spe-17110-pa - modelling of acid fracturing
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7/24/2019 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
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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
SPE Production Engineering, May 1989
------_._---------------------
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
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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
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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
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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
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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
SPE Production Engineering, May 1989
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
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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.
SPE Production Engineering, May 1989