Fluid flow in porous systems driven by capillary pressure is one of the most ubiquitous phenomena in nature and industry, including petroleum and hydraulic engineering as well as material and life sciences. The classical Lucas–Washburn (LW) equation and its modified forms were developed and have been applied extensively to elucidate the fundamental mechanisms underlying the basic statics and dynamics of the capillary-driven flow in porous systems. The LW equation assumes that fluids are incompressible Newton ones and that capillary channels all have the same radii. This kind of hypothesis is not true for many natural situations, however, where porous systems comprise complicated pore and capillary channel structures at microscales. The LW equation therefore often leads to inaccurate capillary imbibition predictions in such situations. Numerous studies have been conducted in recent years to develop and assess the modifications and extensions of the LW equation in various porous systems. Significant progresses in computational techniques have also been attained to further improve our understanding of imbibition dynamics. A state-of-the-art review is therefore needed to summarize the recent significant models and numerical simulation techniques as well as to discuss key ongoing research topics arising from various new engineering practices. The theoretical basis of the LW equation is first introduced in this review and recent progress in mathematical models is then summarized to demonstrate the modifications and extensions of this equation to various microchannels and porous media. These include capillary tubes with nonuniform and noncircular cross sections, discrete fractures, and capillary tubes that are not straight as well as heterogeneous porous media. Numerical studies on the LW equation are also reviewed, and comments on future works and research directions for LW-based capillary-driven flows in porous systems are listed.

X-ray microtomography (micro-CT) provides a nondestructive way for estimating rock properties such as relative permeability. Relative permeability is computed on the fluid distributions generated on three dimensional images of the pore structure of a rock. However, it is difficult to numerically reproduce actual fluid distributions at the pore scale, particularly for a mixed-wet rock. Recent advances in imaging technologies have made it possible to directly resolve a large field of view for arbitrary wetting conditions. Herein, the objective of this study is to evaluate relative permeability computations on imaged fluid distributions under water-wet and mixed-wet conditions. By simultaneously injecting oil and brine on a Bentheimer sandstone before and after wettability alteration, imaged fluid distributions are obtained under steady state conditions. Then relative permeability computations performed on imaged fluid distribution are compared with experimental data obtained on the same rock. We find that relative permeabilities computed directly from imaged fluid distributions show agreement with experimental data in water-wet rock while for mixed-wet rock, the imaged connected pathways provided a poor estimate of relative permeability. Analysis of imaged fluid distributions and connectivity demonstrates that under mixed-wet conditions, increased dynamic connectivity and ganglion dynamics result in non-equilibrium effects at the fluid-fluid interface. These effects result in more energy dissipation during fractional flow in mixed-wet systems and thus lower effective permeability than water-wet rock at the same saturation. Hussain et al. (2014) extended the work of Turner et al. (2004). They conducted steady-state tests on both a small scale (D 5 5mm, L 5 21mm) and a conventional scale sample (D 5 25mm, L 5 53mm) which are homogenous strongly water-wet cores. The image-based computations were compared with laboratory Key Points:Nonwetting phase is less connected under mixed-wet than water-wet conditions Dynamic connectivity is observed more frequently under mixed-wet conditions Energy balance demonstrates higher propensity for interface creation under mixed-wet conditions Supporting Information:Supporting Information S1 Data Set S1

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Summary Wettability is a major factor that influences multiphase flow in porous media. Numerous experimental studies have reported wettability effects on relative permeability. Laboratory determination for the impact of wettability on relative permeability continues to be a challenge because of difficulties with quantifying wettability alteration, correcting for capillary-end effect, and observing pore-scale flow regimes during core-scale experiments. Herein, we studied the impact of wettability alteration on relative permeability by integrating laboratory steady-state experiments with in-situ high-resolution imaging. We characterized wettability alteration at the core scale by conventional laboratory methods and used history matching for relative permeability determination to account for capillary-end effect. We found that because of wettability alteration from water-wet to mixed-wet conditions, oil relative permeability decreased while water relative permeability slightly increased. For the mixed-wet condition, the pore-scale data demonstrated that the interaction of viscous and capillary forces resulted in viscous-dominated flow, whereby nonwetting phase was able to flow through the smaller regions of the pore space. Overall, this study demonstrates how special-core-analysis (SCAL) techniques can be coupled with pore-scale imaging to provide further insights on pore-scale flow regimes during dynamic coreflooding experiments.