media. Transp. Porous Media
**63**, 57–69 (2006)CrossRefGoogle Scholar

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