Khalil Abou-Saleh^*, Jalal Dweik, Youssef Haidar, Abbas Ghaddar

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Department of Physics, Faculty of Sciences V, Lebanese University, Nabatieh, Lebanon.
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Abstract:

Understanding and simulating the underlying microscopic physics of the rock matrix is very useful for determining macroscopic physical properties such as permeability. Matrix diffusion is an important transport parameter controlling the late-time behaviour of breakthrough curves (BTCs). We compute the memory function, implemented in the sink/source term of Mobile-immobile mass transfer by solving the matrix diffusion using a time diffusion random-walk approach. The diffusion is controlled by different parameters like the porosity, tortuosity, mobile-immobile interface and immobile domain cluster shapes. All these properties are investigated by X-ray microtomography that captures the main characteristics of matrix diffusion at three dimensions. We compare the memory function deduced from the field-scale tracer tests well with the computed memory function. Simulation results of the memory function appeared to be coherent with that measured from the tracer test for a large tortuosity value. Probably, the diffusion paths are longer, and they are controlled by the properties mentioned above. From a representative elementary volume of natural reservoirs studied here, we conclude that, microscale diffusion process in the immobile domain play a crucial role to better understand the non-Fickian dispersion measured from the tracer test.

Keywords: Matrix Diffusion, Time Domain Random Walk, Memory Function, Mobile-Immobile Mass Transfer

Cite this paper: Abou-Saleh, K. , Dweik, J. , Haidar, Y. and Ghaddar, A. (2019) Solving Diffusion Time in Heterogeneous Microscale Rock Matrix by 3D Computations: Non-Fickian Dispersion Observed in Porous Media. Journal of Geoscience and Environment Protection, 7, 42-52. doi: 10.4236/gep.2019.712003.

References

[1]   Berkowitz, B., & Scher, H. (2000). Anomalous Transport in Laboratory-Scale, Heterogeneous Porous Media. Water Resources Research, 36, 149-158.
https://doi.org/10.1029/1999WR900295

[2]   Cai, J., Zhang, Z., Kang, Q. et al. (2019). Recent Advances in Flow and Transport Properties of Unconventional Reservoirs. Energies, 12, 1865-1869.
https://doi.org/10.3390/en12101865

[3]   Carrera, J., Sánchez-Vila, X., Benet, I. et al. (1998). On Matrix Diffusion, Formulations, Solutions Methods and Qualitative Effects. Hydrogeology Journal, 6, 178-190.
https://doi.org/10.1007/s100400050143

[4]   Delay, F., Ackerer, P., & Danquigny, C. (2005). Simulating Solute Transport in Porous or Fractured Formations Using Random Walk Particle Tracking: A Review. Vadose Zone Journal, 4, 360-379.
https://doi.org/10.2136/vzj2004.0125

[5]   Delay, F., Porel, G., & Sardini, P. (2002). Modelling Diffusion in a Heterogeneous Rock Matrix with a Time-Domain Lagrangian Method and an Inversion Procedure. C.R. Geoscience, 334, 967-973.
https://doi.org/10.1016/S1631-0713(02)01835-7

[6]   Dentz, M., Gouze, P., Russian, A. et al. (2012). Diffusion and Trapping in Heterogeneous Media: An Inhomogeneous Continuous Time Random Walk Approach. Advances in Water Resources, 49, 13-22.
https://doi.org/10.1016/j.advwatres.2012.07.015

[7]   Dweik, J., Srour, M., Tawbe, M. et al. (2015). Solving Microscale Heterogeneous Matrix Diffusion Based on Two and Three-Dimensional Computing Using X-Ray Tomography Image Data. International Review of Physics, 9, 79-84.

[8]   Gouze, P., Melean, Y., Le Borgne, T. et al. (2008). Non-Fickian Dispersion in Porous Media Explained by Heterogeneous Microscale Matrix Diffusion. Water Resources Research, 44, W11416-W11435.
https://doi.org/10.1029/2007WR006690

[9]   Haggerty, R., Fleming, S. W., Meigs, L. C. et al. (2001). Tracer Tests in a Fractured Dolomite: 2. Analysis of Mass Transfer in Single-Well Injection-Withdrawal Tests. Water Resources Research, 37, 1129-1142.
https://doi.org/10.1029/2000WR900334

[10]   Haggerty, R., Harvey, C. F., Schwerin, C. F. et al. (2004). What Controls the Apparent Timescale of Solute Mass Transfer in Aquifers and Soils? A Comparison of Experimental Results. Water Resources Research, 40, W01510.
https://doi.org/10.1029/2002WR001716

[11]   Haggerty, R., McKenna, S. A., & Meigs, L. C. (2000). On the Late-Time Behavior of Tracer Test Breakthrough Curves. Water Resources Research, 36, 3467-3480.
https://doi.org/10.1029/2000WR900214

[12]   Khrapitchev, A. A., & Callaghan, P.-T. (2003). Reversible and Irreversible Dispersion in a Porous Medium. Physics of Fluids, 15, 2649-2660.
https://doi.org/10.1063/1.1596914

[13]   Le Borgne, T., & Gouze, P. (2008). Non-Fickian Dispersion in PorousMedia: 2. Model Validation from Measurements at Different Scales. Water Resources Research, 44, W06427- W06436.
https://doi.org/10.1029/2007WR006279

[14]   Levy, M., & Berkowitz, B. (2003). Measurement and Analysis of Non-Fickian Dispersion in Heterogeneous Porous Media. Journal of Contaminant Hydrology, 64, 203-226.
https://doi.org/10.1016/S0169-7722(02)00204-8

[15]   McCarthy, J. F. (1993). Continuous Time Random Walks on Random Media. Journal of Physics A: Mathematical and General, 26, 2495-2503.
https://doi.org/10.1088/0305-4470/26/11/004

[16]   Meigs, L. C., & Beauhelm, R. L. (2001). Tracer Tests in a Fractured Dolomite: 1, Experimental Design and Observed Tracer Recoveries. Water Resources Research, 37, 1113- 1128.
https://doi.org/10.1029/2000WR900335

[17]   Salamon, P., Fernàndez-Garcia, D., & Gómez-Hernández, J. J. (2006). A Review and Numerical Assessment of the Random Walk Particle Tracking Method. Journal of Contaminant Hydrology, 87, 277-305.
https://doi.org/10.1016/j.jconhyd.2006.05.005

[18]   Sardini, P., Delay, F., & Hellmuth, K.-H. (2003). Interpretation of Out-Diffusion Experiments on Crystalline Rocks Using Random Walk Modelling. Journal of Contaminant Hydrology, 61, 339-350.
https://doi.org/10.1016/S0169-7722(02)00124-9

[19]   Sardini, P., Robinet, J.-C., & Siitari-Kappi, M. (2007). Direct Simulation of Heterogeneous Diffusion and Inversion Procedure Applied to an Out-Diffusion Experiment. Test Case of Palmottugranite. Journal of Contaminant Hydrology, 93, 21-37.
https://doi.org/10.1016/j.jconhyd.2007.01.011

[20]   Shapiro, A. M. (2001). Effective Matrix Diffusion in Kilometer-Scale Transport in Fractured Crystalline Rock. Water Resources Research, 37, 507-522.
https://doi.org/10.1029/2000WR900301

[21]   Zhan, H., Wang, Q., & Wen, Z. (2019). Advances in Groundwater Flow and Solute Transport: Pushing the Hidden Boundary. Water, 11, 457-460.
https://doi.org/10.3390/w11030457

[22]   Zhou, L., Jing, L., & Cvetkovic, V. (2017). Modeling of Solute Transport in a 3D Rough- Walled Fracture-Matrix System. Transport in Porous Media, 116, 1005-1029.
https://doi.org/10.1007/s11242-016-0810-z

[23]   Zhou, Q., Oldenburg, C.-M., Kneafsey, T.-J. et al. (2018). Modeling Transport of Multiple Tracers in Hydraulic Fractures at the, EGS Collab Test Site. In 43rd Workshop on Geothermal Reservoir Engineering (SGP-TR-213). Stanford: Stanford University.

[24]   Zhoua, Q., Liua, H.-H., Molz, F. J. et al. (2007). Field-Scale Effective Matrix Diffusion Coefficient for Fractured Rock: Results from Literature Survey. Journal of Contaminant Hydrology, 93, 161-187.
https://doi.org/10.1016/j.jconhyd.2007.02.002