Inertial forces affect fluid front displacement dynamics in a pore-throat network model

The seemingly regular and continuous motion of fluid displacement fronts in porous media at the macroscopic scale is propelled by numerous (largely invisible) pore-scale abrupt interfacial jumps and pressure bursts. Fluid fronts in porous media are characterized by sharp phase discontinuities and by rapid pore-scale dynamics that underlie their motion; both attributes challenge standard continuum theories of these flow processes. Moreover, details of pore-scale dynamics affect front morphology and subsequent phase entrapment behind a front and thereby shape key macroscopic transport properties of the unsaturated zone. The study presents a pore-throat network model that focuses on quantifying interfacial dynamics and interactions along fluid displacement fronts. The porous medium is represented by a lattice of connected pore throats capable of detaining menisci and giving rise to fluid-fluid interfacial jumps (the study focuses on flow rate controlled drainage). For each meniscus along the displacement front we formulate a local inertial, capillary, viscous, and hydrostatic force balance that is then solved simultaneously for the entire front. The model enables systematic evaluation of the role of inertia and boundary conditions. Results show that while displacement patterns are affected by inertial forces mainly by invasion of throats with higher capillary resistance, phase entrapment (residual saturation) is largely unaffected by inertia, limiting inertial effects on hydrological properties behind a front. Interfacial jump velocities are often an order of magnitude larger than mean front velocity, are strongly dependent on geometrical throat dimensions, and become less predictable (more scattered) when inertia is considered. Model simulations of the distributions of capillary pressure fluctuations and waiting times between invasion events follow an exponential distribution and are in good agreement with experimental results. The modeling approach provides insights into the rich pore-scale dynamics of displacement fronts; these insights not only improve the basic understanding of these ubiquitous processes, but could shed light on solute dispersion and colloids mobilization at fronts and the mechanical consequences of passing fronts.

Figure 1

Details of pore-throat connection at a node. The pore is volumeless. The red points have the same coordinates. The length of the constricted capillary is $l$ and the pore and throat radii are denoted by $r_p$ and $r_t$, respectively. The virtual node is used for the formulation of force balance and volume conservation.

Figure 2

Schematic representation of the pore-throat network model with constricted capillaries, nodes, and menisci depicted at the position of the displacement front. The wetting phase is marked in blue (gray) and the nonwetting phase in white. The faded area below the displacement front is not considered in the model description. The gray tubes represent idealized connections between the nodes along the interfacial front. Nodes belonging to the front and considered by the governing equations are highlighted in red. The displacement motion is from top to bottom (the network is considered vertically oriented). The length of a capillary $l$, the pore radius $r_p$, and the throat radius $r_t$ are depicted in the schematic. The local coordinate $h$ shows the position of a meniscus relative to the node a throat is connected to and the macroscopic coordinate $z$ represents the elevation relative to the bottom of the network. The macroscopic volumetric withdrawal rate $Q$ is distributed equally among all the nodes along the displacement front.

Figure 3

Four interfacial configurations at a node along the fluid displacement front (dashed circle) considered in the model. The wetting phase is shown in blue (gray) and the nonwetting phase in white. The four cases include (a) a single meniscus, (b) two adjacent menisci, (c) three menisci at a node, and (d) two menisci at opposite orientation across the node.

Figure 4

Liquid phase reorganization after a meniscus reached the bottom of its capillary. (a) The meniscus is just before the breakthrough and will execute the jump with velocity $v_j$ until its position is $h=0$. Volume conservation was applied around all nodes along the front [red (gray)]. (b) Two new menisci evolved and are assigned the velocity $v_j/2$, while the wetting phase in one capillary become disconnected (light gray) and thus contributes to residual saturation. The front then moves from the old node to a new one [red (gray)].

Figure 5

Relative elevation of menisci versus time and distance. The radii of the straight capillaries and the connecting tubes were 1 mm. The initial height was 1 mm and 2 mm for the middle meniscus for the hydrostatic offset. The length of the connecting tubes was 5 mm. The relative elevation relates to the new equilibrium position of the menisci. The dimensionless time is expressed as $(r^2\rho /4\eta )t$.

Figure 6

Range of influence of an interfacial jump (at the origin distance equal to 0 m) marked by changes in menisci elevations with distance for a series of capillaries spaced at 2, 5, and 10 mm. The initial elevation of the menisci was 2 mm for the jumping meniscus and 1 mm for all other menisci in the connected capillaries. The inset shows the relative menisci elevations (scaled by the new equilibrium height) as a function of the relative distance from the origin of the perturbation (jumping meniscus).

Figure 7

Typical (simulated) pressure signal with pressure fluctuation magnitude $\Delta p$ and waiting time $t_w$ for slow drainage processes with constant withdrawal rate (Ca = 5 × ${10}^{-5}$).

Figure 8

Simulated displacement patterns of drainage (constant withdrawal rate) and residual saturation after passage of a drainage front through the same pore-throat network (a) with inertia and (b) without inertia considered in the governing force balance equations. The mean throat radius was 0.7 mm and the standard deviation was 0.2 mm for the log-normal throat-size distribution; the mean front velocity was 1.1 × ${10}^{-3}$ m/s (Ca = 4.8 × ${10}^{-4}$). The relative saturations are shown schematically for every throat from dark blue (full saturation) to white (empty) in the color scheme. The order of throats invaded during the displacement process is marked with time. The entrapped (irreducible) saturations were 0.36 considering inertia and 0.34 without inertia. The blurred regions at the top and bottom were not included in the saturation analysis. The insets show details of the entrapped phase compatible with the pore-throat representation of Fig. 2.

Figure 9

Throat-size distributions. (a) Throat-size distribution of throats invaded during drainage with constant withdrawal rate considering inertia (red dotted line) and without inertia (blue dashed line). The shift to smaller throat sizes invaded with the presence of inertia is highlighted with the hatched area. (b) Throat-size distribution for throats containing the residual (entrapped) liquid phase after the passage of the drainage front with (red dotted line) and without (blue dashed line) inertia. For comparison, the geometrically determined throat-size log-normal distribution with a mean size of 0.7 mm and standard deviation of 0.2 mm is shown by the black solid line in both figures.

Figure 10

Meniscus jump velocities as a function of throat radii simulated during drainage with a constant withdrawal rate through a network with a mean throat radius of 0.4 mm and standard deviation of 0.2 mm (Ca = 4.8 × ${10}^{-4}$). The blue squares show simulations without inertia and the red circles represent the more scattered jump velocity values when inertia was considered (for the same network).

Figure 11

Maximum interfacial jump velocities during drainage through three different pore-throat networks [with a mean throat radius of 0.7 mm and standard deviation (SD) of 0.2 mm (green cricles), mean throat radius 0.4 mm and standard deviation 0.2 mm (blue squares), and mean throat radius 0.4 mm and standard deviation 0.4 mm (red triangles)]. Solid and dashed lines represent the range of expected interfacial jump velocities without inertia for pore-throat networks with mean throat sizes of 0.7 and 0.4 mm, respectively. All drainage simulations were performed with Ca ≈ 5 × ${10}^{-4}$.

Figure 12

Pore-scale jump velocity vs mean front velocity for drainage simulations (without inertia) through the pore-throat network with mean throat radii of 0.7 and 0.4 mm (summarized in the legends) and for experimentally measured velocities during drainage through a regular array of 4-mm glass beads shown as black circles [33] and through irregular monolayer with 2-mm beads (a few data points only) [40] indicated with black asterisks. The accompanied Reynolds number indicates the transition to the turbulent flow regime during rapid interfacial jumps applying flow in a tube with geometrical dimension equal to the pore dimensions $2r_p$ of the network.

Figure 13

Pressure signal associated with a drainage simulation through the network with a mean throat size of 0.4 mm and standard deviation of 0.4 mm with the inertial force considered and omitted. The pressure signal associated with the simulated process with an explicit inertial term indicates simultaneous invasion events with a higher volume of liquid rejection at $t$ = 6 s, whereas it was an individual event when inertia was omitted.

Figure 14

Exponential distribution of pressure fluctuations associated with simulated drainage processes. Green squares and blue circles relate to pore-throat networks with a mean radius of 0.4 mm and standard deviation of 0.2 and 0.4 mm, respectively. Exponents $\beta$ were fitted to −1.4 and −1.2. Closed symbols refer to Ca = 4.9 × ${10}^{-4}$ and 5.1 × ${10}^{-4}$ and individual jumps and open symbols depict faster drainage processes with Ca = 4.9 × ${10}^{-3}$ and 5.1 × ${10}^{-3}$ and simultaneous jumps. The bending at occurs at $\Delta p/\overline{\Delta p}$ ≈ 1.5. The dashed line depicts the exponential distribution found experimentally by [38] with $\beta$ = −1.31.

Figure 15

Exponential distribution of waiting times associated with simulated drainage processes with Ca ≈ 5 × ${10}^{-4}$. Symbols relate to the pore-throat networks. The exponents b were fitted to −1.52 (solid line), −1.39, and −1.27. The dashed line depicts the exponential distribution found in experiments by Måløy et al. [38] with $\beta$ = −1.27.

Figure 16

Four possible configurations at a nonwetting phase node. Blue indicates wetting volume and red (gray) arrow points the meniscus exceeding the dimensions of the capillary. The redistributed liquid volume is shown in red (medium gray) (qualitatively). Light gray shows the wetting volume that was disconnected before the redistribution step.

Figure 17

Relative displacement of meniscus in capillary I versus length of the connecting horizontal tube with the length $\tilde{l}$ (mimicking length along a front) for the numerical calculations (red symbols) and analytical approximation (blue solid line). Here ${\tilde{l}}_{inf}$ marks the predicted length where the jump experiences critical damping. For the calculations shown we used capillaries with a radius of 1 mm and the initial meniscus heights were $h_{\mathrm{I}}(t=0)$ = 2 mm and $h_{\mathrm{II}}(t=0)$ = 1 mm.