Abstract
A crucial part in a two-dimensional discrete fracture network (DFN) model of two-phase flow in a naturally fractured rock is the updating algorithm for the fluid interface positions inside the fractures. In this study, a single fracture situated in an impermeable rock matrix is considered. Displacements of a Newtonian fluid by another Newtonian fluid, by a power-law fluid, or by a Bingham fluid are analysed. It is shown that a significant improvement in accuracy and computational efficiency in all three types of displacement can be achieved by updating the interface position with predictor–corrector scheme instead of forward Euler scheme currently used in such simulators. In order to achieve the same accuracy in the breakthrough time as with the forward Euler method, predictor–corrector can be run with a timestep that is 3 to 50 times larger. The gain in accuracy (or, equivalently, increase of the adjustable timestep) achieved with predictor–corrector is shown to outweigh the extra computational cost of this scheme. It is also shown that further increase in accuracy, e.g. by using the Runge–Kutta 4 scheme, does not pay off. In order to have control over the error in the interface positions, a viable strategy is to choose the timestep equal to a given fraction of the shortest single-fracture breakthrough time in the network. The value of this fraction can be chosen based on the required accuracy in the breakthrough time. In the case of Newtonian/Newtonian displacement, the optimal ratio of timestep to the single-fracture breakthrough time is found to depend only on the mobility ratio and the prescribed accuracy in the numerically computed breakthrough time. In case of a single interface in the fracture, numerical experimentation yields simple equations for recommended timestep in Newtonian/Newtonian displacement based on the fluid mobility ratio and the maximum allowed error in the breakthrough time.
Article Highlights
-
Predictor–corrector provides a reasonable trade-off between accuracy and cost
-
Whenever possible, adjustable timestep should be used
-
Timestep can be chosen based on the mobility ratio of Newtonian fluids
-
For non-Newtonian fluids, timestep can be chosen from estimated breakthrough time
Data availability
All the necessary data are provided in the text of the article.
References
Amadei, B., Savage, W.Z.: An analytical solution for transient flow of Bingham viscoplastic materials in rock fractures. Int. J. Rock Mech. Min. Sci. 38(2), 285–296 (2001)
Bao, K., Lavrov, A., Nilsen, H.M. (2017) Numerical modeling of non-Newtonian fluid flow in fractures and porous media. Computational Geosciences, 1–12
Berre, I., Doster, F., Keilegavlen, E.: Flow in fractured porous media: a review of conceptual models and discretization approaches. Transp. Porous Media 130(1), 215–236 (2019)
Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. B 92, 1337–1347 (1987)
Cacas, M.C., Ledoux, E., de Marsily, G., Tillie, B., Barbreau, A., Durand, E., Feuga, B., Peaudecerf, P.: Modeling fracture flow with a stochastic discrete fracture network: calibration and validation: 1. The Flow Model. Water Resources Research 26(3), 479–489 (1990). https://doi.org/10.1029/WR026i003p00479
Di Federico, V.: Estimates of equivalent aperture for non-Newtonian flow in a rough-walled fracture. Int. J. Rock Mech. Min. Sci. 34(7), 1133–1137 (1997)
Di Federico, V.: Non-Newtonian flow in a variable aperture fracture. Transp Porous Med 30(1), 75–86 (1998)
Fidelibus, C., Lenti, V.: The propagation of grout in pipe networks. Comput. Geosci. 45, 331–336 (2012)
Frigaard, I.A., Ryan, D.P.: Flow of a visco-plastic fluid in a channel of slowly varying width. J. Nonnewton. Fluid Mech. 123(1), 67–83 (2004). https://doi.org/10.1016/j.jnnfm.2004.06.011
Guo, Y., Zhao, P., Zhang, Q., Liu, R., Zhang, L., Liu, Y.: Investigation of the mechanism of grout penetration in intersected fractures. Fluid Dynamics & Materials Processing 15(4), 321–342 (2019)
Hässler, L., Håkansson, U., Stille, H.: Computer-simulated flow of grouts in jointed rock. Tunn. Undergr. Space Technol. 7(4), 441–446 (1992)
Homsy, G.M.: Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19(1), 271–311 (1987)
Hu, M., Rutqvist, J., Wang, Y.: A practical model for fluid flow in discrete-fracture porous media by using the numerical manifold method. Adv. Water Resour. 97, 38–51 (2016). https://doi.org/10.1016/j.advwatres.2016.09.001
Lakhtychkin, A., Eskin, D., Vinogradov, O.: Modelling of transport of two proppant-laden immiscible power-law fluids through an expanding fracture. Canadian J. Chem. Eng. 90(3), 528–543 (2012). https://doi.org/10.1002/cjce.20694
Lavrov, A.: Flow of truncated power-law fluid between parallel walls for hydraulic fracturing applications. J. Nonnewton. Fluid Mech. 223, 141–146 (2015). https://doi.org/10.1016/j.jnnfm.2015.06.005
Lei, Q., Latham, J.-P., Tsang, C.-F.: The use of discrete fracture networks for modelling coupled geomechanical and hydrological behaviour of fractured rocks. Comput. Geotech. 85, 151–176 (2017)
Ma, G.W., Wang, H.D., Fan, L.F., Wang, B.: Simulation of two-phase flow in horizontal fracture networks with numerical manifold method. Adv. Water Resour. 108, 293–309 (2017)
Ma, G.W., Wang, H.D., Fan, L.F., Chen, Y.: Segmented two-phase flow analysis in fractured geological medium based on the numerical manifold method. Adv. Water Resour. 121, 112–129 (2018)
Putz, A., Frigaard, I.A., Martinez, D.M.: On the lubrication paradox and the use of regularisation methods for lubrication flows. J. Nonnewton. Fluid Mech. 163(1), 62–77 (2009). https://doi.org/10.1016/j.jnnfm.2009.06.006
Ren, F., Ma, G., Wang, Y., Li, T., Zhu, H.: Unified pipe network method for simulation of water flow in fractured porous rock. J. Hydrol. 547, 80–96 (2017). https://doi.org/10.1016/j.jhydrol.2017.01.044
Tsang, Y.W., Tsang, C.F.: Channel model of flow through fractured media. Water Resour. Res. 23(3), 467–479 (1987). https://doi.org/10.1029/WR023i003p00467
Unsal, E., Matthäi, S.K., Blunt, M.J.: Simulation of multiphase flow in fractured reservoirs using a fracture-only model with transfer functions. Comput. Geosci. 14(4), 527–538 (2010)
Wrobel, M., Mishuris, G., Papanastasiou, P.: On the influence of fluid rheology on hydraulic fracture. Int. J. Eng. Sci. 158, 103426 (2021). https://doi.org/10.1016/j.ijengsci.2020.103426
Xu, C., Dowd, P.: A new computer code for discrete fracture network modelling. Comput. Geosci. 36(3), 292–301 (2010). https://doi.org/10.1016/j.cageo.2009.05.012
Yang, Z., Méheust, Y., Neuweiler, I., Hu, R., Niemi, A., Chen, Y.-F.: Modeling immiscible two-phase flow in rough fractures from capillary to viscous fingering. Water Resour. Res. 55(3), 2033–2056 (2019)
Zimmerman, R.W., Kumar, S., Bodvarsson, G.S.: Lubrication theory analysis of the permeability of rough-walled fractures. Int J. Rock Mech. Mining Sci. Geomech. Abstracts 28(4), 325–331 (1991). https://doi.org/10.1016/0148-9062(91)90597-F
Zou, L., Håkansson, U., Cvetkovic, V.: Two-phase cement grout propagation in homogeneous water-saturated rock fractures. Int. J. Rock Mech. Min. Sci. 106, 243–249 (2018)
Acknowledgements
Comments and suggestions from three anonymous reviewers are gratefully appreciated; they helped the author improve parts of the manuscript significantly.
Funding
No funds, grants, or other support was received.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lavrov, A. Fluid Displacement in a 2D DFN Fracture: Time Integration of the Interface Position. Transp Porous Med 139, 247–269 (2021). https://doi.org/10.1007/s11242-021-01659-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-021-01659-2