Editorial
The idea of using a network of pores to represent multiphase ¯ow in porous media was ®rst proposed by Fatt in the 1950s [1±3]. He used a 50 by 50 lattice of resistors to compute relative permeability and capillary pressure for a drainage-type displacement (this was a real lattice of resistors, since the numerical computation of the ¯ow was beyond the capability of computers at the time). In the 1980s, pore-scale modeling enjoyed a surge of interest as percolation concepts were used to describe multiphase ¯ow properties (see, for instance [4± 6]). In addition, micromodel experiments were used to observe pore-scale displacement mechanisms for both imbibition and drainage [7]. These displacement pro-cesses were then coded into a three-dimensional network model. With suitable adjustment of network parameters, some experimental data could be matched and the trends in behavior explained in terms of pore-scale phenomena (for one of the best papers on this, see [8]). However, the predictive properties of these models were limited to very simple systems ± namely random close packings of spheres [9,10], and most of the research concentrated on two-phase ¯ow in water-wet media. By the early 1990s, interest in pore-scale modeling had waned. The reasons for this include the move away from fundamental research sponsored by oil companies, and an intellectual recognition that something more than percolation-type models and simple pore-space ge-ometry would be necessary to develop genuinely predictive models and to explain the whole range of ¯ow and transport phenomena in porous media.
Recently, there has been another explosion of interest in pore-scale modeling, with a variety of both petroleum and environmental applications. This is for three rea-sons. First, pore-scale models are no longer limited to simple two-phase ¯ow processes and the computation of relative permeability. They are now used as a platform to explore a huge range of phenomena, including the eects of wettability, three-phase ¯ow, interfacial area, and dissolution ± this special issue has papers that deal with each of these issues. Second, it is now possible to represent the complex geometry of the pore space more adequately, enabling genuine predictions of transport properties to be made. Approaches to this problem in-clude the work of Adler, éren and co-workers [11,12], and two of the contributors to this issue. Third, the continued evolution of computational resources no
longer restricts the detailed simulation of ¯uid ¯ow in those complex pore geometries to only a small number of pores.
The papers in this special issue on pore-scale model-ing cover the full range of topics and dierent ap-proaches in the ®eld. Two papers deal with the description of realistic pore structures and their use in predicting capillary pressure, water relative permeability and dispersion. The ®rst by H.-J. Vogel and K. Roth, ``Quantitative morphology and network representation of soil pore structure'', uses serial sectioning and a new method for determining three-dimensional connectivity to construct a realistic representation of the pore space. M. Hilpert and C.T. Miller, in ``Pore-morphology-based simulation of drainage in totally wetting porous media'', use an erosion±dilation style technique to predict pri-mary drainage capillary pressure-saturation curves for random packings of spheres.
The use of percolation-like displacement models and percolation theory continues to provide insights into ¯ow in porous media. M.A. Knackstedt et al., in ``Pore network modelling of two-phase ¯ow in porous rock: the eect of correlated heterogeneity'', study the eects of power-law (fractal) correlations in the pore-size dis-tribution. The spatial structure of the pore space is a simple but powerful representation of geological systems that have features on all length scales. A.G. Hunt, in ``Applications of percolation theory to porous media with distributed local conductances'', provides a dis-cussion of critical path analysis to determine conduc-tance and other transport properties in highly heterogeneous systems. The paper combines a review of the theory with a large body of new work by the author. R.P. Ewing and B. Berkowitz, in ``Stochastic pore-scale growth models of DNAPL migration in porous media'', discuss the use of dierent types of growth models to represent DNAPL migration and show good agreement between their simulations and experimental results.
The other papers in this issue use pore-scale modeling to explore a variety of dierent phenomena in porous media. Of particular importance is the use of pore-scale physics to derive constitutive relationships at the macroscopic Darcy scale. R.J. Held and M.A. Celia in, ``Modeling support of functional relationships between capillary pressure, interfacial areas and common lines'', compute common lines (lines of contact between phases) Advances in Water Resources 24 (2001) 231±232
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and interfacial areas, which are key properties in recent macroscopic theories of multiphase ¯ow. Since these properties are very dicult to measure experimentally, pore-scale modeling provides an ideal test-bed for ex-ploring dierent functional relationships between them. A plausible range of displacement description par-ameters are found that virtually eliminates hysteresis between these quantities and capillary pressure for drainage and imbibition cycles in a strongly water-wet system. H.N. Man and X.D. Jing, in ``Network model-ing of strong and intermediate wettability on electrical resistivity and capillary pressure'', use a state-of-the-art pore-network model to study the eects of rock wettability on electrical properties in drainage and im-bibition. The eects of wettability are further explored by M.I.J. van Dijke et al. in ``Saturation-dependencies of three-phase relative permeabilities in mixed-wet and fractionally wet systems'' with extensions to three-phase (oil, water and gas) ¯ow. Here dierent generic func-tional forms for three-phase relative permeabilities are proposed. This work can be used as a guide to develop improved empirical models of relative permeability.
All the previous papers have assumed quasi-static displacement at the pore scale, controlled entirely by capillary forces. M.S. Valavanides and A.C. Payatakes challenge this traditional assumption in ``True-to-mechanism model of steady-state two-phase ¯ow in porous media, using decomposition into prototype ¯ows''. Macroscopic properties of the ¯ow are derived semi-analytically from pore-scale parameters. The authors suggest that capillary-controlled percolation-like displacement patterns represent only one of a number of possible types of ¯ow regime, and are only observed for a restricted range of ¯ow rates and satu-rations. The eects of ¯ow rate are accommodated in a perturbative fashion by R.G. Hughes and M.J. Blunt in ``Network modeling of multiphase ¯ow in fractures'' who show how to use a pore-and-throat numerical model to study fracture ¯ow.
We also have two papers that explore an exciting new area in pore-scale modeling, namely simulations of mass transfer between phases. A. Ahmadi et al., in ``Calcu-lation of the eective properties describing active dis-persion in porous media: from simple to complex unit cells'', compute dispersion coecients and dissolution rates of NAPL as a function of Peclet number. The re-sults are compared to experimental trends. A.G. Yiotsis et al., in ``A 2-D pore-network model of the drying of single-component liquids in porous media'', couple solutions for the concentration and pressure ®elds with local displacement to study drying processes. The eects of both capillary and viscous forces are included in the model.
The ®nal paper deals with a quasi-molecular ap-proach for modeling ¯uid ¯ow that is in particular suitable for complex pore geometries. R. Zhang et al., in
``Surface tension eects on two-dimensional two-phase Kelvin±Helmholtz instabilities'', demonstrate how dy-namic ¯uid±¯uid interfaces can be faithfully tracked by a lattice-Boltzmann simulation.
Acknowledgements
We would like to thank all the contributors and re-viewers who worked so hard to keep the issue on time. We hope that you enjoy reading these papers as much as we did.
References
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[2] Fatt I. The network model of porous media II. Dynamic properties of a single size tube network. Trans AIME 1956;207:160±3.
[3] Fatt I. The network model of porous media III. Dynamic properties of networks with tube radius distribution. Trans AIME 1956;207:164±81.
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[6] Heiba AA, Sahimi M, Scriven LE, Davis HJ. Percolation theory of two-phase relative permeability. SPE Reservoir Eng 1992;7:123±32.
[7] Lenormand R, Zarcone C, Sarr A. Mechanisms of the displace-ment of one ¯uid by another in a network of capillary ducts. J Fluid Mech 1983;135:337±53.
[8] Jerauld GR, Salter SJ. Eect of pore-structure on hysteresis in relative permeability and capillary pressure: pore-level modeling. Transp Porous Media 1990;5:103±30.
[9] Bryant S, King PR, Mellor DW. Network model evaluation of permeability and spatial correlation in a real random sphere packing. Transp Porous Media 1993;11:53±70.
[10] Bryant S, Blunt MJ. Prediction of relative permeability in simple porous media. Phys Rev A 1992;46:2004±11.
[11] Adler PM, Jacquin CJ, Quiblier JA. Flow in simulated porous media. Int J Multiphase Flow 1990;16:691±712.
[12] éren PE, Bakke S, Arntzen OJ. Extending predictive capabilities to network models. SPE J 1998;3:324±36.
Martin Blunt
Imperial College of Science, Technology and Medicine Royal School of Mines, T.H. Huxley School Prince Consort Road London SW7 2BP, UK E-mail address:m.blunt@ic.ac.uk
Markus Hilpert
Department of Environmental Science and Engineering School of Public Health University of North Carolnia at Chapel Hill CB 7400, 104, Rosenau Hall Chapel Hill, NC 27599-7400, USA E-mail address:markus_hilpert@unc.edu
