Impact of injection rate on transient oil recovery under mixed-wet conditions : a microfluidic study

Lab-on-a-chip methods were used to visualize the pore-scale distribution of oil within a mixedwet, quasi-monolayer of marble grains packed in a microfluidic channel as the oil was displaced by water. Water injection rates corresponding to microscopic capillary numbers between Ca = 5 × 10-8 and 2 × 10-4 (Darcy velocities between 0.3 and 1100 ft/d) were considered. As expected, early-time water invasion transitions from stable displacement to capillary fingering with decreasing Ca, with capillary fingering observed at Ca ≤ 10-5. End-point oil saturation decreases with Ca over the entire range of Ca considered, consistent with the canonical capillary desaturation curve. In contrast, Sor derived from approximate numerical simulations using reasonable Pc(Sw) do not display a strong dependence on Ca. These results suggest that the Ca dependence of end-point oil saturation is largely due to capillary end effects which, under conditions considered presently, affect the entire length of the packed bed.


Introduction
Invasion of a non-wetting phase into a porous medium saturated with a wetting phase has been studied extensively in numerical simulation (e.g., [1]) and experiments using idealized 2D or quasi-2D microfluidic devices [e.g., 2,3,4,5,6].Most of these studies have considered conditions characteristic of primary drainage that is, the invasion of the non-wetting phase occurs in a porous medium fully saturated with the wetting-phase, and pore-scale displacement mechanisms under these conditions have been stablished [1,2,7,8].In contrast, less attention has been given to secondary drainage relevant to oil recovery from oil-wet reservoirs and groundwater remediation, where the medium is partially saturated with the invading fluid.
In this paper we present laboratory observations of oil displacement from mixed-wet beds of crushed marble by water injection.We focus on injection velocities corresponding to microscopic capillary numbers between Ca = 5 × 10 -8 and 2 × 10 -4 .Contrary to idealized geometrical micromodels, the packed beds used in this study preserve grain surface roughness, intra-grain mineral heterogeneity and a relatively large grain size distribution.Three invasion regimes are identified based on footprints observed at early-time and postbreakthrough displacements.
Front propagation dynamics, "average" oil saturation, and their dependence on flow regime are discussed.The significance of capillary end effects is assessed by comparing observed end-point saturations to residual oil saturation estimated from numerical simulations.
The porosity and permeability of the packed beds were ϕ = 0.27 [9] and k = 690 mD [10], respectively.The length of the packed beds ranged from L = 1540 to 2250 μm (Table 2).Fresh grains were used for each experiment to avoid any ambiguity in wettability.Details of the grain preparation and packing method are reported in Ref. [11] and are thus omitted here.
The displacement sequence consisted of two steps.First, oil was injected into a brine-saturated packed bed to establish initial oil saturation Soi = 0.97 ± 0.03 [10], and left to age between ta = 50 to 81 h.Second, brine was injected using a high-precision microfluidic syringe (pump 11 elite nanomite, Harvard Apparatus) at a constant volumetric flow rate Q w .The combination of the high Soi and large contact angle resulted in a predominantly oil-wet packed bed where waterflood was a secondary drainage process rather than imbibition.
In the present paper, we consider injection rates between Qw = 3.7 × 10 -3 and 15 μL/min, which correspond to Darcy velocities of Uw = Qw/A = 0.95 to 3900 μm/s (Table 2).The corresponding capillary numbers fall between Ca = 5.5 × 10 -8 and 2.2 × 10 -4 .With a viscosity ratio of μ w /μ o = 1.2, these values of Ca extend from the capillary fingering-dominated regime to the transitional regime between stable displacement and capillary fingering as determined for primary drainage in the network of capillaries considered by Lenormand et al. [1] and microfluidic cylinder arrays by Zhang et al. [2] (Fig. 2).

Depth-averaged oil saturation and front progression
The packed bed was back-lit using a variable wavelength light source (Lumen 1600-LED, Prior Scientific), and the depth-integrated fluid distribution was imaged using a high-speed 24-bit colour camera (Pixelink PL-B742F) coupled to an optical microscope (Nikon SMZ745T) in a sequence of RGB images.The exposure time ranged from ∆t = 30 to 100 ms depending on the injection rate (Table 2); the image resolution was 3.0 μm/pix.
Depth-averaged saturation was extracted from the ratio of the blue channel intensity to the red channel intensity, iB(x,y,t)/iR(x,y,t), in each pixel following the protocol proposed by Christensen et al. [16].In brief: 1.The ratio iB/iR was extracted from each image.2. A spatial 3 × 3 median filter was applied.3. iB/iR was further normalized by iB/iR on either side of the microfluidic channel to correct for instantaneous fluctuations in incident light.The mean oil saturation is then given by where angular brackets denote a spatial average over x ± H and L (Fig. 1).All post-processing was performed in MATLAB (Mathworks, Ltd.).

Numerical simulation
Approximate numerical simulations of the experiments were performed using core analysis software CYDAR TM (CYDAREX).Imbibition capillary pressure, P c C (Sw * ), 1 Pressure measurements are not available for these experiments.Because only production volumes were used in the history matching, the extracted best-fit measured on the same rock (Indiana limestone) by centrifuge [20], was used in the simulations after rescaling according to the Leverett-J function to account for differences in ϕ and k: where superscript C denotes properties of the core used in the centrifuge experiment and is water saturation normalized to account for differences in Soi.Irreducible water saturation was taken to be Swi = 0 in all simulations to reduce the number of fitting parameters.S or and relative permeabilities, modelled using Brooks-Corey correlations [21]: where Sor, αw (≥1), αo (≥1), krw(Sor), and kro(Soi) were fitting parameters, were extracted by iterative history matching of simulated packed bed-averaged oil saturation, 〈S o 〉(t), to measured values. 1 At the largest Ca (= 2.2 × 10 -4 ), the invading water front is compact and relatively flat across the entire width of the channel, characteristic of stable displacement as identified by Lenormand et al. [1].

Water invasion dynamics
At water breakthrough, the largest pockets of oil only span ~3dp.After breakthrough, water saturation increases along the entire length of the packed bed, but oil-occupied pores remain largely so.This suggests that additional oil was displaced from already swept pores, at length scales below or comparable to the image resolution.
At low Ca ≤ 2 × 10 -5 , fingers form at the onset of water invasion (bottom two rows).As is typical of drainage, water first bursts through the centre of larger pores, then expands outwards towards corners and into adjacent, smaller pores.Fingers propagate laterally as well as backwards, characteristic of capillary fingering (arrows).After breakthrough, clusters that remain bypassed generally remain so (yellow polygons); presumably the applied (macroscopic) pressure gradient is insufficient to exceed the capillary entry pressures of oil-occupied pores, relative permeability curves may not necessarily be a unique solution.and water simply flows through the network of waterinvaded pores without displacing additional oil.
In between (4.6 × 10 -5 ≤ Ca ≤ 9.1 × 10 -5 ), the invasion behaviour transitions between the two regimes (second and third rows).At early times, the invading water front is less flat, and fingering is observed at the lower limit (arrows, Ca = 4.6 × 10 -5 ).Bypassed oil clusters vary in size from the order a pore diameter to almost half the channel width.
Furthermore, this Ca regime is characterized by water invasion into initially bypassed, large clusters after breakthrough (dotted white polygons).
In the present experiments, capillary fingering emerges at a critical Ca of Cac = O(10 -5 ), which is about an order of magnitude larger than that observed for a Berea sandstone [22].However, Cac is expected to be a function of the porous medium as well as wettability [23].
The packed bed-averaged water saturation, 〈S w 〉, is presented as a function of time in Fig. 4 for selected waterfloods.It is readily apparent that as Ca increases (left to right), end-point 〈S w 〉 increases, i.e., remaining oil saturation decreases, consistent with the classic capillary desaturation curve (e.g., Fig. 5).Fig. 3. Snapshots of the packed bed at ̃= 3b/4, b, and b + 100.An image at the onset of waterflood has been subtracted from each image and a 2 × 2 median filter applied; bright pixels represent invading water.Imposed flow is from left to right.White dotted polygons demarcate regions invaded by water only after breakthrough.Yellow dotted polygons demarcate regions that remain unswept throughout the duration of the waterflood.Yellow arrows depict the direction of finger propagation as determined by visual inspection.From top to bottom: experiment M14, M21, M17, M5, and M7 (Table 2).Adapted from Ref. [16].Note that at Ca = 1.5 × 10 -6 , 〈S w 〉 decreases after breakthrough.This is a salient feature of the lowest Ca waterfloods; visual inspection of the images indicates that this is due to the counter-current imbibition of produced oil from the downstream end of the packed bed.

Capillary desaturation
Figure 5 presents the classic capillary desaturation curve: packed bed (or core)-averaged end-point saturation, normalized by its value at the lowest Ca, as a function of Ca.Superposed are coreflood data from the literature previously compiled by Tanino et al. [24].It is readily apparent that the end-point packed bed-averaged oil saturation, 〈S oe 〉, decays with increasing Ca throughout the range of values considered (red triangles).Interestingly, there is no evidence of a Ca-independent regime at low Ca documented for (uniformly) water-wet conditions.The absence of a Ca-independent regime is, however, consistent with coreflood measurements on Whitestone and Edwards limestone by Tie & Morrow [25] under mixed-wet conditions (open red markers, Fig. 4).Fig. 5. End-point oil saturation normalized by its maximum value as a function of Ca under mixed-wet conditions (red) and uniformly water-wet conditions (blue) measured in Ketton (solid circle) and Indiana (solid square) limestones [24], in Whitestone and Edwards limestones by Tie & Morrow [25], in Berea sandstone by Chatzis & Morrow [23], and in five sandstones by Abrams [26] and 〈S oe 〉 from the present study (red triangles).Expanded from Tanino et al. [24].

Impact of capillary end effects
Finally, we use the numerical simulations to evaluate the significance of capillary end effects.A satisfactory match between simulated and measured saturation is achieved for most experiments, even though early time behaviour in not well captured at low Ca (e.g., Fig. 4).
The best-fit coefficients for the Brooks-Corey correlations derived from history matching are summarized in Table 3. Counter-intuitively, krw(Sor), kro(0), αw, and αo do not display a strong dependence on Ca.However, without pressure drop and five fitting parameters, the best-fit k r (S w ) curves may be one of the many that can match the production data.A more robust interpretation requires measurement of both pressure and oil saturation, which is an ongoing effort.Contrary to 〈Soe〉 measured in the present experiments and in an array of cylinders by Zhang et al. [2], Sor estimated from history matching only displays a weak dependence on Ca, with Sor becoming Ca independent at Ca ∼ 10 -5 (Fig. 7).The difference between 〈Soe〉 and Sor increases with decreasing Ca, which is attributed to two factors.First, physical reasoning suggests that capillary end effects become less significant as Ca increases.Second, premature termination of the waterflood may affect lower Ca.
This experiment was first analyzed in Ref.[9] using a more basic post-processing algorithm.

Fig. 1 .
Fig. 1.The microfluidic setup (left) and scanning electron microscope (SEM) image of the grains (right).The dashed red rectangle demarcates the region downstream of the packed bed used to identify water breakthrough.

Fig. 2 .
Fig. 2. Mobility ratio-Ca regime map.Marker shapes depict flow patterns observed in the displacement experiments: capillary fingering (cross), stable displacement (solid circle), and post-breakthrough sweep (squares), Superposed are regime boundaries for primary drainage proposed in the literature for uniformly wetting micromodels: a network of capillaries ([1], grey solid line) and a uniform cylinder array ([2], grey dotted).Adapted from[16].

Figure 3 4 𝑡
Figure 3 presents snapshots of the packed bed at three instances during five waterfloods:  ̃= 3 4  ̃b,  ̃b, and  ̃b + 100, where  ̃b denotes water breakthrough time and the tilde denotes time normalized by the cumulative volume of water injected,  ̃=  Uw / (L ϕ).At the largest Ca (= 2.2 × 10 -4 ), the invading water front is compact and relatively flat across the entire width of the channel, characteristic of stable displacement as identified by Lenormand et al.[1].At water breakthrough, the largest pockets of oil only span ~3dp.After breakthrough, water saturation increases along the entire length of the packed bed, but oil-occupied pores remain largely so.This suggests that additional oil was displaced from already swept pores, at length scales below or comparable to the image resolution.At low Ca ≤ 2 × 10 -5 , fingers form at the onset of water invasion (bottom two rows).As is typical of drainage, water first bursts through the centre of larger pores, then expands outwards towards corners and into adjacent, smaller pores.Fingers propagate laterally as well as backwards, characteristic of capillary fingering (arrows).After breakthrough, clusters that remain bypassed generally remain so (yellow polygons); presumably the applied (macroscopic) pressure gradient is insufficient to exceed the capillary entry pressures of oil-occupied pores,

Figure 6
Figure 6 presents the simulated equilibrium (end-point) saturation profiles for selected waterfloods.At the downstream end of the packed bed, oil, which is the wetting phase for most of the grain surfaces, is retained by capillary forces giving rise to the sharp drop in Sw near x = L.The extent of this region of rapidly changing Sw broadly increases with decreasing Uw.Nevertheless, under the conditions considered presently, capillary end effects span the entire length of the packed bed.

Table 1 .
Basic properties of the brine and test oil at T = 21∘C.

Table 3 .
Summary of history matching of selected microfluidic waterfloods.We were unable to achieve satisfactory agreement between simulation and measured data for M7 and M6.