Abstract
Multiphase flows in porous media are encountered in several contexts—e.g., hydrocarbon recovery operations, battery electrodes, microfluidic devices, etc. Capillary-dominated flows are interesting due to the complex interplay of interfacial properties and pore geometries. Conventional hydrodynamic flow solvers are computationally inefficient in the capillary-dominated regime, particularly in complex pore structures. The algorithm developed here specifically targets this regime to reduce simulation times. We minimise the fluid–fluid and fluid–solid interaction energies through an approach inspired by the ferromagnetic Ising model. We validate the algorithm on (1) model pore geometries with analytical solutions for capillary action, and (2) rocks with available mercury porosimetry data. We validate its predictions for model geometries and sandstones using (1) curvatures calculated from theories developed by Mayer–Stowe–Princen, Ma and Morrow, and Mason and Morrow; (2) predictions from GeoDict, a commercial software package, which also includes a state-of-the-art drainage simulator; (3) mercury porosimetry data. Drainage capillary pressure curves predicted for Bentheimer and Fontainebleau rocks reasonably match porosimetry data.
Similar content being viewed by others
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Ahrenholz, B., Tolke, J., Lehmann, P., Peters, A., Kaestner, A., Krafczyk, M., Durner, W.: Prediction of capillary hysteresis in a porous material using lattice-Boltzmann methods and comparison to experimental data and a morphological pore network model. Adv. Water Resour. 31, 1151 (2008)
Al-Kharusi, A., Blunt, M.J.: Multiphase flow predictions from carbonate pore space images using extracted network models. Water Resour. Res. 44, 1 (2008)
Andersen, M.A.: Digital core flow simulations accelerate evaluation of multiple recovery scenarios. In: World Oil. p. 50 (2014)
Andra, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rockphysicsbenchmarks—part I: imaging and segmentation. Comput. Geosci. 50, 25 (2013a)
Andra, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rockphysicsbenchmarks—part II: computingeffectiveproperties. Comput. Geosci. 50, 33 (2013b)
Armstrong, R.T., Berg, S.: Interfacial velocities and capillary pressure gradients during Haines jumps. Phys. Rev. E 88, 043010 (2013)
Armstrong, R.T., Berg, S., Dinariev, O., Evseev, N., Klemin, D., Koroteev, D., Safonov, S.: Modeling of pore-scale two-phase phenomena using density functional hydrodynamics. Transp. Porous Med. 112, 577 (2016)
Armstrong, R.T., Evseev, N., Koroteev, D., Berg, S.: Modeling the velocity field during Haines jumps in porous media. Adv. Water Resour. 77, 57 (2015)
Berg, S., Rucker, M., Ott, H., Georgiadis, A., Van der Linde, H., Enzmann, F., Kersten, M., Armstrong, R.T., De With, S., Becker, J., Wiegmann, A.: Connected pathway relative permeability from pore-scale imaging of imbibition. Adv. Water Resour. 90, 24 (2016)
Blunt, M.J.: Pore level modeling of the effects of wettability. SPE J. 2, 494 (1997)
Blunt, M.J.: Flow in porous media: pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 6, 197 (2001)
Blunt, M.J., Scher, H.: Pore-level modeling of wetting. Phys. Rev. E 52, 6387 (1995)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258 (1958)
Dernaika, M., Al Jallad, O., Koronfol, S., Suhrer, M., Teh, W.J., Walls, J., Matar, S., Murthy, N., Zekraoui, M.: Petrophysical and fluid flow properties of a tight carbonate source rock using digital rock physics. In: Paper Presented at the Unconventional Resources Technology Conference, San Antonio
Dias, M.A., Wilkinson, D.: Percolation with trapping. J. Phys. A: Math. Gen. 19, 3131 (1986)
Dullien, F.A.L.: Porous Media: Fluid Transport and Pore Structure. Academic Press Inc, San Diego (1992)
Dvorkin, J., Derzhi, N., Diaz, E., Fang, Q.: Relevance of computational rock physics. Geophysics 76, E141 (2011)
Etris, E.L., Brumfield, D.S., Ehrlich, R.: Relations between pores, throats and permeability: a petrographic/physical analysis of some carbonate grainstones and packstones. Carbonates Evaporites 5, 17–32 (1988)
Feng, J., Rothstein, J.P.: Simulations of novel nanostructures formed by capillary effects in lithography. J. Colloid Interface Sci. 354(1), 386–395 (2011)
Fowkes, F.M.: Attractive forces at interfaces. Ind. Eng. Chem. 56, 40–52 (1964)
Freedman, R., Heaton, N.: Fluid characterisation using NMR logging. Petrophysics 45, 241–250 (2004)
Hazlett, R.D.: Simulation of capillary-dominated displacements in microtomographic images of reservoir rocks. Transp. Porous Med. 20, 21 (1995)
Hilpert, M., Miller, C.T.: Pore-morphology-based simulation of drainage in totally wetting porous media. Adv. Water Resour. 24, 243 (2001)
Hoshen, J., Berry, M.W., Minser, K.S.: Percolation and cluster structure parameters: the enhanced Hoshen–Kopelman algorithm. Phys. Rev. E 56, 1455 (1997)
Hoshen, J., Kopelman, R.: Percolation & cluster distribution I. Phys. Rev. B. 14, 3438 (1976)
Ising, E.: Z. Phys. 31, 253 (1925)
Jettestuen, E., Helland, J.O., Pradanovic, M.: A level set method for simulating capillary-controlled displacements at the pore-scale with nonzero contact angles. Water Resour. Res. 49, 4645–4661 (2013)
Kenyon, W.E., Howard, J.J., Sezginer, A., Straley, C., Matteson, A., Horkowitz, K., Ehrlich, R.: Pore size distribution and NMR in microporous cherty sandstones. In: Paper Presented at the SPWLA 30th Annual Logging Symposium Transactions, Denver
Knackstedt, M.A., Marrink, S.J., Sheppard, A.P., Pinczewski, W.V., Sahimi, M.: Invasion percolation on correlated and elongated lattices: implications for the interpretation of residual saturations in rock cores. Transp. Porous Med. 44, 465 (2001)
Knackstedt, M.A., Sahimi, M., Sheppard, A.P.: Invasion percolation with long-range correlations: first-order phase transition and nonuniversal scaling properties. Phys. Rev. E 61, 4920 (2000)
Knackstedt, M.A., Sheppard, A.P.: Simulation of mercury porosimetry on correlated grids: evidence for extended correlated heterogeneity at the pore scale in rocks. Phys. Rev. E 58, R6923 (1998)
Koroteev, D., Dinariev, O., Evseev, N., Klemin, D., Nadeev, A., Safonov, S., Gurpinar, O., Berg, S., Van Kruijsdijk, C., Armstrong, R.T., Myers, M.T., Hathon, L., De Jong, H.: Direct hydrodynamic simulation of multiphase flow in porous rock. Petrophysics 55, 294 (2014)
Latour, L.L., Kleinberg, R.L., Mitra, P.P., Sotak, C.H.: Pore-size distributions and tortuosity in heterogeneous porous media. J. Magn. Reson. A 112, 83–91 (1995)
Legland, D., Kieu, K., Devaux, M.-F.: Computation of Minkowski measures on 2D and 3D binary images. Image Anal. Stereol. 26, 83 (2007)
Leu, L., Berg, S., Enzmann, F., Armstrong, R.T., Kersten, M.: Fast x-ray micro-tomography of multiphase flow in berea sandstone: a sensitivity study on image processing. Transp. Porous Med. 105, 451 (2014)
Lu, N., Zeidman, B.D., Lusk, M.T., Wilson, C.S., Wu, D.T.: A Monte Carlo paradigm for capillarity in porous media. Geophys. Res. Lett. 37, L23402 (2010)
Ma, S., Jiang, M.-X., Morrow, N.R.: Correlation of capillary pressure relationships and calculations of permeability. In: 66th SPE Annual Technical Conference and Exhibition, Dallas. SPE, SPE (1991)
Ma, S., Mason, G., Morrow, N.R.: Effect of contact angle on drainage and imbibition in regular polygonal tubes. Colloids Surf., A 117(3), 273–291 (1996)
Mason, G., Morrow, N.: Meniscus displacement curvatures of a perfectly wetting liquid in capillary pore throats formed by spheres. J. Colloid Interface Sci. 109(1), 46–56 (1985)
Mason, G., Morrow, N.R.: Coexistence of menisci and the influence of neighboring pores on capillary displacement curvatures in sphere packings. J. Colloid Interface Sci. 100(2), 519–535 (1984a)
Mason, G., Morrow, N.R.: Meniscus curvatures in capillaries of uniform cross-section. J. Chem. Soc., Faraday Trans. 1: Phys. Chem. Condens. Phases 80(9), 2375–2393 (1984b)
Mason, G., Morrow, N.R.: Capillary behavior of a perfectly wetting liquid in irregular triangular tubes. J. Colloid Interface Sci. 141(1), 262–274 (1991)
Mason, G., Morrow, N.R.: Effect of contact angle on capillary displacement curvatures in pore throats formed by spheres. J. Colloid Interface Sci. 168, 130–141 (1994)
Masson, Y.: A fast two-step algorithm for invasion percolation with trapping. Comput. Geosci. 90, 41–48 (2016)
Masson, Y., Pride, S.R.: A fast algorithm for invasion percolation. Transp. Porous Media 102(2), 301–312 (2014)
Mayer, R.P., Stowe, R.A.: Mercury porosimetry—breakthrough pressure for penetration between packed spheres. J. Colloid Sci. 20(8), 893–911 (1965)
Morrow, N.R.: Physics and thermodynamics of capillary action in porous media. Ind. Eng. Chem. 62, 32 (1970)
Nadeev, A., Klemin, D.: Inside the rock. In: GEOExPro. p. 32 (2014)
Nguyen, V.H., Sheppard, A.P., Knackstedt, M.A., Pinczewski, W.V.: A dynamic network model for imbibition. in: Paper Presented at the SPE International Petroleum Conference, Puebla, Mexico
Princen, H.: Capillary phenomena in assemblies of parallel cylinders: I. Capillary rise between two cylinders. J. Colloid Interface Sci. 30(1), 69–75 (1969a)
Princen, H.: Capillary phenomena in assemblies of parallel cylinders: II. Capillary rise in systems with more than two cylinders. J. Colloid Interface Sci. 30(3), 359–371 (1969b)
Princen, H.: Capillary pressure behavior in pores with curved triangular cross-section: effect of wettability and pore size distribution. Colloids Surf. 65(2), 221–230 (1992)
Prodanovic, M., Bryant, S.L.: A level set method for determining critical curvatures for drainage and imbibition. J. Colloid Interface Sci. 304, 442–458 (2006)
Purcell, W.R.: Capillary pressures—their measurement using mercury and the calculation of permeability therefrom. J. Pet. Technol. 1, 39 (1948)
Raeini, A.Q., Blunt, M.J., Bijeljic, B.: Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. J. Comp. Phys. 231, 5653–5668 (2012)
Roof, J.G.: Snap-off of oil droplets in water-wet pores. SPE J. 10, 85 (1970)
Rucker, M., Berg, S., Armstrong, R.T., Georgiadis, A., Ott, H., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Wolf, M., Khan, F., Enzmann, F., Kersten, M.: From connected pathway flow to ganglion dynamics. Geophys. Res. Lett. 42, 3888 (2015)
Rucker, M., Berg, S., Armstrong, R.T., Georgiadis, A., Ott, H., Simon, L., Enzmann, F., Kersten, M., De With, S.: The fate of oil clusters during fractional flow: trajectories in the saturation—capillary number space. In: Paper Presented at the International Symposium of the Society of Core Analysts, St. John’s Newfoundland and Labrador, Canada
Schluter, S., Berg, S., Rucker, M., Armstrong, R.T., Vogel, H.-J., Hilfer, R., Wildenschild, D.: Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media. Water Resour. Res. 52, 2194 (2016)
Sheppard, A.P., Knackstedt, M.A., Pinczewski, W.V., Sahimi, M.: Invasion percolation: new algorithms and universality classes. J. Phys. A: Math. Gen. 32, L521 (1999)
Silin, D., Patzek, T.: Pore space morphology analysis using maximal inscribed spheres. Phys. A 371, 336 (2006)
Silin, D., Tomutsa, L., Benson, S.M., Patzek, T.W.: Microtomography and pore-scale modeling of two-phase fluid distribution. Transp. Porous Med. 86, 495 (2011)
Son, S., Chen, L., Kang, Q., Derome, D., Carmeliet, J.: Contact angle effects on pore and corner arc menisci in polygonal capillary tubes studied with the pseudopotential multiphase Lattice Boltzmann model. Computation 4(1), 12 (2016)
Timur, A.: Pulsed NMR studies of porosity, movable fluid and permeability of sandstones. J. Pet. Technol. 21, 775–786 (1969)
Valvatne, P.H., Blunt, M.J.: Predictive pore-scale network modeling. In: Paper Presented at the SPE Annual Technical Conference and Exhibition, Denver, CO, USA
Van Buuren, A.R., Marrink, S.J., Berendsen, H.J.C.: A molecular dynamics study of the decane/water interface. J. Phys. Chem. 97, 9206–9212 (1993)
Volkov, A.G., Deamer, D.W. (eds.): Liquid–Liquid Interfaces: Theory and Methods. CRC Press, Boca Raton (1996)
Weiss, M.A.: Data Structures and Algorithm Analysis in C, 2nd edn. Addison-Wesley Professional, Boston (1996)
Wilkinson, D.: Percolation effects in immiscible displacement. Phys. Rev. A 34, 1380 (1986)
Wilkinson, D., Barsony, M.: Monte Carlo study of invasion percolation clusters in two and three dimensions. J. Phys. A: Math. Gen. 17, L129 (1984)
Wilkinson, D., Willemsen, J.F.: Invasion percolation: a new form of percolation theory. J. Phys. A: Math. Gen. 16, 3365 (1983)
Xu, J., Louge, M.Y.: Statistical mechanics of unsaturated porous media. Phys. Rev. E 92, 062405 (2015)
Yuan, H.H.: Advances in APEX technology. Log Anal. 32(5), 557–570 (1991a)
Yuan, H.H.: Pore-scale heterogeneity from mercury porosimetry data. SPE Formation Eval. 6, 233 (1991b)
Yuan, H.H., Swanson, B.F.: Resolving pore-space characteristics by rate-controlled porosimetry. SPE Formation Eval. 4, 17 (1989)
Zacharoudiou, I., Boek, E.S.: Capillary filling and Haines jump dynamics using free energy Lattice Boltzmann simulations. Adv. Water Resour. 92, 43–56 (2016)
Acknowledgements
We thank Steffen Berg, Nishank Saxena and Cor van Kruijsdijk from Shell for going through the manuscript and providing valuable suggestions. We also thank Steffen Berg and Nishank Saxena for providing us the sandstone images and associated MICP data. We are also grateful to Shell for funding this work and providing permission to publish this paper.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendix
Appendix
1.1 A1 Link with Work of Invasion
We will now connect our invasion scheme to the thermodynamics of drainage as laid out by Mayer and Stowe (1965) and Morrow (1970). Figure 13 shows the oil meniscus (red) advancing through a water-filled pore with solid walls. A differential displacement of dn leads to changes in the areas of interfaces between oil–water (dAOW), oil-solid (dAOS) and water–solid (dAWS). The dotted red line denotes the new location of the oil–water interface.
If PO is the pressure in the oil phase and PW is the pressure in the water phase, the work expended in advancing the oil–water interface through a volume dV under quasi-static conditions is
where the right-hand side is the energy required to modify the various interfaces in the system. The different interfacial energies are products of the respective interfacial tensions, γ, and the change in interfacial area, dA. As the oil phase moves forward, the increase in the oil-solid contact area equals the decrease in the water–solid contact area, i.e., dAOS = − dAWS. The balance of interfacial forces at the three-phase contact point gives us γOS − γWS = γOW cosθ, where θ is the contact angle measured in the water phase.
The difference in pressure across the oil–water interface is the capillary pressure, i.e., Pc = PO − PW. Applying these 3 simplifications to Eq. A1a gives
Recasting Eq. A1b along the lines of Eq. 2a, 2b, 2c in the main text, we get
where C is the curvature of the meniscus and Pc/γOW is the scaled capillary pressure. Let us focus on the average interfacial energy, \( \bar{E} \). We stated earlier that Pc/γOW scales as \( \bar{E} \). Starting with this notion, we use Eqs. 4–5 from the main text to get
The innermost sum over the neighbourhood of cell i is decomposed into three more sums: one over water-like neighbours, the second over oil-like neighbours, and the third over rock-like neighbours. We denote these constituent neighbourhoods as W(i), O(i) and R(i), respectively.
The spins of water, oil and rock cells—1, − 1 and cosθ, respectively—are inserted in Eq. A2b to facilitate the easy calculation of the three inner sums.
where NX(i) is the number of X-like (X = W, O, R) neighbours near cell i. Using the relation ΔNWO′(i) = NW(i) − NO(i) and teasing apart the fluid and rock terms, we get
A comparison of Eqs. A1c and A2d suggests that
Recall that Nflip is the number of interface cells that are flipped from water to oil during an invasion step. It represents the incremental progress of the interface and hence the differential volume, dV, associated with this progress. On the LHS in Eqs. A3a, A3b, dAOW and dAOS are differential areas that denote changes in the fluid and rock interactions during invasion. Similarly, the RHS in Eqs. A3a, A3b represent the net contributions from the fluid and rock neighbours of the interfacial cells in an invasion step. This completes the equivalence between the thermodynamics of invasion and our computational formalism.
Rights and permissions
About this article
Cite this article
Nair, N., Koelman, J.V. An Ising-Based Simulator for Capillary Action in Porous Media. Transp Porous Med 124, 413–437 (2018). https://doi.org/10.1007/s11242-018-1075-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-018-1075-5