ARTICLE pubs.acs.org/Langmuir
Spontaneous Imbibition of Surfactant Solution into an Oil-Wet Capillary: Wettability Restoration by SurfactantContaminant Complexation Paul S. Hammond and Evren Unsal* Schlumberger Cambridge Research, Cambridge, U.K., CB3 0EL ABSTRACT: For a given type of rock, the effectiveness of oil recovery through wettability alteration is highly dependent upon the nature of the water-soluble surfactant used. Different mechanisms have been proposed by others to explain wettability alteration by surfactants, and understanding the process is crucial to improve recovery performance. Known mechanisms include (1) surfactant adsorption onto the oil-wet solid surface (coating mechanism) and (2) surfactant molecules complexing with contaminant molecules from the crude oil which are adsorbed on the rock surface so as to strip them off (cleaning mechanism). With the second mechanism, the wettability is restored by lifting the contaminant layer away, exposing the rock surface which was originally water-wet. We previously focused on the numerical modeling of the surfactant coating mechanism (Hammond and Unsal Langmuir 2009, 25, 12591; 2010, 26, 6206), and we now present a numerical study for the cleaning process. Our new model shows that when a wettability altering surfactant solution is allowed to imbibe spontaneously and acts by the cleaning process, the meniscus advances more rapidly than when there was wettability alteration by coating alone. In our previous model there was a concentration threshold below which imbibition was not possible. That threshold arose because a finite amount of surfactant needs to be adsorbed onto the oil-wet surface to change the contact angle to a water-wet value, but the maximum amount that can be absorbed is limited by the requirement that it be in equilibrium with the surfactant concentration near the meniscus. In the new model, with the cleaning mechanism there is no such threshold, since the cleaning process is driven by the surfactant flux into the vicinity of the advancing meniscus rather than the surfactant concentration there. As long as there are surfactant molecules present in the aqueous solution, the flux is nonzero and molecule pairs can form and alter the wettability by removing the contaminant from the oil-wet surface. However, under very low surfactant concentrations, the process is extremely slow compared to at higher concentrations.
1. INTRODUCTION The efficiency of oil recovery from naturally fractured reservoirs by spontaneous imbibition of water depends on the degree of water-wetness of the rock. For strongly water-wet rock, the capillary forces allow water to imbibe and push the oil out of the rock pores. In a mixed- or oil-wet reservoir, however, the capillary forces are either weak or act in the wrong direction, and therefore, the spontaneous oil recovery rates are low or nonexistent. The rate of recovery from such reservoirs can be improved by dissolving low concentrations of surfactants in the injected water to alter the wettability of the reservoir rock to be more water-wet. For a given type of rock, the effectiveness of wettability alteration is highly dependent upon the nature of the surfactant used.1,2 For example, it was experimentally observed that anionic surfactants were not very efficient for recovery from oil-wet chalk rocks, but cationic surfactants performed better in changing the rock wettability to a more water-wet state in such cores.35 Several mechanisms have been proposed to explain wettability alteration by surfactants, and understanding such mechanisms is essential to improve the recovery performance. Known mechanisms include (1) surfactant adsorption onto the oil-wet solid surface (coating mechanism) and (2) bonding of surfactant molecules with contaminant molecules from the crude oil which are already attached to the rock surface and stripping them off the surface (cleaning mechanism).2 These two mechanisms are based on completely different principles. During the coating mechanism r 2011 American Chemical Society
the surfactant molecules are adsorbed on to the oil-wet rock surface and form a monolayer through hydrophobic interaction of their tails with the crude oil molecules which are adsorbed on the rock surface. As the tails are attached to the surface, the hydrophilic head groups of the adsorbed surfactant molecules are exposed toward the pore interior, forming a new layer that is water-wet. In contrast, the cleaning process involves a restoration of original wettability that is underneath the oil-wet contaminant layer. The surfactant molecules bond with the crude oil molecules that have been adsorbed on to the rock surface and detach them from the solid surface. The original rock surface, which is water-wet, is thus exposed. The bonding between the oil and surfactant molecules might be due to certain processes, i.e., the ionic nature of the molecules. The adsorption mechanism is believed to be the case between the anionic surfactant molecules and oil-wet chalk rock surfaces, while the cleaning mechanism is the process between the cationic surfactants and such rock surfaces.5 The driving force for the smart water technique, where a correct water composition and salinity can act as a tertiary recovery mechanism, is also believed to be the cleaning mechanism.6 Salehi et al.2 performed some experiments for both types of surfactant and verified the hypothesis of Standnes et al.4,5 Using both synthetic polyethylene and Berea cores, they studied the wettability alteration with different Received: December 6, 2010 Revised: February 14, 2011 Published: March 23, 2011 4412
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Figure 1. Problem geometry. Aqueous surfactant solution displaces oil from a circular cylindrical capillary tube. The walls of the capillary are initially coated with a layer of contaminant, making them oil-wet. Surfactant removes the contaminant, creating a more water wet surface. When clean, the walls of the capillary are water-wet.
surfactants. They observed that the ion-pair formation mechanism between the surfactant molecules and the adsorbed crude oil components on the rock surface was more effective in promoting recovery than the adsorption process. Previously, we published two numerical models for the mechanism of coating by surfactant adsorption onto the solid surface.7,8 We now propose a numerical study for the cleaning mechanism by molecule-pair formation between the surfactant and contaminant molecules. We do not speculate about the reasons for the molecule-pair formation but simply assume that a complexation process of some kind occurs, leading to stripping of contaminant from the rock surface. Our model focuses on what happens once the molecule pairs are formed, how the wettability of the solid surface is then affected, and how spontaneous imbibition progresses.
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 is taken to be pL pR ¼ ΔpðtÞ
ð3Þ
In the case of spontaneous imbibition, this pressure difference will be zero. The capillary pressure at any instant, Δpc, is defined as the difference between oil and water pressures across the meniscus so that pþ p ¼ Δpc
ð4Þ
Upon rearranging eq 2 we find
2. NUMERICAL DEVELOPMENT
ΔpðtÞ þ Δpc ¼
2.1. Geometry of Capillary Flow. We consider displacement
of oil, by an aqueous surfactant solution, from a circular cylindrical capillary of radius a and length L. The meniscus separating oil and aqueous phase is located at position z = X(t), where t is time and z measures axial distance along the capillary. The walls of the capillary are initially oil-wet because of a coating of insoluble contaminant material, but this can be removed by surfactant, thus exposing the underlying water-wet solid surface. Simultaneously, surfactant absorption can also take place on the solid surface. More detail on the effects of this wettability alteration by the cleaning process, and how it is envisaged to take place, will be given below. The problem geometry is sketched in Figure 1. The velocity of the meniscus is U(t), so dX ¼U dt
ð1Þ
subject to X(0) = X0 (which should be greater than zero in order to avoid early time velocity singularities). Ignoring flow processes in the immediate vicinity of the meniscus, the velocity of the meniscus is taken to be linked to pressure differences between various positions in the capillary through the Poiseuille flow formulas,9 U ¼
a2 ðpL p Þ a2 ðpþ pR Þ ¼ 8μw X 8μo ðL XÞ
ð2Þ
8U ðμw X þ μo ðL XÞÞ a2
ð5Þ
which will later be used to find U. 2.2. Adsorption, Contact Angle, and Meniscus Shape. The contact angle θ between surfactant solution and oil at the solid boundary is assumed to satisfy Young’s equation tot tot σow ðΓtot ow Þ cos θ ¼ σ os ðΓos Þ σ ws ðΓws Þ
ð6Þ
where σij is the interfacial tension of the interface between material i and material j, and Γtot ij is the total number of moles of material adsorbed per unit area on that interface (see chapters 1 and 2 of Starov et al.10 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 complete absence of contaminant and adsorbed surfactant σow ð0Þ cos θ0 ¼ σ os ð0Þ σws ð0Þ
ð7Þ
with cos θ0 > 0; the clean capillary surface is water-wet. Adsorption on an interface reduces the interfacial tension, and the surface properties of an ideal monolayer are related to each other by ! Γtot ij tot m ð8Þ σ ij ðΓij Þ ¼ σij ð0Þ þ RTΓij log 1 m Γij 4413
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where R is the universal gas constant, T is the absolute temperature, and Γm ij is the monolayer capacity. Combining 68 we find 0 σow ðΓtot ow Þ cos θ ¼ σ ow ð0Þ cos θ ! ! Γtot Γtot m m ws ðX, tÞ os ðX, tÞ RTΓws log 1 þ RTΓos log 1 Γm Γm ws os
ð9Þ This shows how adsorption of material at the oilsolution and solutionsolid interfaces will change the contact angle. 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 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 Δpc ¼
2σow ðΓtot ow Þ cos θ a
! 2 Γtot 0 m ws ðX, tÞ σ ow ð0Þ cos θ RTΓws log 1 ¼ a Γm ws !! Γtot os ðX, tÞ þ RTΓm ð10Þ os log 1 Γm os using 9. Hence, if the total amounts of material adsorbed on the tot solid surface on either side of the meniscus, Γtot ws (X,t) and Γos (X, t), are known, then eq 10 combined with eq 5 gives " ! 2 Γtot 0 m ws ðX, tÞ ΔpðtÞ þ σ ow ð0Þ cos θ RTws log 1 a Γm ws !# tot Γ ðX, tÞ m þ RTos log 1 os m Γos ¼
8U ðμ X þ μo ðL XÞÞ a2 w
ð11Þ
from which we can determine the speed of advance of the meniscus. This can then be used in eq 1 to update its position. Only positive root(s) for U will be meaningful, since we shall assume throughout that the meniscus moves toward the oil-filled end of the capillary and that movement of surfactant occurs from the oilwater to the liquidsolid interfaces rather than vice versa. 2.3. Surfactant Transport Away from the Contact Line: Outer Region. In order to find the total amount of material adsorbed on the solid surface near the moving meniscus, Γtot ws (X,t) and Γtot os (X,t), we must now consider the processes of transport of surfactant through the bulk aqueous phase, along the meniscus, through the moving three phase contact line onto the solid, and over the surface of the solid and the interactions between surfactant and pre-existing contaminant on the solid surface. Away from the meniscus, surfactant is adsorbed on the walls of the capillary and some contaminant may also be present. We assume that the amount of surfactant adsorbed, Γws(z,t), is in instantaneously achieved equilibrium with the cross-section averaged surfactant concentration in the adjacent fluid, c(z,t),
according to the modified Langmuir isotherm ! Γcon ws ðz, tÞ 1 Kws cðz, tÞ Γm ws m Γws ðz, tÞ ¼ Γws 1 þ Kws cðz, tÞ
ð12Þ
where Γcon ws is the surface concentration of adsorbed contaminant con m so that Γtot ws = Γws þ Γws, Γws is the monolayer capacity, and Kws is a measure of the relative rates of adsorption and desorption from the interface. For 12 to be valid, the initial surfactant concentration must be below the cmc (critical micelle concentration). If we write πa2Ξ(z,t) for the total amount of surfactant per unit length of capillary at location z, then πa2 Ξðz, tÞ ¼ πa2 cðz, tÞ þ 2πaΓws ðz, tÞ
ð13Þ
But Γws(z,t) is related to c(z,t) by 12, and inserting this in 13, we obtain a quadratic equation linking the total and flowing amounts of surfactant. The physically meaningful (i.e., positive) solution of the resulting equation for the flowing concentration is con Ξðz, tÞ ðΓm 1 ws Γws ðz, tÞÞ cðz, tÞ ¼ 2 a 2Kws ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s con Ξðz, tÞ ðΓm 1 2 Ξðz, tÞ ws Γws ðz, tÞÞ þ þ 2 a 2Kws Kws ð14Þ The adsorbed concentration can then be found from eq 12 or 13. When the Peclet number aU/Db is small, and for times large compared to a2/Db, surfactant transport within the capillary is described by ∂Ξ ∂c ∂ ∂c 2Ds ∂Γws þU ¼ Db þ ð15Þ ∂t ∂z ∂z ∂z a ∂z where Db is the bulk diffusivity of surfactant within the aqueous phase, Ds is the surface diffusivity of surfactant on the capillary wall, U is the average axial velocity of fluid within the capillary, which is of course equal to the speed of the meniscus, and c is related to Ξ by 14. The small Peclet number condition ensures that shear-enhanced (Taylor) dispersion contributions to axial spreading are negligible 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 15 is assumed to be subject to the inlet condition cð0, tÞ ¼ cb
so that
Ξð0, tÞ ¼ cb þ
2Γm Kws cb ws ð16Þ a 1 þ Kws cb
and to the initial condition cðz, 0Þ ¼ cb
so that
Ξðz, 0Þ ¼ cb þ
2Γm Kws cb ws ð17Þ a 1 þ Kws cb
where cb is the bulk concentration of surfactant in the initial aqueous solution. The second boundary condition required by the second-order partial differential eq 15 will be addressed below, once we have detailed the model for processes in the vicinity of the moving three phase contact line. 4414
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Figure 2. Sketch of the model for transfer of surfactant from meniscus to solid surface across the moving contact line region, showing the possibility of adsorbed surfactant (orange) on the solid surface on the oil side of the advancing meniscus and adsorbed contaminant (mauve). The underlying solid surface is water-wet when clean. The sketch is drawn in a frame of reference moving with the oilwater meniscus.
The contaminant is taken to be immobile on the solid surface away from the immediate vicinity of the moving three phase contact line, so ∂Γcon is ¼0 ∂t
where
i ¼ w, o
ð18Þ
At large distances downstream of the advancing meniscus con Γcon (¥), a given constant. Upstream of the meniscus we os fΓ con take Γws = 0 initially, corresponding to a clean surface. Together these supply the initial condition for 18. The single boundary condition required will be discussed below as part of the analysis of processes in the vicinity of the advancing meniscus. At the meniscus, the amount of surfactant adsorbed on the oilwater 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
Kow cðX, tÞ 1 þ Kow cðX, tÞ
ð19Þ
where Γm ow is a monolayer capacity and Kow is a measure of the relative rates of adsorption and desorption from the interface. In writing this expression, we are assuming that the surface concentration of surfactant on the meniscus is spatially uniform, consistent with treating the solution concentration as uniform across the capillary radius by virtue of the restrictions on eq 15. For this assumption to be reasonable, we must have diffusive or Marangoni transport processes on the surface of the meniscus strong enough to overcome any effects of surface advection (coupled bulk and surface transport of surfactant is discussed in chapter 2 of Krotov and Rustanov11). We assume that there is no contaminant adsorbed on the meniscus. 2.4. Surfactant Transport near the Contact Line: Inner Region. To complete the system of equations, it is necessary to tot give relations linking c(X,t), Γow, Γtot os (X,t), and Γws (X,t), that is, a description of the transfer of surfactant from the oilwater interface to the liquidsolid interfaces through the moving contact line and its interaction with the contaminant material. These are processes occurring on a length scale thay is very much shorter than the capillary diameter or length scales considered up
to this point. In effect, we now make an analysis valid on a short, or inner, length scale, with the previously considered, outer, variables playing the part of boundary or matching conditions at infinity for the subsequent inner analysis (see chapter 5 of Van Dyke,12 although we do not use the formal machinery of the method of matched asymptotic expansions here). We consider a system in which surfactant molecules adsorb on the solid surfaces and can interact with pre-existing adsorbed contaminant molecules to solubilize these molecules in the oil phase. Thus, the picture is one of wettability modification by cleaning the solid surface. Contaminant is assumed not to be mobile over the solid surface and not to exchange with either of the bulk fluid phases except in conjunction with surfactant. Zhmud et al.13 and Tiberg et al.14 assume that the total time over which transfer of surfactant from the oilwater to the watersolid interface takes place is equal to h/U, where h is the thickness of the surfactant-loaded oilwater interface and U is the speed of the contact line; the configuration is sketched in Figure 2. They further assume that surfactant is transferred at a rate given by the Langmuir kinetic equation (section 2.1.1 of Krotov and Rustanov11). That is, the rate of transfer from the oilwater to the watersolid 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 watersolid interface, either back to the oilwater interface or into the adjacent bulk solution) is taken to be simply proportional to the amount of surfactant adsorbed on the watersolid 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 oilwater interface is ! dΓ ðΓ þ Γcon Þ þ ¼ k Γow 1 ð20Þ k Γ dt Γm ws It seems reasonable to limit the adsorbed amount to be less than the value that would be in equilibrium with the concentration c(X,t) in the bulk aqueous solution immediately adjacent to the contact line. This can be achieved by choosing the value of k such that the net rate of transfer vanishes when the surface concentration attains the appropriate equilibrium value. If we assume that this equilibrium value, Γeq ws, is given by the isotherm 12, then ! Γcon ws ðX, tÞ 1 Kws cðX, tÞ Γm ws eq m ð21Þ Γws ¼ Γws 1 þ Kws cðX, tÞ Inserting 12 into 20 and setting the result to zero, we find k ¼
kþ Γow m Γws Kws cðX, tÞ
It follows, after simple rearrangements, that ! ! dΓ Γcon Γ þ ws ðX, tÞ ¼ k Γow 1 1 eq dt Γm Γws ws
ð22Þ
ð23Þ
On that part of the solid surface exposed to the aqueous solution, surfactant can exchange with the bulk, and we model this 4415
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through
! ! con dΓ Γ ðX, tÞ Γ ¼ hkþþ cðX, tÞ 1 ws m 1 eq dt Γws Γws
ð24Þ
Here kþþ is a rate constant for surface-bulk exchanges, analogous to that for surface-meniscus exchanges, and the overall expression is modified only to take account of the different dimensions of the concentration variables in the two cases. On that part of the solid surface exposed to the oil, we assume that one molecule of surfactant and one molecule of contaminant can complex together, leading to solubilization of the resulting complex into the bulk oil. Hence dΓ ¼ kΓΓcon ð25Þ dt cleaning
and dΓcon dt
¼ kΓΓcon
ð26Þ
cleaning
where Γcon is the surface concentration of contaminant (mol m2), and κ is a rate constant for the combined complexation
and desorption/solubilization process. Beneath the meniscus, and in the region exposed to the aqueous phase, the contaminant is assumed not to desorb or to move over the solid surface, so that dΓcon ¼0 dt
there. It is further assumed that surfactant, but not contaminant, can diffuse over the solid surface, with surface diffusivity Ds. This carries with it the possibility of surfactant being present on the solid surface downstream (i.e., on the oil side) of the advancing oilwater interface. Kumar et al.15 give experimental evidence for the presence of surfactant on the solid surface on the vapor side of an advancing watervapor interface, and Churaev et al.16 present a model for surfactant imbibition including this feature. Kao et al.17 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 too. In a frame of reference moving with the three phase contact line we therefore have
8 ! ! con > Γ ðX, tÞ Γ > > kþþ hcðX, tÞ 1 ws m 1 eq > > > Γws Γws > < ! ! d dΓ con UΓ Ds ¼ Γ ðX, tÞ Γ > dy dy kþ Γow 1 ws m 1 eq > > Γws Γws > > > > con : kΓΓ where y = z X(t), and the surfactant distribution has been assumed to be in a quasi-steady state. Also 8 > 0 in y < 0 < d con 0 in 0 < y < h ð UΓ Þ ¼ ð29Þ > dy : kΓΓcon in y > h For the purposes of evaluating the surface adsorptions in eq 10 we use con Γtot ws ðX, tÞ ¼ Γðy ¼ 0Þ þ Γ ðy ¼ 0Þ
ð30Þ
con þ þ Γtot os ðX, tÞ ¼ Γðy ¼ h Þ þ Γ ðy ¼ h Þ
ð31Þ
and
ð27Þ
in
y h, eqs 28 and 29 may be combined and integrated once, using 32, to obtain
Ds dΓ Γ ¼ Γcon ð¥Þ Γcon U dy
ð41Þ
Taken together with 29c, this constitutes a nonlinear secondorder differential equation system for Γ and Γcon in y > h. If 41 is evaluated at y = hþ, it can be used to simplify the right-hand side of 40, to obtain dΓ ¼ UðΓcon ð¥Þ Γcon ðy ¼ hþ ÞÞ UΓ Ds dy y ¼ h
ð46Þ and A and B are further constants to be determined. Continuity of Γ and its derivative at y = 0 require v λþ R λ λþ
ð47Þ
B¼
λ v R λ λþ
ð48Þ
and
Inserting these values in eq 45, we obtain Γðy ¼ h Þ ¼ Γeq ws
v λ þ λ v Reλ h þ Reλþ h 1þ λ λþ λ λþ
!
ð49Þ and dΓ v λþ eq ðy ¼ h Þ ¼ Γws 1 þ Rλ eλ h dy λ λþ λ v þ Rλþ eλþ h λ λþ
ð42Þ which will prove useful later. We see at once from 29 that Γcon is a constant in y < h, so Γcon(y = 0) = Γcon(y = h) = Γcon(y = hþ) = Γcon ws (X,t).
A¼
ð50Þ
The constant R is determined as part of the numerical solution in y > h of the second-order nonlinear differential 4417
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this system at y = hþ, containing the as yet unknown parameter R:
equation system eq 41 and 29. From eqs 39 and 42, using eqs 49 and 50, we obtain two boundary conditions for
Δr GQ =RT
e
ðΓm os
Γ
con
ðy ¼
hÞΓeq ws Þ
þ
Γðy ¼ h Þ ¼ ðΓm ws
Γ
con
ðy ¼
hÞÞ ð1 eΔr GQ =RT ÞΓeq ws
and UðΓ
con
ð¥Þ Γ
con
ðy ¼ hÞÞ
3 Ds λ λ h Ds λþ λþ h e e ðv λ Þ 1 þ þ ðλ v Þ 1 þ þ 7 6 7 6 U U 7 ¼ UΓeq 1þR ws 6 5 4 λ λþ 2
ðv λþ Þeλ h þ ðλ v Þeλþ h 1þR λ λþ
ðv λþ Þeλ h þ ðλ v Þeλþ h 1þR λ λþ
^con dΓ ^ ^con ¼ δΓΓ dη ^ dΓ ^ ^con ¼ Γþ Γ 1 dη
The third condition boundary, needed in order to fix the value of R, is as
e ^ Γð0Þ ¼ ε
yf¥
Δr GQ =RT
ð53Þ
λ h
þ ðλ v Þeλþ h λ λþ
^con eq con Γtot os ðX, tÞ ¼ Γws ð1 þ RÞ þ Γ ð¥Þ Γ ð0Þ
ð56Þ
^ ^con con Γtot os ðX, tÞ ¼ Γ ð¥ÞðΓ ð0Þ þ Γ ð0ÞÞ
ð57Þ
and
ð54Þ
!
ðv λþ Þeλ h þ ðλ v Þeλþ h ^m ^con ðΓws Γ ð0ÞÞ ð1 eΔr GQ =RT Þε 1 þ R λ λþ 2 3 D s λ λ h Ds λ þ λ þ h ðv λþ Þ 1 þ e þ ðλ v Þ 1 þ e 7 6 6 7 U U ^con 7 Γ ð0Þ ¼ 1 þ ε61 þ R 4 5 λ λþ
where for later convenience we have written ε = Γeq ws/ m con m con ̂ os = Γm (¥), and Γ = Γm (¥). Recall Γcon(¥), Γ̂ ws ws/Γ os/Γ ̂ con(0), as do ν and λ( that by virtue of 21, Γeq ws depends on Γ through eqs 44 and 46. As a result, this system is nonlinear not only through the product term in eq 54 but also through the Γ̂ con(0) dependences within the boundary conditions (eq 55). We solve this system numerically using the Matlab routine bvp5c. In order to get good convergence, it is necessary to give a reasonably accurate initial guess for the solution, and this is done on the basis of various large and small δ asymptotics. These may also be used, where appropriate, in place of bvp5c to speed up the overall calculation. The details of the asymptotics are given in the Appendix. 2.7. Coupling between Inner and Outer Regions. In terms of the solution of the system (eqs 54 and 55), the adsorbed amounts on the watersolid and oilsolid surfaces immediately adjacent to the meniscus are
ð51Þ
where δ = κΓcon(¥)Ds/U2, subject to
ðv λþ Þe ^m ^con ðΓos Γ ð0ÞÞ 1 þ R
^con Γ f1
!
If we introduce a nondimensional coordinate η = (y h)/(Ds/U) ̂ (η) and Γcon = Γcon(¥) Γ ̂ con(η), then eqs 41 and set Γ = Γcon(¥) Γ and 29 may be written as
ð52Þ
Γcon f Γcon ð¥Þ
!
as
!
ηf¥
ð55Þ
When inserted into eq 10, these permit the contact angle and capillary pressure to be computed and hence U. At the outer edge of the inner region, the amount of adsorbed contaminant is Γcon(¥) Γ̂ con(0). This supplies the single boundary condition necessary for eq 18: ^con con Γcon is ðX, tÞ ¼ Γ ð¥Þ Γ ð0Þ
ð58Þ
The final element of the model is a link between the flux of surfactant into the vicinity of the meniscus from the adjacent bulk solution, the fluxes of surfactant away from the three phase contact line along the solid surface, and the rate of change of the amount of surfactant stored on the meniscus itself. In the frame of reference moving with the meniscus, the bulk flux of surfactant onto the meniscus from the main part of the capillary is equal to πa2Db(∂c/∂z)(X,t). The flux of surfactant along the solid surface in the y-direction at any point, in the same frame of reference, is 2πa(UΓþDs(dΓ/ dy)). If we sum the surface fluxes out of the inner region surrounding the three phase contact line with the rate of change of the amount of surfactant on the meniscus, and equate this sum to the outer region flux of surfactant onto the 4418
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meniscus, we find
∂c dΓ πa2 Db ðX, tÞ ¼ 2πa UΓ þ Ds ∂z dy y f dΓ dðAΓow Þ 2πa UΓ þ Ds þ dy y ¼ h dt
This completes the description of the basic mathematical model. The reader is referred to Hammond and Unsal8 for a summary of the major modeling assumptions. ¥
ð59Þ
Table 1. Parameter Values for the Base Case variable
The meniscus area is A. The second term in parentheses on the right-hand side of this expression is nonzero, since a finite flux of surfactant is required to clean the interface of contaminant ahead of the advancing meniscus. Now, matching of inner and outer regions implies that dΓ ∂Γws ðX, tÞ ð60Þ ¼ UΓws ðX, tÞ þ Ds UΓ þ Ds dy y f ¥ ∂z with Γws(X,t) = and eqs 40 and 42 give the analogous quantity at y = h. Equations 60 and 40, when inserted into 59, finally allow us to specify the remaining boundary condition for eq 15: Γeq ws ,
πa2 Db
∂c ∂Γws ðX, tÞ þ 2πaDs ðX, tÞ ∂z ∂z
con con þ ¼ 2πaUðΓeq ws þ Γ ð¥Þ Γ ðy ¼ h ÞÞ
dðAΓow Þ dt ð61Þ
This is best implemented as a condition on the flux at z = X(t), as explained in the numerical solution method section of the Appendix.
value
L (tube length)
1 cm
a (tube diameter)
0.1 mm
h (meniscus thickness) μo (oil viscosity)
4 109 m 1 103 Pa s
μw (water viscosity)
1 103 Pa s
Db (bulk diffusivity)
1 109 m2 s1
Ds (surface diffusivity)
5 1014 m2 s1
cb (initial surfactant concentration)
1 102 mol m3
Kow (adsorption/desorption capacity)
1.3 104 m3 mol1
Kws (adsorption/desorption capacity)
1.3 104 m3 mol1
σow(0) (initial surface tension) cos θ0 (initial contact angle)
30 103 N m1 0.1
R
8.314 J K1 mol1
T
300 K
Q μQ os μws
RT J mol1
Γm ws
4.9 103 mol m2
Γm ws Γm os þ
4.9 103 mol m2
hk
4.9 103 mol m2 1 105 m s1
Figure 3. Contour plot showing the behavior of cos θ as Γcon(¥) and Γ(y=h) are varied. Note the small oil-wet region at the lower right (cos θ < 0) and the large water-wet region to the left (cos θ > 0). Input parameter values corresponding to positions above the bold black line are inadmissible, since m they lead to surface adsorptions, which exceed Γm ws or Γos, or give negative oilwater interfacial tensions. 4419
dx.doi.org/10.1021/la1048503 |Langmuir 2011, 27, 4412–4429
Langmuir
ARTICLE
Figure 4. Results from numerical simulation using base case parameter values. (a) The position of meniscus versus time; (b) bulk surfactant concentration profiles at 10 equi-spaced times between t = 0s and 1000s (colored solid lines); (c) profiles of the total amount of material adsorbed on the capillary wall at the same sequence of times, together with the values of the inner region solution Γws(y f ¥) (dashed line), Γws(y = 0) (dotted line), and Γws(y = hþ) (solid line) evaluated when the meniscus is at that spatial position (i.e., at the time such that X(t) = z, the same data may be seen plotted versus time in the middle track of Figure 6); and (d) the contaminant adsorption profiles at different times.
3. RESULTS 3.1. Parameter Values. The parameter values used for the simulations are listed in Table 1. Values for most of the material parameters needed here were discussed in Hammond and Unsal.8 For the base case simulation, the initial surfactant concentration, cb, is 0.01 mol m3. There are some important differences between the current model and the model of Hammond and Unsal.8 Here we take the material of the capillary to be water-wet, in the absence of any contamination or adsorbed surfactant. This requires that the cosine of the clean surface contact angle be positive, and so we set cos θ0 = 0.1 with σow(0) = 30 103 N m1. For the maximum surfactant adsorption on the capillary surface from water, we set 6 mol m2, which is the same value as was used Γm ws = 1.96 10 in the previous paper. Uniform adsorption of contaminant to a level Γcon(¥) causes the surface to become oil-wet; thus, we require the cosine of the contact angle to be negative in this case, so we must ensure that ! Γcon ð¥Þ 0 m σow ð0Þ cos θ ¼ σ ow ð0Þ cos θ RTΓws log 1 Γm ws ! con Γ ð¥Þ þ RTΓm
