SPE-17110-PA - Modelling of Acid Fracturing

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Modeling of

Acid

Fracturing

K.K.

Lo

SPE, and

R.H.

Dean SPE, Areo Oil Gas Co.

Summary. The paper presents a theoretical framework for modeling acid fracturing stimulations. Starting from the fundamental equations

of fluid mechanics, fracture mechanics, convection, and diffusion, the paper outlines the steps necessary to derive simplified equations

for an acid fracturing model. Unlike some existing models, the coupled problem of fracture geometry, acid transport, and diffusion is

solved simultaneously in this paper. Although an infinite reaction rate is assumed in the solution of the problem, an empirical correlation

is used to account partially for finite reaction rates. Errors in the governing equations

of

some

of

the existing models are identified.

To assess the accuracy of the approximations used in the present model, exact solutions are used for comparison. Predictions from the

present model are compared with a model in the literature, and the results are found to

be

in reasonable agreement. As in all existing

acid fracturing models, not all the phenomena of the acid fracturing process have been incorporated into the present model. Nevertheless,

the present model

is

improved over existing models because it is derived from fundamental equations and thus forms a basis from which

further improvements can be made.

Introduction

Acid fracturing is a common well stimulation technique in the pe

troleum industry for limestone and dolomite formations. n an acid

fracturing treatment, an inert fluid (known as the pad) is injected

into a well under high pressure, creating a fracture in the formation.

As the fracture length is increased with continued fluid injection,

acid is injected into the formation, reacting with the formation on

the fracture surface.

The

acid is transported along the fracture by

convection during fracturing. At the same time, the acid is trans

ferred to the reactive surface by diffusion and by fluid leakoff into

the formation. Once the acid reaches the fracture face, it reacts with

the formation. Because the acid fracturing process

is

complicated,

simplifying assumptions have to be made to make the problem of

modeling the process tractable. On the other hand, several important

features have to

be

retained· to model the physics of the process

properly, including fracture geometry (fracture length, width, and

height), fluid leakoff rate, convection along the fracture, mass

transfer

of

the acid to the rock surface, and the acid reaction rate

on the surface. Of course, these processes occur simultaneously

during the acid fracturing treatment, so they are not independent

of one another. In limestone formations, the acidizing process is

limited by the rate

of

acid transport, not by the reaction rate. As

one

of

the simplifying assumptions in this paper, acid reaction rates

are

ignored at the well and during the stimulation.

This paper describes a model of acid fracturing based on a two

fluid generalization of the Perkins and Kern I model and a one

dimensional

( l0)

approximation

of

the general two-dimensional

(20)

diffusion-convection problem. Several authors

2

-

7

previously

presented acid fracturing models. The model described in this paper,

although in many ways similar to those described in previous pub

lications, is derived directly from the 20 model, and the mass

transfer rate comes directly from the analysis of the

20

diffusion

convection problem.

The model described here consists

of

two parts: a fracturing model

and an acid transport model.

We

first write the governing equations

for a Perkins-Kern fracture model derived by Nordgren. 8 We then

describe the Perkins-Kern approximation and generalize it to two

fluids.

The

two-fluid generalization of the Perkins-Kern approxi

mation is the basis

of

the fracture model proposed in this paper.

For

the acid transport model, the 20 convection-diffusion equation

is used as a starting point for the derivation of the 10 approximation

averaged over the fracture width. The mass-transfer rate obtained

from such an approximation is compared with the full

20

mass

transfer rate obtained from solving the

20

equation.

Governing Equations

Fracture Model. In Nordgr en s8 f racture model, the fracture

height, h, is assumed to be constant. The rate

of

fluid leakoff per

unit length into the formation at any point in the fracture can

be

approximated by

qw=(2hC

L

) I .J t - r ,

1)

Copyright 1989 Society of Petroleum Engineers

194

where

t=time

and C

L

=lea koff coefficient that

is

usually measured

in a static filtration test or a mini fracture test. The factor 2 in Eq.

1 accounts for fluid leakoff rates from both fracture surfaces. In

general, the coefficient is a function of the reservoir properties,

fracturing-fluid properties, and filter-cake buildup. Note that r=r x)

is

the time it takes for the fracture to reach Point

x.

The governing equations for Nordgren s fracture model consist

of the continuity equation and the fracture-width/pressure-elasticity

relationship:

CJqICJx)+ 7rhI4) CJb

x

/CJt)+qw=O (2)

and b

max

= [ 2 l - / t 2 ) h l E ] ~ x), (3)

where h=fracture height,

q=fluid

flux rate, qw=fluid-Ioss rate at

the fracture surface, b

max

= maximum opening at the center of the

fracture cross section,

E=Young s

modulus,

/t=Poisson s

ratio,

and

~ [ = p x ) - a ] = n e t

pressure acting on the fracture surface.

In addition, many fracturing fluids approximately obey a power

law relationship between the shear stress,

s,

and the shear strain

rate, -y

s=K-yn =Klduldy In-1duldy, (4)

a form commonly assumed for non-Newtonian fluids. In Eq. 4,

u is the velocity down the fracture,

nand

K are fluid constants,

and y is in the direction normal to the fracture wall.

The

fracture

fluid is Newtonian

if

n= 1

Guillot and Ounand

9

showed that fracture fluids can exhibit

Newtonian behavior (n=

1

at low shear rates and power-law be

havior n<

1

at high shear rates. They found that an Ellis model

produced a reasonable fit of their experimental data for a wide range

of shear rates (0.01 to 2,000

seconds-I).

In addition, for the high

shear rates normally encountered in fracturing applications, they

found that the power-law model was a suitable approximation for

calculating fracture shapes. The power-law model in Eq. 4 is used

in the next section to calculate fracture shapes.

Perkins and Kern Approximation. As Nordgren observed, an ap

proximate solution of Eqs. 1 through 4 with fluid loss can be ob

tained from the zero leakoff solution. That is, we first obtain the

zero leakoff solution by setting the time derivative and the fluid

loss rate, qw in Eq. 2 to zero and by integrating to obtain an ex

pression for the fracture width, b

max :

[

128 n+ I)K l-/t2)h [ 2n+l)i In }lI 2n+2)

bmax x) = L-x) ,

37r E nh

5)

where

i=injection

rate,

L=fracture

length, and the fracture width

is

required to be zero

at

the fracture tip.

We

then account for the

fluid loss by modifying the fracture volume. In deriving Eq.

5,

we

made use of the velocity profile and its relationship to i to give an

expression for the pressure gradient down the fracture; the volu

metric flow rate is equal to the velocity u integrated over the cross

section between the fracture surfaces. Because

i

is independent

of

SPE Production Engineering, May 1989



rol'---L

2

Oojl '--- L l · L 2 - - - ~ I

FLUID 2

FLUI

1

w

Fig

1-Two

fluid stages along one wing of the fracture

x by assumption, the pressure gradient can be integrated and Eq.

3 used

to

eliminate

Ap

to yield Eq. 5 see Ref. 1 for details).

We

can integrate Eq. 5

to

obtain the fracture volume,

V

both wings):

V=( II /2)h

1max

dx

................................

6)

o

3 11 2 E [nh n 2n+3

= 128 K(1-J. 2)(2n+3)

(2n+

l) i

b

o

, 7)

where

bo=maximum

fracture width at the wellbore. A factor

( 11 /2)

is included in Eq.

6

because the

~ e r g e

fracture width for an el

liptical cross section is given by

b= II b

max

/4 where b

max

is the

maximum width of the cross section. The nonzero fluid loss is ac

counted for in the above approximation by equating the fracture

volume to the actual fluid volume injected minus the fluid loss; Le .,

V=i t -2 ]

L

w

dt

dx ..............................

8)

o

T

=i t -8C

L

hr

J -T(X)

dx. . ........................

9)

o

The factor 2 in the second term

of

Eq. 8 represents the total fluid

loss in both wings of the fracture.

From

Eqs. 7 through 9, one can

form an implicit equation for L

at

a given

t

which means that the

fracture length can be solved as a function

of

time given an injection

rate, i. Eqs.

5

through

9

are known as the Perkins-Kern approxi

mation. Physically, the approximation means that fluid leakoff has

very little effect

on

the fracture shape and fluid leakoff primarily

affects only the overall fracture volume. That is why the zero-leakoff

b

max

-L

relationship in Eq. 5 can be used to approximate the

generalleakoff

case, as long as the global fluid balance in

Eq.

8

is maintained.

To

generalize the Perkins-Kern approximation to two fluid stages

for acid fracturing treatments, one can apply a similar argument

to the

two

stages. In what follows, all quantities with the subscript

j

j

= 1 2)

are

associated with the fluid at Stage

j.

Fig. 1 shows the

arrangement of the two stages. Fluid

2

extends from the wellbore

to a distance

L2,

while Fluid

1

is between

L2

and the end of the

fracture at

L

I

. b

l

is the fracture width

at L

2

, the trailing edge of

Fluid

1,

and b

2

is the fracture width at the wellbore. For Fluid

1,

evaluating Eq. 5

at

L2 gives

b

l

j 1 28

nl+l )K I

(I-J. 2)h

C3 11 E

[

(2n

l

+

)i2ln1 JII(2n1 +2)

X (L

I

-L

2

) . • • • ••

10)

nih

Integrating b

max

as in Eq. 6 but from

L2

to

LI

gives the same re

lation

as Eq.

7, except

b

o

n

and

K

all have the subscript 1

i

is

replaced by i 2, which is the current injection rate when two fluids

are

present in the fracture:

3 11 2 E

[nih

ln

1

V

I

= br

n1

+

3

(11)

128 K 1 1- J. 2)(2nl +

3)

(2nl +

l) i

2


------_._---------------------

Fig

2-Acid

concentration profiles along the plates

For

Fluid

2,

integrating Eqs.

1

through

4

from L2 to

0

and using

the Perkins-Kern approximation gives

[

128 (n2

+ I) K

2

(1-J. 2)h

b

z

=

3 11

E

[

(2n

2

+

l)i2ln2

JII(2n2 +2)

X

L2

+br

n2

+

2

,

12)

n2h

where b

2

= fracture width at the wellbore.

For

widths

at

any point

less than L

2

, Eq. 12 still applies, except

L2

in Eq.

12

is replaced

by

(L

2

-x) .

Integrating Eq.

12

from 0 to

L2

gives the total fracture

volume from

0

to L

2

:

3 11 2

E [n2h

ln

2

V

2

= 128 K2(1-J. 2)(2n2+3) (2n2+1)i2

X(bin2+3-brn2+3) . 13)

We now look at the fluid volumes to obtain a second set of

equations in terms of

L

I

,

L

2

, and

t

time).

The

total volume of

Fluid

1

remaining in the fracture both wings) is

VI

= i It - r I8CLlh.Jt-TI(X)

dx

L2

rL2

- J 8C

Ll

h.J

T2(x)-T,

(x)

dx, 14)

o

where i, = injection rate during Stage

I, t

= time when Stage 1

ended, and T2 x)=time

when

the leading edge of Fluid

2

passed

Point x. Values

of T (x)

and T2(x) are retained at selected points

along the fracture to evaluate the integral expression in Eq. 14.

The

corresponding equation for Stage

2

is

................................... 15)

Eq.

15

assumes that the fluid loss for the second stage obeys Eq.

1

where the leakoff coefficient is

C

2 and T(X) is the time that the

fracture tip reached Point x. Eq.

1

is an adequate approximation

to the leakoff rate for the second stage when C

LI

and C

L2

are of

the same magnitude. Eqs. 14 and

15

can be written as

VI

=

gl

(L

I

.£2,1) and V

2

=g2(L

I

,L

2

,t). Other forms of fluid leakoff rate

other than Eq. I), such as a constant

leakoffrate,

can also

be

used

in Eq.

15

for the acid. In the case of a constant leakoff rate,

C

L

2

and both square-root terms in Eq.

15

are replaced by V2 t -T2) ,

where V2 is the co nstant l eako ff velocity .

Substituting Eqs.

10

and

12

into Eqs. 11 and

13

gives two

equations for the volumes

VI

and V

2

in terms of the unknowns

L l

and L

2

• These two equations can then

be

combined with Eqs. 14

and

15

to produce two nonlinear equations relating the three

unknowns

L

I ,

L

2

and

t. For

a given

t

the

two

nonlinear equations

are solved for

L,

and

L2

with a Newton-Raphson technique with

residual-monitored damping. Thus, the fracture length

L l

and fluid

interface location

L2

can

be

calculated as functions

of

time. As ex-

195



pected, Eqs. 11 through 15 reduce to the one-fluid Perkins-Kern

equations in the literature when the two fluids are the same. Eqs.

11

through 15 are generalized to the case

of

multiple fluids in the

Appendix.

Acid Transport. The general acid transport model can be described

by the 2D continuity equation and the diffusion-convection

equation

2

:

<JuIiJx+<Jvl<Jy=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16)

and u <JCICJx) + v <JCI<Jy) =D J

2

CI<Jy2), . . . . . . . . . . . . . . . . (17)

where

u, v

= lateral and transverse velocity components, C=acid con

centration, and D=ef fect ive diffusivity. We have dropped the time

derivative and the diffusion term in the

x

direction in Eqs. 16 and

17. The effective diffusivity is much larger than the molecular diffu

sivity owing to mixing as the fluid moves down the fracture.

2

In using a 2D convection-diffusion equation as a starting point

of

the derivation, we assume that the z variation in the equation

is negligible. Consistent with this approximation, we also assume

thl t the dOl ain

of

the 2D equation is 2D and can_be defined by

-b12 <y<bI2, where there is no z dependence and b is the average

width across the elliptical cross section of the fracture. Because the

cross-sectional shape varies slowly with respect to

z

we expect this

domain approximation to be valid in a region away from the crack

tips. That is, the solution near the center of the cross section for

the truly 3D problem can be approximated by that

of

a problem

of

a

pair of parallel plates with bbeing the spacing between the plates.

In all the following calculations, it is assumed that the reaction

rate at the walls is much faster than the mass-transfer rate to the

walls

or

through the porous walls. An infinite reaction rate will

be used in all calculations, and all acid reaching the fracture surface

will react immediately at that surface. This infinite reaction rate

causes the acid concentration at the fracture surface to be zero; i.e.,

c=o

at

y= -b12

and

b12 . . . . . . . . . . . . . . . . . . . . . . . . . .

18)

Define the average

of

any function f as

f='; J

l2

f

dy 19)

b -b12

and the average weighted concentration as

C

m

=

_

l2

Cu

dy. (20)

bu -b12

We now derive a

ID

equation from Eq.

17

by

averaging

over

the width of the

fracture-i.e.,

applying the averaging operator in

Eq. 20 to Eqs. 16 and

17:

bu <JC

m

lox)=2[D(<JCloy)l

y

=bl2] +2 C

m

v

w

,

• • • (21)

where

Vw

= leakoff velocity at the wall.

Using the boundary condition in Eq.

18

and combining Eqs.

16

and 17, we obtain

<JCml<Jx

=

2lub

)[vw- Dlb NNU]C

m

, . . . . . . . . . . . . . . . . .

22)

where NNu, the Nusselt number, is defined as

NNu

=

- bIC

m

) oCI<Jy)l

y

=bI2

. . . . . . . . . . . . . . . . . . . . . .

(23)

NNu

is related to the mass-transfer rate of the acid to the wall,

w,

by

w= Dlb)NNu(X)Cm(x) .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .(24)

Equations similar to Eq. 22 have been derived il the literature

2

,7

for the average concentration, but the term Dlb

)NNu(X)

in Eq.

22 (known as the mass-transfer coefficient) has been treated as a

constant that can be determined experimentally.

7

Based on the

above derivation, the exact mass-transfer coefficient varies along

the fracture and depends on the solution

of

the problem. Hence,

care must be taken in applying an experimentally determined con

stant to Eq. 22 because,

if

the wrong value

of

the mass-transfer

coefficient is applied, the right side

of

Eq. 22 may become positive.

This leads to a physically incorrect, increasing concentration profIle

down the fracture.

196

I Approximation. So far, other than the p a ~ a l l e l p l a t e approxi

mation in which the domain

b

is replaced by

b,

the "integrated"

Eq. 22 is an exact equivalent

of

its 2D counterpart, Eqs.

16

through

18. The ID approximation enters when the exact Nusselt number

is replaced by its averaged value down the plates; Le.,

-

_[LaC I

rL

NNu..,NNu=-b J - _ dx J Cmdx .

. . . . . . . . . .

(25)

o Jy y=bl2

0

The Nusselt number approaches a limiting value as we proceed

down the plates so that sufficiently far along the plates the mass

transfer is proportional to

Cm.

To calculate Eq. 25, we make use

of

the solution

of

a similar problem in heat transfer obtained by

Terrill.

to

Terrill studied a 2D problem

of

a pair

of

porous parallel

plates held at a constant temperature. His heat transfer problem is

directly analogous to the convection-diffusion problem considered

here. Translated to the convection-diffusion setting, his problem

can be stated as follows: the fluid is flowing at steady-state con

ditions between the plates while the fluid is simultaneously leaking

off through the porous plates at a constant velocity,

V.

At

x=O

(where the

x

axis is the centerline that runs between the plates),

the acid concentration

is

set to a constant value across the gap be

tween the two plates, and a steady-state concentration profile forms

downstream. The acid concentration is constrained to be zero along

the plate boundaries. The maximum velocity at x=O is U. Fig. 2

shows schematically the evolution

of

the concentration profile down

the plates.

The above problem is more tractable than the acid transport

problem described by Eqs.

16

through

18

because the gap

~ t w e e n

the plates for the above problem is kept constant, whereas

b

in the

fracture problem varies withx. The fracture width is such a slowly

varying function of x, however, that

wOe

assume that the approx

imate Nusselt number derived from the simpler problem

of

the

constant-width parallel plates continues to apply for our acid

transport model. Terrill derived the velocity profile

(u

and

v)

be

tween the porous parallel plates by solving the Navier-Stokes and

continuity equations. This velocity solution was then substituted

into the thermal conduction-convection equation, a direct analog

of Eq. 17, and the temperature distribution was solved by sepa

ration of variables and eigenfunction expansion. Making use

of

Terril l's solution for the convection-diffusion problem considered

here, we can express the first three terms of the expansion fo. C

and Cm. Eq. 25 was then integrated to give an expression

f o ~ u

as a series expansion in terms

of

the Peclet number,

N

Pe

=

VbI2D:

NNU'

4.10+

I 26N

Pe

+O.04Nfe' (26)

which is valid for Peclet numbers

of

order 1. For large Peclet

numbers (>20), Eq. 26 would be replaced by

NNu

",,2N

pe

, as de

rived in Ref. II.

omparisons

Now we compare the solution of the 2D p r o ~ m with that of the

10

equation (Eq. 22) with

NNu

replaced by

NNu

in Eq. 26. The

width between the plates is now

b.

All comparisons will be restricted

to a set of parallel plates with constant fluid leakoff velocity for

simplicity. One can show that the expression on the right side of

Eq. 22 with NNu replaced by NNu is always negative, so the con

centration will always decrease as the acid moves down the fracture.

A program was developed to solve the 2D concentration equation

numerically while the

ID

equation can be solved by quadrature.

The following comparisons assume that the plates are 100

ft

[30

m] in height with a gap of 0.1 in. [0.254 em]. The fluid has a vis

cosity

of

100 cp [100 mPa

s],

a density

of

1 g/cm

3

,

and a diffu

sivity

of

0.0001

2

/s. Fluid is moving down the plates at a rate

of 10 bbllmin [0.0265 m

3

/s] at x=O, and for the first comparison,

fluid

is

leaking off at a rate of 0.001 ft/min [5.08 x 10 -

6

m/s];

the

second comparison has zero leakoff. For these physical parameters,

the Reynolds number down the plate, N

Reu

(

=

UbI2v), is 4.35 and

the Schmidt number,

NSe(

=

vID),

is 10,000. The leakoff Reynolds

numbers, N

Rev

= Vb12v) ,

are 6.45 x

10 -

5

and

0.0

for the leakoff

case and the no-Ieakoff case, respectively. The Peelet number is

0.65 in the 1eakoff case.

SPE

Production Engineering, May 1989



e n - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - - - - - - - ,

NReu = 4.35

NSc = 10000.

.

1·

.

1 ' NRev =6 5£ 5

:i • ',-,---------------.-------,-------

-------

----·0.0 ------

--

o+-____

~ ~ ~ ~ ~

•

10

100

Dlstanc. (II)

Fig. 3 Nusselt numbers along the plates.

The variable Nusselt number in Eq. 23 was calculated by first

solving the 2D equations (Eqs. 16 through 18 numerically and then

evaluating the expression in Eq. 23 for the leakoff and no-leakoff

cases. The Nusselt numbers are shown in Fig. 3 as a function of

distance along the plates.

In

both cases, the Nusselt numbers start

out very large and quickly decrease to a limiting value. One can

show that the Nusselt number goes to infinity like x -

y,

as x ap

proaches zero. After about 30 ft [9 m], the Nusselt numbers ap

proach limiting values of 4.45 and 3.77 for the leakoff and the

no-leakoff cases, respectively. The corresponding Nusselt numbers

predicted by Terrill's analytical result, NNu =3.77+N

e

+0.087

NJe+ , are also 4.45 and 3.77, showing excellent agreement

between the asymptotic results and the 2D numerical solution.

Based on Eqs. 22 and 26, the ID concentration solution was com

puted for the leakoff problem. Its concentration profile is shown

in Fig. 4, along with the C

m

profile computed from the 2D con

centration solution. The concentrations have been scaled to one at

x=O. The ID solution is very similar to the 2D concentration profile

over the entire length of the plates.

The ID concentration is larger than the 2D concentration near

the entrance

of

the plates and smaller than the 2D concentration

farther down the plates because the ID solution uses an average

Nusselt number while the 2D solution has a variable Nusselt number

that begins at infinity

atx=O

and decreases to a limiting value farther

down the plates. The average Nusselt number for this case is 4.93;

the limiting Nusselt number is 4.45.

For the zero-leakoff case, the ID and 2D solutions are compared

in Fig. 5. Again, the agreement is very close over the entire length

of the plates. For this problem, we also performed the lD calcu

lations using the limiting Nusselt number of 3.77 instead of the

N

LEGEND

On Dimensional

Sol

Two

Dimensional

Sol

NRev = 0.0

NReu

= 4 35

NSc= 10000.

O ~ - - - - - T - - - - __ ~ ~ ~ ~.0

00

1 ~ O 200

no

•••

Dlstonc. (II)

Fig. 5-Concentrat lon profile along the plates without leakoff.

SPE Production Engineering, May

1989

N

LEGENO

One DImensIonal Soln

wo imensional SoIn

.

NRev =

6.5E-5

NReu= 4.35

NSc

= 10000

O + . - - - - - - M ~ - - - - - ~ ~ O - - - - - - ~ ~ - - - - ~ . ~ . O ~ ~ ... M . ~ ..

DIstance (II)

Fig.

4 Concentration

profile along the plates

with

leakoff.

average Nusselt number. The results are shown in Fig. 6, along

with the 2D results. As expected, the ID solution in this case is

always larger than the 2D solution asymptoting to the correct value

down the plates.

Eq. 22 models acid transport for a power-law fluid. The ex

pression for the average Nusselt number in Eq. 26, however, is

strictly valid only for a Newtonian fluid

lO

(Le.,

n=1

in Eq. 4).

Bird et al. 12 presented solutions to a variety of convection

diffusion problems involving power-law fluids. On the basis of the

limiting Nusselt numbers presented there for a tube with a circular

cross section, Eq. 26 will underestimate the average Nusselt number

for a typical power-law fluid

of

exponent 0.25 by about 15%.

The

Mode

For the acid transport model, we have from Eqs. 22, 25, al d 26,

with

V replaced by the leakoff velocity,

w

, and N

e

=vwbIW,

ilC

m

lilx= 2/iib)[ -4.1 D

Ib)+0.37v

w

-0.Olv

w

2

bID C

m

,

. (27)

where, unlike the parallel-plate problem, u

b

and w are assumed

functions of x. Except when the Peclet number is of order 1, the

third term on the right side

of

Eq. 27 normally can be ignored in

most applications. The inclusion of the term, however, guarantees

that the right side will always be negative.

Eq. 27, together with Eqs. 10 through 15, constitutes the acid

fracturing model.

For

each timestep, the fracture length and width

are calculated from Eqs. 10 through 15 for a given injection rate.

One can then calculate the fluid velocities and integrate Eq. 27 to

calculate the velocity-weighted average concentration,

Cm.

The

N

..'

LEGEND

, •• •••

••••••••

.

On.

DImensIonal Soln

Two Dimensionaf SoIn

NRe

v

=

0.0

NRe

u

=

4.35

NSc = 10000.

O + O - - - - - - ~ M - - - - - - ' ~ O O - - - - - - ~ ~ - - - - ~ . ~ - - - - - - ~ Z M - - - - - - ~

Dlstanc. (II)

Fig. 6 Concentratlon profile using the limiting Nusselt

number.

197



rate of acid transport to any point on the fracture surface is multi

plied by the timestep size to calculate the amount of acid reaching

that point during that timestep. These amounts are then summed

for the entire fracturing job to calculate the total amount

of

acid

reaching any point along the fracture surface.

The diffusivity,

D

in Eq. 27 will be larger than the molecular

diffusivity because

of

surface roughness at the fracture walls and

the circulation induced by density differences caused by the acid

reaction. The model calculates an effective diffusivity from the

Reynolds number for flow parallel to the fracture from correlations

based on the work of Roberts and Guin

7

and Nierode and

Williams.

3

For finite reaction rates, the model uses the correlation

of

Nierode et al.

4

to calculate an effective diffusivity, although,

as noted by Williams et al. 2 the correlation has little theoretical

basis.

During pumping, we calculate the amount of acid transported to

the fracture surface, but unspent acid still remains in the fracture

when the pumps are shut down. This unspent acid will also react

with the fracture surface to give a productivity enhancement. We

assume that the unspent acid present in a gridblock at the end

of

pumping will react with the fracture surface at that gridblock. This

unspent acid is added to the acid that accumulated at that point during

the fracturing job to arrive at the total amount of acid that reacts

at that point along the fracture surface. Note that

C

m

must be con

verted to an average concentration, C during calculation of the

amount

of

unspent acid remaining in the fracture.

Temperature Dependence

This paper has not addressed the effects

of

temperature variations

along the fracture. Temperature variations could influence the mass

transfer rate, the acid reaction rate, and the fluid rheology. Tem

perature calculations may be included in the model by incorporating

an analytical expression for the temperature distribution or by

solving numerically for the temperature distribution in much the

same way as one solves for the acid distribution. Sinclair

13

dis

cussed the procedure for incorporating an analytical expression of

the temperature distribution, while Lee and Roberts

14

solve nu

merically for the temperature distribution.

Productivity Increase Calculations

For

productivity-increase calculations, the model determines the

amount of rock dissolved within each gridblock. The acidized

fracture width for a gridblock

is

the dissolved fracture volume for

the gridblock divided by the fracture height and gridblock length.

These idealized widths are then used as input for calculating the

corresponding conductivity through each gridblock.

In the actual physical acidizing process, the acid will not dissolve

a uniform fracture width that acts as a single parallel plate for pro

duction. Instead, the acid will create many ridges and valleys along

the fracture face, and it is precisely this heterogeneous distribution

of peaks and valleys that allows portions of the acidized surfaces

to remain separated after the fluid has drained off and the fracture

has tried to close. Because portions of the acidized surface are held

open only by means of irregularities along the fracture surfaces,

however, one would expect that the degree

of

propping should be

a function of the formation softness, the in-situ closure stresses,

and perhaps several other formation parameters. For example,

if

a formation is too soft or the in-situ stresses are too large, the ridges

propping the fracture open become very compressed and very little

of

the fracture will remain open after the acidizing treatment.

Because the relationship between the fracture conductivity and

acidized fracture width will be a function

of

reservoir properties,

one ideally should determine the fracture-conductivity-to-acidized

width relationship experimentally by flowing acid between slabs

of core to create fracture widths. One would then press the two

slabs of core together under in-situ stress conditions and flow fluid

between the slabs to determine the corresponding conductivities.

In this way, fracture conductivity as a function

of

acidized width

for the formation

of

interest could be determined.

In many applications, one will usually not undergo the time and

expense involved in experimentally determining the relationship

-

tween the acidized width and fracture conductivity. In such cases,

correlations that express fracture conductivity as a function

of

198

fracture width can be used. The acid transport model uses the

Nierode and Kruk

15

correlations, which were determined for a

variety

of

core samples under different load levels.

After the fracture conductivities have been determined for each

gridblock, productivity improvements can be calculated. The steady

state productivity calculations based on Raymond and Binder's16

paper were used to calculate the productivity increase after acid

fracturing. In the calculations, radial flow around the wellbore is

assumed and the steady-state radial-flow equations are integrated

out from the wellbore. At any distance

r

from the wellbore, they

assume that the radial conductivity is given by the sum of211 rtimes

the reservoir permeability and two times the local fracture conduc

tivity. n this way, they can account for fmite-conductivity fractures;

however, because of the radial-flow assumption, their results may

not be accurate for fractures extending over a large portion of the

drainage area.

omparison With

Models

In the Literature

The acid fracturing model described in this paper is based on a two

fluid generalization

of

the Perkins-Kern model and uses Eq. 27 to

model acid transport. The fluid velocities and fracture widths

in

Eq. 27 are allowed to vary along the fracture length.

The acid transport model (Eq. 27)

is

a generalization

of

the model

presented by Nierode and Williams, 3 who use Terrill's 10 solution

(for parallel plates) in graphical form to predict the acid penetration

distance. In this paper, Terrill's solution is incorporated into a

fracture model to predict the fracture length and the acid penetration

distance simultaneously. Eq. 27

is

similar to the model presented

by Roberts and Guin.

7

For the case of infinite reaction rate at the

walls, Roberts and Guin's equation becomes

iJCliJx= 2/iib) v

w

K

g

)C, . . . . . . . . . . . . . . . . . . . . . . . . . (28)

where

g

=mass-transfer coefficient. Eq. 28 was also derived in

Refs. 14 and 17. In Refs. 7, 14, and 17, however, no distinction

was made between averaged quantities as defined in Eq. 19 and

velocity-averaged

qulU tities

as defined in Eq. 20. For Eq. 28 to

be rigorously correct, C should be replaced by C

m

. Comparing Eq.

28 with Eq. 22, one can see that the equations are very similar,

except that Eq. 28 is expressed in terms of a mass-transfer coeffi

cient and Eq. 22 uses the Nusselt number.

As shown

in

Ref. 17, a foam acid system is governed by the same

equation as Eq. 28, except that the diffusion coefficient, D (and

hence

Kg)

is replaced by DI I r ) r is foam quality as defined

in Ref. 17). Hence, foam fracture acidizing can be modeled by the

same equations in this paper with a modified mass-transfer

coefficient.

Using Eq.

22

with Eq. 26 has three advantages over using Eq.

28. First, Eqs.

22

and 26 show explicitly how the mass-transfer

rate depends on the fracture width and fluid leakoff, while these

terms are embedded in g

in

Eq. 28.

D

in Eq. 22 will be a function

primarily of the Reynolds number down the plate, while Kg will

be a function of the Reynolds and Schmidt numbers. The second

advantage is that use

of

Eqs. 22 and 26 allows the mass-tran&fer

rate to vary along the fracture. And third, the right side

of

Eq. 22

will always be negative when used with Eq. 26, while the right

side of Eq. 28 may become positive

if

the leakoff velocity is much

larger than the measured leakoff velocity, which occurred during

experimental determination of Kg.

omparison With Published Results

Williams et al presented a set of acid fracturing calculations for

a well completed in a limestone formation at a depth

of

7,SOO ft

[2286 m]. The formation has a SO-ft [lS-m] -thick oil zone with

a permeability

of O.S

md and a viscosity of

O.S

cp [O.S mPa ·s].

Additional reservoir properties are listed in Table I.

The reservoir is stimulated with two fluid stages. A pad fluid

of a given volume is injected followed by a suitable acid volume.

The stimulation is performed with pad volumes of ISO, 300, 4S0,

and 600 bbl [23.8, 47.7, 71.5, and 9S.4 m

3

], followed by acid

volumes of 78 97 121, and IS4 bbl [12.4, IS.4, 19.2 and 24.5

m

3

]. Additional fracturing fluid properties are listed in Table 2.

Using the fracture and the acid transport model presented earlier,

the simulations took less than S seconds on a Macintosh II. The




T BLE

1 RESERVOIR

PROPERTIES USED

N THE MODEL C LCUL TIONS

Fracture gradient, psi/ft

Fluid density, Ibm/ft

3

Fluid compressibility, psi

1

Porosity

Reservoir temperature,

of

Reservoir pressure, psi

Well spacing, acres

Well radius, ft

Young's modulus, psi

Poisson's ratio

0.7

52

0.0001

0.1

200

2,500

40

0.5

6.45

x

10

6

0.25

fluids were injected at a rate of

15

bbllmin [0.0398

m

3

/s]

as op

posed to the 10 bbllmin [0.0265 m

3

/s] used in Ref. 2, because 10

bbllmin [0.0265 m

3

/s] would not support a growing fracture. The

Perkins-Kern fracture model used in this paper predicts longer, nar

rower fractures than the Geertsma-de Klerk model used in Ref. 2,

which necessitates higher injection rates for P-K simulations.

The model presented here predicts productivity increases of3.4,

3.6, 3.8, and 4.0 for the four stimulations. Given the differences

between the current model and the model in Ref. 2, this compares

well with Williams

et al.

s predictions of 3.2, 3.6, 4.0, and 4.4.

The current model predicts productivity variations of 0.2 between

stimulations; Ref. 2 predicts variations of 0.4.

Conclusions

We have shown that the acid fracturing model proposed is derived

rigorously from a consistent approximation of fundamental

equations. The model includes a

10

approximation (averaged over

the fracture width)

of

the 20 acid transport model, and a two-fluid

Perkins-Kern approximation of the 20 fracture model. They were

shown to be excellent approximations

of

the corresponding 20

equations. While the present model is an extension of existing acid

fracturing models, it differs from existing models in that diffusion

and fluid leakoff are explicitly accounted for in the acid transport

equation. The concentration profile and the productivity calculations

based on the model presented here compare well with published

results. Further generalizations of the model are possible. For ex

ample, the model has not addressed the temperature variation from

the wellbore to the fracture tips, which could affect the mass-transfer

rate, the acid reaction rate, and the fluid rheology.

f

he temper

ature variation within the fracture is known, the equations can be

modified to model the phenomenon. The model can also be extended

to include more than two fluids. Because the model is based on

fundamental equations, it provides a sound theoretical framework

from which improvements in modeling the acid fracturing process

can be made.

Nomenclature

b = fracture width, L

Ii = average fracture width, L

b

max

= maximum width of fracture cross section, L

b

o

= maximum fracture width at the wellbore, L

b

1

b

z

= fracture width at

L

and at wellbore

C = acid concentration, m/L3

C

=

average acid concentration,

mIL

3

Cj.CL\ CLZ = leakoffcoefficients for Fluids i 1, and 2, Lltl-Z

C

L

= leakoff coefficient, Lltl-Z

C

m

= velocity-averaged acid concentration, m/L

3

D

=

diffusion coefficient

or

mixing constant,

LZ/t

E

= Young's modulus, m/U

z

fi g g\>gz = known functions, dimensionless

h = fracture height, L

i = injection rate, L3 It

ii il,iz = injection rate for Fluids i 1 and 2, L

3/t

K.Ki.K\>K

z

= fluid constant for fracture fluid and Fluids i 1

and 2,

m/UZ-n

.

Kg

= mass-transfer coefficient, LIt

L = fracture length, L


T BLE 2 FLUID PROPERTIES USED

IN THE MODEL C LCUL TIONS

Pad fluid

Average viscosity, cp

Fluid-loss CL , ftlmin h

Acid

Average viscosity, cp

Acid density,

%

Fluid-loss

eLl

ftlmin'l2

60

0.002

1.2

15

0.002

L

j

= distance from wellbore to interface between Fluids

i I and i

LI

= distance from wellbore to fracture tip, L

L

z

= distance from wellbore to fluid interface, L

n nj n\>nz = fluid constant for fracture fluid and Fluids

i

1 and 2

NNu

= Nusselt number, dimensionless

NNu = average Nusselt number, dimensionless

N

Pe

= Peclet number, dimensionless

NRe.NReu

N

Rev

= Reynolds numbers, dimensionless

NSc = Schmidt number, dimensionless

p

=

fracture pressure,

m/U

2

qw

=

totalleakoffra te (both faces) of one wing per unit

length, LZ/t

r

= distance from wellbore, L

s = shear stress, m/Lt

Z

t = time, t

t

1 = time of injection for Fluid 1 t

u

= lateral velocity component, LIt

Ii = average lateral velocity component, LIt

U = velocity down the plate

v = transverse velocity component, LIt

vw = leakoff velocity at fracture surface, LIt

V2 = constant leakoff velocity, LIt

V = fracture volume, L3

Vi,vI,vZ

= fracture volume for Fluids i, 1 and 2, L3

w

=

mass-transfer rate

x

= distance from the wellbore, L

y

= transverse distance from the centerline or the

channel or fracture, L

y = shear rate,

I

r = foam quality

p. = Poisson's ratio, dimensionless

= viscosity

u = in-situ stress, m/Lt

Z

T = time for acid to reach a particular point in the

fracture, t

Ti, T 1 TZ = time for Fluids i 1, and 2 to reach a point in the

fracture, t

Acknowledgment

We thank R.S. Schechter for helpful discussions in connection with

this work.

References

1. Perkins, T.K. and Kern, L.R.: Widths

of

Hydraulic Fractures, JPT

(Sept. 1961) 937-49;

Trans.

AIME, 222.

2. Williams, B.B., Gidley,

J.L.,

and Schechter, R.S.: Acidizing Fun-

damentals·

Monograph Series, SPE, Richardson. TX (1979) 6.

3. Nierode, D.E. and Williams, B.B.: Characteristics of Acid Reactions

in

Limestone Formations,

SPEI

(Dec. 1971) 406--18;

Trans.

AIME,

251.

4. Nierode, D.E

•

Williams, B.B., and Bombardieri. C.C. : Predict ions

of Simulation From Fracturing Treatments, J

Cdn. Pet. Tech.

(Oct.

Dec. 1972) 31-41.

199



5. van Domselaar,

H.R.,

Schols, R.S., and Visser, W.: An Analysis

of

he Acidizing

rocess in

Acid Fracturing,

SPEJ

(Aug. 1973) 239-50;

Trans. AIME, 255.

6.

Roberts, L.D. and Guin, J.A.: The Effect of Surface Kinetics in

Fracture Acidizing,

SPEJ

(Aug. 1974) 385-95;

Trans.

AIME, 257.

7. Roberts, L.D. and Guin, J.A.: New Method for Predicting Acid

Penetration Distance,

SPEJ

(Aug. 1975) 277-86.

8. Nordgren, R.P.: Propagation of Vertical Hydraulic Fracture , SPEJ

(Aug. 1972) 306-14.

9.

Guillot, D. and Dunand, A.: Rheological Characterization

of

Frac

turing Fluids by Using Laser Anemometry,

SPEJ

(Feb. 1985) 39-45.

10. Terrill, R.M.:

Heat

Transfer

in

Laminar

Flow Between Parallel Porous

Plates,

Inti

J.

Heat Transfer

(1965) 8, 1491-97.

11. Terrill, R.M. and Walker, G.:

Heat

and Mass Transfer in Laminar

Flow Between Parallel Porous Plates,

Appl. Sci. Res.

(1967) 18,

193-220.

12. Bird, R.B., Armstrong,

R.C.,

and Hassager, 0.: Dynamics of Poly

meric Liquids,

Fluid Mechanics

John Wiley Sons, New York City

(1977) I, 470.

13. Sinclair, A.R.: Heat Transfer Effects

in

Deep Well Fracturing,

SPEJ

(Dec. 1971) 1484-92; Trans. AIME,

251

14.

Lee,

M.H. and Roberts, L.D.: Effect of Heat of Reaction on Tem

perature Distribution and Acid Penetration in a Fractur e,

PT

(Dec.

1980) 501-07.

15. Nierode', D.E. and Kruk, K.F.:

An

Evaluation of Acid Fluid Loss

Additives, Retarded Acids,

and

Acidized Fracture Conductivity, paper

SPE 4549 presented at the

1973

SPE Annual Meeting, Las Vegas, Sept.

30-Oct.3.

16. Raymond, L.R. and Binder, G.G. Jr.: Productivity

of

Wells

in

Ver

tically Fractured, Damaged Formations, PT (Jan. 1967) 120-30;

Trans.

AlME, 240.

17. Ford,

W.G.F.

and Roberts, L.D.:

The

Effect

of

Foam on Surface

Kinetics in Fracture Acidizing, SPEJ (Jan. 1985) 89-97.

Appendix Generalization

of

the

Fracture

Model

to

More Than Two

Fluids

Let the number of stages be

m.

Set Lm+

I

=0, b

o

=0, to

=0. The

width relation becomes

b l

ni

+

Z

- b i ~ l + Z

=f(ni,Ki,im)(L

i

-Li+I) , i=2,3

.

.

.

m . . (A-I)

where i corresponds to Fluid i at Stage

i,

and f is a known function

of

the injection rate at Stage m and the fluid constants. The lengths

Li

are defined analogously as LI and Lz (see Fig. 1).

200

The volume of Fluid i, Vi, is

Vi=g(ni,Ki,im)(blni+3 b i ~ 1 + 3 , (A-2)

where g is a known function of the fluid constants and the injection

rate.

The leakoff relations for Fluids m and i (i=

1,2

m-I) corre

sponding to Eqs.

14

and 15 are

J

L +I

- I

8CUh[.JTi+1 -TI(X)-.JTi(X)-TI(X) ]dx

o

and Vm=im(t-tn-I)- J m8CLmh[.Jt-TI X)

o

(A-3a)

.J

Tm(X)-TI x)

]dx,

A-3b)

where Vi is the volume occupied by Fluid i. Equating Eqs. A-3

and A-2 gives

m

equations, which, coupled with the

m

equations

obtained from Eq. A-I, give 2m equations for the 2 m unknowns

L

i

and b

i

for a given

t

51

Metric

onversion Factors

acres x

4.046

873

E+03

m

Z

cp

x 1.0* E-03 Pa's

ft

x 3.048*

E-Ol

m

ft3

x 2.831

685

E-02 m

3

OF

(OF-32)/1.8

°C

Ibm

x 4.535 924

E-Ol kg

md x 9.869233 E-04

J l mZ

psi x

6.894

757

E OO

kPa

• Conversion factor is exact.

SP P

Original SPE manuscript received for review Feb. 12. 1988. Paper (SPE 17110) accepted

for publication Nov.

16. 1988.

Revised manuscript received Oct.

14. 1988.


Documents

SPE-17110-PA - Modelling of Acid Fracturing