A-2315

Imbibition dynamics in interacting capillaries

G. Visavale, S. Ashraf and J. Phirani∗

Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas,New Delhi 110016, India.

visavaleganesh@gmail.com

Abstract

Spontaneous imbibition has long been known to have significant role in recovering oilfrom low permeability fractured reservoirs. Though the fundamental theory of capillarity isknown for about a century by the works of Hagen Poiseuille and then by Lucas and Washburnthey do not entirely describe the phenomena. The capillary action at early times i.e. inthe inertial regime and the dynamics for interacting capillaries remain unexplained. In thepresent work, we report the numerical investigations using volume of fluid (VOF) model forthe spontaneous imbibition in non-interacting and interacting capillaries. Capillaries withradii 0.5 and 1 mm, length 100 mm were selected for studying the effect of non-interactingand interacting nature on the imbibition dynamics. The two interacting capillary cases werewith centre to centre distance of 1.25 and 1.485 mm. The interacting capillary model showsthat if connectivity is more capillaries will behave as single capillary, but if connectivity isless meniscus leads in smaller radii capillary. The study will help us in understanding thephysics of flow in fractured reservoirs.

1 IntroductionIn spontaneous imbibition the wetting fluid (generally water) is sucked in pores due to capillaryforces. The fundamental knowledge of the flow due to capillary forces is known for past century.The theoretical foundations were laid by Hagen Poiseuille for flow in pipes using which Lucasand Washburn1 found the meniscus position variation with time due to capillary forces. Forcapillaries of uniform cross-section the rate of imbibition obeys the law stated by Lucas-Washburnas: l2 = Dt, where l represents the imbibed distance through which the fluid has moved intime t and D is the coefficient that depends on the characteristics of the capillary and thefluids. However these classical equations do not entirely describe the capillary invasion at theinertial regime at very short times, in non-uniform geometries and for interacting nature ofcapillaries i.e connected capillaries that are present in porous media. Pores found in reservoirhave random geometrical orientations, are interacting in nature and the imbibition of the fluidinto these random microchannels cannot be described analytically. A reservoir made of forexample sandstones, limestones, etc. consists of an interconnected, three dimensional (3-D)network of pores or capillaries. As a result, these interconnected capillaries with the fluidspresent in adjacent capillaries interact with each other. Dong et al.2(2005); Dong3 (2006); Ruthand Bartley4 (2011); have developed 1-D models to simulate immiscible displacement in porous

∗corresponding author: jphirani@chemical.iitd.ac.in

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Figure 1: Schematic of interacting and non-interacting capillaries, Z is the centre to centredistance between capillaries of radii 1 and 0.5 mm

media. However their work also do not consider the entry effects (inertial regime) and the degreeof connectivity between the interacting capillaries.

In the present work, we report the numerical investigations of the spontaneous imbibitionin non-interacting and interacting capillaries and investigate the effect of degree of connectivityon the imbibition pattern. These results will help us in understanding the physics of flow infractured reservoirs.

2 Computational modelIn this study, the volume of fluid (VOF) method (Hirt and Nichols5, 1981) was used to simulatethe spontaneous imbibition of a wetting fluid of µ = 0.001 kg/ms, ρ=998 kg/m3, in a capillaryfilled with a non-wetting fluid of identical viscosity. The interfacial tension between the fluidsis considered to be 0.072N/m. The fluid phases are considered incompressible and the flowis assumed to be Newtonian and laminar. In VOF method, the phases are identified by theirvolume fraction (αi, i = 1, 2) in a computational cell6. When αi = 1; the cell is filled with 1st

phase and when αi = 0 cell is filled with the 2nd phase and if 0 < αi < 1; the cell contains theinterface between the 1st and 2nd phases. A single set of Navier-Stokes equation is solved for theincompressible Newtonian flow:

∂

∂t(ρ~v) +∇ · (ρ~v~v) = −δp+ [µ(∇~v + (∇~v)T )] + ρg + F (1)

where F is the surface tension force per unit volume. When a computational cell is not entirelyoccupied by one phase, mixture properties described as:

ρ = α1ρ1 + (1− α1)ρ2 (2)

µ = α1µ1 + (1− α1)µ2 (3)

For the qth phase, advection equation is written as:

∂

∂t(α1ρ1) +∇ · (α1ρ1~v1) = 0 (4)

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Figure 2: Imbibition in capillaries with radii 1 & 0.5 mm and comparative plot of Lucas-Washburnand CFD results (l2 against t) for non-interacting capillaries

The 2nd phase volume fraction is computed from the relation: α1 + α1 = 1. Eqs (1) and (4)are solved using the commercial flow solver ANSYS FLUENT 15.0. The simulations are runfor non-interacting capillaries of radii 1 mm and 0.5 mm and interacting capillary model is alsoinvestigated as the schematic shown in figure 1.

3 Results and DiscussionsFigure 2(a) shows the length travelled by meniscus with time for two capillaries of radii 0.5 and1mm. The meniscus in 1 mm radii capillary leads as compared to the meniscus of 0.5 mm radiicapillary. The viscous resistance is more in 0.5 mm radii capillary which leads to slow movementof fluid. Figure 2(b) shows the comparison of Washburn equation with VOF simulation resultsof length2 vs time. In the figure the length travelled at initial time is used as fitting parameter.This is due to time taken by the fluid to form the meniscus which is not considered in Washburnequation. Figure 3 shows the pressure along the length of the non-interacting capillaries for 0.5and 1 mm radius of capillary, respectively at time 0.4 s. The figures show the pressure jump ofcapillary pressure at the meniscus. Though the capillary pressure is higher in smaller capillaryit also has low permeability that causes the lag as compared to bigger capillary with higherpermeability. However, in case of interacting capillaries shown in figure 1 the leading fluid frontshifts from bigger to smaller capillary as seen in figure 4 and 5. In figure 5 the distance betweenthe capillary centers are 1.25 mm and 1.485 mm. The fluid front is leading in smaller capillarythan in bigger capillary, and more distinctly in Z = 1.485mm case. The pressure equilibriumbefore and after the meniscii in interacting capillaries can be seen in figure 6 as proposed intheories by Dong et al.7. The fluid leading in smaller capillary is due to the exchange of fluidfrom the bigger capillary to smaller capillary in interacting pair as is evident from streamlinesof figure 7. The VOF simulation thus offers great insight to explore the real physics that wouldbe otherwise very difficult to visualize experimentally.

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Figure 3: Pressure against capillary length for non-interacting capillary

Figure 4: Imbibition length vs time for interacting capillary, Z=1.25 & 1.485 mm

Figure 5: Volume fraction of fluid in interacting capillaries, Z = 1.25 & 1.485 mm at 0.4 s

Figure 6: Pressure against capillary length for interacting capillaries, Z=1.25 & 1.485 mm at0.4 s 4

Figure 7: Streamlines near meniscii region showing transfer of fluid from R=1 mm to R=0.5mm for cases Z=1.25 and 1.485 mm

4 ConclusionsWe successfully investigate the spontaneous imbibition in non-interacting and interacting cap-illaries with fluids of equal viscosities. In non-interacting capillaries the VOF simulation showsthat Washburn equation is not able to capture the meniscus formation phenomena at earlytimes. The interacting capillary model shows that if connectivity is more capillaries will behaveas single capillary, but if connectivity is less meniscus leads in smaller radii capillary. VOFprovides insights in analysis of spontaneous imbibition in capillaries enabling good visualizationand tracking of interface (meniscii) between the two fluids.

References1. Edward W. Washburn. The dynamics of capillary flow. Phys. Rev., Mar 1921.

2. Mingzhe Dong, Francis AL Dullien, Liming Dai, and Daiming Li. Immiscible displacementin the interacting capillary bundle model part i. development of interacting capillary bundlemodel. Transport in Porous media, 59(1):1–18, 2005.

3. Mingzhe Dong, Francis AL Dullien, Liming Dai, and Daiming Li. Immiscible displacementin the interacting capillary bundle model part ii. applications of model and comparison ofinteracting and non-interacting capillary bundle models. Transport in Porous media, 63(2):289–304, 2006.

4. Douglas Ruth and Jonathan Bartley. Capillary tube models with interaction between thetubes [a note on “immiscible displacement in the interacting capillary bundle model part i.development of interacting capillary bundle model”, by dong, m., dullien, fal, dai, l. and li,d., 2005, transport porous media]. Transport in porous media, 86(2):479–482, 2011.

5. Cyril W Hirt and Billy D Nichols. Volume of fluid (vof) method for the dynamics of freeboundaries. Journal of computational physics, 39(1):201–225, 1981.

6. Ansys Fluent Ansys. 14.0 theory guide. ANSYS Inc, 2011.

7. Mingzhe Dong, Jun Zhou, et al. Characterization of waterflood saturation profile histories bythe ‘complete’capillary number. Transport in porous media, 31(2):213–237, 1998.

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