Applied Mathematics 014 4(1): 1-11 DOI: 10.593/j.am.0140401.01 A Growth Constant Formula for Generalized Saffman-Taylor Problem Gelu Pasa Simion Stoilow Institute of Mathematics of Romanian Academy Calea Grivitei 1 Bucuresti S6 Romania Abstract The displacement of two immiscible fluids (water and oil) in a horizontal Hele-Shaw cell is linear unstable when the displacing fluid (water) is less viscous. In the multi-layer (or generalized) Saffman-Taylor model a succession of constant viscosity regions between water and oil is considered for minimize the Saffman - Taylor instability. Some upper bounds of the growth constant σ of perturbations were obtained in previous papers. In this paper we give two new results concerning the multi-layer model. a) Some lower bounds of σ which together with the previous upper estimates are giving a range for all values of σ. b) An almost eact formula of σ for a particular three - layer Saffman-Taylor problem in very good agreement with the above lower-upper bounds. Numerical eamples are given to prove the stability improvement obtained in terms of the problem data. Keywords Hele-Shaw immiscible displacement Hydrodynamic stability Growth constant formula and estimates 1. Introduction A known model for secondary oil recovery by displacing it from a porous media with a second fluid (water) is the Hele-Shaw approimation - see [1] []. The displacing fluids are flowing in the gap between two parallel plates at a small distance δ. The velocity average across the plates is verifying a Darcy - type equation for the flow in porous media. In each point of the Hele-Shaw cell we have only one fluid. A sharp interface eists between the two immiscible fluids. The permeability of the corresponding porous medium is function of δ and fluids viscosity. We consider a horizontal Hele-Shaw cell in plane 1 O y filled by two incompressible immiscible fluids and neglect the gravity effect. The velocity U far upstream of the displacing fluid is giving a displacement in the positive direction O 1. In the two regions the flow is governed by the continuity equation and the Darcy law: u 1 v y = 0 p 1 = ηu p y = ηv (1.1) where the low indices denote the partial derivatives (u v) is the velocity p is the pressure η is the viscosity. The Laplace s law holds on the interface: the pressure jump is balanced by the surface tension times the curvature; the interface is a material one. We introduce the moving reference system = 1 U t. The following basic solution with the straight interface * Corresponding author: Gelu.Pasa@imar.ro (Gelu Pasa) Published online at http://journal.sapub.org/am Copyright 014 Scientific & Academic Publishing. All Rights Reserved = U t eists u = U v = 0 P = ηu P y = 0; (1.) η = µ L < U t; η = µ R > U t. Here µ L µ R are the water and oil viscosities. As the basic interface is a straight line the basic pressure P is continuous. The linear stability of the above basic solution was studied in [3]. The growth constant σ and the wavenumber k of perturbations are introduced in the relation (.) below. The well-known formula (.1) of the growth constant σ was obtained giving an important conclusion: the flow becomes unstable if the displacing fluid is less viscous that means k > o s.t. µ L < µ R σ > 0. An intermediate region between the two initial immiscible fluids was considered in [4]. This intermediate region contains a fluid with an a priori unknown viscosity µ which is used to minimize the Saffman-Taylor instability. Such a model can be called a three layer Hele-Shaw model because we have a succession of three immiscible fluids displacing in a Hele-Shaw cell. A numerical optimal viscosity in middle region was obtained in agreement with previous eperimental results -see [5] [6] [7] [8] [9]. The multi-layer Hele Shaw model contains a succession of constant viscosity regions between water and oil and was studied in the papers [10] [11] [1] [13] [14]. Some upper bounds of σ were obtained giving an important stability improvement when the number of intermediate regions is large enough.
Gelu Pasa: A Growth Constant Formula for Generalized Saffman-Taylor Problem In the present paper we improve the previous results concerning the linear stability of multi-layer Hele-Shaw flow and give two new results: a) Some lower-upper bounds for the growth constant. We get two curves σ LOW (k) and σ U P (k) such that σ LOW (k) σ(k) σ U P (k) k. All these curves are similar with a parabola and have a maimum point in terms of k. On this way we obtain a range where the most dangerous values of the growth constants are contained. This range can be minimized in terms of the problem data: the surface tensions on the interfaces the length of the intermediate regions and the total number of intermediate regions. These results are given in section where we study the multilayer Hele-Shaw model. We recall first the Saffman-Taylor result and we give lower- upper bounds of σ for the two layer Saffman-Taylor problem. The first estimate (.19) is based on Remark 1 where the maimum points of the perturbations amplitude f are studied. The integral appearing in the denominator of the growth constant epression (.15) is estimated in Lemma. The usual inequality (.31) and Lemma are used to get the more accurate estimates (.3). Generalizations of Remark 1 and Lemma 1 are used to get lower- upper bounds for σ in the four-layer and N -layer case studied in the last part of section. This is the main point of the paper. The lowest and largest possible values of the growth constant can be computed by using some simple formulas given in this part in terms of N µ R µ L surface tensions intermediate viscosities and intermediate region lengths. b) An almost eact formula of the growth constant for a particular three-layer Saffman - Taylor problem when the difference between the viscosities of the initial immiscible fluids is very large. This result is given in section 3. Numerical eamples show that the two eact growth constant (in this case) are close enough to the upper and lower bounds - see Figure 1. On this way we justify the usefulness of the obtained lower-upper estimates. The considered ratio between the oil and water viscosities is about 100 as in [4]. Some numerical results are given in section 4 where the lower-upper bounds of σ are computed and plotted in terms of the wavenumbers k as function of the number of intermediate layers N µ R µ L the intermediate surface tensions T i on interfaces and the length L of each intermediate region. We conclude in section 5.. The Multi-layer Hele-Shaw Model a) The two-layer Saffman-Taylor problem. The perturbations u v p of the velocity and pressure are introduced in the flow equations (1.1) and we get u v y = 0; p = µ u p y = µ v ; (.1) µ = µ < g; µ = µ > g L where g is the perturbed interface - see the relation (.7) below. For more details see also [10] [11] [1]. Our problem is linear then the horizontal velocity can be decomposed in Fourier modes ( ) ( ) ( σ ) u y t = f ep iky t (.) where k is the wavenumber in the O y direction and σ is the growth constant. The amplitude f is continuous and f has a jump on the interface = 0. The relations (.1) 1 and (.1) 3 are giving the epressions of the perturbations of the second velocity component and pressure v = [ f / ik] ep ( iky σt) ; (.3) p = µ [ f / k ] ep iky σt. R ( ) Figure 1. T 0 = T 1 = 1; U = 1; µ R = 101; µ L = 1; µ = 50; L = 1; N = 1. Black: σ ST - eq.(4.1); Red: σup - eq.(4.1); Blue: σ - eq.(3.16); Orange: σ 1 - eq.(3.15). Green: σ LOW - eq. (4.1)
Applied Mathematics 014 4(1): 1-11 3 Cross derivation of the pressure equations (.1) (.1) 3 is giving ( µu ) = ( µ v ) therefore y µ f k µ f = 0 0 We need far decay solutions then f () = f ( 0 ) ep[k( 0 )] < 0 ; =. (.4) f () = f ( 0 ) ep[k( 0 )] > 0 (.5) and we get the following limit values of the derivatives f in the point 0 : f ( ) kf ( ) = f 0 kf 0 0 0 ( ) = ( ) (.6) where and denote the left and right limits near 0. The perturbed (material) interface near the point 0 is denoted by therefore we obtain = g( 0 y t) with g t = u (.7) g( 0 y t) = [f ( 0 )/σ] ep(iky σt). (.8) The pressure jump in 0 is estimated by using (.3) the basic pressure P given by (1.) and the viscosities jumps as in [4] (recall and are denoting the left and right limits in 0 ): p( ) = P ( y t) 0 0 P ( y t) g( y t) p ( y t) 0 0 0 p( ) = P ( ) 0 0 ( ) µ ( 0) U f ( 0) / σ f ( 0) / k ep iky σt (.9) p( ) = P ( ) 0 0 ( ) µ σ σ ( 0) U f ( 0) / f ( 0) / k ep iky t. Here P ( ) 0 y t P ( ) 0 y t 0 (.10) f ( ) are denoting the left and right lateral derivatives of the basic pressure P and f in 0. The pressure jump in 0 is balanced by the surface tension times the curvature (the Laplace s law) then it follows (µf ) ( 0 ) (µf ) ( 0 ) ={k U [µ ( 0 ) µ ( 0 )] k 4 T ( 0 )} f ( 0 )/σ (.11) where T ( 0 ) is the surface tension in 0 and (f g)() = f()g(). By using the relations (.6) we get the well-known Saffman-Taylor formula: σ ST = {ku (µ R µ L ) k 3 T ( 0 )}/(µ R µ L ) (.1) where µ L µ R are the water and oil viscosities. b) The three-layer Saffman-Taylor problem. The Hele-Shaw cell is filled by three fluids: oil with viscosity µ R contained in the interval 1 > U t; an intermediate fluid with viscosity µ contained in the region U t L < 1 < U t; water with viscosity µ L contained in the region 1 < U t L. All three viscosities are constant. In the mobile coordinate = 1 U t the intermediate region is the segment (L 0). We consider µ L < µ < µ R and use the notations: E 0 = k U [µ (0) µ (0)] k 4 T 0 = k U [µ R µ] k 4 T 0 ; (.13) E 1 = k U [µ (L) µ (L)]k 4 T 1 = k U [µ µ L ]k 4 T 1 (.14) where T 0 T are the surface tensions in the points 1 1 = U t = U t L. 1 We multiply (.4) with f we intergate on (L 0) and get σ = { E f E f }/ NUM 1 1 0 0 NUM = k µ Lf I k µ Rf (.15) o 1 0 I = µ ( f k f ) d. L Remark 1. The local maimum of f can be attained only in the points 0 L. For this we consider the equation (.4) and the following finite-difference approimation for the derivative of f in an interior point y (L 0): f (y) {f (y d/) f (y d/)}/d f (y) {f (y d) f (y) f (y d)}/d (.16) where d is the discretization step. Ma f = f y y { L 0}. In all interior Suppose ( ) points of the interval (L 0) from the equation (.4) and the above finite-difference approimation it follows {f (y d) f (y) f (y d)}/d k f (y) = 0. (.17) As f (y) f (y d) f (y) f (y d) we get (recall d is the discretization step used in (.16)) k d (.18) which is false. Moreover the function f is continuous on R but we have a jump of f in the points L 0 where the equation (.4) is not valid. Therefore the maimum of f () is attained only in the jump viscosity points L 0. Lemma 1. We have the estimate 1 0 ( ) ( ) σ Ma{ E f / kµ ( ) E / kµ. (.19) 0 R } Proof. We use the above Remark 1 the relations (.5)- (.6) the jump relation (.11) in the points 0 = 0 0 = L and the notations (.13)-(.14). The lateral derivatives are approimated as follows: L
4 Gelu Pasa: A Growth Constant Formula for Generalized Saffman-Taylor Problem ( ) ( ) = [ ( ) f 0 [ f ( d) f 0 ]/( d) f f f 0 f ( d)] / d ( L) [ f( L d) f( L)] / d ( L) [ f( Ld) f( L) ] / ( d) (.0) (.1) where d is the discretization step. If Ma f () = f (0) we use the jump relation (.11) in 0 = 0 and get E 0 /σ kµ R µ/d f (0) = µ f (d) /d σ E 0 /(kµ R ). (.) If Ma f () = f (L) we use the jump relation in 0 = L and get E 1 /σ kµ L µ/d f (L) = µ f (L) /d σ E 1 /(kµ L ). (.3) On this way we get only an upper bound of σ but not a lower bound. However if E 0 and E 1 are close enough and µ R >> µ L the difference between the two possible upper bounds is very large. A variational approach and the formula (.15) are used below for obtaining a more accurate upper estimate and also a lower bound for σ. For this we compute the integral I in the denominator of (.15) in terms of the values of f in the middle-region ends. Lemma. We have where b kb ka ( ) ) a (.4) I : = f k f = kd /( e e ka kb ( )( ) ( ) k D= f f e e 4 f fe. (.5) and f a := f (a) f b := f (b). Moreover we have the estimate k f f e e e e I kb ka kb ka ( )( ) / ( ) k f f e e / ( e e ) (.6) kb ka kb ka ( )( ). Proof. The eact solution of the equation (.4) is f () = Ae k Be k then we get I = k[a (e kb e ka ) B (e kb e ka )] = k(e kb e ka )[A e k(ab) B ]. (.7) We have f a = Ae ka Be ka f b = Ae kb Be kb then A = {f a e ka f b e kb }/(e ka e kb ) B = e k(ab) {fbe ka fae kb }/(e ka e kb ). (.8) We insert the above A B in (.7) and obtain the epression (.4) (.5). We get also the following upper and lower estimates of the factor D appearing in the definitions (.4) - (.5): ( ) D= f f e e e ( )( ka k kb ) ( ) (.9) k ka kb ( fa fb) e ( fa fb )( e e ) ; D f f e e f f e ka kb k ( )( ) ( ) ka kb fa fb e e ) = ( )( ( ) (.30) Therefore the estimate (.6) it follows from the equations (.4) (.5) (.9) (.30). In the case a = L b = 0 we have (e kb e ka )/(e kb e ka ) = (e kl 1)/(e kl 1); (e kb e ka )/(e kb e ka ) = (e kl 1)/(e kl 1). As in previous papers [10] [11] [1] for B i i > 0 we have the inequalities Min { A / B } { A }/{ B } 1 i n i i i i i i i= 1 i= 1 Ma { A / B }. 1 i n i i i= n i= n (.31) Then from the formula (.15) it follows the lower-upper estimate σ L σ σ U (.3) where we use the following notations: σ L = Min{E 1 /k(µ L LE) E 0 /k(µ R LE)}; LE = µ(e kl 1)/(e kl 1); σ U = Ma{E 1 /k(µ L U E) E 0 /k(µ R U E)}; U E = µ(e kl 1)/(e kl 1). Remark. Consider the particular case µ R µ = µ µ L T 0 = T 1 (.33) then µ = (µ R µ L )/. The formulas (.13) (.14) are giving E 0 = E 1 and we consider only the most dangereous case when E 0 0 that means E 0 = E 1 = k U (µ R µ) k 4 T 0 ; (.34) k U (µ R µ)/t 0. As µ L < µ R from the relations (.3) - (.34) we get 0 E 0 /k(µ R LE) σ (.35) E 0 /k(µ L U E) The inequality (.31) still holds even if A i < 0. Then we can also consider E 0 < 0. The relation (.3) becomes E 0 /k(µ L U E) σ E 0 /k(µ R LE) 0 (.36) therefore for k > U (µ R µ)/t 0 we have the last above estimate. We can see that the upper (lower) curve in the range k U (µ R µ)/t 0 becomes lower (upper) curve in the range k > U (µ R µ)/t 0 and we have estimates also for negative growth constants. c) The four-layer case. We use the moving reference = 1 U t. The Hele - Shaw cell is filled by four fluids: the oil
Applied Mathematics 014 4(1): 1-11 5 (with viscosity µ R ) contained in the region > 0; the first intermediate fluid with constant viscosity µ 1 contained in the region L < < 0; the second intermediate fluid with constant viscosity µ contained in the region L < < L; the water (with viscosity µ L ) contained in the region < L. All viscosities are constant. The intermediate viscosities are a priori unknown and we considred here the particular case when all regions between water and oil are of the same length L. We use also the assumption µ L < µ < µ 1 < µ R. The two intermediate regions are (L L) and (L 0). Three sharp interfaces eists between the above four immisicble fluids: = L = L and = 0. In the eterior of the interval (L 0) we have the following limit values of f quite similar with the relations (.5): f () = f (L) ep[k( L)]) < L; (.37) f () = f (0) ep(k) > 0. We multiply with f in (.4) we integrate on (L L) and (L 0) then we get L 0 L µ f f µ 1f f µ f L L L 0 L 0 µ 1f k µ f k µ 1f 0 L L L ( ) ( ) = (.38) where (fgh)() = f ()g()h(). The relations (.37) and the jump relations (.11) in the points 0 = L 0 = L 0 = 0 are giving { 1 1 0 0} σ = E f E f E f / NC L 1 R 0 NC = kµ f I I kµ f with the following notations (.39) f = f (L); f 1 = f (L); f 0 = f (0) (.40) E = k U [µ µ L ] k 4 T (.41) E 1 = k U [µ 1 µ ] k 4 T 1 (.4) E 0 = k U [µ R µ 1 ] k 4 T 0 (.43) ( ) L i = µ L i =1. (.44) I f k f d i and T 0 T 1 T are the surface tensions in = 0 L L. The above limit values of the viscosities are µ (L) = µ L µ (L) = µ ; µ (L) = µ µ (L) = µ 1 ; µ (0) = µ 1 µ (0) = µ R. (.45) The denominator NC in the ratio (.39) is positive. We can neglect the (eventual) negative values E i. The most unfavourable case is when all quantities E i are positive and in general we perform our analysis in the corresponding range of the wavenumbers k. Remark 3. The local maimum of f can be attained only in the points 0 L L. For this we use Remark 1 then the maimum can not be attained in an interior point y (L L) (L 0). Moreover we have at least a local maimum point of f () in = 0 or = L or = L. For this consider the interior points i = id in (L 0). Suppose f (0) is not a local maimum then f (0) < f ( 1 ). If f ( 1 ) > f ( ) then f ( 1 ) is a local maimum which is not possible. Therefore it follows f ( 1 ) < f ( ). The same procedure gives f ( ) < f ( 3 ) and so on. This increasing process can be stopped in = L which can be a local maimum for f(). If not f is still increasing until = L then the maimum is attained on the left end of the intermediate region. As a consequence of the above Remark 3 we have the only three possibilities: i) = 0 is a local maimum point; ii) = L is a local maimum point; iii) = L is a local maimum point. In the case i) we use the jump relation (.11) in the point 0 = 0 and get E0 µ 1 f ( 0) µ Rkf ( 0) = f ( 0 ) (.46) σ because f ( 0) kf ( 0) the approimation = - see the relations (.37). We use ( ) ( ) ( ) ( ) ( ) ( ) f 0 f d f 0 / d = f 0 f d / d where d is a discretization step and obtain E 0 /σ kµ R µ 1 /d f(0) = (µ 1 /d) f (d). We have ( ) ( ) f d / f 0 1 (.47) therefore the last two relations are giving 0 de 0 /(dkµ R µ 1 ) σ E 0 /(kµ R ) (.48) because d is very small; recall that we considered E 0 0. In the case ii) we use the jump relation (.11) in the point 0 = L and get E1 µ f ( L) µ 1 f ( L) = f ( L) (.49) σ therefore by using the approimations (.1) it follows (µ /d)[f (L) f (L d)] (µ 1 /d [f (L d) f (L)] = E 1 f (L)/σ; E 1 /σ (µ 1 µ )/d f(l) (.50) = (µ /d) f(l d) (µ /d) f(l d) ; f (L d) / f (L) 1; f (L d) / f (L) 1. From the above relations it follows E 1 /σ 0 (.51) and σ de 1 /[(µ 1 µ )] 0.
6 Gelu Pasa: A Growth Constant Formula for Generalized Saffman-Taylor Problem Then in this case we get only the estimate σ 0 which follows from the hypothesis E i 0 and formula (.48). In the case iii) we use the approimation ( ) ( ) ( ) /( ) f L f L d f L d for the right derivative in the point = L the relation f ( L) = kf ( L) - see (.37) and get 0 de /(dkµ L µ ) σ E /kµ L. (.5) The inequalities (.48) (.5) are giving the estimate σ Ma{E /kµ L E 0 /kµ R }. (.53) which is quite similar with the upper bound obtained in Lemma 1 for the three-layer case. The denominators in the above estimate are depending only on the oil and water viscosities also as in the three-layer case. We can see that the jump relation in the point (L) is not giving any improvement of the growth constant estimate. As before below we use Lemma to obtain more accurate estimates of σ. Remark 4. Consider a = (p 1)L and b = pl then we have (e kb e ka )/(e kb e ka ) = (e kl 1)/(e kl 1) (e kb e ka )/(e kb e ka ) = (e kl 1)/(e kl 1) and from the above Lemma we get i kl kl ( 1 )( 1/ ) ( 1) kl kl ( 1 )( )( ) kµ f f e e I kµ f f e 1 e 1 i= 1 i i (.54) Remark 5. Consider the case E i > 0 i = 0 1 - that means ( µ µ ) k U / T i. (.55) i i i Then σ > 0 and the inequality (.31) the formula (.39) and the above Remark 4 are giving the following general lower-upper estimates for the growth constant: ( 1 1 0 0) ( 1 1 0 0) E f E f E f / NL σ E f E f E f / NR (.56) where we use the definitions (.40) - (.44) and the following new notations ( µ L µ ) ( µ µ ) k( µ R µ ) f 1 1 1 0 ; NL = k Λ f k Λf Λ ( µ L µ ) ( µ µ ) k( µ R µ ) f 1 1 1 0 ; NR = k Ψ f k Ψf Ψ kl kl ( e ) ( e ) Λ= 1 / 1; kl kl ( e ) ( e ) Ψ= 1 / 1. (.57) As before the point is to use the estimate (.56) and the inequality (.31) for obtaining lower-upper bounds for the growth constant σ not depending on the eigenfunctions f i. d) The N-layer case. Consider N intermediate layers [N L N L L] [N L L N L L]... [L L] [L 0] (.58) and the viscosities µ i in the layer [il il L] for i = 1... N such that µ 0 = µ R µ i = µ R i(µ R µ L )/(N 1) µ N 1 = µ L. In the points i = il we have the surface tensions T N T N 1... T i... T T 1 T 0. The same procedure described in the previous section is giving the estimates Min{E N /A N E i / A i E 0 /A 0 } σ (.59) Ma{E N / B N E i / B i E 0 / B 0 } where we use the definitions (.57) 3 (.57) 4 of Λ Ψ and the following notations A N = k (µ L µ N Λ) A 0 = k (µ R µ 1 Λ); (.60) A i = k (µ i1 µ i ) Λ; B N = k (µ L µ N Ψ) B 0 = k(µ R µ 1 Ψ) B i = k(µ i1 µ i )Ψ; E i = k U (µ R µ L )/N k 4 T i 0. We give now a simpler formula for the lower-upper bound of σ when all E i T i are equal (then all viscosity jumps are the same). We use the definitions (.57) 3 (.57) 4 and get the following inequalities Ψ < 1 < Λ (.61) then the largest denominator in the left part of the above estimate (.59) is k(µ R µ 1 )Λ and the smallest denominator in the right part is k(µ L µ N )Ψ. On this way we get the estimates σ LOW σ σ UP (.6) where σ LOW = E 0 /[Λk (µ R µ 1 )] σ U P = E 0 /[Ψk (µ L µ N )] In the formula (.43) of E 0 we have µ R µ 1 = (µ R µ L )/(N 1) and we get lim k 0 σ U P = (µ R µ L )/[(N 1)L(µ L µ N )] (.63) then σ UP is not zero for k = 0. Only for large L and (or) N we epect to have lim k 0 σ UP 0 and this is confirmed by numerical calculations - see Figures 3 4 eplained also in section 4 below.
Applied Mathematics 014 4(1): 1-11 7 Figure. T 0 = T 1 = T = 1; U = 1; µ R = 100; µ L = 1; L = 1; N =. Black: σ ST - eq.(.1); Red: σ UP - eq.(4.); Blue: σ LOW - eq.(4.) Figure 3. T 0 = T 1 = T = 1; U = 1; µ R = 100; µ L = 1; L = 5; N =. Black: σ ST - eq.(.1); Red: σ UP - eq.(4.); Blue: σ LOW - eq.(4.) Figure 4. T 0 = T 1 = T = 1; U = 1; µ R = 100; µ L = 1; L = 10; N =. Black: σ ST - eq.(.1); Red: σ UP - eq.(4.); Blue: σ LOW - eq. (4.)
8 Gelu Pasa: A Growth Constant Formula for Generalized Saffman-Taylor Problem 3. An Eact Formula of σ in Three-layer Case The distance between the above lower-upper estimates (.35) obtained in Remark is large when µ R >> µ L and a comparison with an eact solution could be useful. For this we give below an almost eact formula of σ in this case when the conditions (.33) (.34) are verified. Recall the jump relations (.11) in the points 0 = 0 0 = L and E 0 E 1 given by the relation (.34): (µf) (0) (µf ) (0) = E 0 f 0 /σ (µf) (L) (µf ) (L) = E 0 f 1 /σ (3.1) where f 0 = f (0) f 1 = f (L). We use the solution f () = Ae k Be k of (.4) then the above relations (.5) (3.1) for 0 = 0 0 = L are giving kµ(a B) kµr(a B) = (E 0 /σ)(a B) (3.) kµ L (Ae kl Be kl ) kµ(ae kl Be kl ) = (E 0 /σ)(ae kl Be kl ) (3.3) kµ(1 β)/(1 β) k µr = E 0 /σ; kµ L kµ(1 βc)(1 βc) = E 0 /σ (3.4) where β = B/A; c = ep (kl). Therefore β is verifying the following equation (β 1)/(β 1) (βc 1)(βc 1) = (µ R µ L )/(µ R µ L ). (3.5) In the case µ R >> µ L we have (µ R µ L )/( µ R µ L ) = [1 (µ L )/( µ R µ L )] = (1 ) where = µ L /(µ R µ L ) therefore the equation (3.5) becomes ( c) β (1 )(c 1)β ( ) = 0. (3.6) We consider the case studied in [4]: µ R = 101 µ L = 1 µ = 50 = /(101 1) 0.0. (3.7) When 0 the equation (3.6) can be approimated as ( c) β (c1) β ( ) = 0 (3.8) and we get the solutions β 1 = (c 1± )/ c (3.9) = (c 1) 8 c. After some calculations we obtain the approimate value of the first root β1 = (c 1 )/ c = (8 c)/ [(c 1 ) c] /(c 1). (3.10) We have also [c 1 ]/ c (c 1)/ c = [ (c 1)]/ c = (8 c)/[ c( c 1)] /(c 1) therefore we get the approimate value of the second root β = (c 1 )/ c (c 1)/( c) /(c 1). (3.11) The same value of β can be also obtained by using the relation β 1 β = (c 1)/ c. We introduce the values β 1 in the equation (3.8) and obtain ( c) β 1 (c 1) β 1 ( ) = {4c/(c 1) 1}. (3.1) The maimum value of the function F (c) = c/(c 1) is F ma 0.5 then β 1 are verifying the equation (3.8) with the precision order 0.0 (4 0.5 1) 0.04. We have (1 β 1 )/(1 β 1 ) = (c 3)/(c 1); (3.13) (β c 1)/(β c 1) = βn/βd (3.14) where we use the notations βn = (c 1) (c 1) βd = (c 1) (3c 1) and the equation (3.4) is giving the corresponding growth constants: σ 1 E 0 /D1 D1 = kµ R kµ(1 β 1 )/(1 β 1 ) = kµ R kµ[ep(l) 3]/[ep(L) 1]; (3.15) σ E 0 /D D = kµl kµ(β c 1)/(β c 1) = kµl kµh 1/H (3.16) H 1 = [ep (L) 1] [ep (L) 1] H = [ep (L) 1] [3ep (L) 1]. The estimates (.35) the notations (.57) and the above formulas (3.15) - (3.16) are giving the following inequalities: µ L µψ µ R µ(1 β 1 )/(1 β 1 ) µ R µλ; (3.17) µ L µψ µ L µ(β c 1)/(β c 1) µ R µλ. (3.18) In the net section - Figure 1 - we plot the eact values and the lower-upper bounds of σ in a particular three-lyer case. the approimative value σ is close to ther upper estimate (.35) and σ 1 is close to the lower estimate (.35). Therefore our estimates are close enough to the approimative values of the growth constant. Remark 6. Suitable surface tensions T 0 = T 1 on the two interfaces and the formula (3.16) can give a stability improvement compared with the Saffman-Taylor case. For this we compare with the maimum value (in terms of k) of σ ST appearing in (.1). We have MAX k σ ST = [U (µ R µ L )] 3/ /[3(µ R µ L ) 3T ](3.19) where T is the surface tension on the interface water-oil in the Saffman-Taylor case. In the formula (3.16) we can see that (β c 1)/(β c 1) 1 k 0. On the other hand we have σ 1 E 0 /(kµ R ) then the growth constant is bounded by σ :
Applied Mathematics 014 4(1): 1-11 9 σ σ E 0 /[k(µ L µ)] MAX k σ [U (µ R µ L )/] 3/ / [3(µ L µ) 3T ] (3.0) 0 then a suitable surface tension T 0 for obtaining a stability improvement is T 0 (T/8)[(µ R µ L )/(µ L µ)] = (T/8) (10/51) T/. (3.1) 4. Numerical Results In the Figure 1 we plot the lower - upper estimates (.35) and the almost eact values (3.15) (3.16) of σ in the three-layer case T 0 = T 1 = 1 U = 1 µ R = 101 µ L = 1 µ = 50 N = 1 L = 1. The Saffman-Taylor value (.1) the estimates (.35) and the notations (.57) are giving σ ST = (100k k 3 )/10 σ LOW = (50k k 3 )/(101 50Λ) σu P = (50k k 3 0/(1 50Ψ). (4.1) We can see that σ 1 is almost the same as σ LOW and the maimum values of σ and σ UP are very close. Moreover we have 1.8 σ.5 then the most dangereous growth constant is at least 1.8. In the Figures 3 4 we plot the lower upper and Saffman-Taylor growth constant in the fourth-layer case T 0 = T 1 = T = 1 U = 1 µ R = 100 µ L = 1 N = when L = 1 5 10. Here the viscosty jumps are (µ R µ L )/(N 1) = 99/3 = 33 and µ 1 = 67 µ = 33. The Saffman-Taylor value (.1) and the lower-upper estimates (.6) in this case are σ ST = (99k k 3 )/101 σ LOW = Ψ(33k k 3 )/167 σ UP = Λ(33k k 3 )/34. (4.) For L = 1 the most dangereous growth constant is at least 0.45. For L = 5 we can see a small decrease of the upper bound of σ UP which now is closer to zero for k = 0 and a small increase of σ LOW. For L = 10 we have not a significant change of maimum values of σ UP σ LOW but the upper curve is almost to zero for k = 0 - see the eplanations after formula (.63). Figure 5. T i = 1; U = 1; µ R = 101; µ L = 1; L = 1 a) : N = 4; b) : N = 10; c) : N = 0. Black: σ ST - eq.(.1); Red: σ UP - eq.(4.3) (4.4) (4.5); Blue: σ LOW - eq.(4.3) (4.4) (4.5)
10 Gelu Pasa: A Growth Constant Formula for Generalized Saffman-Taylor Problem Figure 6. T i = 1; U = 1; µ R = 101; µ L = 1; N = 0; L = 5 Black: σ ST - eq.(.1); Red: σ UP - eq.(.6); Blue: σ LOW - eq.(.6) Figure 7. T i =; U=1; µ R = 101; µ L =1; N=0; L=5 Black: σ ST - eq. (.1); Red: σ UP - eq. (.6); Blue: σ LOW -eq.(.6) In the Figure 5a 5b 5c we plot σ LOW σ UP for N = 4 10 0 in the case T i = 1 U = 1 µ R = 101 µ L = 1 L = 1 by using the formulas (.6). The Saffman- Taylor value is given by (.1). We use the notations (.61) then N = 4: σ LOW = Ψ(0k k 3 )/(101 81) σ UP = Λ(0k k 3 )/(1 0); (4.3) N = 10: σ LOW = Ψ(10k k 3 )/(101 91) σ UP = Λ(10k k 3 )/(1 10); (4.4) N = 0: σ LOW = Ψ(5k k 3 )/(101 96) σ UP = Λ(k k 3 )/(1 5). (4.5) The above case 5c is ploted in Figure 6 but for L = 5 then we have T i = 1 U = 1 µ R = 101 µ L = 1 N = 0 L = 5. In the Figure 7 we plot the case U = 1 µ R = 101 µ L = 1 N = 0 L = 5 with all surface tensions T i = (even in the Saffman - Taylor formula). 5. Conclusions We study here the displacement of oil in a Hele-Shaw cell or porous medium by using a graded succession of constant viscosity regions between oil and displacing fluid (water). A basic solution with initial straight interfaces eist. We perform the linear stability analysis of this basic solution and obtain two new results compared with previous papers in this field: 1) Some lower-upper estimates of the growth constant of perturbations in the three fourth and multi-layer Saffman-Taylor problem with equal jumps of intermediate viscosities are given in section. We point out that the eigenfunctions f (which can be considered as the perturbation amplitude) can have a local maimum only in the points of viscosity jumps - ( = 0 L L) for the three-layer case; see Remark 1 and Remark 5. This property is used for obtaining lower-upper estimates of the growth constant starting with the jump relations (.11) - see also Remark 4. On this way we get a lower-upper bound depending only on the water and oil viscosities - see the estimate (.53). More eact estimations (.35) (.56) (.59) are obtained by using Lemma where the integral appearing in the denominator of the growth constant formulas is estimated in terms of the intermediate
Applied Mathematics 014 4(1): 1-11 11 viscosities and amplitude values f in the points of viscosities jumps. We get a range where all possible values of the growth constants are contained. Some numer- ical eamples are given in section 4 where the lower and upper bounds of σ are plotted in terms the number and length of intermediate regions denoted by N and L. The Saffman-Taylor growth constant is also plotted and we can see the stability improvement for large N L. Moreover the obtained upper bounds of the growth constant are zero when the wave-number k is zero if L is large enough - see Figures 3 4 and eplanations after the formula (.63). In general for not very large L the upper bounds of σ are not zero when k = 0 - see [11] [1] [13]. A suitable numerical values of surface tensions acting on the interfaces can give a significant improvement of stability even if the number of the interior layers is not so large - see Remark 6. ) An almost eact formula of the growth constant in the case of three-layer Hele-Shaw cell with a constant viscosity region between the initial displacing fluids (water and oil) - see the section 3. This formula is obtained when the ratio between oil and water viscosities is about 100 - a case studied also in [4]. A numerical eample shows a good agreement between the eact growth constants and the lower-upper bounds. The eact value σ 1 is very close to σ LOW ; the maimum values of the eact value σ and σ U P are also very close. Moreover we have lim k 0 σ U P = 0 even for L = 1 - see the Figure 1. The main conclusions of this paper is following. The distance between lower-upper and the almost eact values of σ is small enough in the three-layer case then the obtained estimates are quite useful. Moreover important stability improvements can be obtained in terms of N L and surface tensions T i on the interfaces. REFERENCES [1] J. Bear Dynamics of Fluids in Porous Media Elsevier New York 197. [] H. Lamb Hydrodynamics Cambridge University Press Cambridge 1933. [3] P. G. Saffman and G. I. Taylor The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. A 45 31-39 1958. [4] S. B. Gorell and G. M. Homsy A theory of the optimal policy of oil recovery by secondary displacement process. SIAM J. Appl. Math. 431 79-98 1983. [5] N. Mungan1971 Improving Waterflooding through Mobility Control The Canadian Jour- nal of Chemical Engineering 49 3-37 1971. [6] R. L. Slobod and S. J. Lestz Use of a graded viscosity zone to reduce fingering in miscible phase displacements Producers Monthly 410 1-19 1960. [7] G. Shah and R. Schecter Improved Oil Recovery by Surfactants and Polymer Flooding. Academic Press New York 1977. [8] A. C. Uzoigwe F. C. Scanlon R.L. Jewett Improvement in polymer flooding: The pro- grammed slug and the polymer-conserving agent J. Petrol. Tech. 6 33-41 1974. [9] J. Wang and M. Dong Optimum effective viscosity of polymer solution for improving heavy oil recovery J. of Petroleum Science and Engineering 67 3-4 155-158 009. [10] P. Daripa and G. Pasa On optimal viscosity profile in enhanced oil recovery by polymer flooding Int. J. Engng. Sci. 4 No. 19-0 09-039 004. [11] P. Daripa Studies on stability in three-layer Hele-Shaw flows Phys. of Fluids 0 11101 008. [1] P. Daripa Hydrodynamic stability of Multi-Layer Hele-Shaw Flows J. Stat. Mech. 008 Art. 11101 008. [13] P. Daripa and Xueru Ding A numerical study of Instability Control for the Design of an Optimal Policy of Enhanced Oil Recovery by Tertiary Displacement Process 01 Transport In Porous Media 93 3 675-703 01. [14] P. Daripa and Xueru Ding Universal Stability Properties for Multi-layer Hele-Shaw flows and Applications to Instability Control SIAM J. Appl. Math. 7 5 1667-1685 01.