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Copyright 2005, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the SPE International Symposium on Oilfield

Scale held in Aberdeen, United Kingdom, 1112 May 2005.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in a proposal submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to a proposal of not more than 300

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Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractDownhole scale inhibitor (SI) squeeze treatments are acommon feature of the scale control plans of many oil

operators. However, reservoir formations are large,

heterogeneous rock bodies in which fluid flow is strongly

determined by the permeability structure. Thus, when a slug

of scale inhibitor is injected into the formation, fluid

placement is an important issue. To design successful squeezetreatments, it is necessary to know where the injected fluid

goes or, even better, we would like to control where the fluid

package is placed in the near-well reservoir formation.In this paper, we go back to basics, in that we re-derive

the analytical expression that describe placement in linear and

radial layered systems for unit mobility and viscous fluids. Initself, this is not new since these equations are well known.

However, we apply them in a novel manner to describe scale

inhibitor placement. We also demonstrate the implications of

these equations on how we should analyse placement both in

the laboratory and by numerical modelling before we apply a

scale inhibitor squeeze. We present an analysis of viscosifiedSI applications for linear and radial systems both with and

without crossflow between the reservoir layers.

Visualisation experimental results are also presented of

simple and viscosified slug placement in layered bead packswith crossflow between layers. It is shown that these agree

very well with the numerical predictions. Additional

calculations on near well placement in radial systems are also

presented showing how the theory carries over into real field

near-well, heterogeneous systems. Some novel ideas are

presented on the application of viscosified scale inhibitor

treatments.

Background and IntroductionChemical scale inhibitors have been applied for many years in

downhole squeeze treatments. The objective is to have an

aqueous phase scale inhibitor (SI) return concentration, [SI],

above some minimum inhibitor concentration (MIC) for as

long as possible [1-8]. The squeeze lifetime is a strongfunction of the SI/rock interaction e.g. by adsorption. In a

homogeneous reservoir layer, adsorption may be the only

retention mechanism governing the SI return from the well.

However, reservoir formations are rarely homogeneous but are

made up of highly heterogeneous rocks which may have a

layered or more complex structure as determined by varioussedimentological, structural and diagenetic factors [9]. Here

we will consider only layered systems where the variouslayers have different permeabilities, k (and porosities, ) in thenear-well formation. In such systems, SI placement within the

formation is an additional aspect of a squeeze treatment that

must be considered since this may affect the SI returns.

In most cases, scale inhibitors are applied as aqueous

solutions at concentration, typically in the range 10,000

150,000 ppm. These solutions usually have a viscosity ()close to that of a normal injection brine; i.e. ~1 cP at 20

oC and

0.3 cP at 100oC. Therefore, apart from a slight temperature

effect, the injected brine displaces formation water (FW) at

unit mobility. Also, for lighter oils, a unit mobility

displacement is often involved although viscosity and relative

permeability effects may be more important in heavier oils. Inunit mobility injection into a heterogeneous layered linear or

radial system, as shown schematically in Fig. 1, the fluid

placement into layer i is governed solely by the (kh)iproduct.

That is, injecting fluid at a total volumetric flow rate of QT

into an N-layer system of the type shown in Fig. 1, then flow

into layer i, Qi, is given by:

( )

( )1

ii N

ii

khQ

kh=

=

(1)

It can easily be shown that this is true for unit mobilitydisplacement in a linear or a radial system with or without

crossflow. However, this well established result might foster

the belief that linear and radial systems are also very similar

under viscous slug injection with and without crossflow and

this is notthe case.

In recent years, the use of viscosified slugs of SI has been

proposed in order to change the placement pattern in a

favourable manner [10 - 13]. In this context, favourable

may mean to place the SI slug entirely in the high permeability

layer from which the water is being produced. However, it

may also mean that we wish to place the inhibitor slug in the

lower permeability layers where it may be stored and flow

back to the well more slowly because of the reduced flows

from these layers. Whatever, our intention, we must clearly

understand the fluid mechanics of viscous slug placement in

SPE 95090

Scale Inhibitor Placement: Back to BasicsTheory and ExamplesK.S. Sorbie and E.J. Mackay, Heriot-Watt U.

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2 SPE 95090

heterogeneous systems to achieve the effect we are after i.e.

most of the SI slug in the high k or low k layer.

At this point, we note that viscosified solutions or othertype of divertor may also be injected to modify the relative

flows in the wellbore and near-well formation in long

horizontal wells [10 - 13]. Viscous fluids are also used in a

similar manner in viscous acidising. However, we will not

consider wellbore effects in this paper. It is the fluidmechanics in layered heterogeneous formations both with andwithout crossflow for linear and radial systems we will

consider here. These layered systems may have N-layers but

for simplicity we consider only 2, as in the simple schematics

in Fig.1. All the analytical results here can be generalized

quite easily to multi-layer systems.

Problem Statement

The linear and radial layered systems considered in this work

are shown schematically in Fig. 1. We will consider the

various combinations in the following order:

(a) L inear heterogeneous systems no crossflow: Analyticalexpressions will be used to show the level of diversion that

occurs when viscous fluid is injected into a layered system.

(b) Radial heterogeneous systems no cr ossflow: Analytical

expressions for viscous injection into no-crossflow radial

systems are also known and are compared with linear results.

(c) L inear heterogeneous systems with crossflow: For this

case, experimental results are presented showing clear viscous

crossflow. Both viscous over-stable cases (p/w> k1/k2) andunstable cases (injected < in situ viscosity) are considered.

(d) Radial heterogeneous systems with cr ossflow: This

case is only amenable to numerical solution and results are

presented following on from the results of case (c) above.

For both linear and radial heterogeneous no-crossflow

cases, analytic expressions for the flows and relative

penetrations of injected fluid have been derived by Seright and

co-workers [14 - 18]. Flow in layered systems with crossflow

has also been studied previous by several workers [19 -23].

Linear Heterogeneous Systems No Crossflow

Formulation: As noted, the flow of a unit mobility fluid is

simply governed by the layer (kA)i. For viscous fluid

placement, we use the notation in Figs. 1 and 2, where 1 and 2

label the high and low k layers, respectively. For constant

volumetric fluid injection rate, QT, of the (viscous) fluid of

viscosity, p(p> w), the cumulative volume injected at timet is given by:

( ) .inj T q t Q t = (2)

The layer flow ratio, ( )1 2Q Q , can be shown to be given by:

( )

1 1 1

2 21 1 1

2 2 2 1

. ( ( ) ).

. .

p w

w inj

w p w

L q t x AAQ k A

Q k A L x

+ = +

(3)

Using the constraint that,1 2TQ Q Q= + , the rate of advance in the

high k layer 1 is given by:

( )

( )

. 1 11

2 21 1 1 1

1 1

;

;

TQ f x t dx

dt k AA f x t

k A

=

+

(4)

where

( )1 1;x t is the expression in the large bracket on the

RHS of Eq. 3. The advance rate in the low k layer can easily

be found by material balance. The following simple algorithm

was used to solve the viscous placement equations above:

1. At t =0, f1= 1 calculate (dx1/dt)

2. Integrate Eq. 4 over t3. Update => x1(t+t) = x1(t) + t.(dx1/dt)4. Calculate (Q1/Q2) from Eq. 3

5. Update t = t + t go to Step 2

Resul ts linear system, no crossflow: Physically, it is clear

that at t = 0, the ratio of flows (Q1/Q

2) will simply be the

permeability ratio since the system is full of fluid of viscosity,

w. Likewise, in the very long time limit, when both layers arefull of the viscous fluid ( =p), then the flow ratio must returnto this original value. However, the final pressure drop across

the system will now be, Pf= (p/w)Pi, where Pi is theinitial pressure drop. It is intuitively clear that the flow ratio

will decrease on injection of viscous material and that the

minimum flow ratio will be reached when the high k layer just

fills with viscous fluid. An example calculation is shown in

Fig. 3 for a permeability contrast of (k1/k

2) = 10 which shows

that increasing p to 10 (w=1) causes the flow ratio (Q1/Q2)to decrease from 10 to ~3.5 at ~0.6 PV injection. The effect of

viscosity ratio, (p/w=5, 10, 50, 100), on the layer flow rate

ratio is shown in Fig. 4 which shows four important features.Firstly, the fact that injecting a viscous fluid does cause

diversion of injected fluid into the low k layer is demonstrated.

Secondly, and unsurprisingly, it is seen that (Q1/Q

2) drops

more rapidly for higher viscosity injected fluids. Thirdly, the

minimum flow ratio is approximately the same (between 3 and4) for all viscosity ratios from 5 to 100, although it is reached

more rapidly for the higher viscosity cases. Fourthly, there is

very little difference in the 50 and 100 viscosity ratio results

suggesting that there is some critical viscosity ratio, (p/w)c,above which little difference is seen in the flow ratio. The

degree of fluid diversion can be seen in Figs. 5 and 6 which

show plots of x2 (penetration into the low k layer), when the

high k layer just fills with viscous fluid, i.e. at x1= L (see Fig.5). Fig. 6 shows quantitatively the penetration of fluid into the

low k layer for viscosity ratios, (p/w) = 1, 20 and 100; thepenetration goes from 0.1 (p= w= 1) to ~0.31 (p20).

Summary:The results for viscous placement in a no crossflow

layered linear heterogeneous system are as follows:

(i) The analytical calculations indicate that using viscous

fluids looks attractive for placing SI in the lower k layer.

(ii) There is a critical viscosity ratio for a given permeability

ratio above which there will be very little additional diversion.

It can be shown that this is given approximately by:

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SPE 95090 3

1

2

2p

w c

k

k

(5)

(iii) Furthermore, as (p/w) increases indefinitely, it can beshown that the flow rate ratio limits as follows:

1 1

2 2as

p

w

Q k

Q k

(6)

(iv) Given the above, it seems reasonable to investigate

viscous placement further by experiment e.g. in parallel cores.

Radial System No Crossflow

Formulation: Now consider the very similar problem in

radial (no crossflow) systems to that in linear layered systems.

We pose the question: regarding placement, are there any

important differences between radial and linear systems?

The radial two-layer system is shown in Fig. 1. We might

suspect that there are differences between the linear and radial

cases because of the logarithmic pressure drop that arises fromthe radial Darcy law [24], shown schematically in Fig. 7,which also gives the notation for the developments below. For

unit mobility placement (p= w) in a radial system, the flowrate ratio is the same as in a linear system:

1 1 1

2 2 2

Q k h

Q k h

=

(7)

However,unlikethe linear system, the radial advance ratio,

(r1/r2), (equivalent to (x1/x2), is given by:

1 1

2 2

r k

r k= (8)

and this is also shown inset on Fig. 7. Using the notation in

Figs. 1 and 7, it can be shown that the flow rate ratio during

viscous placement in a no crossflow radial system is given by:( 1)

max 1

1 1 1

( 1)2 2 2 max 2

ln

.

ln

w

w

R r

rQ k h

Q k h R r

r

=

(9)

where is the viscosity ratio, = (p/w).The algorithm for solving for the frontal advance in each

layer, i.e. r1(t) and r

2(t), is outlined below, where = (Q

1/Q

2):

1. Calculate ( )1 2Q Q from Eq. 9; where initially (t = 0),

( ) ( )1 2 1 1 2 2Q Q k h k h= .

2. Calculate, Q1 as:( )1 1

TQ

Q

=

+ and then calculate,

2 1TQ Q Q=

3. Calculate, ( ) ( )1/ 2

211 1

1 1

.new old

Q tr r

h

= +

and calculate ( )2 newr from a similar expression.

4. Set ( ) ( )1 1old newr r= ; ( ) ( )2 2old newr r= and t = t + tand go to Step 1 above.

Results: The flow ratio, (Q1/Q

2) vs. time, is shown in Figs. 8

and 9 for a no crossflow radial system for viscosity ratios, =(p/w)= 2, 5, 10, 20, 50 and 100. The longer (0 80t ) andshorter ( 0 20t ) time behaviour is shown in Figs. 8 and 9,respectively (t in arbitrary units). System data is: Rw=0.5; Rmax

= 10; k1= 10; k2= 1; h1= h2= 5; 1= 2= 0.2;QT= 2;=(

p/

w) = 1 , 2, 5, 10, 20, 50, 100 (

w = 1). Another way of

representing this fluid diversion in the radial system for thesame data is shown in Figs. 10 and 11. Figure 10 shows the

penetration distance, r2, in the low k radial layer and ratio(r1/r2) when the high k penetration depth is, r1 = 5, as a

function of the viscosity ratio, . Fig. 11 shows the penetrationratio (r1/r2) vs. time for viscosity ratios, = 1 and 10. A closestudy of the results in Figs. 10 and 11 as measured by theactual value of r2 in Fig. 10 or by (r1/r2) indicates that thedegree of diversion in a radial system is much lessthan that in

a linear system. For example, for the (k1/k2) = 10 case, then p> 5 makes little difference since at r1= 5, the value of r2goes

from 1.7 to a maximum of r22.1 as p increases. The mostdirect way to plot the results is to show the actual penetrationin each layer after scale inhibitor slug injection in both linearand radial two-layer systems as shown in Figs. 12(a) - 12(d).

Summary and comparison of no-crossfl ow linear and r adial

systems:The conclusions to this point are as follows:(i) Viscous fluid injection into a layered linear or radial no

crossflow system causes diversion of fluid to the low k layer;

(ii) This diversion is muchlarger in a linear system than ina radial system where it is not very significant;(iii) Implications from these points are: (a) results from

no-crossflow linear systems (such as parallel coreexperiments) are actively misleading in terms of radial

placement; and (b) we should use modelling to examineplacement - using data from 1D core flood experiments tosupply parameters (e.g. on effective viscosities etc.)

Linear Layered System With Crossflow

Uni t mobili ty placement:Fig. 13 shows a schematic two-layer

heterogeneous system withcrossflow but under unit mobility

injection (p=w), showing the (constant) pressure profiles,

P(x), along the layers. This case is essentially identical to

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4 SPE 95090

the no-crossflow case as the frontal advance, rates are constant(x1(t) and x2(t) are linear with time in each layer). This is

shown for the experimental unit mobility flood in Fig. 14

where a blue dyed brine (p=1cP) displaces a clear brine(w=1cP) in a two-layer beadpack. The pack in Fig. 14 is 56cm long by 5.5 cm in vertical height and ~1.3 cm thick; fulldetails are given in Table 1 and ref. [22]. Confirmation of the

linear flows for this unit mobility flood is shown in Fig. 15where x

1(t) and x

2(t) are plotted from Fig. 14 as functions of

PV injected (constant rate). The results in Fig. 15 show thatthe permeability contrast is:

1 1

2 2

13 135 71

2 30

.= .

.

x k

x k= =

(10)

We note that this behaviour is well known but such floods

are necessary to establish the (k1/k2) ratio in Eq. 10 for use inthe viscous slug placement calculations below (Table 1). Fig.16 shows a direct simulation of the unit mobility flood in Fig.

14 using the permeability contrast in Eq. 10 which, as

expected, reproduces the experimental results very well.

Vi scous stable slug in jection: We now consider the injectioninto the two-layer beadpack described above (k1/k2 = 5.71) of a

slug of viscosity, p, greater than that of the resident fluid(brine viscosity, w). A schematic of viscous stable slugplacement is shown in Fig. 17 where the pressure profiles ineach layer at a given time are shown. In this case, the pressure

profiles are evolving as the viscous fluid advances through thesystem. Fig. 18 shows the injection of a viscous slug of clear

fluid (p 10cP) into a blue brine (w=1cP) in the two-layerbeadpack. It is clear by comparison with the unit mobilityresults in Fig. 14 that significant amounts of injected fluid are

diverted into the low k layer [22]. Indeed the layer pressureprofiles for the viscous case in Fig. 17 indicate that: (a)crossflow of viscous injected material occurs behindthe frontfrom high k low k layer; and (b) corresponding crossflowof low viscosity (blue) fluid must occur from the low highk layer in front of the interface. Fig. 19 shows a direct

simulation of the viscous stable flood in Fig. 18. Very goodagreement is seen between the experimental and modelledresults indicating that the crossflows are well predicted.

Fig. 20 shows the experimental and numerically simulatedfrontal advance ratio, (x1/x2) vs. PV injected for the viscousstable slug injection in Fig. 18. This figure also shows some

similar repeat floods from ref. [22]. Fig. 21 shows the

numerically simulated fractional layer recovery vs. PVinjected for this flood. Note that this confirms the direction ofcrossflow shown schematically inset in Fig. 21 as explainedabove, since we see that more fluid is recovered through thewell at the end of layer k

1than was in the layer originally.

This is due to a strong viscous crossflow mechanism asdescribed previously [20 - 22, 25].

Viscous unstable slug i njection: We now take the finalviscous slug placement in the two-layer pack after injection of0.36 PV (Fig. 18) as initial conditions for an unstable

displacement. Fig. 22 shows the injection of a slug of red

brine (w 1 cP) to displace the viscous clear slug (p 10cP). The in situ blue brine has the same viscosity as the

injected red brine (w = 1cP). Viscous fingering of the redbrine into the clear viscous fluid is observed and the (clear)viscous slug is displaced into the low k layer. Fig. 23 shows asimulation of this slug breakdown experiment in Fig. 22. Inthe numerical calculation, fluids are not distinguished bycolour as in the experiment. However, the location of the(clear) viscous slug is evident and it is predicted to be

displaced into the low k layer, as found in the experiment

Summary of stable viscous slug placement and unstable slug

breakdown: The main conclusions for the linear layeredheterogeneous system with crossflow are:

(i) A viscous overstable flood is described here where

(p/w)>(k1/k2), which leads to extensive crossflow in layeredsystem where free communication between layers is allowed.

On increasing the injected fluid viscosity significantly abovethe over stable condition, no further crossflow occurs ratherlike the degree of diversion in no crossflow systems.

(ii) The direction of crossflow in the stable displacement isthat viscous injected fluid is diverted into the low k layer andthere is crossflow of in situ low viscosity fluid from the low to

the high k layer.(iii) There is excellent quantitative agreement between the

experimental observations and numerical simulation for theviscous slug placement and the levels of crossflow.

(iv) The behaviour for viscous slug placement is quitedifferent from that observed for a no crossflow layered system.

Thus, this type of information is only accessible by performinglayer pack studies of the type described here or by carrying outnumerical simulations (which agree with such experiments).

(v) When a viscous slug is further placed into the layeredsystem with crossflow by an unstabledisplacement, fingeringthe high k layer occurs and the viscous slug is displaced into

the lower k layer.

Radial Layered System With Crossflow

Finally, we consider a radial heterogeneous layered systemwith free crossflow between layers. Performing experimentsin radial layered beadpacks with crossflow is not feasible

routinely. For such a system, the best approach is to performdirect numerical simulations. This has been done for a radial

system of laboratory dimensions as described in Table 1.However, the viscous forces scale linearly to real fielddimensions as described elsewhere [22]; i.e. the same resultswould be found by considering a model of the radial formation

with Rmax

= 4.82m etc. Fig. 24 shows the unit mobility andviscous slug placement with and without crossflow at 0.5 PV

of injection. Fig. 25 shows the same cases after a further

0.3PV overflush with = 1 cP fluid in all cases. A summaryof the percentage of the main slug injected which ends up inthe low k layer is given for the radial cases at various stages inTable 2, where a comparison with the linear system is also

given. Placement in the low k layer is the same (17.4%) forthe radial and linear systems both with and without crossflow

for the unit mobility ( = 1cP) main slugs. Table 2 also showsthe % increase in the amount that ends up in the low k layercompared with the unit mobility case (always 17.4%). .

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SPE 95090 5

Summary of radial cases and comparison with l inear system:Taking all cases together, we summarise as follows:

(i) The no crossflow radial 10cP 0.5PV main slug increasesplacement in low k layer by ~36% compared with a ~70%increase for the linear flood;

(ii) The with crossflow 10cP 0.5PV main slug increasesplacement in low perm by ~ 68% for the radial case compared

with a ~125% increase for the corresponding linear case;(iii) The with crossflow 10cP 0.5PVmain slug followed by

a further 0.5PV of 1cP (unstable) overflush increasesplacement in the low k layer in the radial case by ~ 106%

compared with a ~223% increase for the corresponding linearcase.

(iv) The results in Table 2 support and emphasise thosedescribed above The % increase in the amount of diversiongoing from unit mobility viscous floods increases by afactor of ~2 going from a radial to a linear system, for all cases

(figures in parenthesis in Table 2).

Discussion and Conclusions

In this paper, we have revisited the issue of viscous slugplacement in layered linear and radial systems both with andwithout crossflow between the layers. The use of viscosified

slugs for scale inhibitor (SI) placement has been suggested inorder to divert the inhibitor into a given favourable layerwithin the formation. Favourable may either mean to placethe SI slug entirely in the high k or low k layer. Whatever theobjective, we must understand the fluid mechanics ofplacement in heterogeneous systems. In addition, it is

necessary to understand the differences between linear andradial systems as well as between systems with and withoutfree crossflow between layers. This paper does not consider

the effect of shear-thinning systems, which will be discussed

in a future publication.Specific findings and conclusions for each of the 4 cases

(a d) listed in the Problem Statement section are presented inthe appropriate sections above. Here, we summarise the mainbroad conclusions of this work as they apply to placement

using viscosified slugs of scale inhibitor, as follows:1. There are clear differences between linear and radial

heterogeneous systems in that viscous placement results in

more diversion in a linear system. This is the case when thereis either no crossflow or free crossflow between thepermeability layers. More viscous material enters the low k

layer in the latter case. For the cases studied here, the %increase in the amount of diversion from unit mobility viscous floods increases by a factor of ~2 going from a radial

to a linear system.2. When crossflow is present and an unstable postflush

follows a viscous SI slug placement, this results in fingering in

the high k layer and further subsequent diversion of theviscous SI slug into the lower permeability layer. If theobjective is to place the SI in the lower permeability layers,

then this mechanism will do this. However, if the intention isto place all the SI in the high k layer, then the viscosified slugshould contain no SI, and the SI should be present in the low

viscosity post flush.3. In order to assess the levels of diversion caused by a

viscosified SI slug (or indeed a viscosified preflush or

postflush) in a heterogeneous reservoir system, numerical

simulationshould be carried out for a (usually radial) systemwith the correct layer properties and levels of layer-layer

communication (vertical permeability).4. We should avoid performing erroneous assessments

such as carrying out linear parallel core experiments as a

means of assessing levels of viscous fluid diversion. For thereasons detailed in this paper such assessments are actively

misleading. Core flooding in 1D cores should be used simplyto measure the input properties for the simulations describedabove e.g. in situ effective rheology in the porous medium[25], adsorption levels of SI etc.

Acknowledgements

The authors would like to thank the sponsors of the FlowAssurance and Scale Team (FAST) at Heriot-Watt University:FAST I sponsors - Baker Petrolite, BioLab, BP,ChampionServo, ChevronTexaco, Clariant Oil Services,

ConocoPhillips, Exxon/Mobil, Halliburton, Kerr-McGee,Marathon, MI Production Chemicals, Norsk Hydro, Nalco,Petrobras, Rhodia, Saudi Aramco, Schlumberger, Shell,

Statoil, Total and Yukos. We also thank Oscar Vasquez forpreparing Fig. 12.

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17. Liang, J.-T., Lee, R.L. and Seright, R.S.: Gel Placement inProduction Wells, SPE20211, SPE (Production and Facilities),

pp. 276 286, November 1993.18. Seright, R.S.: Improved Techniques for Fluid Diversion in Oil

Recovery, First annual report Contract No. DE-AC22-92BC14880, US DOE, October 1993.

19. Root, P.J, and Skiba, F.F.: Crossflow Effects During anIdealized Displacement Process in a Stratified Reservoir, SPEJ., pp. 229-237, September 1965.

20. Zapata, V.J. and Lake, L.W.: A Theoretical Analysis of

Viscous Crossflow, SPE10111, presented at the 58th SPEAnnual Fall Conference, San Antonio, TX, 5 7 Oct. 1982.

21. Sorbie, K.S., Wat, R.M.S., Hove, A., Nilsen, V. and Leknes, J.:

Miscible Displacements in Heterogeneous Core Systems:Tomographic Confirmation of Flow Mechanisms, SPE18493,

presented at the SPE International Symposium on OilfieldChemistry, 8-10 February 1989.

22. Sorbie, K.S., Sheb, M., Hosseini, A. and Wat, R.M.S.: ScaledMiscible Floods in Layered Beadpacks Investigating ViscousCrossflow, the Effects of Gravity and the Dynamics of Viscous

Slug Breakdown, SPE20520, presented at SPE Annual Fall

Conference, New Orleans, LA, 23-26 Sept. 1990.23. Sorbie, K.S. and Seright, R.S.: Gel Placement in

Heterogeneous Systems with Crossflow, SPE/DOE24192,presented at the SPE/DOE 8thSymposium on EOR, Tulsa, OK,

22-24 April 1992.24. Dake, L.P.: Fundamentals of Reservoir Engineering,

Developments in Petroleum Science No. 8, Elsevier,

Amsterdam, 1978.25. Sorbie, K.S.:Polymer Improved Oil Recovery, CRC Press, Boca

Raton, FL, 1991; discussion of viscous crossflow p. 274ff.

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SPE 95090 7

Table 1: Properties of the two-layer beadpack [22] and thesimulation model for both the beadpack and the radial model.

Linear Pack Details

Pack Length = 56.0 cm x-Grid: 122 cells; x = 0.5 cmPack Width = 1.3 cm y-Grid: 1 cell; y = 1.3 cm

Pack Height = 5.5 cm z-Grid: 22 cells; z = 0.25 cmLayer 1(2.5 cm): High permeability; Average k1= 9707 mD(uniform random distribution - range +/- 4500 mD)

Layer 2(3.0 cm): Low permeability; Average k2= 1700 mD

(uniform random distribution - range +/- 500 mD)

Permeability contrast, (k1/k2) = 5.71

Porosity, =0.38Total pore volume (PV) = 152.2 cm

3

Total fluid injection rate, QT= 60 cm3/hour

Key to experimental floods:

Fig. 14:Unit mobility flood:Injected blue fluid, = 1 cP; In situ clear fluid, = 1 cP

Fig. 18:Viscous stable slug injection:

Injected clear fluid, = 10 cP; In situ blue fluid, = 1 cP

Fig. 22:Viscous unstable slug breakdown flood:

Injected red fluid, = 1 cP; In situ clear fluid, = 10 cP; Insitu blue fluid, = 1 cP.

Radial Model Details

Radius, Rmax= 4.82 cm; 122 cells increasing logarithmicallyAngle = 360o 2 cells x 180o

System Height = 5.5 cm 22 cells, z = 0.25 cm)Layer 1(2.5 cm): High perm same properties as linear pack

Layer 2(3.0 cm): Low perm same properties as linear pack

Porosity (), Pore volume (PV) and Flow rate (QT) as forlinear pack.

Table 2: Percentage of the injected slug in low k layer, k2, atvarious stages after 0.5PV of injection and after a further

0.5PV of (= 1 cP) overflush. Simulated for both the linearand radial systems; data in Table 1.

Radial System Linear System

NoCrossflow

WithCrossflow

NoCrossflow

WithCrossflow

Unit mob.0.5 PVslug inj.

17.4%1 17.4% 17.4% 17.4%

Viscousstable 0.5

PV sluginjected

23.6%

(35.6%)2

29.3%

(68.4%)

29.6%

(70.1%)

39.2%

(125.3%)

Unit mob.0.5PV inj.+ 0.5PV

overflush

17.4% 17.4% 17.4% 17.4%

Viscousstable 0.5PV sluginj.+

0.5PVunstableoverflush

23.6%(35.6%)

35.9%(106.3%)

29.5%(70.1%)

56.2%(223.0%)

Notes: 1. The 17.4% value for all unit mobility cases is a simulated number;

the theoretical exact answer is 17.37%; 2. The numbers in parenthesis refer to

the % increase ending up in the low k layer relative to the unit mobility case.

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SPE 95090 9

Penetration (X2) when high perm (k1) just fills

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50 60 70 80 90 100

VIscosity ratio (m_p/m_w)

X2(atX

1=L)

p=1

p=20

p=100

Viscosity ratio (p/w)

Penetration (X2) when high perm (k1) just fills

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 10 20 30 40 50 60 70 80 90 100

VIscosity ratio (m_p/m_w)

X2(atX

1=L)

p=1

p=20

p=100

p=1

p=20

p=100

Viscosity ratio (p/w)

Fig. 6:Penetration into the low k layer, x2, when the high k

layer just fills with viscous fluid (i.e. at x1= L) as a function of

viscosity ratio, (p/w). Inset shows quantitatively thepenetration of fluid into the low k layer for the 3 viscosity

ratios, (p/w) = 1, 20 and 100. Data as in Fig. 4 caption.

rw r

P(r)

Q

Pw

fP(r) = Pwf- P(r)

1 1 1

2 2 2

Q k h

Q k h

=

p=

w- tracer

For UNIT MOBILITY case

1 1

2 2

r k

r k=

In radial system (1= 2)=>(tracer inj. ) cf. linearcase

rw

Rmax

r

r1

r2

tracer

NO CROSSFLOW

rw r

P(r)

rw r

P(r)

Q

Pw

fP(r) = Pwf- P(r)

1 1 1

2 2 2

Q k h

Q k h

=

p=

w- tracer

For UNIT MOBILITY case

1 1

2 2

r k

r k=

In radial system (1= 2)=>(tracer inj. ) cf. linearcase

rw

Rmax

r

r1

r2

tracer

NO CROSSFLOW

rw

Rmax

r

r1

r2

tracer

NO CROSSFLOW

Fig. 7:Slug placement in a radial layered system showing the

schematic of the radial pressure profile, P(r) and the resultsfor a unit mobility injection; additional notation in Fig. 1.

(Q1/Q2) vs. time for various viscosity ratios

0

2

4

6

8

10

12

0 20 40 60 80

Time

(Q1/Q2)

= ( p/ w)

= 2

= 5

= 10

= 20

= 50 100

= (p/w)

(Q1/Q2) vs. time for various viscosity ratios

0

2

4

6

8

10

12

0 20 40 60 80

Time

(Q1/Q2)

= ( p/ w)

= 2

= 5

= 10

= 20

= 50 100

= (p/w)

Fig. 8: Volumetric flow ratio, (Q1/Q2) vs. time, in a radial

system with no crossflow for various viscosity ratios, =

(p/w) = 2, 5, 10, 20, 50 and 100. Longer time behaviour, t inarbitrary units, 0 80.t System data: Rw= 0.5; Rmax= 10;k1= 10; k2= 1; h1= 5; h2= 5; 1= 2= 0.2; QT= 2;= 1 ,2, 5, 10, 20, 50, 100 (w= 1).

(Q1/Q2) vs. time for various viscosity ratios

0

2

4

6

8

10

12

0 2 4 6 8 10

Time

(Q1/Q

2)

= 2

= 5

= 10

= 20

= 50

= (p/w)

= 100

(Q1/Q2) vs. time for various viscosity ratios

0

2

4

6

8

10

12

0 2 4 6 8 10

Time

(Q1/Q

2)

= 2

= 5

= 10

= 20

= 50

= (p/w)

= 100

Fig. 9: Volumetric flow ratio, (Q1/Q2) vs. time, in a radial

system with no crossflow for various viscosity ratios, =(p/w) = 2, 5, 10, 20, 50 and 100. Shorter time behaviour, t inarbitrary units, 0 10.t System data: as in Fig. 8.

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SPE 95090 11

PV= 0.00

PV= 0.0629

PV=0.171

PV= 0.306

PV= 0.441

PV= 0.540

PV= 0.607

PV= 2.00

PV= 0.00

PV= 0.0629

PV=0.171

PV= 0.306

PV= 0.441

PV= 0.540

PV= 0.607

PV= 2.00

Fig. 14: Unit mobility displacements in a 2-layer pack; detailsin text and ref. [22].

Dist. along high perm vs. time

y = 13.127x + 0.1256

R2= 0.9984

y = 2.2982x - 0.0574

R2= 0.9959

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5

PV injected

Dist.along

layer

High perm

Low perm

X1(t)

X2(t)

Distance travelled along each layer vs. timeX1(t) vs. time (PV) .. X2(t) vs. time (PV)

Dist. along high perm vs. time

y = 13.127x + 0.1256

R2= 0.9984

y = 2.2982x - 0.0574

R2= 0.9959

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5

PV injected

Dist.along

layer

High perm

Low perm

X1(t)

X2(t)

Distance travelled along each layer vs. timeX1(t) vs. time (PV) .. X2(t) vs. time (PV)

Fig. 15: Frontal advance in each layer, x1(t) and x2(t), vs. PVinjected in the unit mobility flood shown in Fig. 14.

Fig. 16:Simulation of the unit mobility floods in Fig. 14.

LINEAR SYSTEM WITH CROSSFLOW

Q2

P

k2 (w)

x2

(p)

QTQ1

k1

p > w

(w)

x1

FIXED(p)

0 L

P(x)

LAYERPRESSUREPROFILES(at t1)

High perm layer 1

Low perm layer 2

VISCOUS FLUIDINJECTION, p > w

LINEAR SYSTEM WITH CROSSFLOW

Q2

P

k2 (w)

x2

(p)

QTQ1

k1

p > w

(w)

x1

FIXED(p)

0 L

P(x)

LAYERPRESSUREPROFILES(at t1)

High perm layer 1High perm layer 1

Low perm layer 2Low perm layer 2

VISCOUS FLUIDINJECTION, p > w

Fig. 17: Schematic of a two layer heterogeneous system with

crossflow under viscous stable injection (p>w) showing the

(now evolving) pressure profiles, P(x), along the layers at afixed time.

PV= 0.065

PV= 0.00

Viscous Stable Floods in Layered Pack

PV= 0.225

PV= 0.157

PV= 0.36

PV= 0.065

PV= 0.00

Viscous Stable Floods in Layered Pack

PV= 0.225

PV= 0.157

PV= 0.36

Fig. 18:Injection of a viscous slug of clear fluid (p 10 cP)

into a blue brine (w= 1cP) showing that significant amountsof injected fluid are diverted into the low k layer, k2. Detailsin text and ref. [22].

PV= 0.065

PV= 0.157

PV= 0.225

PV= 0.36

Fig. 19: Direct simulation of the viscous stable flood in the

two-layer bead pack in Fig. 18. Details of the pack and fluidproperties in text and ref. [22].

PV= 0.0629

PV= 0.306

PV= 0.54

PV= 2.00

PV= 0.0629

PV= 0.306

PV= 0.54

PV= 2.00

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12 SPE 95090

1

2

10

= 5.71

p

w

k

k

Frontaladvanceratio,(x1/x2)

PV fluid injected

k2

x2

k1

x1

1

2

10

= 5.71

p

w

k

k

Frontaladvanceratio,(x1/x2)

PV fluid injected

k2

x2

k1

x1

Fig. 20: Experimental and numerically simulated frontaladvance ratio, (x1/x2) vs. PV injected for the viscous stableslug injection in Fig. 18 also showing some similar repeatfloods from ref. [22].

PV fluid injected

Layerfractionalrecovery

k1

k2

PV fluid injected

Layerfractionalrecovery

k1

k2

Fig. 21: Numerically simulated fractional layer recovery vs.PV injected for the viscous stable slug injection in Fig. 18.

PV= 0.00

PV= 0.0168

PV= 0.0508

PV= 0.1384

PV= 0.1924

PV= 0.2498

PV= 0.3511

PV= 0.00

PV= 0.0168

PV= 0.0508

PV= 0.1384

PV= 0.1924

PV= 0.2498

PV= 0.3511

Fig. 22:Injection of a slug of red fluid (w 1 cP) to displacethe viscous clear slug (p 10 cP). The blue brine has thesame viscosity as the red brine (w= 1cP). Viscous fingeringof the red brine into the blue is observed and the (clear)viscous slug is displaced into the low k layer, k2. Details intext and ref. [22].

PV= 0.017

PV= 0.05

PV= 0.14

PV= 0.19

PV= 0.25

PV= 0.35

Fig. 23: Simulation of the slug breakdown experiment inthe layered beadpack experiment in Fig. 22. The injected

blue brine has the same viscosity as the resident brine (w =1cP); they are not distinguished by colour as in theexperiment. The (clear) viscous slug is predicted to be

displaced into the low k layer, k2. Details in text and ref. [22].

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SPE 95090 13

(a) Unit mobility - with/without crossflow, 0.5PV

(b) 10 cP viscous slug no crossflow, 0.5PV

(b) 10 cP viscous slug with crossflow, 0.5PV

Fig. 24: Unit mobility and viscous slug placement with andwithout crossflow at 0.5 PV of injection (Table 1).

(a) Unit mobility - with/without crossflow, 0.8PV

(b) 10 cP viscous slug no crossflow, 0.8PV

(c) 10 cP viscous slug with crossflow, 0.8PV

Fig. 25: Unit mobility and viscous slug placement with and

without crossflow at 0.8 PV of injection (Table 1).