Pore-scale dynamics and the multiphase Darcy law

Synchrotron x-ray microtomography combined with sensitive pressure differential measurements were used to study flow during steady-state injection of equal volume fractions of two immiscible fluids of similar viscosity through a 57-mm-long porous sandstone sample for a wide range of flow rates. We found three flow regimes. (1) At low capillary numbers, Ca, representing the balance of viscous to capillary forces, the pressure gradient, $\mathbf{\nabla }P$, across the sample was stable and proportional to the flow rate (total Darcy flux) $q_t$ (and hence capillary number), confirming the traditional conceptual picture of fixed multiphase flow pathways in porous media. (2) Beyond ${\text{Ca}}^{*}\approx {10}^{-6}$, pressure fluctuations were observed, while retaining a linear dependence between flow rate and pressure gradient for the same fractional flow. (3) Above a critical value $\text{Ca}>{\text{Ca}}^i\approx {10}^{-5}$ we observed a power-law dependence with $\mathbf{\nabla }P\sim q_t^a$ with $a\approx 0.6$ associated with rapid fluctuations of the pressure differential of a magnitude equal to the capillary pressure. At the pore scale a transient or intermittent occupancy of portions of the pore space was captured, where locally flow paths were opened to increase the conductivity of the phases. We quantify the amount of this intermittent flow and identify the onset of rapid pore-space rearrangements as the point when the Darcy law becomes nonlinear. We suggest an empirical form of the multiphase Darcy law applicable for all flow rates, consistent with the experimental results.

Figure 1

Experimentally measured pressure gradient, $\mathbf{\nabla }P$, as a function of capillary number, $\text{Ca}=\mu q_t/\sigma$ plotted on base-10 logarithmic axes. (1) For $\text{Ca}\le {\text{Ca}}^{*}\approx {10}^{-6},\mathbf{\nabla }P\sim \text{Ca}$ with fixed flow paths of the phases, consistent with the traditional multiphase Darcy law, Eq. (1). (2) For ${\text{Ca}}^i\ge \text{Ca}>{\text{Ca}}^{*}$ we observe the onset of dynamics, defined as when there is some intermittency but the Darcy law remains valid. (3) For ${\text{Ca}}^v>\text{Ca}>{\text{Ca}}^i\approx {10}^{-5}$ intermittent flow occurs where $\mathbf{\nabla }P\sim {\text{Ca}}^a$, Eq. (6), with an empirical exponent $a=0.60\pm 0.01$. The dotted lines are fits to the data assuming Eq. (1) for the first five points and Eq. (6) for the three points with the highest flow rate. The error bars reflect the standard deviation in the pressure measurements, evident in Fig. 2, and the measurement accuracy.

Figure 2

The pressure differential as a function of time; the capillary numbers are also indicated. At the lowest flow rates, in the capillary-dominated regime (1), the trace is constant, indicating a fluid configuration that does not change over time, or when the flow rate is altered. At ${\text{Ca}}^{*}\approx {10}^{-6}$ we see the onset of pressure fluctuations and pore-scale dynamics, regime (2). A significant impact on flow conductance is seen for $\text{Ca}>{\text{Ca}}^i\approx {10}^{-5}$, regime (3). These fluctuations are of a magnitude equal approximately to a characteristic capillary pressure, indicative of local pore-scale displacement, shown in the inset.

Figure 3

Segmented images of selected regions of the pore space. (a) A two-dimensional cross-section of a three-dimensional image in the capillary-dominated regime, $\text{Ca}=2.1\times {10}^{-7}$, with oil shown in red, solid in gray and brine in black. (b) An example three-dimensional configuration where only oil is shown in red, also for $\text{Ca}=2.1\times {10}^{-7}$ in the capillary controlled regime 1. The oil is constrained to occupy only the larger portions of the pore space and has insufficient energy to connect the two regions shown by the ellipse. (c) The same portion of the sample is shown at a higher flow rate, with intermittent flow, regime (3), where $\text{Ca}=4.2\times {10}^{-5}$. Here the oil has sufficient energy to make the short-cut to open up a more conductive flow path. The colors indicate different isolated clusters of oil. (d) This pathway is not fixed, however, and an image, taken 10 min later, at the same flow rate, $\text{Ca}=4.2\times {10}^{-5}$, shows that the connection has been temporarily broken. The numbers in brackets refer to the flow regime, while the times shown refer to when the scan is taken; see Fig. 2 for the corresponding pressure signal.

Figure 4

(a) The fraction of void-space voxels for which the fluid occupancy changes from one scan to the next (type 1, left axis) and during a scan (type 2, right axis) as a function of capillary number Ca. The scans were taken every 60 s. The values are averaged over the final 30 images at each Ca. The error bars indicate the standard deviation. Regime 1 has no rapid oscillations (type 2) and very few type 1 changes. Regime 2 again has no type 2 intermittency, but more sustained low-frequency rearrangements, while in regime 3 we see significant fluctuations of both types, leading to a change in flow conductance. (b) The fraction of void-space voxels for which the fluid occupancy changes from one scan to the next (type 1) as a function of time during the final half hour of the scans. In regime 1 there are periods with no changes in fluid configuration, while in regime 2 there are always some low-frequency periodic rearrangements.