Acid Placement in Long Horizontal Wells

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Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the European Formation Damage Conferenceheld in Scheveningen, The Netherlands, 30 May–1 June 2007.

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acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435.

AbstractIn several places around the world, notably the North Sea andthe Middle East, carbonate reservoirs are being accessed with

very long horizontal wells (2000 to 20,000 feet of reservoir

section.) These wells are often acid stimulated to remove drill-ing fluid filter cakes and to overcome formation damage

effects, or to create acid fractures or deep matrix stimulation to

enhance productivity. Good acid coverage with a relatively

small acid volume is required to economically obtain the

desired broad reservoir access.We have developed a model to predict the placement of

injected acid in a long horizontal well, and to predict the

subsequent effect of the acid in creating wormholes,

overcoming damage effects, and stimulating productivity. Themodel tracks the interface between the acid and the

completion fluid in the wellbore, models transient flow in the

reservoir during acid injection, considers frictional effects in

the tubulars, and predicts the depth of penetration of acid as afunction of the acid volume and injection rate at all locations

along the completion.

We have used this model to simulate treatments that are

typical of those performed in the North Sea and in the MiddleEast. We present a hypothetical example of acid placement in

a long horizontal section and an example of using the model to

history match actual treatment data from a North Sea chalkwell.

IntroductionHorizontal wells are drilled to achieve improved reservoir

coverage, high production rates, and to overcome water

coning problems. Acid stimulation is a cost effective methodto enhance the productivity of horizontal wells in carbonate

reservoirs. Acid can be injected using many acid placement

methods including bullheading down the production tubing,injection from coiled tubing, injection with or following a

diverting material, injection into intervals isolated by packers,

and injection from acid jetting tools. Effective stimulation

requires that a sufficient acid volume be placed in all desiredzones. The model presented here is aimed at predicting the

acid distribution and subsequent stimulation for a variety of

placement methods used in long horizontal wells.Eckerfield et al.

1 concluded in their work that movement o

interfaces formed between acid and completion fluid is

significantly affected by uneven reservoir flow distribution

which ultimately leads to nonuniform volume of acid injected

into the formation. Wellbore hydraulics were found to havemuch less impact because of the small wellbore volume

relative to the volume of acid injected. Gdanski2 described

recent advances in carbonate stimulation stating that zonacoverage of long carbonate sections remains a challenge and

most of the acidizing treatments are designed on the basis of

of rules devised on the basis of past experience.

Davies and Jones3 presented an acid placement model for

horizontal wells. The model was for barefoot completions in

sandstone formations and the simulator used a pseudo-steady

state reservoir model. They concluded that variations in

reservoir properties along the treatment interval significantly

impacted the acid placement. The need to include wellbore phenomena was also emphasized in their work.

A new model is presented in this paper to study the aciddistribution and evolution of skin during acidizing treatmentsin horizontal wells in carbonate reservoirs. The acid placemen

model couples models of wellbore flow, including interface

tracking, a wormhole model to predict the effect of the acid

injection on local injectivity, a skin evolution model that

combines the stimulation effect of the acid with other skineffects, and a transient reservoir inflow model. The mode

predicts the bottomhole pressure response during an acid

treatment, the distribution of acid along the treated sectionand the resulting distribution of stimulation.

Model description

In a typical matrix acidizing process, the acid is being injectedinto the wellbore through production tubing, coiled tubing, ordrill pipe. The acid emanating from the tubing (whichever

type), or from ports in the tubing, displaces the residen

wellbore fluid, creating one or two interfaces between these

fluids. The acid behind the front flows into the formation andcreates wormholes in the reservoir rock, increasing the

injectivity of the contacted portions of the formation. The

effect of the acid on the formation injectivity at any location

along the well is accounted for with a local skin factor that ischanging in response to the acid injected at that point. Loca

injectivity is simultaneously affected by the transient nature o

the process – injection of any fluid will cause a pressure build

SPE 107780

An Acid-Placement Model for Long Horizontal Wells in Carbonate ReservoirsVarun Mishra, SPE, D. Zhu, SPE, and A.D. Hill, SPE, Texas A&M U., and K. Furui, SPE, ConocoPhillips



SPE 107780 2

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up in a porous medium. The transient pressure build up due to

injection and the acid stimulation that is increasing injectivity

are competing effects that must both be considered to properly predict acid placement.

This acidizing model for a long horizontal well integratesseveral sub models which are coupled. These include a

wellbore model which handles the pressue drop and material

balance in the wellbore; an interface tracking model to predict

the movement of interfaces between different fluids in thewellbore; a transient reservoir flow model; a skin factor modelaccounting for partial penetration and well completion effects;

and, an acid stimulation model that predicts wormhole growth

and the effects these have on local injectivity. Each model is

discussed in this paper separately.

Wellbore flow modelThis model incorporates the wellbore material balance andwellbore pressure drop.

Fig. 1 Schematic of a wellbore during an acidizing process

Figure 1 shows a part of the wellbore during an acid injection

process. For flow of an incompressible fluid in a horizontal

wellbore, we have

5

2)],([2),(

d

t xq f

x

t x p w f w ρ

−=∂

∂ (1)

),(),(

t xq x

t xq R

w −=∂

∂ (2)

Equation 1 describes the frictional pressure drop in thewellbore. The material balance, Eq. 2 shows that the change in

the wellbore flow rate is equal to the flow rate into the

formation at that point in the well.

Model for tracking fluid interfacesA model to track the interfaces created between various

injected fluids was presented by Eckerfield et al.1

Our acid placement model uses a discretized solution approach which is

integrated with the reservoir flow, wormhole, and skin models.

Figure 2 depicts a part of the wellbore where the interface

created between injected acid and wellbore fluid is traveling to

the right. The velocity of an interface located at xint is simply,

int

int

x x

w

A

q

dt

dx

== (3)

In jected acid

X in t |t=t X int | t=t+ t

Aq w

t

Fig. 2 Interface movement inside the wellbore

We solve this equation by discretizing the wellbore into smal

segments and assuming constant qw over each segment.

Reservoir flow modelDuring the acidizing process, the wellbore rate and the

reservoir inflow at any location are changing with time so

transient effects are occurring in the reservoir. A transien

inflow equation with variable injection rate is4

nn

j D

n

n

j

j

R

n

w sqt t pq p

i pkl

D D+⎥⎦

⎤⎢⎣⎡ −∑ ∆=−−

−=)()(2

11µ π (4)

where

1−−=∆ j

R

j

R

j

Rqqq (5)

)80907.0(ln2

1+≈ D D t p (6)

2

610395.4

wt

Dr c

kt t

φµ

−×= (7)

After dividing through by l, the length of a reservoir segment

and rearranging, qR n, the transient injection rate per unit length

of wellbore at time tn can be written as

Jxnw R Jx

n R b p paq −−−= )( (8)

where

])([

1091816.4

1

6

nn Dn D D

Jx st t p

k a

+−

×=

−

−

µ (9)

])([

)()(

1

111

11

nn Dn D D

n Dn D Dn

R

n

j j Dn D D

j R

Jx st t p

t t pqt t pq

b+−

−−⎥⎦

⎤⎢⎣

⎡−∆

=−

−−−

= −∑

(10)The constant in Eq. 9 is for oilfield units of bpm/ft for

injection rate, md for permeability, and cp for viscosity.

Wormhole modelWe have implemented two empirical models of the

wormholing process that occurs in carbonate acidizing. Thefirst of these is the volumetric model5, 6, which is based on the

assumption that a constant fraction of the rock volume is

dissolved in the region penetrated by wormholes. For radial

flow, the volumetric model is



SPE 107780 3 _________________________________________________________________________________________________________________________

bt wwh

hPV

V r r

πφ += 2

(11)

The key parameter in this model is PV bt, the number of porevolumes of acid needed to propagate a wormhole through a

core sample. The PV bt can vary from as low as one, or even

slightly lower, when acid is injected at near the optimal rate in

limestone, to as high as 50 when the wormholing process isnot efficient. We also note that the wormholing model

presented by Gdanski2, 7, 8, which is presented by Glasbergen

et al.8 as

h

V r wh

φ 25.035.27= (12)

for units of cm for r wh and m3/m for V/h, can be approximated

as a special case of the volumetric model with PV bt set to 1.1.

This can be derived by equating the right hand sides of Eqs. 11

and 12 and neglecting the r w2 term in Eq. 11.

An improved empirical model of the wormholing process is

that presented by Buijse and Glasbergen9. In this model, thewormhole propagation rate varies with the acid flux in a

manner based on the commonly observed “optimal flux” behavior. The user of this model supplies the optimal acid flux

and the optimal PV bt based on laboratory tests. We have

implemented both the volumetric and the Buijse models ofwormhole propagation in our acid placement simulator. If the

PV bt input to the volumetric model is close to the value

determined by the Buijse model at the acid flux occurring in

the simulated acid treatment, the results from these two

models are similar.

Skin factor and well completion model

The changing injectivity during acid injection is accounted forwith a local skin factor, s(x), which includes the effects of thecompletion, possible formation damage, and the stimulation

effect of the acid. In addition, injectivity of individual zones

along a long horizontal well are affected by a partial penetration effect which can be treated as a skin effect. This

partial penetration effect is described separately in the next

section.

The effects of the completion, formation damage, and

stimulation are all coupled and depend on the completion type.For a cased, perforated completion, we used the perforation

skin factor model of Furui et al.10, 11 For this type of model, we

assume that wormholes propagating from the tips of

perforations can be considered as extensions of the effectivelengths of the perforations.

For an openhole or slotted/perforated liner completion, weassume radial flow of the acid through a possibly damaged

zone that extends to a radial distance, r d. For this case, if the

pressure drop in the region penetrated by wormholes is small,

the evolving skin factor is

For r wh<r d :

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −⎟⎟

⎠

⎞⎜⎜⎝

⎛ =

w

d

wh

d

d r

xr

xr

xr

xk

k x s

)(ln

)(

)(ln

)()( (13)

And for r wh>r d :

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −=

w

wh

r

xr x s

)(ln)( (14)

The radius of the region penetrated by wormholes, r wh, is

obtained from the wormhole model.

Partial penetration skin model Acid injection in long horizontal wells is often into relativelyshort, isolated sections of the well. Because the section treated

is connected to the entire reservoir, the injectivity is higher

than it would be if the reservoir ended at the end of the

completion interval. A partial penetration skin factor, which

will be negative, can be used to account for this effect. This partial penetration effect is important when injecting into

relatively small intervals of horizontal wells and is not widely

recognized, so a brief review is in order.

Fig. 3 A partially completed vertical well

The effect on productivity of completing a vertical well inonly a portion of the reservoir has been described numerous

times, beginning with Muskat12. For a well completed along a

thickness, hw, in a reservoir of thickness h (Fig. 3), and in the

absence of any other skin effects, the steady-state productivity

index is

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

=

cw

e sr

r B

kh J

ln2.141 µ

(15)

where sc is the partial completion (also called partia

penetration) skin factor. When hw is less than h, sc is positive

accounting for the lessened productivity of the partially

completed well. Models to calculate sc have been presented in

many studies, including those of Cinco-Ley et al.13, Odeh14

and Papatzacos15. The productivity index could also be written

using the completed thickness in the inflow equation:

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

='

ln2.141 cw

e

w

sr

r B

kh J

µ

(16)

h

Zw

hw

hw = CompletionthicknessZw = Elevation

r w



SPE 107780 4

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If hw is less than h, sc’ must necessarily be negative to give the

same productivity index as Eq. 15.

When hw is relatively small compared with h, these partialcompletion effects are large. For example, when hw/h is 0.25,

sc is 8.8 using the Papatzacos model when the completion iscentered in an isotropic reservoir. If ln(re/r w) is 8, a typical

value, the corresponding sc’ is -3.8. Thus, when calculating

productivity or injectivity based on the completion zone

thickness, the well appears to be stimulated because thereservoir is thicker than the completed interval.

The corresponding situation for acid injection into a short

interval of a horizontal well is shown in Fig. 4. Because we

are assuming radial flow from the completed interval in our

reservoir flow model, there will be a large partial penetrationeffect which we can account for with a negative skin factor.

Fig. 4 Horizontal well partially open to the reservoir

Fig. 5 Ellipsoidal flow geometry

We have developed a simple model to calculate this type of

skin factor as follows. Consider a horizontal well partially

open to the reservoir as in Fig. 4. Ellipsoidal flow exists due tothe partial opening of the wellbore in the reservoir as in Fig. 5.

The ellipsoidal inflow equation is

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

−

+=∆

1

1ln

)2(

2.141ξ

ξ µ

e

e

ak

q p (17)

where

)(sinh1

Dr −=ξ (18)

ar r D /= (19)

The radial flow equation based on a completed interval of

length 2a is

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛ =∆ c

w

sr

r

ak

q p 'ln

)2(

2.141 µ (20)

Equating the pressure drops given by Eqs. 17 and 20 givesthe horizontal well partial penetration skin factor as

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−+++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+=

)11(

)11(2ln)1()1(2ln'

2

2

D D

D Dwwc

r r h

r r r eher sξ

ξ

(21)

Solution approachThe models for wellbore flow, partial penetration and

completion skin factor, front tracking, reservoir inflow

wormhole growth, and skin evolution were incorporated into anumerical simulator. The solution method for these coupled

models is described in the Appendix.

ResultsWe illustrate the predictions of the horizontal well acid

placement model presented here with two contrastingexamples. In the first example, acid is injected at a relatively

low rate into a long section of a horizontal well. This is the

situation where wellbore flow conditions are most likely to be

significant. The second example, the simulation of an actual

North Sea acid treatment, is a case of high rate injection into avery short interval.

Example 1 – Small volume injection into a long interval. Inthis case, we investigate the effects of acid volume and acid

injection rate on the placement of injected acid and the

resulting distribution of acid along the well. The conditions for

this case are presented in Table 1. The volumetric model ofwormhole growth was used in this example.

Table 1 Data for Case 1

Well length 1000 ft

Number of grid blocks 50

Grid block length 20 ft

Completion Open hole

Damage radius 0.5 ft

Permeability 2 md

Index of anisotropy 1

Permeability impairment

ratio

0.5

Reservoir rock Limestone

Acid 15 % Hcl

Reservoir pressure 3200 Psi

Wormhole model Volumetric

Pore volume for

breakthrough (PV bt)

2

Injection rate 2 bpm

Duration of pumping 100 Min

Assuming that the acid is being injected from a tubing tai

located at one end of the completed interval, the progressionof acid placement with time is shown in Fig. 6. By the end of

200 barrels of acid injection at 100 minutes of pumping time

acid has not yet reached the far end of the completed interval



SPE 107780 5 _________________________________________________________________________________________________________________________

For better acid coverage with this small volume treatment (thetotal volume pumped in 100 minutes is only 8.4 gal/ft), some

method of diversion is required.

0.00

0.05

0.10

0.15

0.20

0.25

0 200 400 600 800 1000

Position along well (ft)

A c i d v o l u m e ( b b l

/ f t )

10 min

40 min

80 min

100 min

Fig. 6 Acid coverage over the entire length of wellbore

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 200 400 600 800 1000Position along well (ft)

W o r m h o l e l e n g t h ( i n )

10 min

40 min

80 min

100 min

Fig. 7 Wormhole length distributions at different times

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 200 400 600 800 1000


A c i d v o l u m e ( b b l / f t )

500 bbls acid (21 gal/ft)

200 bbls acid (8.4 gal/ft)

Fig. 8 Acid placement profiles for 200 and 500 bbls of acid

The distribution of wormhole lengths along the wellbore

created by this acid injection is shown in Fig. 7. By 100

minutes of acid injection, wormholes had extended 6 inches

into the formation at the heel of the completed intervalInjection of larger volumes of acid improves the coverage of

acid in this long interval. With 500 bbl of acid injected, the far

end of the completed interval has received a significan

amount of acid injection, with good acid coverage along mostof the interval (Fig. 8). For a well with only minor damage, as

was assumed for this case, although the acid is increasing the

local injectivity, and thus retarding the progress of the acid

down the wellbore, the injectivity is changing slowly, and thusdoes not have a strong effect on the acid placement. Another

illustration of this is obtained by changing the efficiency of the

acid treatment by changing the PV bt parameter used in thevolumetric model. Figure 9 compares the acid placement for

cases ranging from PVbt of 0.5 (very rapidly propagating

wormholes) to inert fluid (no wormholes, hence no change in

injectivity during injection). The acid coverage changes a little

depending on how efficiently the acid is increasing injectivity but it is not a large effect.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 200 400 600 800 1000



PVbt=0.5

PVbt=2

PVbt=10

Inert fluid

Fig. 9 Acid placement profiles for different values of PV bt

One of the interesting predictions of this model is the

downhole pressure response during acid injection. Bottomhole

pressure measurements are becoming more and more commonduring acid injection and can provide very useful diagnostic

information about the treatment. The predicted pressure

responses for a wide range of PV bt are shown in Fig. 10. When

an inert fluid is injected, the pressure builds up because of thetransient nature of the reservoir flow. With acid injection, the

simultaneous stimulation is tending to decrease the injection

pressure. Thus, depending on how efficiently the acid is

increasing the near-well permeability, the injection pressure

may rise or fall, as shown in Fig. 10. Comparison of actuatreatment response with predictions like these provide a means

of diagnosing the effectiveness of acid stimulation, and if donein real time can be used to optimize a treatment on the fly.

The final aspect of this hypothetical case that we studied

was the effect of the wormhole model on the predicted acid

placement. Figure 11 shows the wormhole length distributionfrom the volumetric model with PV bt set to 2.5 compared with

the predicted placement using the Buijse model with the PV bt

opt equal to 1.5. With the Buijse model, the wormhole

propagation is varying with acid flux, with the maximum

wormhole propagation being at the optimal injectioncondition. In this particular case, the acid fluxes are near the



SPE 107780 6

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optimum, but somewhat higher. For the range of acid fluxes

occurring in this treatment, the PV bt from the Buijse model

varies from about 2 to about 2.5. The volumetric model, whichassumes a constant PV bt independent of acid flux, gives a

similar prediction of acid placement, and hence, wormholedistribution, when a value of 2.5 was used for PV bt.

3100

3200

3300

3400

3500

3600

3700

3800

0 20 40 60 80 100 120Time (min)

P r e s s u r e ( p s i )

PVbt=0.5

PVbt=2

PVbt=10

Inert fluid

Fig. 10 Pressure response during acid injection

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0 200 400 600 800 1000Position along well (ft)

W o r m h o l e l e n g t h ( i n )

Buijse's Model for PVbt-opt = 1.5

Volumetric model for PVbt = 2.5

Fig. 11 Comparison of wormhole distributions from the

volumetric and Buijse’s models

Example 2 – North Sea short interval, high volume acid

treatment. In this case, we present predictions for an actual North Sea horizontal well completed in a chalk formation. The

6000 ft-long horizontal well was completed with sixteen

individual 10 foot-long perforated intervals spaced along thewell. Each interval is perforated with one shot per foot with

the perforations oriented downward. In this stimulation

treatment, each zone was isolated with packers and

individually treated with 15% HCl. The treating string was

equipped with pressure gauges between the packers and oneither side of the packers enabling the operator to monitor the

downhole treating pressure and to determine if the packers

were set and not leaking. We used our acid placement model

to history match the treating pressure response for one of thezones treated.

The pressure records from the three downhole gauges are

shown in Fig. 12. There is a clear indication of the packers

being set. The pressure gauge on the heel side of the firs

packer shows no pressure response to injection, indicating tha

it is set. Then at about 22 minutes, the second packer is set, asindicated by the rapid pressure falloff recorded by the gauge

beyond the second packer. We began simulation of thetreatment at the 22 minute time, when both packers were se

and acid injection into the isolated interval began.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 10 20 30 40 50

Time (min)


BHP-Zone

BHP-Below

BHP-Above

Isolation of bottom

zone achieved

Isolation of top zone

achieved

Start of acid injection in

the formation

Fig. 12 Pressure response of downhole gauges in Case 2

To history match the pressure response during this

treatment, we input the actual injection rate schedule recorded

to our model – Fig. 13 shows how we approximated the

changing rate schedule as a series of discrete rate changesAdditional data used in the model is given in Table 2.

Table 2 Input data for Case 2

Casing ID 6.625 inches

Coiled tubing OD 2.55 inches

Pipe roughness 0.0001

Zone length 10 ft

Reservoir pressure in zone 5350 psi

Reservoir compressibility 5E-06 psi-1

Permeability 5 md

Porosity 0.38

Initial formation damage none

Perforation length 7 inches

Perforation diameter 0.264 inches

Perforation spacing 1 spf

Perforation phasing 0 degree

Perforation orientation 90 degreeAcid type HCl

Acid density 69.91 lbm/ft3

Acid viscosity 1 cp

Acid concentration 15%

Wormhole model Volumetric

Number of grid blocks 10

Reservoir thickness 200 ft



SPE 107780 7 _________________________________________________________________________________________________________________________

0

2

4

6

8

10

12

14

16

0 10 20 30 40 50

Time (min)

I n j e c t i o

n r a t e ( b p m )

Treatment rate

Simulated rate

zonal isolation achieved

at 6 bpm

Pzone=9050 Psi

Acid Injection

Started

at 5 bpm

Response of

the whole well

(packers are

not set)

Acid injection

stopped

Fig. 13 Rate schedule for Case 2

From the data given about the well, we calculated the initial

skin factor as follows. For the given perforating conditions, we

obtained a perforation skin factor of 4.6 using the Furui etal.10,11 model. For this very short interval in a large reservoir,

we calculated a partial penetration skin factor with Eq. 21 of -

5.5. Combining these, and assuming no formation damage was

present initially, we use an initial total skin factor of -.9. Wethen adjusted the reservoir permeability and the PV bt in the

volumetric model to obtain a match of the actual treating

pressure (Fig. 14). This match ws obtained by setting the PV bt

to 4.5, which means the acid is propagating wormholesrelatively slowly into the matrix and that a large volume of

rock is being dissolved in the treated region. With PV bt of 4.5,the wormhole front is moving 4.5 times slower than the

injected fluid (spent acid) front.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 20 40 60 80

Time (min)


Treatment pressure

Simulated pressure

Fig. 14 History match of treatment pressure

For the high rate injection into such a short interval, acid

placement is not an issue, as shown in Fig. 15. More

importantly for this type treatment is what the model can tell

us about the effects of this large volume acid treatment from

the predicted depth of acid penetration into the formation. Notice that this interval has received 120 barrels of acid, about

500 gal/ft.

0

2

4

6

8

10

12

0 2 4 6 8 10



120 bbls acid injected (504 gal/ft)

Fig. 15 Acid placement for Case 2

From the history-matched pressure response using a PV bt o

4.5, we predict that a radial region of wormholes has

propagated about 40 inches into the formation. The volumetricmodel presumes that the acid is dissolving a fixed fraction of

rock, given by6

bt Ac PV N =η (22)

Where the Acid Capacity No., NAc, is

( ) rock

HCl Ac N

ρ φ

ρ φβ

−=

1

15 (23)

For this high porosity chalk formation, η is 0.22, meaningthat in the regions where wormholes have formed, 22% of the

rock has been removed. With the initial porosity in this chalkformation being 38%, after this amount of dissolution, the

porosity would be 0.52. It is likely that this amount odissolution would result in the collapse of some of the

remaining rock in this region, leaving a large cavern.

Based on the dissolving power of 15 % HCl reacting withcalcite, 12 bbl of acid injection into a single perforation wil

dissolve 5.5 ft3 of solid. Assuming that the dissolution region

extends 40 inches from the wellbore, as predicted by the

volumetric model with PV bt = 4.5 as used in this history

match, the acid has likely dissolved a sufficient amount ofrock out to at least this distance to make the remaining rock

unstable.

ConclusionsWe have developed an acid placement model for horizonta

wells in carbonate reservoirs which combines a wellbore flowmodel, including interface tracking, a wormhole model to

predict the effect of the acid injection on local injectivity, a

skin evolution model that combines the stimulation effect of

the acid with other skin effects, and a transient reservoir

inflow model. With this model, we find that

• Small volume treatments in long horizontal intervals

result in non-uniform acid placement, but that the

placement improves with increasing acid volume;



SPE 107780 8

_______________________________________________________________________________________________________________________________________

• Partial penetration effects are important when injecting

into relatively short intervals of long horizontal wells;

• The parameters in a wormholing model can be adjusted to

history match (or predict) the pressure response of an acid

treatment in a horizontal well;

• History matching of an acid treatment in a North Sea well

completed in a chalk formation required a relatively high

value of the pore volumes to breakthrough parameter,

suggesting that the acid is propagating slowly into therock, creating a cavity around the wellbore.

AcknowledgementsThe authors thank the sponsors of the Middle East CarbonateStimulation joint industry project at Texas A&M University

for support of this work.

Nomenclaturea = half length of open interval, ft

a jx = parameter in inflow equation, bbl/min-psi

A = cross-sectional area of wellbore, ft2

Ai = coefficients in solution matrix

b jx = parameter in inflow equation, bbl/min B = formation volume factor, dimensionless

Bi = coefficients in solution matrix

ct = total compressibility, psi-1 C i = coefficients in solution matrix

d = internal diameter of wellbore, ft

f f = fanning friction factor, dimensionlessh = reservoir thickness, ft

hw = length of completed interval, ft

J = productivity index, bbl/day/psi

J s = specific productivity index at any point in wellbore,

bbl/day/psi/ftk = permeability of reservoir rock, md

k d = permeability of damaged region, mdl = length of reservoir segment, ft L = length of wellbore, ft

N Ac = acid capacity number, dimensionless

p D = dimensionless pressure

pi = initial reservoir pressure, psi

pw = pressure at any point in the wellbore, psi

PV bt = pore volume for break through, dimensionless

q R = reservoir inflow rate per unit length of wellbore,

bbl/min/ft

qw = wellbore flow rate at any point, bbl/minr d = radius of damaged zone, ft

r D = dimensionless radius

r e = reservoir drainage radius, ft

r w = wellbore radius, ftr wh = radius of wormhole region, inches

s = skin factor, dimensionless sc = partial completion skin factor, dimensionless

sc’ = partial completion skin factor using hw for thickness

t = time, minutes

t D = dimensionless time

V = volume, ft3

x = position of any point along the wellbore length, ft

xint = location of interface from the heel of the well, ft

Z w = elevation of completed interval, ft

β 15 = gravimetric dissolving power of 15% HCl

dimensionless

ζ = pressure drop function, psi/ft/bbl/minη = wormholing efficiency, dimensionless

µ = viscosity of fluid, cp

ξ = ellipsoidal coordinate dimension

ρ = density of fluid in wellbore, lbm/ft3

ρ HCl = density of HCl, lbm/ft3

ρrock = density of rock, lbm/ft3

φ = porosity of the reservoir rock, fraction

∆q R = change in rate, bbl/min

∆t = time step, minute

References1. Eckerfield, L. D., Zhu, D., Hill, A. D., Thomas, R. L.

Robert, J. A., and Bartko, K.: “Fluid Placement Model forHorizontal-Well Stimulation,” SPE Drilling &

Completions, Volume 15, Number 3, September 2000.

2. Gdanski, R.: “Recent Advances in Carbonate

Stimulation,” SPE paper 10693 presented at the 2005International Petroleum Technology Conference, 21-23

November, Doha, Qatar.3. Jones, A. T. and Davies, D. R.: “Quantifying Acid

Placement: The Key to Understanding Damage Removain Horizontal Wells,” SPE paper 31146 presented at the

1996 SPE Formation Damage Control Symposium, 14-15

February, Lafayette, Louisiana.4. Lee, J., Rollins, J.B., and Spivey, J.P.: Pressure Transien

Testing , SPE Textbook Series, Vol. 9, SPE, Richardson

Texas (2003).

5. Hill, A. D., Zhu, D., and Wang, Y.: “The Effect ofWormholing on the Fluid-Loss Coefficient in Acid

Fracturing,” SPE Production and Facilities, 10, No. 4, p

257-263, November 1995.

6. Economides, M. J., Hill, A. D., and Ehlig-Economides

C.: Petroleum Production Systems, Prentice HallEnglewood Cliffs, NY, 1994.

7. Gdanski, R.: “A Fundamentally New Model of AcidWormholing in Carbonates,” SPE 54719 presented at the

European Formation Damage Control Conference, The

Hague, The Netherlands, May 31 – June 1, 1999.

8. Glasbergen, Gerald, van Batenburg, Diederik, van

Domelen, Mary, and Gdanski, Rick: “Field Validation ofAcidizing Wormhole Models,“ SPE 94695 presented at

the 6th European Formation Damage Conference

Scheveningen, The Netherlands, May 25-27, 2005.

9. Buijse, M. and Glasbergen, G.: “A Semiempirical Modelto Calculate Wormhole Growth,” SPE paper 96982

presented at 2005 SPE Annual Technology ConferenceOctober 9-12, Dallas, Texas.

10. Furui, K., Zhu, D., and Hill, A. D.: “A New Skin Factor

Model for Perforated Horizontal Wells,” SPE 77363

presented at the SPE Annual Technical Conference and

Exhibition, Sept. 30 – Oct. 2, 2002, San Antonio, Texas.

11. Furui, K., Zhu, D., and Hill, A.D.: “A ComprehensiveSkin-Factor Model of Horizontal-Well Completion

Performance,” SPE Production and Facilities, Volume

20, Number 3, August 2005.



SPE 107780 9 _________________________________________________________________________________________________________________________

12. Muskat, M.: Flow of Homogeneous Fluids Through Porous Media, McGraw Hill Book Co., New York, N. Y.,

1937.

13. Cinco-Ley, H., Ramey, H. J., Jr., and Miller, F. G.:

“Pseudoskin Factors for Partially-PenetratingDirectionally-Drilled Wells,” SPE 5589 presented at the

SPE-AIME Annual Conference, Sept. 28-Oct. 1, 1975,

Dallas, Texas.

14. Odeh, A. S.: “An Equation for Calculating Skin FactorSue to Restricted Entry,” JPT , June 1980, p. 964-965.

15. Papatzacos, P.: “Aproximate Partial-Penetration

Pseudoskin for Infinite-Conductivity Wells,” SPE Reservoir Engineering , May 1987, p. 227-234.

Fig. A-1 Schematic of a segmented wellbore

Appendix A: Solution approach

),(),(

t xq x

t xq R

w −=∂

∂ (A-1)

Jxnw R Jx

n R b p paq −−−= )( (A-2)

),()()],([2),(

5

2

t xqqd

t xq f

x

t x pww

w f w ζ ρ

−=−=∂

∂ (A-3)

Eq. A-1 is the wellbore material balance. Eq. A-2 is achievedfrom the reservoir flow model and Eq. A-3 represents the

pressure drop in the wellbore, where ζ i is a function of qw.

])([ ,,,2/1,2/1, i Jxiw Ri Jxiiwiw b p pa xqq +−∆=− −+ (A-4)

For i=1, 2, 3, 4, 5

2/1,

1

,1, 2

)(

+

+

+

∆+∆

−=− iwi

ii

iwiw q

x x

p p ζ (A-5)For i=1, 2, 3, 4

Fig. A-1 is a schematic of segmented wellbore. These

equations are to be discretized in this domain and will be

solved simultaneously. Eq. A-1 and Eq. A-2 are coupled and

can be written in discretized form as Eq. A-4. Eq. A-5 iswritten as discretized form of pressure drop equation, Eq. A-3

Initial and boundary conditions can be applied on this domain

When the injection rate is specified at the heel, 9 equationscan be written for 5 segments. This set of 9 equations reduces

to a tri-diagonal matrix system as in Eq. A-6, where

coefficients Ai , Bi , and C i are defined by Eqs. A-7, A-8 and A

9 respectively.

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+

+

+

++

=

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−

−

−

−

−

−

−

55

44

33

22

11

5,

2/9,

4,

2/7,

3,

2/5,

2,

2/3,

1,

5

4

4

3

3

2

2

1

1

0

0

0

0

10000000

1)(1000000

01100000

001)(10000

00011000

00001)(100

00000110

0000001)(1

00000001

B p A

B p A

B p A

B p A

Q B p A

p

q

p

q

p

q

p

q

p

A

qC

A

qC

A

qC

A

qC

A

i

i

i

i

wi

w

w

w

w

w

w

w

w

w

(A-6)

i Jxii a x A ,∆= (A-7)

i Jxii b x B ,∆= (A-8)

2/)( 1 iiii x xC ζ ∆+∆= + (A-9)

Documents

Acid Placement in Long Horizontal Wells