OF SECONDARY OIL RECOVERY
GELU PAS¸A
The secondary oil recovery is a process used to obtain oil from a porous medium, by pushing it with a second immiscible fluid. Useful results were obtained when the second fluid is a polymer with surfactant properties. We consider here the following models for porous media: the multi-layered Hele-Shaw model, the Darcy model with relative permeabilities and the Buckley-Leverett model. For the Hele- Shaw and Darcy models, a “sharp interface” exists between the immiscible fluids, where a surface tension can exists. We give some results concerning the linear stability of this interface. Previous numerical and experimental results show that the surface tension minimizes the interface instability. This result is confirmed by our analysis. We obtain a new formula for the growth constant of perturbations, for a multi-layered Hele-Shaw flow. A stability criterion is given for this model, in terms of the viscosity ratio of fluids considered; a strategy for minimizing the instability is given. In the case of Darcy model with relative permeabilities, the stability criterion is given in terms of the fluids “mobilities”. Also, for a particular case of the Buckley-Leverett model, we give a stability analysis related to a “constitutive” function which characterizes the porous medium and fluids interactions.
AMS 2000 Subject Classification: 35K30, 76Txx.
Key words: immiscible flow in porous media, stability of interfaces.
1. INTRODUCTION
The secondary oil recovery is a procedure to obtain oil from a porous medium, by displacing it with a second immiscible fluid. The Hele-Shaw ap- proximation modelizes the flow in a porous medium by using a fluid between two closed parallel plates; the filtration velocity is the average of the real ve- locity of the fluid (see [6]). A sharp interface exists between the two fluids.
The main property of this model is the possibility to visualize the flow (the evolution of the interface), if the plates are transparent (see [1]). In a real porous medium, the interface between two immiscible fluids is very compli- cated and in general is impossible to describe it. Moreover, if a surface tension exists on this surface, we need his curvature, to use some relations concerning the pressure drop (for example Laplace’s law); this is not possible at the pore
MATH. REPORTS10(60),2 (2008), 169–183
level. In fact, the Hele-Shaw model gives a “macroscopic” interface whose equation can be used to describe the effect of the surface tension. By using the Hele-Shaw model, in [8] a basic steady solution with a constant velocityU and straight linear interface is considered. It is proved that if the second fluid is less viscous, the interface between the fluids is unstable. Moreover, if the gravitation is considered, it is proved that a “critical” basic velocity exists, such that behind this value, the interface becomes unstable. The importance of a surface tension on the interface was also pointed out in [1].
Starting with this result, a new model can be considered (see [4]): a polymer surfactant is introduced between the displacing fluid and oil. The main point is to use the unknown viscosity in this intermediate region “I.R.”
as a parameter, for improving the stability. Numerical and experimental re- sults indicated an exponential optimal viscosity in I.R. (see [10], where a large number of references is given). Carasso an Pasa obtained in [2] a formula which confirms these results, by using some numerical approximations. A more simple formula was given in [3], without numerical approximation. In this paper we give a new formula for the growth constant of the perturba- tions, for a multi-layer Hele-Shaw model. In a particular case, a significative improvement of stability can be obtained. As in the case studied by Saffman and Taylor, all above analysis gives us a stability criterion in terms of the ratio of fluids viscosities.
A slightly different approach was used by Marle (see [1]), considering an extension of the Darcy model, which uses the relative permeabilities. Here the stability criterion is given in terms of the mobility ratio. We emphasize that the Hele-Shaw model is useful only for homogeneous porous medium, because the filtration velocity is the average of the real velocity of the fluid between parallel plates, which must be at a constant distance. The relative permeabilities can describe a nonhomogeneous behavior of the fluids, given by the relation of the each fluid with the medium.
In the last part we consider the Buckley-Leverett model (see [1], [7]), which is nonlinear. The relationship between the fluids and the porous medium is given by the relative permeabilities, which are functions of the oil (or water) saturation. This model is based on a hyperbolic transport equation and we can use the characteristic methods. In some particular cases, we give an analysis of a steady basic solution. The stability criterion is given in terms of the
“fractional flow function”, which is in fact a constitutive relation between the fluids and the porous medium.
The paper is laid out as follows. In Section 2 we recall the Hele-Shaw model and the Saffman-Taylor result. A three-layer Hele-Shaw model is stu- died in Section 3. The multi-layer Hele-Shaw model is studied in Section 4.
In Section 5 we give a result concerning the generalized Darcy model with
relative permeability functions. In Section 6 we give some results concerning the Buckley-Leverett model.
2. THE HELE-SHAW MODEL
A very useful model for a porous medium was introduced by Hele-Shaw in 1898. Two immiscible viscous fluids flow between two vertical parallel plates; the distance between the plates is “small enough”. This distance is used to define a “permeability”, and the filtration velocity is the average of the real velocity of the fluids. Therefore only a homogeneous medium can be considered – that means the permeability is constant. The Darcy law is obtained fort this averaged velocity. For details see [6]. The main point is following: an abrupt interface exists between the two immiscible fluids. This interface can be visualized if the plates are transparent. We emphasize that in a real porous medium, the real interface between two immiscible fluids is very complicated at the level of each pore, and can not be described in an useful manner. Then in fact the interface pointed out by the Hele-Shaw model is a “macroscopic” interface. On this interface we can consider a surface tension. This second main property of the Hele-Shaw model can be used for describing some phenomena, related with the Laplace’s law: if the the interface’s curvature is not zero, then the pressure drop is balanced by the surface tension times the curvature.
Consider the porous medium contained in the vertical plane (x0y), and two immiscible fluids with constant viscosity contained in this medium. The fluids flow due the gravity. The upward (pushing) fluid 1 is contained in the halfplane x <0, therefore ρ1> ρ2, where ρi are the densities. The fluids flow is given by the continuity equation for velocity and Darcy law
(1) u1 =−K µ1
grad (p1+ρ1gx) = gradφ1, ∇u1 = 0, x <0, (2) u2 =−K
µ2
grad (p2+ρ2gx) = gradφ2, ∇u2 = 0, x >0,
where g is the gravity acceleration, K is the permeability of the medium, x is the vertical height above a given horizontal line, µi, ui, pi are viscosities, velocities and pressures of the two fluids. For the sake of simplicity, we consider K = 1. A steady basic solution is considered for the above system. The two fluids are running with the velocity V and the interface is x = 0 in a mobile coordinate system running with this velocity. The viscosity is µ1 for x < 0 and µ2 forx >0.
Suppose the separation interface is slightly deformed as (3) x(y, t) =a·exp(iky+σt).
This assumption is possible because our problem is linear and we can decom- pose the perturbation in a Fourier series. Here, kis the positive wavenumber of perturbation and σ is the growth constant (in time). From the continu- ity equation and from the condition of continuity of normal velocity to the interface, on the unperturbed interface x= 0 we obtain
(4) ∂φ1
∂x = ∂φ2
∂x =V +a·σ·exp(iky+σt).
The disturbances must vanish at infinity. Then we get (5) φ1=V x+a·σ
k exp(iky+kx+σt), x <0, p1 =µ1φ−ρ1gx, (6) φ2 =V x−a·σ
k exp(iky−kx+σt). x >0, p2=µ2φ−ρ2gx.
The boundary condition on the interface is given by Laplace’s law: the pressure drop is balanced by the surface tension times the surface’s curvature.
Consider first the case without surface tension. Therefore, in the first approximation, near the unperturbed interface, the pressure is continuous and we have
(7) σ
k(µ1+µ2) =V(µ2−µ1) + (ρ2−ρ1)g.
As usual, our flow is unstable when the growth constant σ is positive. The main result obtained by Saffman and Taylor is that (recall ρ1> ρ2)
(8) µ2 > µ1 and V > g(ρ1−ρ2) µ2−µ1
⇒σ >0, ∀k >0.
We can also consider the case of equal densities, then the flow is given only by the velocityV of the displacing fluid (upstream); in this case the interface is unstable for µ1 < µ2. This last situation appears when the oil is displaced by water in a horizontal plane and the gravity influence can be neglected.
Finally, we consider a surface tension T on the interface. The first or- der approximation of the curvature of (3) is xyy = −k2aexp(iky+σt) and Laplace’s law yields
(9) σ
k(µ1+µ2) =V(µ2−µ1) + (ρ2−ρ1)g−T k2.
From this relation we deduce that there exists a wavenumberk0 such that for k > k0 the growth constant σ becomes negative. This is the main effect of surface tension.
3. THE THREE-LAYER HELE-SHAW MODEL
Consider an immobile coordinate systemOx1yzt1, wheret1 is the time and x1Oy is the horizontal plane. The negative direction on the Oz axis is the direction of the gravitational force, therefore we neglect the effect of the gravitational force for the flow in the plane x1Oy.
Gorell and Homsy [4] introduced in the following three-layer Hele-Shaw model. A porous medium is considered in the immobile x1y plane. For the sake of simplicity, the permeability is considered 1. The medium is filled by water, water with polymer (the intermediate region I.R.) and oil. As in previous section, a sharp interface exists between the three fluids. The flow of the three fluids is due to the upstream water velocity U at infinity, in the positive 0x1 direction and we shall study a basic steady solution corresponding to this flow. Then the displacing fluid is water, as in previous section, but we have a third fluid between water an oil. The main point is the following: the third fluid (between water and oil) has an unknown variable viscosity, denoted by µ. On the other hand, the viscosity is constant in the water and oil region, with values µ1, respectively µ2. It is well known that µ1 < µ2. Therefore, in the absence of the third fluid, the interface between water and oil should be unstable, as we proved in the previous section. The following “control”
problem can be considered: find the unknown viscosity of the third fluid such that the corresponding growth constant becomes as small as possible. To solve this problem, we shall consider some simplifying hypothesis, related with the interaction of the third fluid with the porous medium. We consider the third fluid as a polymer with surfactant properties and whose concentration is directly related with the viscosity.
We suppose that an invertible relation exists between the unknown vis- cosity µof the third fluid and the polymer concentration. On the other hand, we neglect the diffusion, adsorption and absorption of the third fluid in the porous medium. The amount of polymer in I.R. is assumed to be constant.
Then we have a conservation law for the concentration of the third fluid.
Therefore, the above assumption gives us a “conservation” equation for the unknown viscosity µof the third fluid.
Consider the velocity (u, v) and the pressure P of the fluids. As in the previous section, the flow in the medium is given by the continuity equation for velocity and the Darcy law. Moreover, we have a conservation law for the unknown viscosity in I.R. The flow equations are
(10) ∂u/∂x1+∂v/∂y= 0,
(11) ∂P/∂x1 =−µu, ∂P/∂y=−µv,
(12) Dµ/Dt1 = 0.
The above system can be solved by using some boundary conditions. We shall consider Laplace’s law on the two interfaces water-third fluid and third fluid-oil. The last equation is equivalent to
(13) ∂µ/∂t1+u∂µ/∂x1+v∂µ/∂y= 0.
Consider the steady basic solution
(14) ub =U, vb = 0, µb =µb(x1−U t1),
(15) Pb =−U
Z x1
xc
µb(s−U t1)ds,
where xc is a fixed point. Using this steady solution, we can consider an intermediate region I.R. Let the length of I.R. bel, situated at the left of the origin. The intermediate region I.R. runs with the water velocity U. In the regionx1 < U t1−l, the medium is filled by water, the I.R. is contained in the part U t1−l < x1 < U t1 and the region U t1 < x1 is filled by oil. Two sharp interfaces exists between the above three regions, namely,
(16) x1=U t1−l and x1 =U t1.
Consider that the moving reference frame is defined by x=x1−U t1, y=y, z=z,t=t1. The linear stability of the above steady solution is now studied;
in fact we study the stability of the straight initial interfaces (16). In the following, u0, v0, P0, µ0 are the perturbations of the basic solution (14)–(15).
We consider the solutions u=U+u0, v=v0, P =Pb+P0;µ=µb+µ0 in the system (10)–(12). In the frame of linear elasticity, we neglect the products of the form u0v0 etc, and obtain the system (17)–(19) below
(17) ∂u0/∂x+∂v0/∂y= 0,
(18) ∂P0/∂x=−µ0U −µbu0, ∂P0/∂y=−µbv0.
We have ∂/∂x1 =∂/∂x,∂/∂t1 =∂/∂x(−U) +∂/∂t,and (13) yields
∂(µb+µ0)/∂t1+ (U +u0)∂(µb+µ0)/∂x1+ (0 +v0)∂(µb+µ0)/∂y= 0, whence∂µ0/∂t1+u0dµb/dx1+U ∂µ0/∂x1 = 0.Then, in the moving coordinate system, we have
(19) ∂µ0/∂t+u0·dµb/dx= 0.
The perturbations can be developed into Fourier components. We use here a different approach, compared with Saffman-Taylor (1958). We do not perturb the interface separations, but perturb the basic steady velocity. However, we shall see that the results obtained are equivalent. We start with the pertur- bation of the velocity
(20) u0(x, y, t) =f(x)·exp(iky+at),
in the Oxdirection, whereais the growth constant andkis the wavenumber.
As usual, we consider only amplitudes which decay far upstream and down- stream. Using the perturbation system (17)–(19), for others perturbations we get
(21)
v0 =−1
ik ·df(x)/dx·exp(iky+at), µ0 =−1
a f(x)·dµb(x)/dx·exp(iky+at), (22) P0 =−µb(x) 1
k2 ·df(x)/dx·exp(iky+at).
By cross differentiating the pressure expression and using the abovev0 and µ0, we obtain the differential equation governingf(x) and µb(x) in I.R.:
(23) −(µb(x)fx(x))x+k2µb(x)f(x) = k2U
a (µb)x(x)f(x), x∈(−l,0).
In (23), above, fx is the derivative of the function f with respect to x. We recall that in the intermediate region I.R. the viscosity µb is variable and unknown. Out of I.R. the viscosity is constant. Then, for x <−l and x >0, equation (23) becomes fxx−k2f = 0. The amplitude f is continuous and f(|x| → ∞) = 0. We then get (recall k >0)
(24) f(x) = exp(k[x+l])f(−l), x <−l, f(x) = exp(−kx)f(0), x >0.
We need two boundary conditions to solve equation (23). In the Hele- Shaw model, each point of the medium contains only one fluid and a sharp interface exists between the immiscible fluids. We can consider that a surface tension acts on this interface. Then Laplace’s law is considered as boundary condition between the two fluids: the pressure drop on the interface should balance the surface tension times the curvature. Moreover, the velocity u0 is continuous across the interface. At this point we emphasize that the above form of Laplace’s law is only “a first approximation”. In fact, we must also consider the tangential and normal stress given by the viscous terms. Here, we neglect this part, as in [1]. A very interesting analysis is given in [5], concerning the validity of this approximation.
Let a sharp initial interface at x = x0. The perturbed interface is x(y, t) =x0+g0(y, t), whereg0is small compared withx0. We have to compute the perturbation g0. For this, we suppose that a fluid element initially at the interface will remain there (that means the interface is a material one). At the first order of approximation, we obtain
(25) ∂g0/∂t=u0(x0+g0, t).
We assume thatu0(x0+g0)≈u0(x0) and use the continuity of u0 atx0. Then (25) gives usg0(y, t) =f(x0) (1/a) exp(iky+at). The pressure on the interface
is approximated by
(26) P(x0+g0(y, t)) =Pb(x0+g0(y, t)) +P0+(x0), (27) Pb(x0+g0(y, t)) =Pb(x0) +g0(y, t)·∂Pb+/∂x(x0).
From (15) and (21)–(22) we get
(28) ∂Pb+/∂x(x0) =−U µ+b (x0), (29) P0+(x0) =−µ+b (x0) 1
k2 ·df /dx+(x0)·exp(iky+at).
Therefore, relations (26)–(29) yield (30) P+(x0) =Pb(x0)−µ+b(x0)
df /dx+(x0) k2 +U
af(x0)
exp(iky+at),
(31) P−(x0) =Pb(x0)−µ−b(x0)
df /dx−(x0) k2 +U
af(x0)
exp(iky+at), where df /dx+(x0), df /dx−(x0), P+,P− and µ+b,µ−b are the “right” (“left”) derivatives and limit values at x0.
Laplace’s law at x0 is P+(x0)−P−(x0) = Tens·gyy0 , where Tens is the surface tension at x0. We have gyy0 = −k2f(x0)(1/a) exp(iky +at). Then relations (30) and (31) and Laplace’s law yield:
(32)
µ−b (x0)
df /dx−(x0) k2 +U
af(x0)
−
−µ+b (x0)
df /dx+(x0) k2 +U
af(x0)
=−Tensk2f(x0) a .
By multiplying by f(x) in (23) and integrating on the interval (−l,0), we get
(33) −
Z 0
−l
(µbfxf)x+ Z 0
−l
µbfx2+k2 Z 0
−l
µbf2= k2U a
Z 0
−l
(µb)xf2,
(34)
µ+b fx+(−l)f(−l)−µ−b (0)fx−(0)f(0) + Z 0
−l
µbfx2+k2 Z 0
−l
µbf2 =
= 1 ak2U
Z 0
−l
(µb)xf2.
Consider the surface tensions S, T on the interfaces x = −l, x = 0 and the notation
(35) E=k2U[µ−(−l)−µ+(−l)] +Sk4, G=k2U[µ−(0)−µ+(0)] +T k4.
Relations (32) and (33)–(35) yield:
(36) a= −Ef2(−l)−Gf2(0) +k2R0
−lµxf2 µ1kf2(−l) +µ2kf2(0) +R0
−lµ fx2+k2R0
−lµ f2, where we omit the subscriptb for the basic viscosity. We have (37) E, G <0⇒a >0,
that is the most dangerous situation. The above condition is verified if the wavenumber kis in the range
(38) k2 ≤max
µ+(−l)−µ−(−l)
S , µ+(0)−µ−(0) T
. We neglect the positive termR
µ|fx|2 at the nominator of (36) and obtain the following inequality (recall that µis an increasing function)
(39) a≤max
−E kµ1; −G
kµ2; D µ1
, D= sup
x
{µx(x)}.
The above estimate is obtained by using the inequality A+B+C
X+Y +Z ≤MAX A
X, B Y, C
Z
. valid for positive numbers A, B, C, X, Y, Z. We see that
M E= max
k
−E
kµ1 = 2 µ13√
3S{U[µ+(−l)−µ−(−l)]}3/2,
(40) M G= max
k
−G
kµ2 = 2 µ23√
3T{U[µ+(0)−µ−(0)]}3/2.
We use (39) and (40) to obtain anoptimal viscosity profile in I.R. given by
(41) D/µ1 ≤max{M E, M G}.
If the limit viscosities µ+(−l) and µ−(0) are close enough to water and oil viscosities, there the positive growth constants become arbitrary small. So, we are nearneutral stability.
Remark 1. The Saffman-Taylor case (without I.R.) is a “limit” case of formula (36). We can consider small values forl, µx, a continuous viscosity on x=−land a surface tension only atx= 0. In this case we havef(−l)≈f(0), µ+(0) = µ2, µ−(0) = µ1. Formulae (35) give E = 0, −G = k2U(µ2−µ1)− k4T. Moreover, we can neglect the terms containing the integrals on (−l,0).
Therefore, from (36) we get
(42) a≈ −Gf2(0)
µ1kf2(−l) +µ2kf2(0) = kU(µ2−µ1)−k3T µ1+µ2 .
The last term is the Saffman-Taylor formula (9) in the absence of gravity. •
4. THE FOUR-LAYER HELE-SHAW MODEL
Consider the Hele-Shaw model with four regions. The fluids and viscosi- ties are water (µL), second fluid (µa), third fluid (µb), oil (µR):
x <−2L:µ=µL, x∈(−2L,−L) :µ=µa, x∈(−L,0) :µ=µb, x >0 :µ=µR.
—————————————————————————–
| | |——µR
| Second fluid :µa | Third fluid :µb |
µL——-| | |
—————————————————————————–
−2L −L 0 x
Fig. 1. The four-layer Hele-Shaw model.
We study the basic solution (14)–(15) given by the water velocityU far upstream. This time, the I.R. length is 2L. We have three interfaces located at x =−2L,x=−L,x= 0. Consider the surface tensions S, Q, T at x=−2L, x =−L,x = 0. The function f verifies the second order equation (23). Out of I.R. we have conditions similar to (24): f(x) = exp[k(x+ 2L)]f(−2L),
∀x < −2L and f(x) = exp(kx)f(0), ∀x > 0. The limit values of fx at the point x =−L are unknown. We multiply by f(x) and integrate (23) on the interval x ∈ (−2L,0). We have a jump of µ(x)fx(x) at x = −L, therefore we split the integral of this term into two parts, on the intervals (−2L,−L) and (−L,0). The jump condition in at x=−L is given by (32) with surface tension Q. We get
(43)
− Z −L
−2L
(µafxf)x− Z 0
−L
(µbfxf)x+ Z 0
−2L
µfx2+k2 Z 0
−2L
µf2 = k2U a
Z 0
−2L
µxf2; where the subscript a, and b are omitted in the last three integrals. Let (f gh)(x) =f(x)g(x)h(x). The above relation implies
(44) −(µ−afx−f)(−L) + (µ+afx+)f(−2L)−(µ−bfx−f)(0) + (µ+bfx+)f(−L)+
+ Z 0
−2L
µfx2+k2 Z 0
−2L
µf2= 1 ak2U
Z 0
−2L
µxf2.
We have f(x) = exp(k(x + 2L))f(−2L), x < −2L; µ−(−2L) = µL. Let M = −2L. Then (32) with the surface tension S yields (we omit again the
subscripts aand b) (µ+fx+f)(M) =
nk2U
a [µ−(M)−µ+(M)] + Sk4 a
o
f2(M) + (µ−fx−f)(M) =
= nk2U
a [µL−µ+(M)] + Sk4 a
o
f2(M) +µLkf2(M).
Also, f(x) = exp(−kx)f(0) for x >0, µ+(0) =µR. LetN =−L. From (32) with the surface tension T we get
−(µ−fx−f)(0) = nk2U
a [µ−(0)−µR] +T k4 a
o
f2(0) +µRkf2(0).
We use relation (32) at N =−L with surface tensionQ and get (µ+fx+f)(N)−(µ−fx−)f(N) =f2(N)
nk2U
a [µ−(N)−µ+(N)] +Qk4 a
o . Equation (44), the last three relations and (45)–(47) below yield formula (48) for the growth constant a:
(45) E4 =E(−2L) =k2U[µ−(−2L)−µ+(−2L)] +k4S, (46) F4 =F(−L) =k2U[µ−(−L)−µ+(−L)] +k4Q, (47) G4 =G(0) =k2U[µ−(0)−µ+(0)] +k4T, (48) a= −(E4)f2(−2L)−(F4)f2(−L)−(G4)f2(0) +k2R0
−2Lµxf µLkf2(−2L) +µRkf2(0) +R0
−2Lµ fx2+k2R0
−2Lµ f2 . Remark 2. At the numerator of (48) there appearsf2(−L), which does not appear at the denominator. We can obtain an estimate, if the term in f2(−L) is negative, that is,F4>0. We have
(49) F4>0⇔µ−(−L)> µ+(−L).
Consider the above case (49) and the estimate of aonly in terms of E4, G4, as in the previous section. We can obtain a positive upper bound of a which is close to 0. Using (48) and (49), the upper bound of the growth constant can be negative. Therefore, the four -region case can give us astable flow if on the interface x = −L the second fluid (after water) is more viscous than the third fluid (before oil). It is not surprising that a more viscous displacing fluid gives us a stable flow. The new aspect follows: we need a more viscous fluid onlyon the interface x=−L. •
Remark3. In the case of multi-layer Hele-Shaw flow, we obtain a formula similar (48). We have more terms of the form (F4)if2(−Li), where −Li are the “interior” interfaces in the region between water and oil. It is not necessary
to have equal “interior regions. The jump relations are similar to the previous results. •
5. THE GENERALIZED DARCY MODEL
We give here a very elegant and simple analysis due to Marle (see [1]), of the stability of immiscible flow in porous media, using the generalized Darcy model.
In the Hele-Shaw model, at an arbitrary point of the medium, we have only one fluid while an interface exists between immiscible fluids. The same situation exists in the generalized Darcy model, but here the flow of each fluid is characterized by the so called “relative permeability” functions. Then the “discharge” given by Darcy law is given not only in term of the medium permeability, but also in terms of the particular interaction of each fluid with the medium. In general, the relative permeabilities are functions of the fluid’s saturations, but in this section we do not use this dependence.
Consider the one-dimensional “discharges”qi and filtration velocities Vi
of two immiscible fluids in a horizontal porous medium, governed by the gen- eralized Darcy law
(50) qi =−ki
µi
∂pi
∂x, Vi = qi
n; i= 1,2.
Here,nis the porosity of the medium andkiare the corresponding relative per- meabilities multiplied by the medium permeabilityK. Suppose a stable front of separation exists between the fluids, with filtration velocity VF. Therefore, we have
(51) VF =V1= q1
n =V2 = q2
n.
Suppose that a “finger” of fluid 1 advances in fluid 2. This can be given by a non-homogeneity of the medium or by a perturbation. Therefore, this finger of fluid 1 will be sorounded by fluid 2, then in the finger we have pressurep2. Denote byVD the filtration velocity of the top of the finger. Therefore, we use (50)–(51) and get
(52) VD =−k1
µ1
∂p2
∂x 1
n, VD−VF =− k1
µ1n
∂p2
∂x + k1
µ1n
∂p1
∂x, (53)
VD−VF =− k1 µ1n
n−∂p1
∂x +∂p2
∂x o
=− k1 µ1n
nµ1nVF
k1 −µ2nVF k2
o
=nλ1 λ2−1o
VF, where λi =ki/µi are the mobilities of the two fluids.
The stability criterion is as follows: the flow is stable if VD < VF and unstable in the opposite case. According to this criterion, we have
(54) λ1 < λ2 ⇒VD < VF ⇒stable interface.
Consider the case k1 =k2, that is, of the same permeability for both fluids.
Then λ1 < λ2 ⇔ µ1 > µ2 and we obtain again the Saffman-Taylor result concerning the Hele-Shaw model (neglecting the gravity).
6. THE BUCKLEY-LEVERETT MODEL
We now consider a more realistic model for two-phasic flow in porous medium: at every point of the medium both fluids exist. This is a crucial difference as compared with Hele-Shaw and Darcy models, where each point of the medium contains only one fluid.
The two fluids are at different pressures. Then at every point exists the so called “capillary” pressure, given by the pressure difference of the fluids.
As in the Darcy model, the flow of both fluids are characterized by relative permeability curves, which are given by experiment.
Consider a porous medium filled by water and oil. Let S be the water saturation. The relative permeabilities ki are usually given as functions of S and are used to obtain the “flow fraction function” fW(S) which appears in the water saturation equation given by Buckley and Leverett (see [1])
(55) n∂S
∂t +U∂fW(S)
∂x = 0,
wherenis the porosity medium,U =u1+u1is the total velocity of flow,uiare the filtration velocities of the two fluids. U can depend on t, but we consider here U = constant. This equation gives us a good enough description of the flow, in the case of large oil fields, where the effects of the capillary pressure can be neglected.
We have only graphical forms of the fractional flow functionfW(S), given by the experimental data. The values offW(S) are in the interval [0,1]. There exists a domain S where fW is convex, a region S where fW is quite linear and a domain S wherefW is concave.
Equation (55) is hyperbolic and we can use the characteristic method to study a problem with specific initial and boundary conditions.
We consider the following problem. Let a porous medium in the half- plane x≥0 of the planex0y, and the initial and boundary conditions
(56) S(x,0) =SR= const., ∀x >0; S(0, t) =SL= const., ∀t≥0.
Therefore, at the initial momentt= 0 the regionx >0 is filled by water with saturation SR (and oil with saturation 1−SR). At the pointx= 0 we inject
water with saturation SL. Consider the particular case
(57) SL> SR, S∈[SR, SL]⇒fW00 (S)>0, fW0 (S)>0.
Moreover, suppose that [SR, SL] is strictly included in the domain where fW
is convex. The solution of (55) is constant on the characteristics dx/dt = (U fW0 (S(x(t), t))/n, then on the lines
(58) dx
dt = U fW0 (SL)
n , dx
dt = U fW0 (SR)
n .
We have (59)
SL> SR⇒fW0 (SL)> fW0 (SR)⇒ 1
fW0 (SL) < 1
fW0 (SR), dt dx
x=0< dt dx x>0. Therefore, in the plane x0t(with horizontal axis 0x) the characteristics slope is increasing for t≥0. A shock front appears (see [9]) and we have a traveling wave solution S(x−tVS) of problem (55)–(56)–(57), given by the formula (60) S(x.t) =SL, x < VSt, S(x.t) =SR, x > VSt, VS=U
n
fW(SL)−fW(SR) SL−SR . We have the following result.
Proposition 1. The solution (60) of problem (55)–(56)–(57) is stable in the Lyapunov sense.
Proof. We perturb the initial and boundary conditions. Then for small δ we consider
(61) Sδ(x,0) =SR+δ, ∀x >0, Sδ(0, t) =SL+δ, ∀t≥0.
As before, the solution of problem (55)+(61) is given by
Sδ(x, t) =SL+δ, x < VSδt, Sδ(x, t) =SR+δ, x > VSδt,
(62) VSδ= U
n
fW(SL+δ)−fW(SR+δ) SL−SR .
The interval [SR, SL] is strictly included in the convexity domain offW. Then we also have fW00 (S) >0 for S ∈[SR+δ, SL+δ]. The function fW is smooth enough and we get
(63) VSδ =VS+O(δ)U
n(SL−SR).
Therefore, the “difference” between the basic solution (60) and the perturbed solution (63) Sδ is of order O(δ). •
Acknowledgments. This work was supported by Contract CEx06-11-12.
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Received 3 July 2007 Romanian Academy
“Simion Stoilow” Institute of Mathematics Calea Grivitei 21, Bucharest, Romania
Gelu.Pasa@imar.ro
