Dynamic Fluid Connectivity Controls Solute Dispersion in Multiphase Porous Media Flow
Abstract
Solute transport in multiphase flow through porous media plays a central role in many natural systems and geoengineering applications. The interplay between fluid flow and capillary forces leads to transient flow dynamics and phase distributions. However, it is not known how such dynamic flow affects the dispersion of transported species. Here, we use highly resolved numerical simulations of immiscible twophase flow to investigate dispersion in multiphase flows. We show that repeated activation and deactivation of different flow pathways under the effect of capillary forces accelerates the spreading of solutes compared to single phase flow. We establish the transport laws under dynamic multiphase flows by linking the dispersion coefficient to the Bond number, the ratio of the force driving the flow and the surface tension. Our results determine the controlling factors for solute dispersion in porous media, opening a range of applications for understanding and controlling transport in porous geological systems.
Key Points

Solute dispersion in multiphase flow is significantly amplified by dynamic fluid connectivity

We derive a scaling law for the solute dispersion in multiphase systems applicable to a wide range of subsurface geosystems

We propose a phase diagram for the dispersion coefficient in terms of Capillary and Péclet numbers
Plain Language Summary
When a single fluid flows through porous media such as soils or geological reservoirs, the transport of contaminants, nutrients, microorganisms, and chemicals is fairly well understood. When two or more fluids flow together, these transport phenomena have largely not been considered despite their importance in many natural systems. Forces between the flowing fluids and the solid boundaries may create large variations in the local flow rates and form timevarying flow pathways, which can in turn accelerate solute spreading. Here, we use extensive computer simulations of flow to suggest a new theory for how solute spread in systems of two fluids flowing through porous media which may help us understand and control transport properties in natural systems.
1 Introduction
The dispersion and mixing of elements transported by fluids in porous media are key processes controlling the fate and transformation of contaminants, nutrients, microorganisms, chemicals, and gases in numerous natural and engineered systems in the subsurface (Bochet et al., 2020; Chen et al., 2017; De Gennes, 1983; Hartmann et al., 2021; Harvey et al., 2002; Kirchner et al., 2000; Osselin et al., 2022; Rolle & Le Borgne, 2019; Szulczewski et al., 2012). Under fluidsaturated conditions, models, and theories that relate the dispersion properties to the porous media structure and heterogeneity are well established (Berkowitz et al., 2006; Bijeljic et al., 2011; Bordoloi et al., 2022; De Anna et al., 2013; Dentz et al., 2011; Kang et al., 2014; Le Borgne et al., 2008; Meigel et al., 2022; Neuman & Tartakovsky, 2009; Puyguiraud et al., 2021). However, in many porous systems, several fluid phases, including water, gas, and different immiscible fluids, coexist in the pore space (Blunt, 2017; Holtzman et al., 2020; Zhao et al., 2016). Dispersion in such multiphase flows affects several important processes, including contaminant and nutrient transport in soils and catchments (e.g., Kirchner et al., 2000; Sebilo et al., 2013), brine and nutrient transport in snow and sea ice (e.g., Golden, 2001; Waldner et al., 2004), CO_{2} sequestration in geological reservoirs (e.g., Szulczewski et al., 2012; Zheng et al., 2017), underground hydrogen storage (Lysyy et al., 2022; Tarkowski, 2019), and solute dispersion in gas seepage sites (e.g., Aiken et al., 2022; Scandella et al., 2016).
The increase of computational power and imaging capabilities allow to explicitly resolve flow and transport over an increasing range of scales (e.g., Yeh et al., 2023). However, for highly nonlinear processes, such as multiphase flow, current numerical methods can resolve the transient pore scale flow over tens to hundreds of grains at most (e.g., Ferrari et al., 2015). Therefore, it remains an important task to derive effective transport parameters quantifying unresolved subscale processes.
The control of multiphase flow properties on transport dynamics has been debated in the last two decades (Bromly & Hinz, 2004; Harter & Yeh, 1996; Hasan et al., 2020; JiménezMartínez et al., 2015, 2017; Nützmann et al., 2002; Padilla et al., 1999; Triadis et al., 2019; Vanderborght & Vereecken, 2007; VelásquezParra et al., 2022). Consistently with stochastic transport theories (e.g., Dentz et al., 2011), the general trend is that dispersion is larger in unsaturated flows compared to saturated flows due to an increase in flow heterogeneity. However, there is currently no quantitative model able to predict this phenomenon from the basic nondimensional parameters that describe multiphase flow dynamics. Recent work on dispersion in multiphase flows has focused on the particular case where the distribution of fluid phases is “frozen” in time (JiménezMartínez et al., 2015, 2017; VelásquezParra et al., 2022). The fixed distribution typically occurs after a transient regime when a nonwetting fluid remains trapped by capillary forces. This requires that the pressure difference at the interface of the two phases is everywhere below the capillary threshold.
In many multiphase flow systems this condition is not fulfilled, resulting in a highly intermittent flow field (Gao et al., 2020; Reynolds et al., 2017; Rücker et al., 2015). The intermittent flow is characterized by largescale avalanches in the mobility of the fluid phases (Dougherty & Carle, 1998; Planet et al., 2009; Primkulov et al., 2022; Tallakstad et al., 2009a; Zhao et al., 2016), which fundamentally modify the transport dynamics compared to singlephase and frozen multiphase flows. Yet, how the nature of multiphase flow influences solute dispersion remains unknown. Intermittent flows are wellestablished in coflow systems, where two phases flow simultaneously under a globally steadystate flow (Datta et al., 2014; Reynolds et al., 2017; Tallakstad et al., 2009a). Coflow results in dynamic connectivity (persistent fluctuations) of flow pathways under a large range of flow rates and saturations (Datta et al., 2014; Gao et al., 2021).
Here, we use numerical simulations in two and three dimensions under coflow conditions to establish how dynamic multiphase flow controls solute dispersion in a wide range of multiphase flow systems in the subsurface. We explore a large range of Capillary and Péclet numbers. The Capillary number characterizes the relative effect of viscous drag versus surface tension γ, Ca ≡ μ_{w}〈v〉/γ where μ_{w} is the dynamic viscosity of the wetting fluid and 〈v〉 the average flow speed. The Péclet number characterizes the relative effect of advection and diffusion, Pe ≡ ℓ〈v〉/D_{m}, where ℓ is a characteristic length scale (given by the pore scale) and D_{m} is the molecular diffusion constant.
While solute dispersion along the primary flow direction (longitudinal dispersion) is controlled mainly by the solid structure, transverse dispersion becomes strongly enhanced relative to single phase flow when the Péclet number is larger than unity. This is a consequence of dynamic fluid connectivity, where local pathways are constantly reorganized by the movement of the boundaries between the fluid phases (Figure 1). We demonstrate that the relevant scale driving this enhanced dispersion follows from a nonlinear function of the ratio between the surface tension and the body force driving the flow. We thus establish the laws of dispersion in dynamic multiphase porous media flows, as a function of both Capillary and Péclet numbers.
2 Methods
Based on the freeenergy Lattice Boltzmann Method (Connington & Lee, 2013), we perform simulations in two systems, a twodimensional (2D) cell of size L_{⊥} × L_{‖} = 2,048 × 768 filled with nonoverlapping cylindrical obstacles, radius r = 8 and a threedimensional (3D) cell of size L_{⊥,1} × L_{‖} × L_{⊥,2} = 256 × 384 × 256 filled with overlapping spheres, radius r = 8. All lengths are here given in lattice units. The porosity is measured as the fraction of area/volume not covered by cylinders/spheres and is 53% in 2D and 21% in 3D. Note that the porosity of the 2D cell is large compared to natural porous media, which is needed in 2D for good connectivity (De Anna et al., 2013). See Text S1 and Figure S1 in Supporting Information S1 for a description of how we generate the porous systems and for a description of how our dimensionless simulation parameters can be converted to real parameters. The SI also contains information about our numerical code available at Ref. (Mathiesen et al., 2023). Movies S1 and S2 present movies of two and threedimensional simulations.
The two fluid phases have a kinematic viscosity ratio ν_{w}/ν_{nw} = 0.57 and a density ratio ρ_{w}/ρ_{nw} = 1.25 equivalent to a system filled with water and silicone oil. The density difference is not significantly influencing the flow and, in the considered regime, both phases are mobilized (Text S1 and Figure S3 in Supporting Information S1, we present results from a system with two fluids of equal density and viscosity). The wetting angle is imposed to be 60° and the nondimensionalized surface tension γ is varied in the interval [0.005, 0.040]. In addition to the multiphase simulations, we have computed the corresponding singlephase flow field satisfying the Stokes equations. All simulations are initiated in a state where the two phases are arranged in alternating stripes of width 2r transverse to the main flow direction. Following (Ramstad et al., 2012), we drive the flow by a body force density ρ_{w∕nw}g pointing along L_{‖}. Our system has periodic boundary conditions in all directions, that is, when a fluid phase exits on one side, it reenters on the opposite side of the cell. In this way we avoid capillary endeffects.
3 Results
In order to investigate the impact of dynamic multiphase flow on solute dispersion, we shall here consider simulations of coflow of two immiscible fluids where the flow field is stationary. Based on our results of solute dispersion in a broad range of flow regimes, we derive a general transport law, which also offer fundamental insights into more general and transient multiphase flow systems.
In our simulations, we control the driving force, which we express by the dimensionless Bond number, the ratio between the gravitational (body) force g ≡ g and the surface tension at the interface of the two fluid phases, $$\mathrm{B}\mathrm{o}\equiv {\ell }^{2}\bar{\rho }g/\gamma $$, where $$\bar{\rho }=\left({\rho }_{w}+{\rho }_{nw}\right)/2$$ is the mean of the two mass densities. In our twodimensional system, the dimensionless characteristic length scale, ℓ = 8.2, is the average pore size between neighboring obstacles and is calculated by Voronoi tessellation. In three dimensions, we use the same scale as an approximation. We describe the flow rate by the Capillary number Ca ≡ μ_{w}〈v〉/γ using the dynamic viscosity of the wetting fluid μ_{w}. For simplicity, and without lack of generality, we take the average of the flow speed over both phases. Alternatively we could define a Capillary number specific to each phase (see Figure S3 in Supporting Information S1).
To test the generality of our findings, we have followed two simulation protocols of varying the Bond number: one where we vary the body force and hold the surface tension fixed, Bo(g), and one where the surface tension is varied and the body force is held constant, Bo(γ). Overall, our simulations are performed for Bond numbers in the range 0.1–30 and with resulting Capillary numbers in the range 10^{−4}–10. This corresponds to a dynamic fluid connectivity regime where both phases are mobile, as observed in several experiments in bead packs and rocks in the range Ca ≥ 10^{−7} (Gao et al., 2021; Reynolds et al., 2017; Tallakstad et al., 2009a). In Figure S3 in Supporting Information S1, we confirm the persistence of dynamic fluid connectivity down to Ca ≈ 10^{−4} from an additional set of simulations in a larger system. Whereas we focus on the twodimensional system, we present results from a threedimensional system to confirm the generality of our findings.
For large Bond numbers (Bo > 1), the average flow rate is proportional to the applied gravitational force, following the Darcy regime. For small Bond numbers (Bo < 1), the flow is significantly influenced by surface tension forces and crosses over to a nonDarcian behavior where 〈v〉 ∝ Bo^{β} with β = 1.31, both in 2D and 3D (Figure 2a) and consistent with experimental observations (Chevalier et al., 2015; Tallakstad et al., 2009a; Zhang et al., 2021). For large Bond numbers, the velocity probability density function is close to that of the singlephase and follows closely an exponential distribution (Figure 2d), driven by the random redistributions at pore junctions (Alim et al., 2017). For small Bond numbers, we observe a broadening of the velocity distributions, which in the range of intermediate velocities become more powerlawlike. This leads to an increase of the flow velocity variance relative to the mean velocity, following a similar behavior in two and three dimensions (Figure 2b). All the distributions are characterized by a marked exponential cutoff at large values and a plateau at low values (Figures 2c and 2d).
In order to quantify the dispersion in the fluid flow, we consider tracers over a large range of Péclet numbers Pe = ℓ〈v〉/D_{m} by varying D_{m} in Equation 1. In general, the dispersion resulting from the fluid flow becomes orders of magnitude larger than the pure molecular diffusion in the fluid. In the infinite Péclet regime Pe^{∞}, the particles follow the flow and are subject to fluctuations through spatiotemporal heterogeneity of the flow velocity field. In the simulations, the flow evolves through a transient stage and then reaches a stationary regime (when the mean and the second moment of the flow rate reach a plateau) where we start to collect the statistics of dispersion. In Figure 3a, we show the Pe = 10^{6} (diffusion effectively zero) particle trajectories for a singlephase and two multiphase simulations in two dimensions. As the Bond number is lowered, the velocity fluctuations grow relative to the mean flow and lead to particle paths that look more jagged and wander further transversely, producing an enhanced dispersion compared to single phase flow. In the following, we compute the particle dispersion from the spatial variance of the position of passive tracers.
In Figure 3b, we plot the diagonal components of the position variance tensor of the individual tracer positions x^{k} as function of time for Pe = 10^{6} (asymptotically close to Pe∞ for the time scales considered here) and for different Bond numbers, σ^{2}(Δτ) = $${\langle \left[{\mathbf{x}}^{k}({\Delta }\tau +t){\mathbf{x}}^{k}(t)\right]{\left[{\mathbf{x}}^{k}({\Delta }\tau +t){\mathbf{x}}^{k}(t)\right]}^{T}\rangle }_{k,t}$$. The average is performed over the ensemble of particles and over time. We denote the diagonal components of the position variance tensor by $${\sigma }_{\perp }^{2}$$ and $${\sigma }_{\Vert }^{2}$$ in the transverse and longitudinal directions, respectively (in threedimensions we use $${\sigma }_{\perp }^{2}={\sigma }_{\perp ,1}^{2}+{\sigma }_{\perp ,2}^{2}$$). The longitudinal dispersion behaves similarly to that of a singlephase flow, which is apparent from Figure 3b where the longitudinal variances for multiphase flow for different Bond numbers all collapse on the singlephase flow when normalizing time by the mean velocity. The spatial variance shows a transition from the ballistic regime $${\sigma }_{\Vert }^{2}(t)\sim {t}^{2}$$, that develops below the characteristic advection time τ_{a} = ℓ/〈v〉, to a slightly superlinear growth. The broadly distributed velocity distributions together with the almost uniform velocity distribution at low velocities (Figures 2c and 2d), result in a nonFickian longitudinal dispersion (Dentz et al., 2016) given by $${\sigma }_{\Vert }^{2}(t)\stackrel{t > {\tau }_{a}}{\approx }\left(t/{\tau }_{a}\right)\,\mathrm{ln}\left(t/{\tau }_{a}\right)$$. The asymptotic scaling is verified in Figure 3b, for times larger than the characteristic advection time τ_{a}. As discussed in the following, the addition of diffusion induces a transition to Fickian dispersion in the longitudinal direction after a time corresponding to the time needed to diffuse a distance of the characteristic velocity correlation length (pore size for singlephase flow and fluid cluster size for multiphase flow). We note that in multiphase flow systems composed of water and air, with much larger viscosity and density ratios, the emergence of trapped clusters and dead end zones can lead to broader velocity distributions and strongly anomalous transport (VelásquezParra et al., 2022).
In addition to the large Pe regime, we have investigated the effect of diffusion on dispersion in multiphase flows over a large range of Péclet numbers, 10^{−3} ≤ Pe ≤ 10^{3} (Figure 3d). For any finite Péclet number, longitudinal dispersion becomes asymptotically Fickian as diffusion introduces a cutoff time in the travel time distribution (Bijeljic et al., 2011; Puyguiraud et al., 2021). The longitudinal dispersion coefficient follows the classical transition from D_{‖} = D_{m} for Pe < 1 to D_{‖} ∼ Pe for Pe > 1 (Dentz et al., 2011). In single phase flow, transverse dispersion is much smaller than longitudinal dispersion for moderate to high Péclet numbers, as usually assumed in transport models (Dentz et al., 2011). In contrast, dynamic fluid connectivity in multiphase flows significantly increases transverse dispersion, which can reach a similar amplitude as longitudinal dispersion for low Capillary numbers. At Bo = 0.73, the enhancement of transverse dispersion compared to the single phase case in twodimensions is thus a factor 8 for Pe = 10 and a factor 50 for Pe = 100 (Figure 3d).
Whereas dispersion is classically described to be a function of the Péclet number (Bijeljic et al., 2011; Dagan, 1984; Dentz et al., 2011; Gelhar & Axness, 1983), our results show that it can vary largely for any given Péclet number depending on the multiphase flow properties (Figure 3d). In single phase systems, the dispersion versus Péclet function is uniquely related to the pore structure (Puyguiraud et al., 2021). Therefore, we emphasize that dispersion must be considered in the combined Péclet number—Capillary number space, consistent with our theoretical prediction (Equation 5). For Pe < 1, solute dispersion is diffusion dominated and we observe little to no difference between single and multiphase flows (Figure 5a). For lower Capillary numbers, the dynamic fluid connectivity controls dispersion, which is greatly amplified relative to single phase flow. Whereas, we only observe an increase in dispersion as we lower the Capillary numbers, there might be a lower critical value (likely system size dependent) where capillary forces dominate the viscous forces to an extent that the distribution of fluid phases is frozen (Gao et al., 2021). In this regime and depending on the boundary conditions, the two phases can flow in parallel (Datta et al., 2014) or one phase flows around a trapped phase (JiménezMartínez et al., 2015). For different porous media structures, the transition has been found to vary between 10^{−7} and 10^{−4} (Gao et al., 2021). When viscous forces dominate over capillary forces, linear Darcy flow is recovered, which is expected to happen for Capillary numbers in the range 10^{−2}–10^{−1} (Yiotis et al., 2013), consistent with our results (Figure 2a).
While we have studied here dynamic coflow systems under global steadystate conditions, the results are relevant for a broad range of transient multiphase flow systems where dynamic flow connectivity occurs as the saturation and flow rate evolve in space and time. Transient multiphase flows play an important role in solute transport in unsaturated soils (Vereecken et al., 2022), gas and brine flow in porous rock (Reynolds et al., 2017), CO_{2} and brine flow in subsurface reservoirs (Zheng et al., 2017), water and hydrogen flow in porous media (Lysyy et al., 2022), and water and air in snow and ice (Waldner et al., 2004). A typical range of dimensionless numbers for multiphase flow in porous media systems is Ca = {10^{−7}–10^{−3}} and Pe = {1–10^{2}}, which overlaps with the enhanced dispersion regime described here (Figure 5). This phenomenon hence likely affects the transport, mixing, and reaction of dissolved elements, such as contaminants, nutrients, CO_{2}, hydrogen, salts, and ions, in a broad range of natural and geoengineered systems.
4 Conclusion
Transient dynamics and flow intermittency is a key property of multiphase flow in porous media, occurring over a broad range of Capillary numbers above the threshold for mobilization of the fluid phases (Armstrong et al., 2014; Gao et al., 2021). We have shown that the repeated activation and deactivation of flow paths, dynamic connectivity, induces a large variability of the flow field, with a strong enhancement of transverse solute dispersion. While previous works have associated the enhancement of dispersion in unsaturated flows to a greater variability of velocity, we have shown here that the main driver for this phenomenon is the change of the size of water/air clusters with saturation. We have derived a fundamental scaling law for dispersion in dynamic multiphase flow in porous media emphasizing that dispersion must be described in combined Capillary number—Péclet number space. To establish these laws, we have here considered multiphase flow under global steady state conditions. However, our findings will be relevant for understanding the role of dynamic connectivity on solute transport in a broad range of transient systems, hence impacting a large range of phenomena occurring in natural and engineered porous media systems.
Future work should explore solute transport dynamics over a range of wetting contact angles, viscosities, and densities and explore the impact of transient flows. Our findings can be used to tune solute dispersion in porous media by altering the ratio between the surface forces and the force driving the flow. Strikingly, dispersion can be controlled by changing the surface tension between fluids without modifying the mean velocity. Our results also suggest that transverse dispersion could be used as a macroscopic metric for characterizing the dynamics of twophase flows. This opens new perspectives for characterizing, modeling and controlling multiphase flow and transport processes in natural systems.
Acknowledgments
This study received funding from the Akademiaavtaalen between the University of Oslo and Equinor (Project MODIFLOW to F. Renard). Joachim Mathiesen acknowledges support from the ESS lighthouse on hard materials in 3D, SOLID, funded by the Danish Agency for Science and Higher Education, Grant 814400002B.
Open Research
Data Availability Statement
The numerical code used to generate all the data presented in this manuscript is available at https://doi.org/10.5281/zenodo.8212058.