Effective rheology of bubbles moving in a capillary tube

Effective Rheology of Bubbles Moving in a Capillary Tube Santanu Sinha,1, ∗ Alex Hansen,1, † Dick Bedeaux,2, ‡ and Signe Kjelstrup2, § arXiv:1208.4538v1 [physics.flu-dyn] 22 Aug 2012 2 1 Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Department of Chemistry, Norwegian University of Science and Technology, N-7491 Trondheim, Norway (Dated: October 12, 2018) We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train from considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles. PACS numbers: 47.56.+r, 47.55.Ca, 47.55.dd, 89.75.Fb Multiphase flow in porous media plays a pivotal role in a vast range of applications in different fields such as oil recovery, soil mechanics and hydrology [1–4]. In spite of its relevance in these important fields, fundamental questions still linger on. In particular, this is true in connection with steady-state multiphase flow [5–11], which sets in after the initial instabilities such as viscous fingering are over in e.g. flooding experiments. A way to study this flow in the laboratory, is to simultaneously inject the two immiscible fluids into the porous medium and let them mix until a steady state where clusters and bubbles break up and merge but in such a way that their averages remain constant [10, 11]. Recently, the relation between average volumetric flow and excess pressure drop across the system has been investigated both experimentally and theoretically [10–13]. The conclusion from these studies is that the volumetric flux depends quadratically on the excess pressure. This is in contrast to the assumptions of linearity commonly made when considering such systems, e.g. in connection with invoking the concepts of relative permeability in reservoir simulations at the flow rates where capillary and viscous forces compete [14]. One aim of this work is to derive the volumetric flux versus excess pressure drop for a single capillary tube. We find that there is a square root singularity in this relation. This is in contrast to a the situation for network of pores, i.e., a porous medium. We base this calculation on the equation of motion of a bubble train in a long capillary tube with varying diameter, the Washburn equation [15]. We derive this equation following a different route than the now 91 year old original derivation. Lastly, we derive the probability distribution of the configuration of bubbles in the tube. This makes it possible to calculate the average of any quantity associated with the flow. We assume a long tube of length L oriented along an x axis. The radius of the tube, r, varies with the position along the tube as r= r0 , 1 − a cos (2πx/l) (1) xi ∆x xf ro P1 1 0 Pio Water Piw Pfo Oil Water Pfw P2 1 0 x l FIG. 1: Shape of tube and corresponding pressure drops. where r0 is the average radius of the tube, l is the wavelength along the tube and a is the dimensionless amplitude of the oscillation, see Fig. 1. We assume L ≫ l. We now imagine a bubble in a tube segment with length l limited by interfaces at xi and xf . The fluid in the bubble (“oil”) is less wetting with respect to the tube walls than the fluid outside the bubble (“water”). The capillary pressure drop across the interface between the two fluids at xi is [2]    2σwo 2πxi 2σwo , (2) 1 − a cos = r(xi ) r0 l and across the interface at xf , 2σwo 2σwo =− − r(xf ) r0  1 − a cos  2πxf l  . (3) Here, σwo is the surface tension between the two fluids. We sum the two capillary pressure drops to get     4σwo a 2πxb π∆xb pc (xb ) = sin , (4) sin r0 l l 2 where we have defined xb = (xi + xf )/2 and ∆xb = xf − xi . We assume the length of the bubble ∆xb to be smaller than the wavelength l, which is also equal to the length of the tube segment. Furthermore we have chosen the origin of the x-coordinate such the whole bubble is located between 0 and l. We assume the viscosities of the two fluids to be µw (“water”) and µo (“oil”) respectively. In order to derive a constitutive equation between volumetric flux q and pressure drop along the tube, ∆P = (L/l)∆p, we consider the entropy production of a tube segment with an average cross-sectional area πr02 and length l [16], dS = dt Z l dx ṡ(x) , (5) 0 where ṡ(x) the entropy production in the tube per unit of length at position x. We assume the temperature to be constant along the tube. The entropy production in the fluid sections times the temperature is equal to −q∂p(x)/∂x, where q is the volumetric flux and p(x) the pressure at x. Because of the incompressible nature of the flow, the volumetric flux is independent of position. There is no excess entropy production at the interfaces between the fluid and the bubble, assuming equilibrium between the two immiscible fluids, cf. Eqs. (2) and (3). Substituting ṡ(x) into Eq. (5) and integrating gives Θ dS = −q (piw − p1 ) − q (pf o − pio ) − q (p2 − pf w ) dt (6) = −q (∆p − pc (xb )) piw − p1 = −Rw xi q πr04 (∆p − pc (xb )) . 8µav l (10) 1 [(l − ∆xb ) µw + ∆xb µo ] l (11) q=− where µav = is the average viscosity of the two fluids, which we note is independent of the position of the bubble. One may obtain a similar expression directly from Eq. (7). Eqs. (10) and (11) were first derived by Washburn [15], but by a different route. The present procedure clarifies why Eq. (10) should contain the average viscosity and how the contribution due to the capillary pressure arises. The center of mass coordinate of the bubble moves as πr02 ẋb = q. Hence, we have the equation of motion    2πxb r02 , (12) ∆p − γ sin ẋb = − 8lµav l where we have defined γ= 4σa sin r0  π∆xb l  . (13) We introduce the angle variable θ = 2πxb /l and the time scale τ = γtπr02 /(4l2 µav ). We assume the pressure drop to be negative, ∆p = −|∆p|. The equation of motion (13) then becomes dθ |∆p| = + sin θ , dτ γ (7) where Θ is the temperature, ∆p = p2 − p1 is the pressure drop across a length l and pc (xb ) = pio − piw + pf w − pf o is the capillary pressure difference given in Eq. (4). The resulting linear force-flux relations for the pressure differences in the three sections are written in the form pf o − pio = −Ro ∆xb q p2 − pf w = −Rw (l − xf ) q The sum of the pressure differences gives the constitutive equation for motion of a single bubble (14) which is nothing but the equation of motion for the overdamped driven pendulum [17]. The period Tτ (measured in the same units as τ ), needed for the bubble to move from one end of the tube with length l to the other end for a given ∆p, is Z 2π Z Tτ 2πγ dθ , (15) = p dτ = Tτ = dθ/dτ ∆p2 − γ 2 0 0 In seconds this time is (8) T = where Rw and Ro are the Onsager resistivities per unit of length for the flow of water and bubble phases, respectively. Using Poisseulle flow the Onsager resistivities are equal to 8µ/(πr04 ), where µ is the viscosity of the fluid considered. The resulting pressure differences are 4l2 µav 8l2 µav . Tτ = 2 p 2 γπr0 r0 ∆p2 − γ 2 (16) piw − p1 = − We then define the time-averaged angular speed as Z Tτ Z 2π dθ 1 1 dθ h i = dτ = dθ dτ Tτ 0 dτ Tτ 0 1 p 2 2π = = ∆p − γ 2 . (17) Tτ γ pf o − pio Now, transforming back from dθ/dτ to ẋb to volumetric flux, q, we find the time-averaged flux equation p2 − pf w 8 xi µw q πr04 8 = − 4 ∆xb µo q πr0 8 = − 4 (l − xf ) µw q πr0 (9) hqi = − p πr04 sgn(∆p)Θ(|∆p| − γ) ∆p2 − γ 2 , 8µav l (18) 3 where sgn is the sign function and Θ is the Heaviside function. Hence, for |∆p| < γ there is no flow through the tube. Furthermore, if there is flow, T hẋb i = l, as one would expect. We see that we can write Darcy’s law for the time averaged volume flux only if γ = 0, which is the case if the tube has a constant radius (a = 0). The deviation from this law for the time average is due to a capillary pressure which varies as a function of the position of the bubble along the tube. The effective flux equation (18) has a threshold pressure that must be overcome to induce flow, γ defined in Eq. (13). Close to this threshold, when |∆p| − γ ≪ γ, the average flow equation becomes hqi = − p πr04 p 2γsgn(∆p)Θ(|∆p|−γ) |∆p| − γ , (19) 8µav l i.e., there is a square root singularity. As shown in Ref. [17], this square root is a consequence of the quadratic extremum of r in Eq. (1) leading to a saddle-node bifurcation, and therefore it does not depend on the specific sinusoidal shape of the profile chosen. The smaller the value of r0 , the larger is the threshold value. A radius of 10 micrometer can give a threshold of the order of 80 bar, which is non-negligible for most practical purposes. We may generalize these considerations to a tube segment with length L = (2N + 1) l in which there are 2N + 1 bubbles numbered from −N to N . Each bubble j may be characterized by a center of mass position xj and a width ∆xj . In view of the incompressible nature of the flow the volume flux q is independent of the position. This implies that the velocity x˙j of all the bubbles is the same, x˙0 = x˙j . The equation of motion (12) may then be generalized as follows,     +N X r02  2π x˙0 = − (x0 + δxj )  , ∆P − γi sin 8Lµav l It should be noted that Γs and Γc are proportional to the number of segments 2N + 1 with length l and therefore to the total length L. On non-dimensional form, Eq. (22) becomes   dθ Γc |∆P | cos θ . (25) = + sin θ + dτ Γs Γs Choosing this form, we have assumed Γs > Γc . Hence, the saddle-node bifurcation will occur in the sine term and not in the cosine term, which may be ignored. Working through the arguments leading to (18), we end up with the same effective flux equation as for the onebubble case, but with γ substituted for Γs . If, on the other hand, Γc > Γs , we may shift θ by π/2, and we are back to Eq. (25), but with Γs and Γc interchanged. Hence, again we find an effective flux equation as (18), but with γ substituted for Γc . Hansen and Ramstad [18] proposed to approach steady-state immiscible two-phase flow in porous media using the methods of statistical mechanics. A central quantity in this context is the configurational probability Π{configuration}. Possessing this quantity makes it possible to calculate the average of any quantity associated with the flow. We now derive the configurational probability for bubbles in a capillary tube. The derivation is based on mapping the time average on to a configurational average. Time averaging assumes that the states in each time interval are equally probable. The state of the tube at time t is characterised by the position xb (t) of the bubble. The time average of a function f (xb (t)) is therefore given by hf i = 1 T j=−N (20) where ∆P is the total pressure drop over the whole tube segment and δxj ≡ xj − x0 . Furthermore we define   π∆xj 4σa . (21) sin γj = r0 l By using trivial trigonometric identities, we may rewrite this expression as      r2 2πx0 2πx0 x˙0 = − 0 − Γc cos ∆P − Γs sin 8Lµav l l (22) where   +N X 2πδxj Γs = γj sin , (23) l j=−N and Γc = +N X j=−N γj cos  2πδxj l  . (24) Z T f (xb (t))dt . (26) 0 Using the relation between xb and t we may write this average as an average over xb in the following way: 1 T Rl 1 0 f (xb ) dxb /dt dxb p R l (xb ) dxb . = 1l ∆p2 − γ 2 0 |∆p|+γfsin(2πx b /l) hf i = (27) This may be written as hf i = Z l Π(xb )f (xb )dxb , (28) 0 where p 1 ∆p2 − γ 2 Π(xb ) = = T (dxb /dt) l (|∆p| + γ sin (2πxb /l)) (29) is the probability that the bubble has the position xb . This probability distribution can be interpreted as the probability distribution of an ensemble of tubes. Hence, it is the configurational probability. 4 The configurational probability depends on the manner the flow is controlled. If q is kept constant Π(xb ) = (T ẋb )−1 = 1/l. If ∆p is kept constant one finds the value given on the right hand side of Eq. (29). Whether q or ∆P is kept constant is comparable to the choice of an ensemble. It is interesting to calculate a few averages in the constant ∆P ensemble. For the average velocity we find using Eqs.(16) and (29) that Z hẋb i = is non-linear and controlled by a square-root singularity. We have shown this by deriving the equation of motion from considering the capillary forces and the entropy production associated with the viscous flow. We have also derived the configurational probability, i.e., the probability of bubble configurations from the equation of motion. From this probability the average of any quantity associated with the flow may be calculated. S. S. thanks the Norwegian Research Council for financial support. l Π(xb )ẋb (xb )dxb p r02 ∆p2 − γ 2 l . = = T 8lµav 0 (30) We calculate the average potential energy stored associated with the capillary forces, using Eqs. (10), (29) and pc (xb ) = γ sin (2πxb /l), finding hpc qi = Z † ‡ § [1] l Π(xb )pc (xb )q(xb )dxb [2] 0 = ∗ πγr04 p 2 ∆p 8µav l2 − γ2 Z 0 l sin  2πxb l  [3] dxb (31) [4] The average potential energy associated with the capillary forces is therefore zero as it must be. We proceed to consider an ensemble of single tube segments. For the ensemble we may interprete Π(xb ) as the probability that the bubble has the position xb . For the ensemble of tubes this contributes kB ln lΠ(xb ) to the entropy density per unit of length along the tube. Together with the other entropy contributions in the single tube we then have [5] S = S 0 + πr02 [∆xb so + (l − ∆xb ) sw ] Z l −kB Π(xb ) ln lΠ(xb )dxb . [10] = 0. (32) 0 From thermodynamics it follows that the entropy densities per unit of volume of the single component bulk phases are given by so = −∂po /∂Θ and sw = −∂pw /∂Θ. 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