Mixing in the downward displacement of a turbulent wash by alaminar spacer or cement slurry

Schlumberger Dowell

6th to 9th April 1999

1 Introduction

When drilling an oil well, the well is lined by sinking a steel casing, or liner, into the drilling mud(see Figure 1). The inside of the liner forms a central pipe, and leaves an annular gap between theoutside of the liner and the surrounding rock. The inside and outside of the liner are initially filled withdrilling mud, which is displaced by pumping a sequence of fluids down inside the liner from the surface.Typically, a chemical wash is pumped down first, followed by a spacer, and finally a cement slurry. Thewash, which is usually water based, is less dense than the spacer, a water-based suspension, which isitself less dense than the cement slurry.

The scenario of a light wash being forced down a pipe by a denser spacer fluid is liable to instabilitiesof Rayleigh–Taylor type. The two fluids are miscible, so mixing between the wash and the spacer islikely to take place. This may impair the intended efficiency of the wash in displacing mud from thewalls of the annulus.

Schlumberger Dowell asked the study group to estimate the amount of mixing between the wash andthe spacer, and its dependence on parameters such as the pumping rate, diameter of the pipe, viscosityof the fluids etc. In particular, they asked us to consider the following questions:

• Is there any physical mechanism that might prevent mixing between the wash and spacer/slurry?

• Can we give a reliable estimate of the length of mixed region?

• How does the above vary with pump rate, tube diameter, etc?

• What happens if we stop pumping during downward displacement?

2 Flow regimes

Concentrating on the downward flow inside the pipe, the study group identified four different flowregimes of interest.

1. Initial mixing across the pipe.

2. Spanwise mixing completed, what streamwise stratification can be stabilised against Rayleigh–Taylor instabilities?

3. Widening of the mixed layer by streamwise dispersion.

4. Effects of halting pumping.

The study group tried to estimate typical lengthscales and timescales associated with each regime inturn.

1

Figure 1: Stages in drilling an oil well. The study group considered the fifth stage, in which the wash isbeing displaced by denser fluids pumped from above.

2April 6th, 1999

Primary Cementing of a Casing

Drill new stage

Trip-outdrillpipe

Trip-insteelcasing

Pump chemicalwash

Pump spacer & slurry

Displacemud in annulus

2.1 Typical parameters

To fix ideas we considered some typical parameters, namely a pipe radiusa = 0.1 m, a pumpingvelocity of u = 1 ms−1, a spacer viscosity ten times that of water (ν = 10−5 m2 s−1), and a densityratio∆ρ/ρ = 0.3. We also assumed that the pipe would be nearly vertical, since shall find that most ofthe mixing takes place near the top of the pipe. See figure 2(a). A wider range of typical parameters aregiven below.

Typical material properties:

densityρ/kg m−3 viscosityν/m2 s−1

wash 900 to 1100 9× 10−7 to 5.5× 10−6

spacer 1200 to 1600 1.8× 10−5 to 4.1× 10−4

Typical pipe radii and pumping rates:

radiusa/m mean velocityu/ms−1 Re (ν = 10−5 m2 s−1)0.05 0.63 to 3.2 2000 to 160000.11 0.52 to 0.79 5700 to 87000.16 0.24 to 0.5 3800 to 8000

The Reynolds number in the wash may be expected to be somewhat higher due to its lower viscosity.

Pipes are between500 m and5000 m in length. A typical volume of spacer corresponds to between100 m and2000 m of pipe.

2

2.2 A simple model for turbulent pipe flow

The Reynolds number based on the pumping velocity,Re = au/ν, is high enough for the flow to beturbulent;Re = 104 for the above typical parameters (u = 1 m s−1, a = 0.1 m, ν = 10−5 m2 s−1).A ‘universal’ model for turbulent flow of a homogeneous Newtonian fluid in a straight cylindrical pipemay be found in§154 of Goldstein (1938) or§5.6 of Townsend (1976). A key parameter is the frictionvelocity u? =

√

σwall/ρ, whereσwall is the stress on the pipe wall, andρ is the density. The meanstreamwise velocityu is related to the friction velocityu? via the equation

uu? =

1k

log Re? + 5.5, whereRe? =au?

ν= Re

u?

u. (1)

Re? is the Reynolds number based on the friction velocityu?, k = 0.4 is von Karman’s constant, and5.5 is another empirical constant. Equation (1) is often rewritten in terms of the Fanning friction factorfr = 2u?2/u2,

fr−1/2 = 4 log10(Re fr1/2)− 0.4, (2)

as in Table A3 of Nelson (1990) or equation (19) of Goldstein (1938,§154), though the latter usesγinstead offr. ForRe<∼ 105, Blasius proposed an approximate explicit formula forfr, (equation 20 inGoldstein (1938)§155)

fr = 0.0665(ua/ν)−1/4 = 0.0665Re−1/4. (3)

The associated eddy viscosityD is given by

D = ku?(a− r), (4)

close to the pipe walls, wherer is the radial coordinate (e.g.Townsend (1976)§5.6). The eddy viscosityis expected to be uniform in the bulk of the flow, where we used the valueD = ku?a.

Although the study group only considered Newtonian fluids, analogous equations exist for power-law and Bingham fluids (e.g. tables A-4 and A-5 of Nelson (1990)).

3 Initial mixing across the pipe

We supposed that the denser spacer fluid would tend to form a finger running down the side of the well,driving an upward return flow of the less dense wash, as shown in figure 2(b). This scenario resembles aturbulent gravity current running down the lower side of the pipe. We assumed that this flow would reacha terminal finger velocityuf in which buoyancy is balanced by the turbulent stressσwall on the wall. Weshall see below that this terminal velocity is an order of magnitude larger than the pumping velocity,so we ignored the pumping velocity in this calculation. Takingg = 10 m s−2, ∆ρ = 300 kg m−3 anda = 0.1 m, the wall stress is

σwall = ga∆ρ/2 ∼ 150 Pa, (5)

and the friction velocity is

u? =√

σwall/ρ =√

ga∆ρ/2ρ ∼√

150 Pa/1000 kg m−3 ∼ 0.4 m s−1. (6)

From equation (1), the friction Reynolds numberRe? = au?/ν ∼ 4000, and the finger velocityuf(equated withu) is around10 m s−1. This is an order of magnitude larger than the pumping velocity,which is itself large enough for the flow to be turbulent. The corresponding eddy viscosity is

D = kau? = 0.2√

2ga3∆ρ/ρ ∼ 10−2 m2 s−1, (7)

which is about1000 times the molecular viscosityν. We assumed thatD is also the eddy diffusivity formixing between the spacer and the wash. The diffusion timeT for turbulent mixing across the pipe isthus

T = a2/(2D) = 1.25√

2aρ/(g∆ρ) ∼ 1 s, (8)

3

Figure 2: (a) A typical well. We assume most of the mixing takes place near the top where the pipe isnearly vertical. (b) Initial mixing via a turbulent gravity current. (c) Streamwise mixing of a horizontallyuniform mixture.

(a) (b) (c)

which is independent of the molecular viscosities of the two fluids. In fact, this estimate holds evenfor the worst case scenario (a = 0.16 m, ∆ρ = 100 kg m−3). We conclude that turbulent mixing willerase any variations in density across the pipe within a few seconds, by which time the flow will havedescended at most a few tens of metres from the inlet.

4 Stabilising streamwise stratification

As the suspending fluid in the spacer is miscible with water, we modelled the spacer/wash mixture as asingle fluid with a spatially varying concentrationC of heavy particles. Thus pure spacer fluid wouldcorrespond toC = 1, say, and pure wash toC = 0. As argued above, we expect turbulent mixing toestablish a uniform concentration across the pipe within a few tens of metres from the inlet. We thusconsidered a mean concentration profileC(z) depending only on the streamwise coordinatez, as shownin figure 2(c).

This concentration profile appears at rest in a frame moving with the pumping velocityu ∼ 1 m s−1.The governing equations in this moving frame, in the Boussinesq approximation, are

∂C∂t

+ u · ∇C = ∇ · (κ∇C), (9a)

∂u∂t

+ u · ∇u = −∇(p/ρ) + αCg +∇ · (ν∇u), ∇·u = 0. (9b)

Hereu is the fluid velocity,α = (1/ρ)dρ/dC is the expansion coefficient, andκ andν are the possi-bly space-dependent concentration diffusivity and kinematic viscosity respectively. In the Boussinesqapproximation we neglect variations in density, and other material properties, except for the variation indensity in the buoyancy termαCg in the momentum equation. See Chandrasekhar (1961)§8 or Drazin& Reid (1981)§7.2.

4

As the flow is turbulent, according to§2.2, we followed Reynold’s analogy and used the sameturbulent valueD for bothκ andν. Equivalently, we assumed the Prandtl number to be one. Havingreplacedκ andν by a turbulent effective valueD, we assumed that (9a,b) describe the mean part ofthe turbulent flow in some locally averaged sense. Making the usual decompositionu = 〈u〉 + u′ andC = 〈C〉+ C ′, we identifiedu andC in (9a,b) with their local averages〈u〉 and〈C〉, and assumed thatthe small scale fluctuationsu′ andC ′ only influence the mean flow by contributing to the turbulent eddydiffusivity D.

In the Boussinesq approximation we also assume that the molecular diffusivity, needed in (1), isspatially uniform. As the turbulent diffusivityD depends only appears logarithmically on the molecularviscosityν via the definition ofRe? in (1), or asD ∝ ν−1/8 from Blasius’ formula (3), this approxima-tion should be reasonable even thoughν itself will vary according to the local concentration of particlesfrom the spacer.

In the frame moving with the pumping velocityu, the noslip boundary condition implies that themean fluid velocity should be−uz on the pipe wall. Thus we expect a mean profile of the formu =U(r)z with U ≈ 0 over most of the pipe, but with a shear layer near the wall so thatU = −u at thewall. However, we ignored the shear layer and assumed plug flow,u = 0 in the moving frame, so wecould formulate a tractable stability problem. As the eddy diffusivityD = ku?(a − r) vanishes at thepipe wallr = a, this provides some justification for ignoring the shear layer on the wall, and adoptingwhat are effectively free slip boundary conditions.

Thus the pumping velocity enters the problem only in supplying the mean velocityu for the turbu-lence model, from which we determineu? andD by solving equation (1).

4.1 Uniform streamwise stratification

By analogy with the Rayleigh-Benard problem for thermal convection (e.g.Chandrasekhar (1961)) weconsidered the linear stability of a uniformly stratified layer,C = C0+z/L, whereL is the scale height.The key dimensionless parameter is the Rayleigh number,

Ra = g∆ρρL

a4

D2 = g∆ρρL

a2

k2u?2 , (10)

wherea is the pipe radius, andD the turbulent diffusivity. The stratified layer is expected to be stableprovided the Rayleigh number is below some critical valueRa < Racrit. We expect that the lengthscaleL for the mixed layer will be that for which the Rayleigh number defined in (10) attains this criticalvalue,

L =1

Racrit

g∆ρρ

a4

D2 =1

Racrit

g∆ρρ

a2

k2u?2 . (11)

This idea seems to have been proposed first by Taylor (1954b), and is supported by experiments reportedby Lowell & Anderson (1982) and Taylor (1954b).

LettingC = C0 + z/L + C ′ andu = u′, and neglecting quadratic terms inu′ andC ′, we obtain

∂C ′

∂t+ w′/L = ∇ · (D∇C ′), (12a)

∂u′

∂t= −∇(p′/ρ) + αgC ′ +∇ · (D∇u′), ∇·u′ = 0. (12b)

In dimensionless variables, and with a spatially uniformD,

∂C ′

∂t+ w′ = ∇2C ′, and

∂u′

∂t= −∇p′ −Ra zC ′ +∇2u′, (13)

where the Rayleigh number is defined in (10). The pressure and the horizontal velocity componentsmay be eliminated by taking∇×∇× of the perturbation momentum equation,

∂∂t∇2u′ = Ra∇×∇×( zC ′) +∇4u′, (14)

5

and then taking the vertical component [Drazin & Reid (1981)§8.1, Chandrasekhar (1961)§9],

∂∂t∇2w′ = −Ra∇2

⊥C ′ +∇4w′, (15)

where∇2⊥ = ∂xx + ∂yy = ∇2 − ∂zz is the horizontal Laplacian. In other words, vertical density

gradients may be balanced by pressure gradients, so only horizontal density gradients remain to drivean instability. Equations (13) and (15) now form a closed system forC ′ andw′, equivalent to equations(8.8,8.13) of Drazin & Reid (1981), or (74,76) of Chandrasekhar (1961)§9, on replacingC ′ by−θ′.

4.2 An eigenvalue problem for the Rayleigh number

We assumed that these equations become unstable via an exchange of stabilities, as in Rayleigh-Benardconvection between parallel planes [Drazin & Reid (1981)§9.1, Chandrasekhar (1961)§11]. In otherwords, the time derivatives vanish at the critical Rayleigh numberRacrit.

The onset of Rayleigh-Benard convection in a cylindrical pipe has been studies by Yih (1959) andBatchelor & Nitsche (1993). The most unstable mode is independent ofz [Proctor 1993] and pro-portional tocos θ in azimuth [Yih 1959]. A more complex mode, such as one withz dependence,would encounter more dissipation, and would only become unstable for Rayleigh numbers larger thanthe critical Rayleigh number for the mode considered here. Perhaps surprisingly, the same is true foraxisymmetric modes [Yih 1959].

The assumption of lack ofz dependence, and also the boundary conditions for the eigenvalue prob-lem, only hold for a vertical pipe, so thatz andg are antiparallel. We assumed that a stable mixed layerwill be established close to the top of the pipe, after the initial horizontal mixing of§3, where the pipeis still very close to vertical (see figure 2(a)).

With these assumptions, (13) and (15) reduce to an eigenvalue problem for the perturbationsw′ andC ′ in the streamwise velocity and concentration respectively,

LC ′ = w′, andLw′ = Ra C ′, whereLf =1r

ddr

(

rD(r)dfdr

)

− fr2 . (16)

The lack ofz-dependence makes∇2⊥ identical to∇2, so we may ‘cancel’ one Laplacian in (15).

The boundary conditions arew′, C ′ bounded asr → 0, dC ′/dr = 0 at r = 1 (no flux of particles)and eitherw′ = 0 at r = 1 (rigid) or dw′/dr = 0 at r = 1 (free slip). This eigenvalue problem has ananalytical solution for rigid boundaries [Yih 1959],

w′ = [J1(kr)I1(k)− I1(kr)J1(k)] cos θ, with Racrit = k4 = 67.94. (17)

This critical Rayleigh number first appeared in Taylor (1954b). The corresponding solution for free slipboundaries is

w′ =[

J1(kr)I ′1(k)− I1(kr)J ′1(k)]

cos θ, with Racrit = k4 = 11.49. (18)

We also obtained the upper boundsRacrit ≤ 71.68 andRacrit ≤ 11.51 respectively by considering thetrial functionsC ′ = r − r3/3, andw′ = r − r3 or w′ = r − r3/3, in a variational formulation of (16)(see§4.4).

Although we allow for a spatially dependent eddy diffusivityD(r) in (16), the study group onlyconsidered uniform diffusivities, for which the eigenproblem is analytically tractable. The free slipboundary conditions were an attempt to compensate for the eddy diffusivity vanishing at the wallsaccording to (4).

4.3 Stability estimates

We used (10) to obtain an estimate for the minimum lengthL, above which a mixed layer shouldbe stable against Rayleigh–Benard convection. Using Blasius’ approximate formula (3) to obtain anexplicit expression foru?, we find that

6

L =1

Racritg∆ρρ

a2

k2u?2 ≈1

Racritg∆ρρ

a2

k2u22

0.0665

(

uaν

)1/4for Re =

uaν

<∼ 105. (19)

Using the free-slip critical Rayleigh number,Racrit = 12, and a vertical pipe (g = 10 m s−2), thisbecomes

L = 160∆ρρ

a2

u2

(

uaν

)1/4metres≈ 1600

∆ρρ

a2

u2 metres, (20)

with a in metres andu in metres per second. The latter approximation is based on the Reynolds numberbeing around104, and highlights the main dependence ona and u. Substituting typical parametersu = 1 m s−1, a = 0.1 m andν = 10−5 m2 s−1 from §2.1, together with∆ρ/ρ = 0.3, we find

L = 160× 0.3×(

0.11.0

)2×

(

0.1× 1.010−5

)1/4metres≈ 4.8 metres. (21)

For a worst case estimate, we take a wide pipea = 0.16 m, a slow pumping velocityu = 0.24 m s−1,and a large density contrast∆ρ/ρ = 0.7, for which

L = 160× 0.7×(

0.160.24

)2×

(

0.16× 0.2410−5

)1/4metres≈ 390 metres. (22)

For a best case estimate, we take a narrow pipea = 0.05 m, a fast pumping velocityu = 3.2 m s−1,and a small density contrast∆ρ/ρ = 0.1, for which

L = 160× 0.1×(

0.053.2

)2×

(

0.05× 3.210−5

)1/4metres≈ 0.04 metres. (23)

Apart from the worst case estimate (22), these lengths are all much smaller than the total length of spaceror wash in the pipe, between100 m and2000 m.

The estimates are all proportional toν−1/4 via Blasius’ approximation (3). Thus using the highestspacer viscosityν = 4 × 10−4 m2 s−1, or the lowest wash viscosityν = 9 × 10−7 m2 s−1, instead ofν = 10−5 ms−1, produces only about a factor of two variation either way in these lengths.

4.4 A variational principle for the Rayleigh number

We also considered a variational principle for estimating the critical Rayleigh number, which has theadvantage of allowing for a spatially dependent eddy diffusivityD(r). A solution of the eigenvalueproblem for the critical Rayleigh number satisfies

0 = −Ra zC ′ −∇p′ +∇·(d∇u), (24a)

0 = −w +∇·(d∇C ′), (24b)

whered(x) is a position dependent dimensionless diffusivity. Multiplying (24b) byC ′, taking the innerproduct of (24b) withu, and integrating by parts over the domain, we obtain

Ra〈wC ′〉 = −〈d |∇u|2〉, 〈wC ′〉 = −〈d |∇C ′|2〉, (25)

where〈·〉 denotes a volume integral,

〈·〉 =∫ 2π/k

0dz

∫ 1

0rdr

∫ 2π

0dθ (·). (26)

The integration by parts involves discarding surface terms of the formC ′ ∂C′∂n , u·n p′ andu · ∂u

∂n , whichvanish on the pipe wall due to the boundary conditions of no flux, no penetration, and either no slip orfree slip. The eigenmode is assumed to be periodic inz with period2π/k, so the contributions from thetwo end surfacesz = 0 andz = 2π/k cancel.

7

Rearranging (25) we find [Proctor 1993],

Ra =〈d |∇C ′|2〉〈d |∇u|2〉

〈wC ′〉2≥ 〈d |∇C ′|2〉〈d |∇w|2〉

〈wC ′〉2≥ 〈d |∇⊥C ′|2〉〈d |∇⊥w|2〉

〈wC ′〉2= S, (27)

where∇⊥ = (∂x, ∂y, 0) denotes the horizontal gradient operator. Proctor (1993) showed that the so-lution of the Euler–Lagrange equation for minimisingS is a z-independent solution of the originaleigenvalue problem (24a,b). The right hand inequality in (27) thus justifies the assumption made in§4.2that the most unstable mode is independent ofz.

For the uniform diffusivity cased = 1 we obtained the upper boundsRacrit ≤ 71.68 andRacrit ≤11.51, for rigid and free slip boundary conditions respectively, by considering the trial functionsC ′ =r − r3/3, and eitherw = r − r3 or w = r − r3/3. Takingd = 1 − r as in (4), we obtain the boundRacrit ≤ 2.2 for free slip boundary conditions respectively, which is in reasonable agreement withthe valueRacrit ≈ 1.84 obtained by solving a discrete form of the eigenvalue problem numerically.Adopting this new value gives about a factor of six increase in the lengths of the mixed layers estimatedin §4.3

5 Streamwise dispersion

For a typical well of lengthLp = 3000 m, and a pumping speedu = 1 ms−1, fluid is resident in thepipe for a timetres = 3000 s. During this time the mixed layer between pure wash and pure space willtend to broaden due to turbulent mixing, and also due to Taylor dispersion [Taylor 1953], the effectivestreamwise diffusion due to variations in streamwise velocity across the pipe,

Taylor (1954a) proposed the valueD = 10.1au? for the effective streamwise diffusivity of a passivetracer in turbulent pipe flow, based on the ‘universal’ model outlined in§2.2. Experimental evidence insupport of this value may be found in Taylor (1954a,b). This value is comparable with the one obtainedby substituting the eddy viscosityD = kau? into the Taylor dispersion formulaD = a2u2/48ν [Taylor1954b] for laminar pipe flow with molecular viscosityν, which leads toD = 1.57au?Re1/4 with theaid of Blasius’ approximation (3).

An order of magnitude estimate for the spreading of the mixed layer due to Taylor dispersion is

LTaylor =√

2Dtres =√

20.2au?Lp/u = 1.92√

aLpRe−1/16 ≈√

aLp, (28)

where the residence timetres = Lp/u. Blasius’ formula (3) has been used for the ratio between the fric-tion velocityu? and the mean pumping velocityu. For the three cases considered in§4.3 the Reynoldsnumber lies in the range3800 ≤ Re ≤ 16000, for which1.92Re−1/16 ≈ 1 to within 10%.

In other words, the mixed layer grows to a length which is roughly the geometrical mean of thepipe radius and the pipe length, with a very weak dependence on the pumping velocity and materialproperties. Pumping faster leads to a shorter residence time which almost exactly compensates for thelarger effective diffusivity. For the three cases considered in§4.3, and a worst case5000 m pipe, theestimated lengths after Taylor dispersion are22 m, 28 m, and16 m respectively. Again, these lengthsare all somewhat smaller than the total length of spacer or wash in the pipe, which is between100 mand2000 m.

Taylor (1954a) reported some experiments which suggest that the dispersionD may be up to a factorof two larger in a slightly curved pipe, even one where the radius of curvature is a hundred times thepipe radius. This would increase the above estimates by

√2.

We also considered some more sophisticated one-dimensional models of two phase flow based uponaveraging over the pipe’s cross-section. These models all reduced to advection-diffusion equationsfor the volume fraction of spacer, or of particles, where the advection velocity is comparable to thepumping velocityu and the diffusivity is comparable with the turbulent diffusivityD. Thus the aboveconclusion based on dimensional analysis seemed robust — the streamwise diffusivity is too small toallow substantial broadening of the mixed layer before it reaches the end of the pipe.

8

6 Effects of a halt in pumping

The above estimates are all based upon an effective turbulent diffusivityD many times larger than eitherthe molecular viscosity or the molecular diffusivity of particles. This large diffusivity is maintainedby wall turbulence generated by the pumping velocityu. If pumping were to cease, the diffusivitiesappearing in (10) would return to their molecular values, leading to a large increase in the maximumstable density gradient calculated in§4.3. Thus the mixed layer, which had previously been stabilised byturbulent diffusion, would be become unstable and start to grow. However, the growth of an instabilitywould itself drive turbulence which would help to stabilise the layer. To obtain a ‘worst case’ estimate,we supposed that the turbulence driven by pumping would decay quickly, and tried to estimate howrapidly the mixed layer would spread in its the absence.

We tried to model the resulting turbulent convection by assuming that the spanwise mixing modelin §3 only “sees” the effective density contrast over a distance comparable with the pipe radius. In otherwords, the effective density contrast isa∆ρ/L instead of∆ρ. Noting, from (6) and (3), that the fingervelocity uf is approximately proportional to∆ρ1/2, we first considered a simple model in which themixed layer grew with this modified finger velocity,

dLdt

= uf

√

aL

⇒ L3/2 = L3/20 + tuf

√a. (29)

Hereuf is the finger velocity calculated in§3, anduf√

a/L the modified value based on a reduceddensity contrast. Taking typical valuesuf = 10 m s−1, a = 0.1 m, L0 = 30 m, we find thatL hasdoubled after50 s. It seems that the mixed layer will not grow substantially so long as the pumping isnot halted for more than a few seconds.

6.1 A nonlinear diffusion model

A more sophisticated approach is to replace∆ρ by a∂ρ/∂z in equation (6) for the friction velocity,

u? =√

σwall/ρ = a(

gρ

∂ρ∂z

)1/2. (30)

As above, this replacement is motivated by a mixing length approach, in which the effective densitycontrast is that seen over a mixing length of approximately one pipe radius. In the absence of a meanflow due to pumping, we expect the concentrationC of the dense component to diffuse with the eddydiffusivity D = kau?. This leads to a nonlinear diffusion equation for the concentrationC,

∂C∂t

=∂∂z

(

D∂C∂z

)

= ka2(

αgρ

)1/2 ∂∂z

(

∂C∂z

)3/2, (31)

whereα is the expansion coefficient,ρ = ρ0 + αC, as in§4, andk is von Karman’s constant.Holmeset al. (1991) and Bairdet al. (1992) performed experiments on turbulent convective mixing

between salty and fresh water in tubes. Their experimental setup was quite similar to the problemconsidered here, except their tubes were somewhat narrower (a < 3.3 cm) and much shorter (less than1.3 m). They proposed an effective diffusivity of the same form,

D = `2(

gρ

∂ρ∂z

)1/2, (32)

based on dimensional analysis and an empirical study of their experimental data. The parameter` is aturbulent mixing length, which they found to be somewhere between their pipe radius and pipe diameterby fitting their experimental concentration profiles.

9

6.2 Diffusion timescales

The change of variablesC(z, t) = (∆ρ/α) c(z/L, t/T ) puts the nonlinear diffusion equation (31) intothe dimensionless form

∂c∂τ

=∂∂x

(

∂c∂x

)3/2, (33)

wherex = z/L andτ = t/T are dimensionless independent variables. The diffusion timescaleT isgiven by

T =23k

(

La

)2 (

ρLg∆ρ

)1/2. (34)

Eliminating the lengthscaleL using the mixed length estimate (20) from§4.3, the diffusion timescaleTmay be rewritten as

T =20483k

(

∆ρρ

)2 g2a3

u5

(

uaν

)5/8. (35)

We found in§4.3 thatL varied by four orders of magnitude between the best and worst cases considered.Since the diffusion timescaleT is roughly proportional toL2, we find an even more sensitive dependenceof T upon the problem parameters, in particulara, u and∆ρ/ρ. For the three cases considered in§4.3,we findT ≈ 1200 s, T ≈ 7 × 107 s, andT ≈ 0.2 s respectively. For a more plausible estimate in thelast case, corresponding to (23), if we take the lengthL = 16 m based on Taylor dispersion from§5 thetimescale becomesT ≈ 4× 105 s.

Some neglected constants may change these estimates by a factor of three or four. For instance, theexperiments of Holmeset al. (1991) and Bairdet al. (1992) suggest that the mixing length` should becloser to2a than toa, which would reduce the timescales in (34) and (35) by a factor of four. In thelower stages of the pipe where the pipe axis is some way from verticalg should perhaps be replaced bythe reduced gravityg cosβ, though it is not clear what effect a slanted pipe would have on the mixinglength. A numerical solution of (33) with initial conditions

c(x, 0) =

0 if x < −1,

x +12

if |x| < 1,

1 if x > 1,

(36)

shows that the time to double the width of the profile (i.e. halve the maximum gradient) is only0.3T .Conversely, the doubling time is25/2T ≈ 5.7T for the similarity solution derived below.

However, despite all these possible modifications the timescales in (34) and (35) for realistic initialmixed layer thicknesses all suggest that the mixed layer would not grow substantially if pumping werehalted for a few seconds or even a few minutes.

6.3 Approach to a similarity solution

Equation (33) has a similarity solution of the formc(x, τ) = f(x/τ2/5), where

f(η) =

0 if η < −η0,(

3η5 − 10η3η20 + 15ηη4

0)

/3375 +12

if |η| < η0,

1 if η > η0.

(37)

The constantη0 = 153/52−4/5 ≈ 2.916 is determined by the requirement thatf(η) = 0, 1 at the twoturning pointsη = ±η0. As with the more familiar similarity solution to the porous medium equation,deviations from the background statesc = 0, 1 are confined to a compactly supported region, in thiscase|η| < η0. However, unlike the similarity solution to the porous medium equation,f , f ′ andf ′′ areall continuous atη = ±η0. Figure 3 shows a numerical solution of (33) with piecewise linear initialconditions (36) approaching the similarity solution. Byt = 1 the numerical solution is indistinguishablefrom the similarity solution with the time origin offset by0.04.

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Figure 3: Numerical solution of the nonlinear diffusion equation (33) with initial conditions (36) com-puted using the NAg routineD03PGF. The solutions att = 0.3 andt = 1 are indistinguishable fromthe similarity solution (37).

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3x

c(x,t)

t=0.0t=0.1t=0.3t=1.0

Equation (33) may also be rewritten as a porous medium (Barenblatt) equation for the density gra-dienth,

∂h∂τ

=∂2

∂x2

(

h3/2)

=32

∂∂x

(

h1/2 ∂h∂x

)

, whereh =∂c∂x

. (38)

The above similarity solution then corresponds to a solution of (38) forced by the boundary conditionh = h0τ−2/5 at x = 0, and satisfyingh → 0 asx → ±∞. The constantsη0 andh0 are related byη40 = 225h0.

7 Conclusions

We found in§4.3 that the length of the initial mixed layer is proportional to the density contrast∆ρ/ρ,proportional to the pipe radius squared, and approximatelyinverselyproportional to the pumping ve-locity squared. Pumping faster causes more turbulence and so, somewhat counterintuitively, allowslessmixing between the wash and the spacer. This favours a narrow pipe and a high pumping velocity. Asmall density contrast is also helpful.

The subsequent growth of the mixed layer due to Taylor dispersion was found in§5 to be propor-tional to the geometric mean of the pipe radius and pipe length, but almost independent of the pumpingrate. Pumping rapidly produces enhanced turbulent dispersion which almost exactly compensates forthe reduced residence time. For two of the three cases considered, Taylor dispersion was found to be thedominant spreading mechanism. This also favours a narrow pipe.

Halting pumping for a few seconds or even a few minutes seems unlikely to cause substantial growthof the mixed layer, but again the growth rate would be minimised by a narrow pipe and a high pumpingvelocity.

The critical Rayleigh number and the Taylor dispersion coefficient are both sensitive to the meanflow profile, which could presumably be modified by changing the rheology of the spacer fluid.

8 Contributors

The problem was posed by Ian Frigaard and Guiliano Sona. Academic contributors included P.J. Dellar,G. Duursma, A. Fitt, E.J. Hinch, O.G. Harlen, J. King, J.R. Ockendon, D. Parker, C. Please, N. Stokesand S. Wilson. The report was prepared by P.J. Dellar. Figures (1) and 2(a) were provided by IanFrigaard.

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9 References

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Batchelor, G.K. & Nitsche, J.M., (1993) Instability of stratified fluid in a vertical cylinder,J. FluidMech.,252, 419–448.

Chandrasekhar, S., (1961) Hydrodynamic and Hydromagnetic Stability,Oxford University Press(Dover reprint 1981).

Drazin, P.G. & Reid, W.H., (1981) Hydrodynamic Stability,Cambridge University Press.

Goldstein, S. (ed), (1938) Modern Developments in Fluid Dynamics,Oxford University Press.

Holmes, T.L., Karr, A.E. & Baird, M.H.I., (1991) Effect of unfavorable continuous phase densitygradient on axial mixing,AIChE J.,37, 360–366.

Lowell, M.E. & Anderson, J.L., (1982) Stable concentration gradients in a vertical tube,Chem.Eng. Commun.,18, 93–96.

Nelson, E.B. (ed), (1990) Well Cementing,Elsevier: Amsterdam.

Proctor, M.R.E., (1993) Proof thatκ = 0 gives the minimum critical Rayleigh number in the caseof constant density gradient,J. Fluid Mech.,252, 447–448. (Appears as an appendix to Batchelor &Nitsche (1993).)

Taylor, G.I., (1953) Dispersion of soluble matter in solvent flowing slowly through a tube,Proc. Roy.Soc. Lond.,A219, 186–203. Also inScientific Papers of G. I. Taylor, vol. IV, pp 225–243, ed.G. K.Batchelor, Cambridge University Press.

Taylor, G.I., (1954a) Dispersion of matter in turbulent flow through a pipe,Proc. Roy. Soc. Lond.,A223, 446–468. Also inScientific Papers of G. I. Taylor, vol. II, pp 466–488, ed.G. K. Batchelor,Cambridge University Press.

Taylor, G.I., (1954b) Diffusion and mass transport in tubes,Proc. Phys. Soc. Lond.,B67, 857–869.

Townsend, A.A., (1976) The Structure of Turbulent Shear Flow,Cambridge University Press.

Yih, C.-S., (1959) Thermal instability of viscous fluids,Quart. Appl. Maths.,17, 25–42.

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