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Spontaneous and Forced Imbibition of Aqueous Wettability Altering Surfactant Solution into an Initially Oil-Wet Capillary Paul S. Hammond and Evren Unsal* Schlumberger Cambridge Research, Cambridge, CB3 0EL, United Kingdom Received May 19, 2009. Revised Manuscript Received July 17, 2009 Unforced invasion of wettability-altering aqueous surfactant solutions into an initially oil-filled oil-wet capillary tube has been observed to take place very slowly, and because this system is an analogue for certain methods of improved oil recovery from naturally fractured oil-wet reservoirs, it is important to identify the rate-controlling processes. We used a model for the process published by Tiberg et al. (Tiberg, F.; Zhmud, B.; Hallstensson, K.; Von Bahr, M. Phys. Chem. Chem. Phys. 2000, 2, 5189-5196) and modified it for forced imbibitions. We show that when applied pressure differences are not too large invasion rates are controlled at large times by the value of the bulk diffusion coefficient for surfactant in the aqueous phase and at early times by the resistance to transfer of surfactant from the oil-water meniscus onto the walls of the capillary. For realistic values of the bulk diffusion coefficient, invasion rates are indeed slow, as observed. The model also predicts that the oil-water-solid contact angle during unforced displacement is close to π/2, and so, the displacement occurs in a state of near-neutral wettability with the rate of invasion controlled by the rate of surfactant diffusion rather than a balance between capillary forces and viscous resistance. Under forced conditions, the meniscus moves faster, but the same kinds of dynamical balances between the various processes as were found in the spontaneous case operate. Once the capillary threshold pressure for entry into the initial oil-wet tube is exceeded, the effect of pressure on velocity becomes more significant, there is not sufficient time for the surfactant molecules to transfer in great quantity from the meniscus to the solid surface, and wettability alteration is then no longer important.
1. Introduction The inhomogeneities in porous rock already make the oil recovery process very difficult, but the capillary forces within the pore space increases this difficulty further. These forces are usually described in terms of energies of the phase boundaries present, that is, oil-water, water-rock, and rock-oil. Wettability is defined as the tendency of one fluid to spread on or adhere to a solid surface in the presence of other immiscible fluids. In rock/oil/brine systems, it is a measure of preference that the rock has for either the oil or the water. When the rock is water-wet, there is a tendency for water to occupy the small pores and to contact the majority of the rock surface. Similarly, in an oil-wet system, the rock is preferentially in contact with the oil; the location of the two fluids is reversed from the water-wet case, and oil will occupy the small pores and contact the majority of the rock surface.1 The wettability of the rock/ fluid system is important because it is a major factor controlling the location, flow, and distribution of fluids in a reservoir. When the system is in equilibrium, the wetting fluid will completely occupy the smallest pores and be in contact with a majority of the rock surface (assuming of course that the saturation of the wetting fluid is sufficiently high). The nonwetting fluid will occupy the centers of the larger pores and form connected regions and clusters that extend over several pores. During water-flooding of strongly wet core samples, except for high oil/water viscosity ratios, most of the recoverable oil is typically displaced before the water breakthrough, and oil production ceases soon thereafter. The residual oil (the nonwetting phase) remains trapped by capillary forces as *To whom correspondence should be addressed. Tel: þ44 1223 325231. Email:
[email protected]. (1) Anderson, W. G. J. Pet. Technol. 1986, 38(10), 1125–1144.
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discontinuous droplets or irregular bodies of oil separated by continuous water.2 Changes in wettability have effects on capillary pressure, waterflood behavior, relative permeability, recovery, irreducible water, and residual oil saturations.3 Wettability of some rocks can be altered by crude oil contact or by changing the brine composition, that is, adding surfactants and/or lowering brine salinity.4-8 The alteration can mobilize the trapped oil and enhance the oil recovery by affecting the rock-fluid properties, such as capillary pressure. It is the wetting condition at the oil/water/solid junction that controls the interface’s behavior and the capillary pressure acting across it. Most reservoirs have a mixed-wet condition where the wetting is altered by adsorption of crude oil components on some surfaces, and more water-wet condition is preserved on the others. Spontaneous imbibition only occurs when the pore surfaces are effectively water wet so that water imbibes into the rock matrix and oil is expelled. Introduction of surfactant into the brine phase can improve the oil production by lowering the oil-water interfacial tension and by altering the wettability of the rock surfaces from oil-wet to more water-wet. Consequently, imbibition of water can be promoted. Although the exact relationship between waterflood efficiency and capillary forces is not (2) Salathiel, R. A. J. Pet. Technol. 1973, 25(10), 1216–1224. (3) Anderson, W. G. J. Pet. Technol. 1987, 39(10), 1283–1300. (4) Morrow, N. R.; Lim, H. T.; Ward, J. S. SPE Formation Eval. 1986, 1(1), 89– 103. (5) Buckley, J. S.; Bousseau, C.; Liu, Y. SPE J. 1996, 1(3), 341–350. (6) Buckley, J. S.; Liu, Y.; Monsterleet, S. SPE J. 1998, 3, 54–61. (7) Austad, T.; Milter, J. International Symposium of Oilfield Chemistry; 1997; pp 257-266, SPE 37236. (8) Austad, T.; Matre, B.; Milter, J.; Saevareid, A.; Oyno, L. Colloids Surf., A 1998, 137, 117–129.
Published on Web 08/12/2009
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completely understood, its importance has long been recognized.9-14 In the development of enhanced oil recovery systems based on water flooding, the generation of a very low surface tension is often important. However, it is also recognized that a system with a low surface tension is not by itself certain of success. Other parameters are also relevant to the dynamic process of recovery, that is, adsorption of surfactants on minerals, which is the result of favorable interactions of the surfactant molecules with the chemical species on or near the solid surface. Molecular and ionic forces such as covalent bonding and electrostatic attraction can contribute toward the adsorption process and enhance the recovery. Recently, interest in surfactant injection has increased mainly due to increasing oil prices. The economics of the surfactant injection is critical due to the high cost of the chemicals and injection procedure.15 Inexpensive surfactant methods are being developed for low interfacial tension, wettability alteration, and reduced adsorption.16 Two distinct types of surfactant flooding methods are possible, injecting either high or low concentration surfactant solutions. High concentration solutions are more costly, and a possible drawback with low concentration is that there is always a risk of loss of surfactant from solution by adsorption onto the rock surface. This may seriously affect the oil recovery.17 The development and optimization of chemical oil recovery systems can best be achieved on the basis of a good understanding of the mechanisms by which oil is displaced from a porous material and of the parameters which control those mechanisms. Under water injection or aquifer drive in an oil-wet fractured reservoir, the subsequent recovery of oil from the rock matrix, if any, is mainly dependent on the spontaneous imbibition of water, a very slow process.14 Surfactant solutions may speed up this process since they decrease the capillary pressures, so that solution can invade the pore space easier. Better insight into the displacement processes is required than is afforded by experiments such as tests in rock cores, where it is impossible to see individual pores. It is therefore necessary to carry out laboratory experiments concerned with specified system parameters and to link the results of these experiments with displacement tests.
2. Imbibition into Oil-Wet Capillaries In spontaneous imbibition, the pressure is the same at both ends of the capillary. The driving force is the capillary pressure, which depends on the interface curvature, surface tension, and capillary radius. Imbibition will occur only if the contact angle, θ, between aqueous phase and capillary surface is less than π/2. Pure water does not penetrate spontaneously into a hydrophobic porous capillary because if the contact angle is greater than π/2, it needs to be forced. However, surfactant solutions might alter this behavior, through their effect on the contact angle. If the (9) Austad, T.; Standnes, D. C. J. Pet. Sci. Eng. 2003, 39, 363–376. (10) Gupta, R.; Mohanty, K. K. SPE/DOE Symposium on Improved Oil Recovery; Tulsa, OK, 2008; 13407-MS. (11) Skalli, L.; Buckley, J. S.; Zhang, Y.; Morrow, N. R. J. Pet. Sci. Eng. 2006, 52, 253–260. (12) Babadagli, T. J. Colloid Interface Sci. 2002, 246, 203–213. (13) Babadagli, T. J. Pet. Sci. Eng. 2006, 54, 43–54. (14) Morrow, N. R.; Mason, G. Curr. Opin. Colloid Interface Sci. 2001, 6, 321– 337. (15) Salager, J. L.; Bourrel, M.; Schechter, R. S.; Wade, W. H. SPE J. 1979, 19 (5), 271–278. (16) Rao, D. N.; Ayirala, S. C.; Abe, A. A.; Xu, W. SPE/DOE Symposium on Improved Oil Recovery; Tulsa, OK, 2006; 99609-MS. (17) Morgan, J. C.; Schechter, R. S.; Wade, W. H. Recent advances in the Study of low interfacial tensions. Improved Oil Recovery by Surfactant and Polymer Flooding; Schah, D. O., Schechter, R. S., Eds.; Academic Press: New York, 1977; p 101.
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Figure 1. Meniscus in the capillary with surfactant solution. Surfactant molecules accumulate at the solid/liquid/liquid intersection and at the meniscus.
Figure 2. (a) Adsorption mechanism at the interface, surfactant molecules at the water/solid surface (Γws), oil/solid surface (Γos), and oil/water interface (Γow). (b) The contact angle decreases as the meniscus advances in the capillary (θI > θII > θIII > θIV).
advancing contact angle is decreasing as a function of the surfactant concentration, then at some critical concentration, C*, θ becomes equal to π/2 or lower. Spontaneous imbibition takes place, and the penetration rate depends on the concentration of the surfactant. Experiments showed that spontaneous imbibition of a wettability altering surfactant solution into hydrophobic capillaries is slow even at high surfactant concentrations.18-20 Because this phenomenon is an analogue for certain methods of improved oil recovery from naturally fractured oilwet reservoirs, it is important to identify the rate-controlling processes. Displacement of the oil phase from a porous medium or capillary tube under an externally applied pressure difference is known as forced imbibition. The pressure at each end of the capillary is then different. The water, or surfactant solution, is applied under a force, which gives rise to a pressure gradient. This type of imbibition is potentially much faster than the spontaneous case, at least when the applied pressure is large enough to overcome the capillary pressure threshold for entrance into the oil-wet capillary. However, when the applied pressure is less than the threshold pressure for entry, the meniscus can only penetrate if surfactant acts to change the contact angle. Diffusion and adsorption of surfactant molecules are very slow processes. Applied differential pressure does not make these two processes faster; it only speeds up the meniscus. Under very high differential pressures, the surfactant might lag behind the moving meniscus; consequently, it may not have the desired effects on wettability alteration. At low applied differential pressures, the meniscus must move in slowly, so that the surfactant needed to change the contact angle can keep up. We consider displacement of oil by aqueous surfactant solution from a circular cylindrical capillary of radius a and length L (Figure 1). The inner walls of the capillary are initially oil-wet but on contact with surfactant become water-wet. Surfactant molecules move with the cross-section average speed of the fluid in the region between the tube entrance and the meniscus and experience an extra transport due to diffusion on top of this (Figure 2a). The meniscus also moves with the cross-section average speed of the fluid, and so, it is only through the effects of diffusion that a surfactant molecule can catch up with and be transferred onto the meniscus. Once at the meniscus, surfactant molecules are transferred onto the solid surface through the three-phase contact line. (18) Tiberg, F.; Zhmud, B.; Hallstensson, K.; Von Bahr, M. Phys. Chem. Chem. Phys. 2000, 2, 5189–5196. (19) Starov, V. M. J. Colloid Interface Sci. 2004, 270, 180–186. (20) Starov, V. M. Adv. Colloid Interface Sci. 2004, 111, 3–27.
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This transfer from bulk solution to solid surface results in a decrease in the bulk surfactant concentration in the vicinity of the meniscus and hence a concentration gradient starting from the capillary inlet in the direction of the moving meniscus. Adsorption of surfactant on the solid surface causes the wettability to alter, and so, the capillary becomes more water wet. The contact angle and the surface tension decrease (Figure 2b). Below, we present a model where these phenomena are studied under both spontaneous and forced conditions. The remainder of the note is laid out as follows. The basic mathematical model is first described. Then, two approximate asymptotic solutions are derived, one valid for late dimensionless times and the other for early. These reveal the governing parameters of the problem and make plain the rate-limiting role of bulk diffusion at late times and of resistance to mass transfer of surfactant through the three-phase contact line region at early times. A sequence of illustrative numerical solutions is then presented. The results are entirely consistent with the late and early time asymptotics. Lastly, key findings are summarized, and a few suggestions are made concerning means to increase invasion rates and future work.
3. Model We consider displacement of oil, by aqueous surfactant solution, from a circular cylindrical capillary of radius a and length L. The meniscus separating oil and water is located at position z = X(t), where t is time and z measures the axial distance along the capillary from the entrance. X(t) is the position of the meniscus at the contact line. The walls of the capillary are initially oil-wet but on contact with surfactant become water-wet. More detail on the effects of this alteration process, and how it is envisaged to take place, will be given below. Our model for the process closely follows that of Tiberg et al.,18 modified to treat forced imbibition cases. The problem geometry is sketched in Figure 1. The model links together a number of submodels for individual elements of the overall process, and to help clarify the description below, we briefly list these and point out any major assumptions made. The fluids within the capillary are incompressible, and so, the speed at which the meniscus moves along the tube is equal to the cross-section averaged axial velocity of the fluid at any point in the capillary, and this in turn can be related to axial pressure differences by Poiseuille’s law. The capillary pressure jump across the meniscus, which together with the applied pressure difference between the ends of the capillary, drives the flow and is linked to the oil-water interfacial tension and contact angle, and these in turn are related to the surface energies of the oil-solid and aqueous solution-solid interfaces by Young’s equation. The surfactant can adsorb onto any of the interfaces, the adsorbed quantities are related by Langmuir isotherms to the concentrations in the adjacent fluid, and the various interfacial tensions or surface energies are related to the amount of surfactant adsorbed by the van Laar equation of state. Combining all of these elements, given knowledge of all of the adsorbed surfactant concentrations, it is possible to compute the speed of the meniscus and so update its position. To find these surfactant concentrations, it is necessary to solve for the concentration distribution within the aqueous phase and then consider how surfactant is transported and exchanged between the aqueous phase, the meniscus, and the two solid-liquid surfaces. The overall speed of motion in the meniscus is assumed to be sufficiently slow and bulk diffusion is assumed to be sufficiently strong, so that the surfactant concentration is uniform across each capillary crosssection. The surfactant concentration within the aqueous phase Langmuir 2009, 25(21), 12591–12603
then satisfies a simple one-dimensional advection diffusion equation (ADE), with the concentration at the capillary inlet prescribed. The advective velocity in the ADE is the mean axial velocity of the fluid, which is the same as the velocity of the meniscus, and the diffusivity is the bulk value for surfactant in solution. Up to this point, the modeling is, in the authors’ view, essentially conventional and uncontroversial. The next steps, however, involve the modeling of the processes of transfer of surfactant from bulk solution, to oil-water meniscus, and thence onto the water-solid and oil-solid surfaces where it changes the surface energies and thence influences the capillary pressure. Insufficient information is known about these processes for the modeling to be other than speculative. We assume a uniform adsorbed concentration on the oil-water meniscus, in equilibrium with the adjacent aqueous solution. The surfactant is assumed to transfer from the oil-water meniscus to the solid surface, through the moving contact line, at a rate proportional to both the concentration on the meniscus and the density of unoccupied adsorption sites on the solid. Once transferred, surfactant can diffuse on the solid surface and so is present both on the oil-solid interface ahead of the advancing meniscus and on the water-solid surface behind. The surfactant concentrations on the solid surface are computed by solving the appropriate surface diffusion equation, with source terms representing the transfer from the oilwater meniscus. Once this is done, we have all of the information necessary to compute the current speed of advance of the meniscus, as outlined above. The remainder of this section puts these ideas into mathematical form. For simplicity, we concentrate on situations where the presence of surfactant on the oilsolid interface can reasonably be neglected, leaving a consideration of the effects of surfactant ahead of the meniscus to a future publication. If the velocity of the meniscus is denoted U(t), then dX ¼U dt
ð1Þ
subject to X(0) = X0 (which in the absence of inertia effects must be greater than zero to avoid early time velocity singularities if the oil viscosity is zero). Ignoring flow processes in the immediate vicinity of the meniscus, the velocity of the meniscus is taken to be equal to the cross-section averaged axial velocity of the fluid in the capillary (which for an incompressible fluid is the same at every location z). The average fluid velocity is linked to pressure differences between various positions in the capillary through the Poiseuille flow formulas,21 and so U ¼
a2 ðpL - p - Þ a2 ðpþ - pR Þ ¼ 8μw X 8μo ðL - XÞ
ð2Þ
Here, μw and μo are the viscosities of surfactant solution and oil, respectively, pL is the aqueous solution pressure at the entrance to the capillary, pR is the oil pressure at the exit, p- is the pressure in the aqueous solution immediately adjacent to the meniscus, and pþ is the pressure in the oil adjacent to the meniscus. The difference between the externally applied pressures at the ends of the capillary, ΔPd, is taken to be pL - pR ¼ ΔPd
ð3Þ
In the case of spontaneous imbibition, this pressure difference will be zero. The capillary pressure at any instant, ΔPc, is defined (21) Batchelor, G. K. An Introduction to Fluid Dynamics; CUP: Cambridge, 1967.
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as the difference between oil and water pressures across the meniscus, so that pþ - p - ¼ ΔPc
ð4Þ
2σow ðΓow Þ cos θ a 2 ! 24 Γws 0 m σ ow ð0Þ cos θ - RTΓws log 1 - m ¼ a Γws !# Γos m þ RTΓos log 1 - m Γos
ΔPc ¼
Upon rearranging eq 2, we find ΔPd þ ΔPc ¼
sphere and the capillary wall is θ, then simple geometry shows that the radius of curvature of the sphere cap is a/cos θ. It follows that the pressure difference across the meniscus is
8U ½ μ X þ μo ðL -XÞ a2 w
ð5Þ
which will later be used to find U. The contact angle θ between surfactant solution and oil at the solid boundary is assumed to satisfy Young’s equation σ ow ðΓow Þ cos θ ¼ σ os ðΓos Þ - σ ws ðΓws Þ
ð6Þ
where Γij is the number of moles of surfactant adsorbed per unit area at the interface between material i and material j and σij(Γij) is the interfacial tension of that interface. (See chapters 1 and 2 of Starov et al.30 for a discussion of the foundations of Young’s equation.) The suffices i and j can each take the value o, denoting oil, w denoting aqueous solution, or s denoting the solid of the capillary wall (thus, σow is the interfacial tension of the oilaqueous solution meniscus etc.). In the absence of surfactant σow ð0Þ cos θ0 ¼ σos ð0Þ - σ ws ð0Þ
ð7Þ
using eq 10. Hence, if the amounts of surfactant adsorbed on the solid wall near the meniscus are known, then eq 11 combined with eq 5 gives the speed of advance of the meniscus. To find the amount of surfactant adsorbed on the solid surface near the moving meniscus, we must consider the process of transport of surfactant through the bulk aqueous phase, along the meniscus, and through the moving three-phase contact line to the solid. When the Peclet number Pe = aU/Db is small, so that the effective diffusion coefficient is Db rather than the Taylor dispersion value Db (1 þ Pe2/48), and for times large as compared to a2/ Db, surfactant transport within the capillary may be described by Dc Dc D2 c þ U ¼ Db 2 Dt Dz Dz
whence σow ðΓow Þ cos θ - σ ow ð0Þ cos θ0 ¼ σws ð0Þ - σ ws ðΓws Þ - ½σos ð0Þ - σos ðΓos Þ
ð8Þ
where the last term in square brackets on the right-hand side vanishes when no surfactant is ever adsorbed on the oil-solid interface. Adsorption of surfactant on an interface reduces the interfacial tension, and the surface properties of an ideal monolayer are related to each other by the van Laar equation (Krotov and Rustanov, section 1.6)22 σij ðΓij Þ ¼ σij ð0Þ þ
RTΓm ij
Γij log 1 - m Γij
! ð9Þ
where R is the universal gas constant, T is the absolute temperature, and Γm ij is the monolayer capacity. Combining eqs 8 and 9, we find
cð0, tÞ ¼ cb
! ! Γws Γos m log 1 - m þ RTΓos log 1 - m Γws Γos ð10Þ
This shows that adsorption of the surfactant at the oil-solution and solution-solid interfaces will change the contact angle. Adsorption on the oil-water interface influences the contact angle too, through its effect on σow. For low capillary number flows, that is, slow flows in which viscous stresses at interfaces are much weaker than stresses due to interfacial tension, menisci are surfaces of constant curvature. In the present circular cylindrical geometry, this means that the meniscus is a cap of a sphere. If the contact angle between the (22) Krotov, V. V.; Rustanov, A. I. Physicochemical Hydrodynamics of Capillary Systems; Imperial College Press: London, 1999.
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ð12Þ
where c(z, t) is the cross-section averaged surfactant concentration, Db is the bulk diffusivity of surfactant within the aqueous phase, and U is the average axial velocity of fluid within the capillary, which is of course equal to the speed of the meniscus, and we have assumed that the rate of adsorption of surfactant onto the wall from bulk fluid within the capillary away from the meniscus is negligible. (A finite rate of adsorption is easily added to the model; see Starov et al.,30 section 5.2.) The small Peclet number condition ensures that shear-enhanced (Taylor) dispersion contributions to axial spreading are negligible as compared to ordinary diffusion, and the large time requirement ensures that diffusion has had sufficient time to homogenize concentrations across the capillary cross-section in the region of nonaxial streamlines in the immediate vicinity of the meniscus. Equation 12 is assumed to be subject to the inlet condition
σ ow ðΓow Þ cos θ ¼ σ ow ð0Þ cos θ0 ¼ -RTΓm ws
ð11Þ
ð13Þ
and to the initial condition cðz, 0Þ ¼ cb
ð14Þ
where cb is the bulk concentration of surfactant in the initial aqueous solution. On the assumption that there is an instantaneous balance, in a frame of reference moving with the meniscus, between the rate of transport of surfactant into the oil-water interface by diffusion from the bulk aqueous phase and the convective transport of surfactant out of the region of the moving contact line on the water-solid interface, we have -πa2 Db
Dc ðX, tÞ ¼ 2πaUΓws ðX, tÞ Dz
ð15Þ
The approximations being made here are that changes in the amount of surfactant stored on the meniscus are negligible and Langmuir 2009, 25(21), 12591–12603
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that in a frame of reference moving with the three-phase contact line, surface diffusive fluxes of surfactant along the oil-solid interface are in balance with the convective flux of material with the surface back toward the aqueous side. The amount of surfactant adsorbed on the oil-water interface is assumed to be in Langmuir isotherm equilibrium with the concentration of surfactant in the immediately adjacent aqueous phase, so that Γow ¼ Γm ow
KcðX, tÞ 1 þ KcðX, tÞ
ð16Þ
where Γm ow is a monolayer capacity and K is a measure of the relative rates of adsorption and desorption from the interface. (See Krotov and Rustanov,22 section 1.6, for the linkage between the van Laar equation, the Langmuir isotherm, and the Gibbs adsorption equation.) In writing this expression, we have also assumed that the surface concentration of surfactant on the meniscus is spatially uniform and, hence, that diffusion on the surface of the meniscus is strong enough to overcome any effects of surface advection. [The surfactant concentration in the bulk aqueous phase is uniform across the cross-section, by virtue of the restrictions on eq 12. The paper (Bain, C.; Manning-Benson, S.; Darton, R. J. Colloid Interface Sci. 2000, 229, 247-256) shows that for stagnation point flow, there is an exact solution of the coupled equations of bulk and surface transport in which surface concentration is uniform even in the absence of surface diffusion. However, the velocity field on the meniscus cannot completely correspond to a stagnation point flow, as this violates the no penetration boundary condition at the solid wall.] Coupled bulk and surface transport of the surfactant is discussed in chapter 2 of Krotov and Rustanov.22 To complete the system of equations, it is necessary to give a relationship linking Γow, Γws(X, t), and Γos(X, t), that is, a description of the transfer of surfactant from the oil-water interface to the water-solid and oil-solid interfaces through the moving contact line. Zhmud et al.23 and Tiberg et al.18 assume that the total time over which transfer from the oil-water to the water-solid interface takes place is equal to h/U, where h is the thickness of the surfactant-loaded oil-water interface and U is the speed of the contact line; the configuration is sketched in Figure 2a. They further assume that surfactant is transferred at a rate given by the Langmuir kinetic equation (section 2.1.1 of Krotov and Rustanov).22 That is, the rate of transfer from the oil-water to the water-solid interface is taken to be proportional to the product of the amount of surfactant on the oil-water interface and the number of unoccupied adsorption sites on the watersolid interface, with rate constant kþ. The rate of backward transfer (i.e., desorption from the water-solid interface) is taken to be simply proportional to the amount of material adsorbed on the water-solid interface, with rate constant k-. Thus, at a fixed point on the solid surface, the time rate of change of the amount of adsorbed surfactant, Γ, due to exchanges with the oil-water interface is dΓ Γ ¼ kþ Γow 1 - m - k - Γ dt Γ
ð17Þ
Outside the transfer zone, it is assumed that any further changes in Γ are due to surface diffusion of surfactant, with surface (23) Zhmud, B. V.; Tiberg, F.; Hallstensson, K. J. Colloid Interface Sci. 2000, 228, 263–269.
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diffusivity Ds. This carries with it the possibility of the surfactant being present on the solid surface downstream (i.e., on the oil side) of the advancing oil-water interface. Kumar et al.24 give experimental evidence for the presence of surfactant on the solid surface on the vapor side of an advancing water-vapor interface, and Churaev et al.25 present a model for surfactant imbibition including this feature. Kao et al.26 observe changes at/near the solid surface on the oil side of the advancing interface as a surfactant solution displaces an oil drop from a surface, and these are strongly suggestive of the presence of surfactant on the downstream side of the advancing interface in that case as well. In a frame of reference moving with the three-phase contact line, we have
¼
d dy
8
h 0 Γ : k Γow 1 - m - k Γ in 0 < y < h Γ þ
ð18Þ
where y = z - X(t) and the surfactant distribution has been assumed to be in a quasi-steady state. Γ and its y-derivative are continuous everywhere. At great distances from the meniscus, we have boundary conditions Γ f 0 as y f ¥, and dΓ/dy f 0 as y f -¥. We take Γws(X, t) = Γ(y = 0) and Γos(X, t) = Γ(y = h). Equation 18 is a second-order linear differential equation and is easily solved analytically by standard means. The full solutions are complicated, and so for simplicity, we only state results valid for k- = 0, in the approximation that DskþΓow/U2Γm ws is small. At y = 0, we find 2
!3 þ k Γ h ow 4 5 Γðy ¼ 0Þ ¼ Γm ws 1 -exp Γm ws U þterms of order Ds or smaller
ð19Þ
and at y = h Γðy ¼ hÞ 2 !39 = þ Ds kþ Γow Uh D k Γ s ow 5 4≈ 1 exp 1 þ ; Ds U2 : U 2 Γm ws 8
h, Γ decays with distance as Γ(y = h) exp[-U(y - h)/Ds]. Dropping small terms, from eq 19, we find that 2 Γws ðX, tÞ ¼
4 Γm ws 1 - exp
!3 kþ Γow h 5 - m Γws U
ð21Þ
The surfactant concentration on the oil-solid interface just downstream of the advancing meniscus, Γos(X, t), is seen from eq 20 to be nonzero but small when the surface diffusivity of (24) Kumar, N.; Varnasi, K.; Tilton, R.; Garoff, S. Langmuir 2003, 19, 5366– 5373. (25) Churaev, N. V.; Martynov, G. M.; Starov, V. M.; Zorin, Z. M. Colloid Polym. Sci. 1981, 259(7), 747–752. (26) Kao, R.; Wasan, D.; Nikolov, A.; Edwards, D. Colloids Surf. 1988, 34(4), 389–398.
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surfactant is small. Furthermore, as U increases, Γos(X, t) decreases. For the study of forced flows at least, for which U will be greater than the low value obtained in spontaneous unforced imbibition, it therefore seems reasonable to neglect the presence of surfactant downstream of the advancing meniscus. It is, however, clear from eq 10, which for small values of the interface adsorptions may be linearized to give σow(Γow) cos θ - σow(0) cos θ0 ≈ RT(Γws - Γos), that nonzero values of Γos act to reduce the effect of Γws in generating a positive capillary pressure driving imbibition [i.e., a positive value of σow(Γow) cos θ]. This must have an effect on spontaneous imbibition, as has indeed been demonstrated in Churaev et al.,25 which predicts a finite bulk surfactant concentration below which no imbibition occurs. We shall see below that the present calculation, neglecting the downstream presence of surfactant, predicts spontaneous imbibition, albeit at a rate tending to zero as the concentration is reduced, for any nonzero value of surfactant concentration. From here on, we focus exclusively on the situation Γos 0. Equation 21, when combined with eq 16 and inserted into eq 15, completes the specification of boundary conditions for eq 12: Dc -πa Db ðX, tÞ Dz 0 " #1 k þ h Γm m@ ow KcðX, tÞ A ¼ 2πaUΓws 1 - exp U Γm ws 1 þ KcðX, tÞ
chemical equilibrium between the surfactant concentration in the bulk and that on the three surfaces. This completes the specification of the basic mathematical model. For computational purposes, it is most convenient to work on a domain of fixed size. We therefore introduce a new coordinate ξ ¼
z XðtÞ
ð24Þ
in terms of which eq 12 becomes Dc U Dc Db D2 c þ ð1 - ξÞ ¼ 2 2 Dt X Dξ X Dξ
ð25Þ
which may be written in conservation form as Dc D U Db Dc Uc ¼ ð1 - ξÞc - 2 Dt Dξ X X X Dξ
ð26Þ
(Recall that U is the time-varying velocity of the meniscus, equal to the cross-section averaged fluid velocity.) The boundary and initial conditions are
2
ð22Þ
cð0, tÞ ¼ cb
ð27Þ
cðξ, 0Þ ¼ cb
ð28Þ
and Inserting eq 21 into eq 11, using eq 16, and then putting the result into eq 5 gives a quadratic equation for U: ΔPd ¼
2 R KcðX, tÞ σ ow ð0Þ cos θ0 þ RT Γm a U ow 1 þ KcðX, tÞ
8U ¼ 2 ½ μw X þ μo ðL -XÞ a
The condition at the meniscus is
ð23Þ
where we have written kþh = R, which has the dimensions of velocity. The meaningful root is the positive one, since we have throughout assumed movement of surfactant from the oilwater to the water-solid interface. The terms within the round brackets on the left-hand side of this expression are equal to σow(Γow) cos θ. Bain27 presents a model for this process in which the transfer of surfactant from the meniscus to the solid surface occurs by convection. Rame28 points out that convective transfer of surfactant through the contact line is not consistent with removal of the stress singularity there using a slip boundary condition; when the slip conditions are used, the interface velocities are zero at the contact line and so any mass transfer must be by nonconvective processes. Chesters and Elyousfi29 treat the singularity by imposing an inner cutoff length on the Stokes equations rather than a slip boundary condition; as a result, interfacial velocities are nonzero as the contact line region is approached and convective transfer is possible. Starov et al.,30 section 5.2, following Churaev et al.,25 give yet another different treatment of the relationship between the surfactant concentration on the meniscus, solid, and in the bulk. Nonzero surfactant adsorption is allowed on the oilsolid interface ahead of the advancing contact line, and concentrations are determined through applying the conditions for (27) Bain, C. D. Phys. Chem. Chem. Phys. 2005, 7, 3048–3051. (28) Rame, E. J. Fluid Mech. 2001, 440, 205–234. (29) Chesters, A. K.; Elyoufsi, A. B. A. J. Colloid Interface Sci. 1998, 207, 20–29. (30) Starov, V. M.; Velarde, M. G.; Radke, C. J. Wetting and Spreading Dynamics; Surfactant Science Series; CRC Press: Boca Raton, 2007; Vol. 138.
12596 DOI: 10.1021/la901781a
-πa2 Db
Dc ð1, tÞ Dξ
8 " #9 < m R Γow Kcð1, tÞ = ¼ 2πaXUΓm ws 1- exp : U Γm ws 1 þ Kcð1, tÞ ;
ð29Þ
The domain size X(t) evolves according to eq 1, with U given by eq 23. Equation 26 is easily solved numerically on a uniform grid; centered differences suffice for the diffusive term, while for stability the advective term is upwinded. 3.1. Approximate Solutions. 3.1.1. Late Times. We seek an approximate asymptotic solution, valid as t f ¥, in the special case with μo = μw = μ and introduce the dimensionless time τ = R2t/Db. We assume that as τ f ¥ the solution may be expanded in the form "
cð1Þ ðξÞ cðξ, tÞ ∼ cb c ðξÞ þ 1=2 τ
#
ð0Þ
ð30Þ
Db ð0Þ 1=2 X τ R
ð31Þ
with XðtÞ ∼ and UðtÞ ∼ R
U ð0Þ τ1=2
ð32Þ
Using these expressions, we see from eq 1 that X(0) = 2U(0). Also, from eq 27, c(0)(0) = 1 and c(0)(1) = 0. Turning now to eq 23 with Langmuir 2009, 25(21), 12591–12603
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μo = μw, we see that we must impose c(0)(1) = 0, for otherwise there is an unbalanced term growing as τ1/2 arising from the factor of 1/U. Equation 25 may be written as Dc Dc D2 c X 2 þ XUð1 - ξÞ ¼ Db 2 Dt Dξ Dξ
dcð0Þ D2 cð0Þ ¼ dξ Dξ2
ð34Þ
which is easily demonstrated to have solution h i ð0Þ ð0Þ exp - X 2U y2 dy h i cð0Þ ðξÞ ¼ R 1 X ð0Þ U ð0Þ 2 y dy 0 exp 2 R 1 -ξ 0
ð35Þ
satisfying the boundary conditions previously found for c(0) at the two ends. It remains to determine U(0). From eq 23 at O(τ0), we have 2 RTΓm 0 ow Kcb ð1Þ σ ow ð0Þ cos θ þ c ð1Þ ¼ 0 ΔPd þ a U ð0Þ
ð36Þ
Obtaining this balance is the reason for selecting the time dependence of the second term in eq 30. We note for future use that the terms within round brackets in eq 36 are the leading order expression for σow(Γow) cos θ. Also, from eq 29 at O(t0), we have dcð0Þ ð1Þ -πa2 cb dξ 8 " #9 < = m Kc Γ b ow ð1Þ ¼ 2πaX ð0Þ U ð0Þ Γm ws 1 -exp - ð0Þ m c ð1Þ : ; U Γws
ð37Þ
which may be rewritten using eq 36 as dcð0Þ -πa2 cb ð1Þ dξ 8 " #9 aΔPd 0 = < þ σ ð0Þ cos θ ow 2 ð38Þ ¼ 2πaX ð0Þ U ð0Þ Γm ws 1 -exp : ; RTΓm ws Differentiating eq 35 and inserting the result into eq 38, we find h
1
i ð0Þ ð0Þ exp - X 2U y2 dy 8 " #9 aΔPd 0 = ð0Þ ð0Þ m < þ σ ow ð0Þ cos θ X U Γws ¼4 1 -exp 2 ; 2 acb : RTΓm ws R1 0
ð39Þ
which is a transcendental equation for X(0)U(0)/2. Rewriting the integral in terms of the error function, Abramowitz and (31) Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions; Dover: New York, 1972.
Langmuir 2009, 25(21), 12591–12603
1 qffiffiffiffiffiffiffiffiffiffiffiffiffi
ð33Þ
and unless the leading behavior of c(ξ, t) is independent of t, the factor of X2 generates a term that cannot be balanced by the other terms in the equation; this is part of the justification for the form assumed in eq 30. We thus have, at O(τ0), X ð0Þ U ð0Þ ð1 - ξÞ
Stegun31 (section 7.4.32), we find after simple rearrangements that
erf
X ð0Þ U ð0Þ 2
8 " #9 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aΔPd 0 = < pffiffiffi X ð0Þ U ð0Þ Γm þ σ ð0Þ cos θ ow ws 1 -exp 2 ¼2 π ; 2 acb : RTΓm ws ð40Þ The left-hand side is a monotonic decreasing function of X(0)U(0)/2, infinite at the origin and finite at infinity, while when the quantity A defined in eq 41 is positive the righthand side is a monotonic increasing function of this group, zero at the origin and infinite at infinity. There is thus a unique positive root for X(0)U(0)/2, which we may write as F(A ), where the dimensionless group 8 " #9 aΔPd 0 = < pffiffiffi Γm þ σ ð0Þ cos θ ow A ¼ 2 π ws 1 -exp 2 ; acb : RTΓm ws
ð41Þ
It is easy to show that F(A ) ∼ (π1/2)/2A when A is large, and F(A ) ∼ 1/A 2 when A is small. When ΔPd = 0, the group of constants in the argument of the exponential is negative because cos θ0 < 0 (the capillary is initially hydrophobic). As ΔPd increases, the argument of the exponential becomes positive and so A can become negative, and no solution exists. This sets a limit to the values of ΔPd for which this approximate solution is valid; we must have ΔPd < -2σow(0) cos θ0/a. At large times, X dX/dt = XU ∼ DbX(0)U(0) = 2DbF(A ), so X(t) ∼ [4DbF(A )t]1/2. The meniscus advances with square root of time kinetics, the velocity being proportional to the square root of the surfactant bulk diffusivity Db and also dependent on A (which does not contain the bulk diffusivity or R). The smaller A , the faster the invasion (for fixed Db); broadly speaking, small A corresponds to a large inlet surfactant concentration cb or to small monolayer capacity Γm ws or to a value of ΔPd close to the maximum allowed. The concentration profile eq 35 can be written as
cð0Þ ðξÞ ¼
erf
hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i F ðA Þð1 -ξÞ hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii erf F ðA Þ
ð42Þ
This is near-linear when F(A ) is small, that is, when A is large and is approximately equal to 1 except in a thin boundary layer near ξ = 1 when F(A ) is large, that is, when A is small. Once the root of eq 40 is found, the value of c(1)(1) follows, if required, from eq 36 using X(0) = 2U(0). It remains to consider the conditions for the validity of the large time approximation. Central to this approximation is c(1, t) ≈ 0, but the initial conditions have c(1,0) = cb. It seems reasonable that large times should be understood as meaning times greater than the time scale for c(1, t) to decay from its initial value to near-zero. Now, eq 25 evaluated at ξ = 1 gives (∂c/∂t)(1, t) = (Db/X2)(∂2c/∂ξ2)(1, t), from which we estimate that the time T for decay of c(1, t) scales as T ∼ X2/Db [at least when A is not small so there is no concentration boundary layer and the characteristic length scale of c(ξ, t) is 1]. From eq 23, we see that DOI: 10.1021/la901781a
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U ¼O to
Hammond and Unsal
RRTΓm ow
aΔP - 2d
Kcb -σ ow ð0Þ cos θ0 1þKcb
estimate
T ∼Db =U ¼ O 2
at early times. If we use this value
X ∼ UT , ( aΔP Db R2
-
then, it follows that ) 2 2 0 d 1þKcb 2 -σ ow ð0Þ cos θ . Thus, the RTΓm Kcb
Rt . Db
"
τ2 X ð1Þ
ow
long time asymptotic solution can be expected to be valid provided 2
Using eq 1, we find X(1) = U(1), and from eq 13, we have c(1)(0) = 0. Substituting the expansions for X(t) and U(t) into eq 33, we find
d - aΔP 2
# 0 2
- σ ow ð0Þ cos θ RTΓm ow
1 þ Kcb Kcb
2
Dc Dc D2 c þ τ2 X ð1Þ U ð1Þ ð1 - ξÞ ¼ 2 Dτ Dξ Dξ
ð47Þ
and thence, for small values of τ, D2 cð1Þ ¼0 Dξ2
2 ð43Þ
ð48Þ
The concentration perturbation c(1) is thus linear in ξ. Turning now to eq 23, we find
The smaller R, the longer we must wait for the similarity solution to set in. For the base case parameters used in the numerical solutions reported below, the right-hand side of this expression is of order one. An important feature of this solution is that the parameter R does not appear anywhere in the leading order problem, influencing only the value of c(1)(1) and thence higher order corrections and the time at which the large time approximation becomes valid. Thus, at large times, the rate of spontaneous imbibition predicted by the Tiberg model is controlled by the rate of bulk diffusion to the advancing meniscus (i.e., by Db) but is not dependent at leading order on the rate of transfer of surfactant from the meniscus onto the solid surface through the three-phase contact line. This property can be seen at large times in numerical solutions of the complete problem, as will be illustrated in the next section. Also significant is the fact that c(0)(1) = 0. At large time, the surfactant concentration vanishes, to leading order, at the position of the advancing meniscus. Given that the surfactant concentration in the immediate vicinity of the meniscus is zero to leading order, we must conclude that the implication of eq 36, that [(2ΔPd/a) þ σow(Γow) cos θ] ≈ 0, should be interpreted as meaning cos θ = aΔPd/2σow(Γow) to leading order. When ΔPd = 0, this feature can be seen in the numerical solutions presented below; contact angles are near π/2 for all cases studied, at all times. Although we often speak of wettability-altering surfactants as bringing about oil recovery by changing an oil-wet rock into a water-wet state and so generating strong interfacial forces that draw water into the pore space, in fact, the change is to a state of neutral wettability where capillary forces are near-negligible. When ΔPd > 0, a dynamic balance is set up in which the contact angle is close to cos-1[aΔPd/2σow(Γow)], and ΔPc ≈ ΔPd. 3.1.2. Early Times. It is also possible to find a simple asymptotic solution that is valid for small values of the dimensionless time τ = R2t/Db, and we sketch out that calculation here. Take μo = μw = μ and set X0 = 0 for convenience. Pose expansions of the form cðξ, tÞ ∼ cb ½1 þ τcð1Þ ðξÞ
ð44Þ
Db XðtÞ ∼ 0 þ τ X ð1Þ R
ð45Þ
UðtÞ ∼ RU ð1Þ
ð46Þ
ΔPd þ
2 RTΓm 8RU ð1Þ μL ow Kcb σ ow ð0Þ cos θ0 þ ¼ ð49Þ ð1Þ a a2 1 þ Kcb U
at lowest order in τ. This is a quadratic equation for U(1), but because, using the parameter values quoted below, the term on the right-hand side is small as compared with the first and second terms on the left, the former may be discarded to give a linear equation. We then find U ð1Þ ¼
RTΓm Kcb ow 0 1 þ Kc -ðaΔPd =2Þ - σow ð0Þ cos θ b
ð50Þ
We see again that (aΔPd)/2 < -σow(0) cos θ0 for U(1) to be positive. This sets a limit on the ΔPd values for which this solution is good. Another restriction on ΔPd arises relating to the neglect of the term on the RHS of eq 29. Lastly, substituting the proposed expansions into eq 29 and taking the leading terms for small values of τ gives -
ð1Þ
Dc ð1Þ ¼ 2X ð1Þ U Dξ
8 " #9 < m 1 Γ Kcb = 1 - exp - ð1Þ ow m acb : U Γws 1 þ Kcb ;
m ð1Þ Γws
ð51Þ which supplies the second boundary condition needed to fully define c(1)(ξ). This expression may be rewritten, using eq 51, to give " #2 Dcð1Þ RTΓm Kcb ow ð1Þ ¼ 2 Dξ -aΔPd =2 - σ ow ð0Þ cos θ0 1 þ Kcb 8 " #9 < 0 = Γm aΔP =2 þ σ ð0Þ cos θ d ow ow 1 -exp ; acb : RTΓm ws ð52Þ The crucial points about this early time solution are that the velocity is constant and of order of R, and X(t) grows linearly in time. This solution is valid provided that τ = R2t/Db is small. Because R can be very small, this can correspond to quite large dimensional times and to significant dimensional values of X(t).
with
4. Results
and
12598 DOI: 10.1021/la901781a
Using the mathematical model described above, spontaneous and forced imbibitions into an oil-wet capillary are studied. The surfactant solution was applied at one end, and the other end is open to an infinite reservoir of oil at fixed pressure. A pressure difference was imposed between the ends of the capillary, and simulations were performed at a series of differential pressures, Langmuir 2009, 25(21), 12591–12603
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Table 1. Model Variables for the Reference Case variable
value
L (tube length) a (tube diameter) μo (oil viscosity) μw (water viscosity) Db (bulk diffusivity) cb (initial surfactant concentration) K σow(0) (initial surface tension) cos θ0 (initial contact angle) R T Γm ws Γm ow R
1 cm 0.1 mm 10-3 Pa s 10-3 Pa s 10-9 m2 s-1 10-2 mol m-3 1.3 104 m3 mol-1 30 10-3 N/m -0.1 8.314 J K-1 mol-1 300 K 2 10-6 mol m-2 2 10-6 mol m-2 10-5 m s-1
starting from some negative values. The pressure was increased stepwise at each simulation up to a value that is greater than the capillary threshold pressure {ΔP0c = [2σow(0) cos θ0/a], eq 11}. Investigated were the following questions: • How does the differential pressure affect velocity of the moving meniscus? • How does the surfactant concentration at the interface change as the interface advances in the capillary? • How does the contact angle change as the interface advances? • How does the concentration of surfactant on the solid surface vary with applied pressures? • How is the capillary pressure affected? • What happens if the applied pressure is negative? The results are analyzed for three cases, (1) the spontaneous case (ΔPd = 0 Pa), (2) ΔPd is smaller than capillary threshold pressure ΔP0c , and (3) ΔPd is larger than ΔP0c . 4.1. Effect of Differential Pressure on Velocity. Simulations were run at different differential pressures, ΔPd, and the velocity of the moving meniscus was noted. Variable values are listed in Table 1. The capillary threshold pressure (ΔP0c ) of the capillary is calculated as 60 Pa using eq 11. We will refer to this case as the “reference case”, to which further results will be compared. The values of the parameters listed in Table 1 require little comment, being typical of small scale laboratory oil-water systems. The bulk diffusivity is of the same order of magnitude as those shown in Figure 4 of Lin et al.32 The bulk concentration chosen is about 1/10 of the critical monolayer concentration (CMC) value of 0.11 mol m-3 reported for C12E10 by Kumar et al.;24 this value cannot be much increased with the present model, as the oil-water interfacial tension predicted by eqs 9 and 16 can become negative. Kumar et al.24 also measured a value -6 mol m-2 for the adsorption of C12E10 onto of Γm ws = 2.2 10 a hydrophobically treated surface, which leads (approximately) m to the value of RTΓm ws quoted above. A value for Γow = 2.0 10-6 mol m-2 is found in Table 1 of Lin et al.32 for C12E10 at the air-water interface. The same table yields the value of K (equal to 1/a in their notation). Figures 3-5 show the results of a numerical integration of the system consisting of eqs 1 of 26. The numerical grid had 100 points. The simulation was repeated with differential pressures (ΔPd) varying from -55 to 105 Pa. A plot for the velocity, U, at 1 s for different ΔPd is shown in Figure 3. The reason why 1 s is used is because at high pressures the meniscus moves very fast and reaches the other end of the capillary very quickly. (32) Lin, S.; Tsay, R. Y.; Lin, L. W.; Chen, S. Langmuir 1996, 12, 6530–6536.
Langmuir 2009, 25(21), 12591–12603
Figure 3. Velocity at the meniscus vs differential pressure at t = 1 s.
When the differential pressures are negative, the meniscus is still able to imbibe within the capillary (Figure 3). This is because the surfactant molecules are adsorbed at the capillary walls and alter the wettability of the capillary, lowering the contact angle and the surface tension. However, the process is extremely slow, and the velocity values are tiny. As the pressure increases, a slow increase at the front velocity is observed until the threshold pressure. At this point, the pattern changes, and velocity increases dramatically. Further results are presented at three representative differential pressures, spontaneous imbibition (ΔPd = 0 Pa), one pressure value below (ΔPd = 50 Pa), and one above the threshold pressure (ΔPd = 75 Pa). Although additional analysis was made at different pressures, they are not illustrated as the results are broadly similar. From the upper track of Figure 4, we see that the surfactantdriven imbibition process is slow for the spontaneous case; with these parameter values, it takes around 1 day for the meniscus to travel the 1 cm length of the capillary. Under forced conditions with ΔPd > ΔPc0, the process is much faster; it takes around 5 s, and the meniscus position versus time is almost linear. Using eq 5 and setting the capillary pressure to 2σow(0)/a, we find that the time for spontaneous imbibition into a strongly water-wet tube is only a few tenths of a second. The lower track of Figure 4 shows the concentration profiles at a set of times equispaced between the start of the simulation and the instant the meniscus arrives at the far end of the capillary (the initial state is a very short horizontal line at c = 0.01 and is almost invisible at the scale of the plot). With the present parameters, the concentration is almost linear in z for spontaneous case and is near zero at the position of advancing meniscus. At 75 Pa, the concentration stays unchanged and is equal to the initial surfactant concentration almost everywhere apart from a thin boundary layer adjacent to the advancing meniscus. At applied pressures larger than the threshold for entry into the oil-wet capillary, there is no need for wettability alteration to occur, and the speed of advance of the meniscus is controlled by a balance between the viscous forces and the applied pressure, which leads to a comparatively rapid motion. At this speed, there is little time for any surfactant on the meniscus to transfer to the solid surface, and as a result, the concentration on the solid surface remains small, and so, there is little change in wettability. Furthermore, because the flux of surfactant off the meniscus and onto the solid is small, only a small flux of surfactant from the bulk onto the meniscus is required to maintain the situation. Away from the meniscus, the surfactant molecules move with the average speed of the fluid, as does the meniscus. Diffusion simply depletes the bulk surfactant concentration in the immediate vicinity of the meniscus and so replenishes the material transferred from the meniscus to the underlying solid. DOI: 10.1021/la901781a
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Figure 4. Meniscus position of surfactant concentration profiles under different differential pressures of ΔPd = 0, 50, and 75 Pa. The upper track shows the location of meniscus. The lower track plots the concentration profiles at a set of times equispaced between t = 0 s and t = 105 s. The units are x (m), t (s), and z (m). 12600 DOI: 10.1021/la901781a
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Figure 5. Plot of meniscus properties of the moving interface under different pressures. Langmuir 2009, 25(21), 12591–12603
Hammond and Unsal
For times larger than around 1 s, we see in Figure 5 that features characteristic of the large time similarity solution have set in. The surfactant concentration near the meniscus is small for all cases. The normalized surfactant concentration at the meniscus decreases gradually from 1 to 0 by time t ≈ 1 s. For higher pressure (i.e., ΔPd = 75 Pa), the meniscus reaches the end of the tube before the large time structure is fully developed. The number of surfactant molecules at the interface (Γow) is also decreasing in time, with a larger relative change at 75 Pa. This results in an increase in oil-water surface tension in all cases. The number of adsorbed surfactant molecules at the solid surface (Γws) is initially constant at very early times but then decreases and reaches a constant value at later times. At lower pressures, the initial value is higher as compared to higher pressures. This is because there is more time for the surfactant to transfer from the meniscus to the solid surface at lower speed. The change in capillary threshold pressure was also observed as the solution imbibes. Surfactant adsorbs on the solid surface and so changes the wettability and also affects the surface tension and the contact angle. This reflected in the capillary pressure. The applied differential pressure and the threshold pressure balance each other (Figure 6). To understand this, we note that if ΔPd < ΔP0c , then we must have some wettability (and/or surface tension) alteration if the meniscus is to move into the capillary. This requires that surfactant is present on the meniscus, transferred to the water solid interface. Now, only diffusion transfers surfactant onto the meniscus. Diffusion is weak because Db is small. Because diffusion is weak, it only supplies a small flux of surfactant onto the meniscus. Therefore, there is only a small flow of surfactant onto the water solid interface through the three-plane contact line. The meniscus cannot move forward rapidly, because if it did the adsorbed concentration on the water solid surface would be small. So, there would be little change in wettability. Hence, the meniscus moves slowly. Because the meniscus moves slowly, viscous pressure losses are small; hence, the pressure jump across the meniscus is approximately equal to the applied pressure difference across the entire capillary. So, (ΔPd - ΔPc) ≈ 0, as seen in Figure 6 (ΔPd < ΔPc). When ΔPd > ΔP0c , we do not need to alter wettability for the meniscus to move into the capillary, so 6 0. fast movement is possible at (ΔPd - ΔP0c ) ¼ The present calculation implies that diffusion controls the rate at which surfactant is transported to the vicinity of the advancing three-phase contact line, and for realistic parameter values, diffusion is weak. It follows from this observation that during spontaneous imbibition the advancing contact angle in a flow driven by wettability alteration must be close to π/2, at least when oil-water interfacial tension is not reduced to ultralow values or there is no localized mechanical resistance to advance of the contact line. For if this were not so, a large capillary driving force would be set up that would drive a fast flow requiring a large flux of surfactant through the contact line region to be sustained to support the process of wettability alteration. Realistic, weak diffusion cannot generate such a large flux, and so, contact angles significantly different from π/2 are not possible within the scope of the present model. Hence, if slow flow and a contact angle significantly different from π/2 are observed in a surfactantdriven invasion with ΔPd = 0, then we must conclude that either the interfacial tension has been reduced to near-zero or there is an additional source of mechanical resistance to flow operating at the moving contact line. A direct observation of contact angle during a surfactant-driven imbibition would be a strong test of the present theory. 4.2. Effect of Bulk Diffusivity (Db). We used a realistic Db value in the reference case. However, for the sake of model Langmuir 2009, 25(21), 12591–12603
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Figure 6. Change in the capillary pressure as the meniscus advances in the capillary. The units are P (Pa) and t (s). At 75 Pa, the meniscus reaches the end of the capillary at around 30 s; therefore, there are no more data points after this time.
analysis, the behaviors at different Db values are also investigated. The other variables are kept the same as the reference case. DOI: 10.1021/la901781a
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Figure 8. Simulation with a reduced value of R = 10-7. All other parameters are unchanged. The solution sets in after 104 s.
Figure 7. Plot of meniscus properties of the moving interface under higher Db for the spontaneous case.
The displacement is faster with a larger value of surfactant diffusitivity and slower with a smaller value (Figure 3). For a comparison at 1 s (where all cases are in the late time regime), Db values had to be below 5 10-9. As Db is increased, the similarity solution sets in later in times (eq 43), at ≈60 s for Db = 10-8 and ≈500 s for Db = 10-7 (around 1 s at the reference case) (Figure 7). 12602 DOI: 10.1021/la901781a
4.3. Effect of r. Resistance to mass transfer between meniscus and solid surface, as represented by R, limits the rate of advance of the meniscus at early times. At late times, the resistance to mass transfer associated with diffusion through bulk aqueous phase dominates. The value chosen for R in the reference case is somewhat arbitrary since no direct measurements are available. However, the short time approximate solution presented above shows that R sets the cutoff value, limiting the meniscus velocity at early times, and this observation can be used to guide the selection of the value of R. Because velocities can be Langmuir 2009, 25(21), 12591–12603
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expected to decay monotonically in time for invasion processes, R determines the maximum velocity in the process and can be used to determine an upper bound on dimensionless groups such as the Peclet number, Pe = aU/Db. For the validity of eq 12, we require this parameter to be small at all times; thus, R should satisfy aR/Db < 1. In fact, aR/Db = 1 for the parameters given in the reference case, so for these values the representation of surfactant transport in the bulk by eq 12 is strictly valid only at times much greater than Db/R2 = 10 s, since only then has the velocity decayed from its early time value of order R. (For the reference case parameters a2/Db = 10 s also, and so both conditions for the validity of eq 12 are satisfied after that time.) Figure 8 shows the solution computed with R = 10-7 ms-1, and all other parameters are as in the reference case. The similarity solution sets in much later in time (after 104 s, as opposed to 1 s at the reference case; this scaling with R is consistent with eq 43).
5. Conclusions The calculations presented above show that the spontaneous invasion of aqueous surfactant solution into an initially oil-wet and oil-filled capillary tube is a slow process. At early times, the rate of penetration is controlled by the dynamics of transfer of surfactant from the meniscus to the solid surface through the three-phase contact line, whereas at late times the rate of penetration is controlled by the rate of diffusion of surfactant in the bulk. Under positive applied differential pressures, the process is faster, with the velocity becoming substantial once the applied pressure exceeds the capillary threshold pressure for entrance into the oil-wet capillary. For lower values of differential pressure, the surfactant solution will imbibe slowly into the capillary, and indeed, through the wettability altering action of the surfactant, solution will imbibe even at negative differential pressures provided these are not too large in magnitude. Increased pressures result in higher surfactant concentrations at the meniscus, because as the meniscus moves faster there is less time for molecules to be transferred through the contact line region onto the solid wall. Consequently, the wettability does not change a great deal at higher velocities. Because the penetration of the meniscus into the capillary at subthreshold values of the applied pressure depends on substantial wettability alteration, with low solution concentrations and ΔPd < ΔP0c , the meniscus speed is low. Because it is unlikely that the diffusivity of a species in aqueous solution can be greatly increased, the only scope for increasing the rate of invasion in the late time regime is to reduce the value of the parameter A . This can be done by increasing the surfactant concentration (although once the CMC is exceeded, the concentration of surfactant monomers ceases to increase, and a further
Langmuir 2009, 25(21), 12591–12603
potentially rate limiting process, the break-up of micelles, becomes relevant), by reducing the monolayer capacity Γm ws or of course by applying a driving pressure ΔPd. In early time, the rate of invasion is controlled by the parameter R, and so, higher initial invasion rates will be achieved if the rate of transfer from meniscus to solid can be increased in some way. The model also predicts that contact angles during unforced flow, ΔPd = 0, will be close to π/2. A contact angle significantly smaller than π/2 is not consistent with slow invasion, unless strong additional frictional processes occur at the moving contact line. Fast invasion, on the other hand, is not consistent with realistic values for surfactant diffusion. The model predicts that unforced invasion of wettability-altering solution takes place at conditions of near-neutral wettability, with the displacement rate set by the rate of diffusive transport of surfactant. This is completely different from the naive view in which the flow is driven by a surfactant-induced change to strongly water-wet conditions, and the rate is controlled by a balance between capillary pressure at the advancing meniscus and viscous resistance in the bulk flowing fluids. Similar arguments lead us to expect a balance between the applied differential pressure and the capillary pressure when 6 0, provided ΔPd < ΔP0c . Again, the meniscus can only ΔPd ¼ penetrate into the capillary if wettability alteration occurs, which requires that surfactant be transported to the meniscus in sufficient quantity to develop a significant concentration on the solid surface. Such transport occurs by diffusion, and for small values of Db, diffusion is slow. Hence, the overall speed of the meniscus must be slow, viscous forces will be negligible as a result, and so the capillary pressure must adjust to be in balance the applied pressure, ΔPd ≈ ΔPc. Bain,27 Starov et al.,30 and Tiberg et al.18 each set up a different model for invasion of an aqueous surfactant solution into an initially hydrophobic oil-filled capillary. The most profound differences are in the modeling of the transport of surfactant in the immediate vicinity of the moving three-phase contact line. This is a reflection of the fact that rather little is known about such processes, especially from an experimental point of view. Turning to oil recovery processes in rocks, even such basic points as whether wettability-altering surfactant acts to form a hydrophilic coating over previously adsorbed hydrophobic material or cleans that material off to reveal the original hydrophilic mineral surface is the subject of present investigation.33 There is certainly great scope for experimentation and molecular dynamics simulation to shed light on the action and transport of wettability altering surfactants near moving contact lines in idealized systems, on mineral surfaces, and in rock cores. (33) Salehi, M.; Johnson, S. J.; Liang, J.-T. Langmuir 2008, 24(24), 14099– 14107.
DOI: 10.1021/la901781a
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