Theory of Mobilization Pressure Gradient of Flowing
Foams in Porous Media
I1. Effect of Compressibility
W I L L I A M R. R O S S E N l
Chevron Oil Field Research Company, P.O. Box 446, La Habra, California 90633-0446
September 29, 1988; accepted July 31, 1989
This report extends the theory of the minimum pressure gradient (Vp)m~"for flowingfoams in porous
media to include the effects of gas compressibility.The compressibilityof the gas phase raises (Vp) rain
because, in compressiblefoams, lamellaetend to lodge in pore throats, where capillaryresistanceto flow
is greatest. The effect of compressibilitydepends on bubble size, gas compressibility,pore-throat geometry,
compressibilityof adjacent trapped-gas bubbles, capillary pressure Pc, and bubble-train length, i.e., the
number of consecutive bubbles between slugs of liquid. With reasonable parameter estimates, theory
predicts that medium-textured steam foams (bubbles 400 um in diameter in bulk) have (Vp) rainof 1.2
MPa/m (55 psi/ft). For nearly incompressibleCO2 foams, the compressibilityeffect depends strongly
on capillarypressure and (Vp)minis reduced by lowergas-liquid surfacetension. Theory predicts (Vp)rain
= 125 and 20-40 kPa/m (6 and 1-2 psi/ft) for medium-textured CO2 foams at low and high capillary
pressure, respectively. Finer textured CO2 foams have a still higher (xup)min. Clearly, limited foam coalescence is essential for deep foam penetration away from the wellbore in an oil reservoir. Modified for
the effect of compressibility,this theory of (~Tp)min fits the beadpack data of Falls et al. (SPE Reservoir
Eng. 4, 155, 1989). ©1990AcademicPress,lnc.
INTRODUCTION
where ( A p ) t oavgt 1S
- the average capillary Ap per
Foams are a promising means to improve
reservoir sweep and oil recovery in gas-injection enhanced oil recovery ( E O R ) projects
(1). However, the existence of a substantial
m i n i m u m pressure gradient to create and sustain a flowing foam in porous media could
limit the successful application of foams to
lamella in the train a n d nL is the n u m b e r of
lamellae per u n i t length of the train i n the direction of flow. By c o n v e n t i o n , Ap is positive
if the forward b u b b l e is at a lower pressure.
Analysis of the passage of a single lamella
t h r o u g h a single pore gives the time-average
Ap
EOR (2).
At low pressure gradients, foams flow as
" b u b b l e trains" along s i n u o u s paths t h r o u g h
regions of trapped b u b b l e s (3, 4). I n a comp a n i o n paper (2) we derive a theory for the
m i n i m u m pressure gradient (Vp) rain required
to keep a train of incompressible b u b b l e s
flowing. T h e f u n d a m e n t a l e q u a t i o n of this
theory is
(VP) rain = ( An'~avg~
/J]tot 'tC,
(Ap) avg =
f0 ' Apd-r,
[21
where Ap is the pressure difference between
bubbles and r is the fraction of the pore transit
time. Analysis also gives ¢, the standard deviation of the distribution Ap(r). The curvature of the lamella governs Ap,
43,
zXp --- Pl - P2 = - - ,
[1]
[3]
rl
where 3' is the liquid/gas surface tension and
rl is the radius of the spherical lamella at a
Present address: Department of Petroleum Engineering, The University of Texas at Austin, Austin, TX 78712.
17
0021-9797/90 $3.00
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
Copyright © 1990 by Academic Press, Inc.
All rights of reproduction in any form reserved.
18
WILLIAM R. ROSSEN
given position. By convention, we define q and of linear compressibility [4], though an apAp as positive when the lamella bulges forward proximation, greatly simplifies the analysis to
and negative when it bulges backward.
follow. Plugging the equation of state ( c f [ 4 ] )
(Ap)ta'g depends both on (Ap)avg and on
for each bubble into Eq. [3] gives
(2). Large values of (Ap) avg means lamellae
~_ = 2 7 k ( rl ]-l + v_!
require, on average, large positive pressure difrtVp \ rt ]
Vp
ferences between bubbles in order to flow.
Large values of a mean that the random fluctuations in pressure between bubbles cause
large excursions in pressure along the train and
frequent bubble separation and snap-off in the with
interior of the train. Both bubble separation
K ~ ~'~T"el"Ov°
[6]
and snap-off increase (Ap) ~vg and (~Tp)min.
Up
In this paper we extend our theory to account for bubble compressibility. In addition,
Pc~ 23/
[7]
we consider several effects that magnify the
rt
effective compressibility of the foam: first, the
contraction and expansion of trapped-gas where vl is the volume behind the lamella in
bubbles and liquid films to accommodate the the pore of interest; vp is volume of a pore;
compression and decompression of moving and p~ is the capillary entry pressure of the
bubbles, and, second, the autocorrelation of pore. For a given pore geometry, both (rffrt)
lamella movement in trains of multiple bub- and (v~/Vp) are functions of dimensionless lables. We show good agreement between theory mella position ( x / l ) , so it is convenient to
and the data of Falls et al. (3) when these ef- solve Eq. [5] for r as a function o f x / l (2).
fects are taken into account. A preliminary Portions of the curve z ( x / l ) where T decreases
with increasing x are replaced by jumps in poreport of this work was given elsewhere (5).
sition (6).
Equation [ 6 ] reveals that the effective comTHEORY AND RESULTS
pressibility is proportional to gas compressiEFFECT OF GAS COMPRESSIBILITY
bility, bubble volume, and the capillary entry
pressure of the given pore. High gas compresTheory for One Lamella
sibility and large bubble size both contribute
to
large values of K; for a given pore shape,
We employ the same conceptual model as
that in Ref. (2), that is, a single lamella tra- so does a large capillary entry pressure,
versing a single pore, impelled by the steady whether due to large surface tension 3' or small
flow produced by moving pistons upstream throat radius rt.
If the pressure either upstream or downand downstream. Here, however, we allow the
gas in both forward and rearward bubbles to stream of the bubble is constant, then Eq. [ 5 ]
be compressible. We assume that the pressure still obtains, with K twice the value given by
Eq. [ 6 ]. If constant-pressure conditions apply
in each bubble is linear in volume
both upstream and downstream, then either
Ov
- - =- - k ~- - k xo vbo ~-- constant,
[4] the lamella is stranded at a position where Eq.
Op
[ 3 ] is satisfied or there is no position at static
equilibrium
at any time.
where k is the linear coefficient of compressibility and kT and V0b are, respectively, the
conventional coefficient of compressibility
( - 0 In V/Op)T and bubble volume at the average pressure of the foam. The assumption
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
Results for One Lamella
Figure 1 shows the progress of the lamella
through a conical pore with dimensionless
19
FLOWING FOAMS IN POROUS MEDIA, II
a)
K =0
/
1.0
r
b)
/
i
1.0 I
./,v
0.5
K = 0.2
/V
T 0.5
I
0.5
1II
0
1.0
01.5
x/~
c)
1.0
x~
K = 0.6
d)
K = 2.0
1.0
7- 0.5
[
"._.,;'
0
0
I /
T 0.5b ~,
/
¢"
/
L
0.5
1.0
x/~
ii
/
OL
o
0,5
x/,(
//
'\
1.0
FIG. 1. Effect of compressibility factor K on lamella transit through a conical pore with Pb
Pt = 0.1.
pore-body radius P b ~--- rb/l = 0.5 and dimensionless throat radius pt = rt/l = 0.1 as a function of K. Horizontal segments correspond to
j u m p s in position. For K = 0 (Fig. l a ) , there
is one j u m p as the lamella reaches the corner
at the midpoint of the pore and no change in
bubble volume at the j u m p (2). For K = 0.2,
the rearward bubble compresses and the forward bubble decompresses for a time, until
the rearward bubble has enough pressure to
force its way through the pore throat. Then it
jumps to a position at which it is decompressed
enough to be at equilibrium at the lower capillary ~xp. There is the j u m p again at the pore
body, this time including some expansion of
the rearward bubble as a result of the j u m p
and change in sign of ~xp. Finally, there is a
third j u m p as increasing capillary force pulls
the lamella into the converging pore throat.
For higher K(Figs. lc and ld), the j u m p s become greater until they merge into one jump:
the lamella hops from the position of maxim u m ~xp in the pore throat to the next pore
=
0.5,
throat, near the position of m a x i m u m Ap. A
similar progression is obtained for a sinusoidal
pore with Pb = 0 . 5 , Pt = 0 . 1 , although larger
values of K are required to induce the multiple
jumps.
Figure 2 shows for the conical pore how the
frequency distribution of Ap (2) changes with
K. As K increases, the lamella spends a large
fraction of its time in the diverging pore throat
until, for sufficiently large K, the lamella is
always in the throat with ~xp near its m a x i m u m
value, (Lxp) ma~. For uniform, conical pores,
(Ap)max/p e = 2 s i n [ t a n - l ( 2 ( P b -- Pt))].
Figure 3 shows how (Lxp) avg and ¢ depend
on K for conical and sinusoidal pores with pt
= 0.5, pt = 0.1. For conical pores, (Lxp) avg approaches the incompressible limit for K less
than 0.1 and approaches (~xp) maxfor Kgreater
than 10. As K increases beyond 1.0, ¢ approaches 0 because all lamellae are at the pore
throat with Lxp near (~xp) maX. For sinusoidal
pores, although there is a small j u m p in position for K as low as 0.12, (Ap) ,vg is insignifJournal of ColloM and Interface Science, Vol. 136, No. 1, April 1990
20
WILLIAM R. ROSSEN
b) K = 0 . 2
a) K = 0
Diverging
~
COrn:B r g ~ _ ~
e
Body
Diverging
Pore Body
Converging
Pore Body
Diverging
Converging /
~t Por~Throat
I
- 1.0
0
1.0 (Ap) max
. ~p/pc. ~
_ (~p) .
.
c) K = 0.6
(Ap)max
A p/pce
Diverging
Pore Body
Converging
Pore Throat
Diverging
Pore Throat
Diverging
Pore Throat
\
/
. . . . . . . . . . .
1-'1.o
_ (~p) .
0
d) K = 2.0
/
_
1,0 (AP) max
- 1.0
I
o
.
~p/pc e
.
1.0 (Ap) max
.
,
l- .o
(~p)max
o
v
,
/,4
1.o
,x
zXp/pce
FIG. 2. Frequency distribution of Ap as a function of K for a conical pore with Pb = 0.5, Pt = 0.1.
icant for K less than 3. (Ap) avg approaches its
maximum value as K increases beyond 100.
These results show that for long or highly
compressible bubbles, only the geometry of
the pore throat, which determines (Ap) max,
matters to (Vp) rain (Eq. [1]). In contrast, for
incompressible bubbles in smooth symmetric
pores, only the geometry of the pore body,
where the jump in position occurs, matters to
(Vp) mi" (2).
Multiple Pore Shapes
If a train of highly compressible bubbles
passes through a variety of pore shapes, the
pores with the highest (Ap) max determine
(Ap) "~g for the train: after compressing to
overcome (Ap) max at the tightest throats, lamellae pass completely through pore throats
with lower (Ap) m"x. Figure 4 illustrates this
phenomenon for a train passing through conical pores with the same length l and body
Journal of Colloid and Interface Science, Vol. 136,No. 1, April 1990
radius Pb = 0.5 in all pores but with porethroat radius pt alternating between pt = 0.1
and 0.25. Also pictured is (Ap) avg for uniform
conical pores with pt = 0.1 and with ot = 0.25.
For comparison, we have normalized all the
curves by using pe for the tighter pore throat,
pt = 0.1. For the sequence of alternating pore
types, in the incompressible limit K --~ 0,
(Ap) "vgis simply the volume average of(Ap) avg
for the two pore types. As K increases, however, the lamella spends more of its time in
the tighter pore throats; as K ~ oe, (Ap) avg
reflects only the presence of the tighter throats.
Thus, (Ap) avgfor a train of highly compressible
bubbles reflects only the tightest pore throats
in the train.
The train follows the path of least resistance,
however. Therefore, out of all the pores in the
porous medium, the train samples the percolation-threshold (7-10) fraction (about
15%) of pores with the largest pore throats
(2). As K approaches infinity, (Ap) a v g reflects
FLOWING
FOAMS IN POROUS
i
21
M E D I A , II
i
j
fs.
1.0
c°n7
(Ap) avg
pce
0.5
"
I
i
0,1
1.
~
]
10.
100.
10.
100.
K
\
pce
C o n i al
0.1
1.
K
FIG. 3. Effect of compressibility factor K o n ( A p ) avg a n d ~r for conical a n d sinusoidal pores with/0 b - 0.5,
pt = 0.1.
the tightest throats in this fraction; i.e., it reflects the throats at the percolation threshold
itself--the throat size measured at the
"threshold pressure" in conventional mercury
porosimetry (9, 10).
EFFECT OF BUBBLES ALONGSIDE
TRAIN
Theory
Trapped-gas bubbles alongside the flowing
bubble train expand and contract in response
to fluctuations in pressure of individual bubbles in the train. This increases the effective
compressibility of the bubbles in the train.
We augment our model to allow for movement of lamellae separating trapped-gas bubbles from the train as follows. For each bubble
i in the train, we assume there are nsi trapped
bubbles surrounding it numbered j = 1
through n~i. We assume initially that the surrounding bubbles are isolated from each other
and fixed in volume except for the m o v e m e n t
of the lamellae they share with the bubble
train. The m o v e m e n t of side lamella ij produces a change Avsi; in the volume of the side
bubble and an opposite change in the volume
of bubble i. By convention we call 2xvs o > 0 if
the volume of bubble i increases and that of
side bubble ij decreases. For each lamella
alongside the train we choose a reference position such that 2xv~u = 0 at r = 0. The pressure
in side bubble ij is P w , and we define ps°j
=-
p ( AVsij = 0 ) .
From the bubble equations of state (cf. Eq.
[4]) and the capillary Ap conditions (cf. Eq.
[ 3 ] ) we obtain
47
- 2(vor-
vl)/k
rl
+
47
'~PS O' - -
rso
(-~
Vsu + ~ vs2j)/k
[81
- Pi - Ps°j - Avsij/k, o,
i = 1,2;
j=
1. . . . .
nsi.
[9]
Journal of Colloidand InterfaceScience, Vol. 136, No. 1, April 1990
22
WILLIAM R. ROSSEN
1.0
/
pe
Pt = 0"1/
(Ap)avg
/Alternating
Pore Types
0.5
~
01
0
~
I
Pt = 0.25
/
0.1
I
1.
K
10.
FIG. 4. Effect of compressibility factor K on (Ap)aVg for conical pores with pb
p~ = 0.25, or an alternating sequence of the two pore types.
Here, ks o ~- (OAvsij/Op~ij), a function of the
initial volume and pressure of side bubble ij
(cf. Eq. [4]). For the side bubbles we define
rsij as the radius of the spherical lamella separating bubble ij from the train, with r~ij > 0
if the lamella bulges into the side pore. We
assume here for simplicity that the simple
conical and sinusoidal pore shapes adequately
describe both the pores of the bubble train and
the side-pore throats.
One could solve Eqs. [8] and [91 simultaneously for positions of all lamellae as functions of time given their initial positions and
knowledge of how r~ varies with Vl and how
the r~ij vary with AvsO. Instead, we offer here
an approximate solution.
100.
=
0.5 and either pt = 0.1,
keff
keffdpl = d(vpr - 1)1)
[11]
with
1
keff/ k - ~ 1 + ~c
]"
[121
Differentiating Eq. [ 9 ] for i = 1 with respect
to P l g i v e s
dAl)slj
dpl
[
dApslj] -1
- kslj 1 + ksl j dAvslj J "
[13]
The capillary entry pressure of the side throat
is
2X
Pc~sU~
[141
rts lj
A p p r o x i m a t e Solution
From the definition of k =- - ( O v / O p ) and
the equation of state for the rearward bubble
1,
kdpl +
d ~ AVsij
dpl
dpl = d(vpr - Vl).
with rtslj the side throat radius.
Combining Eq. [ 13 ] with Eq. [ 12 ], noting
k ~ U 0b k 0T and kslj ~- l ) 0b s k T0 ,
Gfr/k ~- 1
V°s [
d-(APslHP~slj)] -I
+ ~ V--~D 1 + Ks d(AVslj/Vp ) j
[151
[101
with
The effect of movement of side lamellae is to
increase the effective compressibility to a value
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
V°s
Ks =- peslj - - g°T.
Vp
[161
FLOWING
FOAMS IN POROUS
Here we have assumed that k ° has roughly the
same value in the side bubbles and in the bubble train and that all side bubbles have roughly
the same volume, v°s. Equation [15] indicates
that the m o v e m e n t of side lamellae increases
the effective compressibility of bubbles in the
train. If the large term in brackets in Eq. [ 15 ]
is 1, then bubble compressibility in the train
increases as though each flowing bubble included the trapped bubbles around it. The
term in brackets thus indicates the capillary
resistance to m o v e m e n t of the side lamellae.
If this term is near 1, then the first row of side
lamellae does little to d a m p the expansion and
contraction of bubbles in the train. In a real
porous m e d i u m this effect would then propagate through m a n y rows of trapped bubbles
and greatly increase effective compressibility.
This capillary resistance is proportional to (Eq.
[ 16 ] ) the capillary entry pressure of the side
pores, the volume of side bubble relative to a
pore in the bubble train, gas compressibility,
and (Eq. [ 15 ] ) the dimensionless change in a
capillary Ap per unit change in bubble volume.
From the definition of compressibility factor
K, we have
Keff/K = 1
V°s [
d(Aps*J/P~slJ)]-'
[171
+ ~ V~b 1 +Ks d(AVslffVv) j "
If we l e t m s ~- ~ 1)bs//)
0 0b be the total volume
of side bubbles relative to the volume of the
flowing bubble, and replace the elements in
the summation by a representative average
value, Eq. [ 17 ] becomes
Keff/K = 1 + m s 1
d(Aps/P~s)]- 1
+Ks~j__
.
[18]
Two Examples
Beadpack. The effect of the side bubbles depends on the geometry of the pore throats
alongside the train. We consider two examples.
The first is a pack of beads of radius R at the
m a x i m u m packing density. There are two pore
23
M E D I A , II
types in such a pack: for one there are eight
pore throats, l = 1.73R, Pb --~ 0.24, and Pt
- 0.086; for the other there are four pore
throats, l - R, Pb ~ 0.23, and Pt ~ 0.155 ( 11,
12). We assume that there are both pore types
in the train; the train must alternate pore types
if the beads are in a hexagonal-close-pack arrangement. Using a conical-pore model for
pore bodies and taking the average for the two
pore types, we estimate Vp = 0.59R 3.
All throats in such a pack have the same
diameter (rts = rt = 0.155R), and, thus, when
gas invades the interior of the pack, it fills virtually all the pores. We assume there are, on
average, four gas-filled side bubbles per pore
in the train, and, if each side bubble occupies
one "average" pore, v°s/Vp = 1 and ms = 4.
Equation [ 18 ] is, for this case,
[
Kerf/K~- 1 + 4 1 + 0 . 1 5 5 ~
d(Aps/pes)] -1
×k ° ~ ~ j
.
[19]
For instance, in the 3 - m m beadpack experiments of Falls et al. (3) at 1 atm, R = 0.3 cm,
kT -~ ( 14.7 psi) -1 = 9.86 × 10 -7 ( d y n / c m 2) 1,
3' = 30 d y n / c m , and
Kecf/ K = 1
+4
d(Apslpes)] 1
1+0.0013 ~
j
.
[20]
To estimate the derivative in Eq. [ 20 ], we
approximate the pore throats by sinusoidal
pores with Pb = 0.177, Pt = 0.0896; this approximates the curvature in a toroidal pore
throat with the same geometry as pore throats
in the beadpack. The result is shown in Fig.
5a. In the center of the pore throat (Aps = 0),
the derivative is large but as the side lamella
is displaced from the throat, the derivative
falls, passing through zero for large displacements. If Ks is sufficiently large (side bubble
size very large or gas compressibility very low),
as the derivative falls below zero the denominator in Eq. [ 18 ] also m a y pass through zero
for some lamellae: these side lamellae j u m p
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
24
W I L L I A M R. R O S S E N
a)
i
15
i
Pore Throat
10
d Aps/pes
d AVs/V p
0
Pore Body ~
_/2
I
-1
-3
J
0
Pore Body
I
i
1
2
APs/P~
b)
i
i
i
i
i
Pore Throat
5000
dZ~p~/p e
d AVs/Vp
I
,I
0
-1000
-3
II
I
't1
Pore Body
I
:
ZXPs/pg
FIG. 5. F u n c t i o n used to estimate effect of trapped-gas bubbles alongside train. (a) Bead pack. ( b ) Rock.
from their throats and the effective compressibility is, briefly, infinite. We employ a generous estimate of the derivative in Eq. [20]
for all side lamellae, 15 (cf. Fig. 5a). Inserting
this value into Eq. [20] gives
Keer/K = 1 + 4 [1.02] -1.
[21]
Equation [21] indicates that the effective
compressibility is quintupled due to the expansion and contraction of the first row of
trapped-gas bubbles alongside the train. It is
a conservative estimate because it excludes the
possibility of discontinuous jumps by side lamellae. More important, the value near 1
within the brackets in Eq. [ 21 ] means that the
first row of side lamellae damps little of the
expansion and contraction of bubbles in the
train: in other words, what Falls et al. called
the "breathing" effect (3) propagates through
many rows of lamellae around the train, effectively engaging the entire beadpack in the
pulsating pressure fluctuations of the train itself, as Falls et al. observed. If we allowed for
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
many rows of bubbles in our model, Keffwould
increase by orders of magnitude.
In general, pore geometry, bed size, and gas
compressibility together determine the magnitude of this effect in beadpacks. Rounded
pore throats facilitate breathing (Fig. 5a). On
the other hand, "breathing" would be less
pronounced in a pack of smaller beads, while
it would be magnified by lower gas compressibility (Eq. [ 16 ] ).
Rock. Our second example is oilfield rock.
We approximate both pore shape in the train
and shape of the side throats by angular, conical geometry. Rocks can have a wide distribution of pore-throat sizes (13). Since the
bubble train selects the largest throats for its
path, most throats alongside the train are significantly tighter than those in the path of the
train. Therefore, we describe pore geometry
in the train by a conical pore with Pb = 0.5,
Ot = 0.1, and pore-throat geometry alongside
the train by a conical pore with Pb = 0.5, 0t
= 0.05. We assume initially, as before, that
FLOWING
FOAMS
IN POROUS
each side bubble occupies one pore (VObs/Vp
1) and 3' = 30 d y n / c m . We take kT ~ (50
psi) -I = (350 kPa) -1 = 2.9 × 10 -7 ( d y n /
cm2) -1 as in a steamflood away from the nearwellbore region (14). We assume that there
are four gas-filled pores beside each pore in
the train; this generous estimate corresponds
to a Bethe-tree pore network of coordination
number 7 ( 15, 16) with one side pore occupied
by water and two of the throats, of course,
taken by the path of the train. We assume a
pore length l of 150 #m. Equation [18] becomes
d(Aps/P~s)] -1
geff/g
~ 1 -]-4 1 + 0.023 ~
j
beyond the first row of bubbles, and Ke~ would
be larger than estimated here. A reasonable
estimate is that "breathing" increases Kerr by
a factor of about 10 for CO2 foams.
Side lamellae make a smaller contribution
to Kerr in rock than in beadpacks primarily
because the derivative in Eq. [18] is small in
narrow, angular pore throats. The wider distribution of pore-throat sizes in rock than in
beadpacks means that P~s is relatively small
as well (Eq. [ 16 ] ). "Breathing" is expected to
be less significant in rock than in beadpacks
unless the pore throats in a given rock are large,
rounded, and of uniform size.
EFFECT OF LIQUID FILMS AND LIQUIDFILLED PORE THROATS
Figure 5b shows how (d(2xpJpec~)/
d(2XVs/Vv)) varies with 2xps for the given
side-pore-throat geometry. If 2xps exceeds
+2.7pce, then the derivative discontinuously
changes sign: if Ks is large enough, the denominator term in Eq. [22] may fall below zero,
and the lamella may jump from the throat.
For 2xps between +_2.7p~, a reasonable value
for the derivative is 5500. Equation [22] is
then
[23]
The effective compressibility increases by only
a few percent. Moreover, because the first row
of side lamellae damps almost all of the
"breathing" effect at the first row of bubbles,
the effect does not propagate further into the
surrounding rows of trapped bubbles as it does
in beadpacks.
According to Eqs. [ 15 ] and [ 16 ], if k ° and
3' are lower, the effect of side lamellae is somewhat greater. For instance, using the same parameters as above except k ° = 6.7 × 10-4 psi-i
= 9.7 × 10 -9 ( d y n / c m 2 ) -~ and 3' = 5 d y n /
cm (reasonable values for CO2 floods ( 17, 18 ))
gives
Liquid Films
Liquid films filling corners and crevices in
pores can swell and shrink to accommodate
fluctuations in pressure in the bubble train.
An important practical consideration is
whether these films have time to respond to
the rapid changes in pressure as bubbles cascade from pore to pore. Flow along films presumably is slower than the bulging in and out
of trapped lamellae analyzed above. Here,
nevertheless, for simplicity, we assume that
these films maintain equilibrium with the surrounding liquid at constant (though spatially
nonuniform) pressure. An alternative asymptotic assumption would treat the films as isolated from surrounding liquid and fixed in
volume. In that case, of course, they would
have no effect on compressibility.
By arguments similar to those in the previous section (cf. Eq. [ 12 ]), one can obtain
Kerr~K= 1 -
[24]
Again, the relatively small value in brackets
means the "breathing" effects extends outward
(1)d(2xvf/vp)
~f d(pi/p~)
[25]
with
Kf
Kaf/K= 1 + 4 [ 1 + 0 . 7 ] -1 = 3 . 4 .
25
II
.
[221
Keff/K~ 1 + 411 + 130] I = 1.03.
MEDIA,
e 0
=- Pckx,
[26 ]
where Avf is the change in the volume occupied by films in a single pore. The swelling
and draining of liquid films can give a finite
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
26
W I L L I A M R. ROSSEN
( d y n / c m 2 ) -l, then Kf -~ 0.012. For low capillary pressures, Keff increases by about a factor
of 5; at high capillary pressures, by only
about 9%.
For virtually incompressible CO2 foams, the
effect of liquid films depends on bubble size
as well as on capillary pressure (Eq. [27]).
For instance, assuming a medium foam texture, with bubbles about 400 #m in diameter
in bulk (2) and v ° / v v = 30, and low capillary
pressure Pci = 0.5p~, gives Ke~ = 1.5. According to Fig. 3, such a foam would behave as
though highly compressible, with (Ap) avg
about half of (~xp)max. Finer foam texture and
higher capillary pressure would reduce Kef~,
however.
Real pores in rocks may have larger crevices
and more substantial liquid films than assumed in this model, especially if the surfaces
are deeply roughened. For example, other
simple geometric pore shapes, such as cubes
and tetrahedra, have about double the edge
length per unit volume of the conical pore,
and this difference can increase the effect
of liquid films on effective compressibility.
Moreover, as capillary pressure increases,
smaller scale roughness (20) retains liquid on
surfaces that are effectively smooth at lower
capillary pressure. This mitigates the reduction
compressibility effect even if k ° -- 0 for the
gas phase. Rearranging Eq. [ 25 ] gives
K~ff = K -
pt
V ° d(Avf/Vp)
Vp d(pi/p~) "
[27]
In deriving the relation between Avf and Pi,
we consider a conical pore and, more specifically, the liquid film filling the edge encircling
the pore body. We assume that this film is not
in contact with a lamella and that the axial
cross section of the film surface is circular. This
second assumption, though an approximation,
greatly simplifies the calculation of ~vf. One
can determine vf/v v as a complicated algebraic
function (19) of P b , P t , and Of ~ rf/l. The
capillary condition on rf is
_3'_
Pci ~- P i
-
-
Pliq
ot
rf
e
[28]
2pf pc"
Figure 6 plots the derivative in Eq. [25 ] for
a conical pore with 0b = 0 . 5 , P t = 0.1 as a
function of dimensionless capillary pressure
(Pci/P~). We expect Pci/P~ to take values between 0.5 and 2 (2). At the low end of this
range, the derivative is about 0.05; at the high
end, about 0.001. If we take the same parameter values as above, i.e., 3/ = 30 d y n / c m , l
= 150 urn, and kT = (50 psi) -1 --- 2.9 X l 0 -7
1.0
i
i
I
~
i
0.1
0.01
d (A Vf/Vp)
d (pi/pce)
0.001
0.0001
0.00001
0
5
Pci/Pce
10
I~G. 6. Function used to evaluate effect of liquid films on effective compressibility for a conical pore with
Pb = 0.5, Pt = 0.1.
Journalof Colloid and InterfaceScience, Vol. 136, No. 1, April 1990
FLOWING FOAMS IN POROUS MEDIA, II
in the effect of liquid films predicted above at
high capillary pressures. Thus, the effect of
liquid films on effective compressibility of
foams in rock may be somewhat larger than
estimated here.
Liquid-Filled Throats
A similar analysis covers the intrusion of
gas into liquid-filled throats. By arguments
similar to those above, one can obtain
Keff/K = 1 +
(~f)
d(Al)t/I)P)
ml d ( p i / p ~ )
'
[29]
where m~ is the average number of liquid-occupied throats adjacent to each pore in the
bubble train, Art is the change in bubble volume due to movement into or out of a liquidfilled throat, Kf is defined by Eq. [ 26 ], and rt,
Vp, and p~ apply to the pores and throats in
the path of the bubble train. For virtually incompressible foams
v°
d(Avt/vp)
Vp
d ( p i / p e) "
Keff= K + p~ - - ml
[30]
Three examples. We consider three cases.
In each case we use conical pore geometry for
both the pore shape in the train and the liquidfilled pore throats alongside the train. The derivative in Eq. [ 29 ] can be determined for this
geometry as a function of pe given values for
0.1
~0.01
d(vttVp)
Wide, ~
Uniform
\
27
parameters, Pt, Pb, and Pts, the throat radius
of liquid-filled throats. We assume that initially
there are one or two liquid-filled pore throats
adjacent to each bubble in the bubble train
( m j -- 1 . 5 ) .
In the first case, a wide distribution of porethroat sizes means that liquid-occupied pore
throats are significantly narrower, on average,
than those occupied by gas: P b = 0 . 5 , P t = 0 . 1 ,
Pts = 0.05. We believe this model is appropriate for reservoir rock. Figure 7 shows the derivative in Eq. [ 29 ] as a function of capillary
pressure in bubble i (curve labeled "narrow,
varied throats"). This factor is large in two
instances: if capillary pressure is low, liquid
fills the throat completely but small changes
in pressure cause significant liquid drainage.
As capillary pressure approaches 2p~, gas invades the liquid-filled throat. For Pci > 2P e,
gas breaks through the throat and fills the liquid-filled pore on the other side. The effective
compressibility is, briefly, infinite. If snap-off
does not create a lamella in the newly invaded
throat during invasion (11, 21), one will be
placed there by lamella division ( 11 ) when the
next lamella in the train passes by. In either
case, the liquid-filled side pore is replaced by
a trapped bubble. If Pci falls below about
0.5pc~, on the other hand, bubble separation
or snap-off will occur in the bubble train (2)
and disrupt the train's movement: therefore
we do not expect Pci to fall below 0.5p~. If we
[
]
Throats ~' I
Narrow,
/Uniform
~/
"'"-'
~~ThroatSNarrow,
VariedJ
0.001
0.00010
I
(Pci/Pe
FIG. 7. Function used to evaluateeffectof liquid-filledthroats on effectivecompressibility.
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
28
WILLIAM
take Kf = 0.012 as above, then swelling and
drainage of liquid-filled pore throats can increase Keg significantly only if bubble capillary
pressure is much higher or lower than 1. For
Pci = 0.5p¢e, Keg is increased by a factor of
about 3.
For virtually incompressible foams, Keg
again depends on bubble size V°Vp(Eq. [ 30] ).
For v°/vp = 30, Ken is as large as 1.3 at Pci
= 0.5 pc~. Keg is smaller for finer textured foams
and at higher capillary pressures.
K~f~is still larger for other throat geometries
shown in Fig. 7. The curve labeled "narrow,
uniform throats" corresponds to conical pores
with Pb = 0.5, Pt = Pts = 0.1. The final case in
Fig. 7 has uniform, relatively wide pore
throats, a model we believe is more appropriate for a beadpack, although for simplicity here
we use a conical model with Pb = 0.5, Pt = Ors
= 0.25. In this case (Fig. 7), if v°/Vp = 30,
Keg ~> 5 regardless of the values ofpci, k°t, and
K. Wider side throats experience significant
liquid drainage upon small changes in capillary pressures.
With the random fluctuations in pressure
expected in a bubble train (2), it is likely that
all of the larger pore throats alongside the
bubble train are rapidly occupied by gas. This
suggests that within a short time ml approaches
zero and Keg approaches K. If pore-throat sizes
are widely distributed, then it is the narrowest
of throats that remain occupied by liquid and
their effect on capillary Kefrwould be moderate
at best.
This analysis, and that for "breathing" of
trapped-gas bubbles, is complicated when, at
low capillary pressures, the lamellae separating
trapped-gas bubbles from the train swell and
isolate the trapped-gas bubbles from the train.
This is expected to occur at or below about
one-half the capillary entry pressure of the side
throats (2). Thus, at low capillary pressures,
"breathing" of side lamellae stops: bubbles in
the train push against the swollen liquid lenses
occupying the side throats instead. At these
low capillary pressures, Eq. [ 29 ] then applies
with m~ representing the throats filled with liqJournal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
R. R O S S E N
uid, whether the pores across the throats are
occupied by liquid or by gas.
Summary
In summary, the drainage and swelling of
liquid films in the crevices and comers of pores
and of the liquid in liquid-filled pore throats
can help relieve the compression and decompression of gas bubbles in a train and raise
the effective compressibility. At low capillary
pressures, the effect ofliquidfilms can increase
Keg by orders of magnitude, especially for otherwise incompressible foams. For porous media of uniform throat size, we expect the liquid-occupied pores alongside the train to fill
with gas quickly. Then the effect of side lamellae depends on capillary pressure. If capillary pressure is relatively high, these lamellae
move back and forth in response to bubbletrain pressure and the process of "breathing"
covered by Eq. [ 17 ]. If the capillary pressure
is low, these lamellae swell into lenses that
swell and drain with changes in capillary pressure as described by Eq. [29]. This effect can
raise Kerr by orders of magnitude. For porous
media of varied pore sizes, the effect of the
narrow throats that remained filled with liquid
even at high capillary pressure would increase
Kerr more moderately.
BACKWARDS LAMELLA MOVEMENT
AS noted, when pressure builds exceptionally high in rock (Fig. 5b) or builds more
moderately in a beadpack (Fig. 5a), a lamella
may jump from the side pore throat into the
body of the side pore. In that event, the affected
bubble decompresses discontinuously; the lamella behind it may pull forward and the lamella ahead may pull back. Conceivably, the
lamella ahead could pull back to a position it
had previously jumped over, e.g., into one of
the dotted "inaccessible" branches in Fig. 1.
This is the only means we are aware of by
which a lamella in an isolated bubble train
could take a position it had previously jumped
over. We believe this happens infrequently.
29
FLOWING FOAMS IN POROUS MEDIA, II
Side lamellae jump only if the side bubbles are
highly compressible, i.e., if Ks is large in Eq.
[26]. In addition, side lamellae jump only if
the bubble in the train is compressed to a relatively high pressure. This high pressure is
most likely if there is a large, positive Ap for
the forward lamella, i.e., a position in the pore
throat. A sudden decompression behind this
lamella would merely cause it to retreat slightly
and spend more time in the throat, not to jump
back into the preceding pore body.
Individual lamellae do retreat when, at
higher pressure gradients, the bubbles in multiple bubble trains converge on the same pore
body (22, 23 ). We do not expect this to be a
c o m m o n occurrence at the threshold of flow
at low Vp, where isolated bubble trains move
separately through the porous medium ( 3, 4).
A U T O C O R R E L A T I O N OF
L A M E L L A POSITIONS
In a real bubble train, the lamellae act as
the "pistons" for each other. In a compressible
foam, when one lamella pauses in a pore
throat, it stops pushing and pulling, respectively, the lamellae ahead of and behind it. As
a result, these lamellae advance more slowly,
and they in turn displace the lamella between
them from its pore throat more slowly. Thus,
lamellae in real bubble trains spend more time
in pore throats than they do in the model of
relentless pistons, behaving, in effect, as if they
were more compressible. When jumps do occur, all the lamellae tend to jump together.
Model for Multiple Lamellae
Our model for a train of many bubbles envisions steadily advancing pistons separated
by (n + 1 ) bubbles and n lamellae. While relentless pistons on each end of the train drive
its overall movement, in the interior of the
train lamella movement is governed by the
movement of adjacent lamellae. Our algorithm selects random and uncorrelated bubble
volumes at a fixed reference pressure and then
tracks bubble progress through a chain of uniform pores or repeating sequences of pores.
In the results below we assume uniform conical pores with Pb ----"0.5, Pt = 0.1.
For solving for lamella positions, we selected
a slow, but, for this application, sure algorithm:
successive substitution. That is, after each advance of the pistons, we allowed one lamella
at a time to relax with all others fixed. Over
many iterations, the entire train relaxed to
equilibrium, and then we advanced the pistons
again. Other algorithms, more efficient for
most applications, cannot preclude overshoot
of lamella position in an iteration followed by
backwards "correction" into a position the lamella would not have taken in forward flow.
An additional problem with derivative-based
algorithms such as Newton-Raphson (24) is
that the functions required to form the Jacobian possess sharp corners and discontinuities
(see, e.g., Figs. la and 8 below). Even taking
small time steps cannot preclude large overshoots because there are times when lamellae
1 Lamella AP/Pe
+'t--o
_lqO
~----~1~
r
10 hamellae
NO.
1
Ap/pee
~/JGi'
~
2
!
3
!
4
0--
t
5
o-
t
6
o
7
8
o~
o ~
10
0
~_t
!
~]..__.._._
I ~''-'~--
+10
Total
_,°r
~
_
~
~/z
FIG. 8. Capillary Ap over time for trains of 1 and 10
lamellae. K = 0.
Journal of Colloid and Interface Science, Vol. 136,No. 1, April 1990
30
W I L L I A M R. R O S S E N
must make large jumps across these corners
and discontinuities.
1 Lamella
+1
zxP/Pc 0 4
-1 u
10 Lamellae
Results
Figure 8 contrasts how Ap varies with time
~- for trains of 1 and 10 lamellae between incompressible bubbles ( K = 0). The top curve,
for one lamella, is the same as in Fig. 3 of Ref.
(2), except that here we have not assumed
that the lamella passes through the pore throat
at ~- = 0. The dot and arrow to the right of the
Ap curve in Fig. 8 indicate the time-average
#i and standard deviation ag of Ap for each
lamella during transit through the pore. The
rapid rise in Ap corresponds to the rapid
movement of the lamella through the pore
throat, from ( - - ( A p ) max) to (Ap) max. The
smaller, discontinuous drop corresponds to the
lamella j u m p at the pore body.
In the train of 10 lamellae, the fluctuations
in ~p are uncorrelated because the bubble
volumes and hence lamella positions are uncorrelated, as we assumed in Ref. (2). The
time-average total Ap is 10 times the time-average Ap for the 10 individual lamellae, and
the standard deviation over time of total Ap
across the train, o-, is 1/~ times the standard
deviation over time of ~xp for the individual
lamellae. Thus, the model of relentless pistons,
and the random-walk model based theoreon
(2), is reasonable for a long train of incompressible bubbles, if the movement at the ends
of the train is steady.
Figure 9 illustrates how £xp varies with time
for trains of 1 and 10 lamellae with K = 1.
The steady rise in Ap for one lamella corresponds to slow lamella movement through the
pore throat (cf. Figs. lc and 2c for K = 0.6).
The discontinuous drop in Ap corresponds to
the jump from the pore throat to the diverging
pore body, and the nearly steady Ap thereafter
(actually, a slight decline) corresponds to
steady lamella movement to the corner at the
pore body. Another discontinuous jump from
the pore body to the pore throat begins the
steady buildup of ~p to its maximum again.
Journal of Colloid and Interface Science, Vol. !36, No. 1, April 1990
r
°i
NO. 1 /XP/P e 0
-1~
2
0
4
0
5
0
6
0
7
0
8
0
9
0
~
~z/'t'
I-!
I--I
t
o
+10
Total
~
0
3
lO
t"~l
+1
•
0
-10
FIG. 9. Capillary Ap over time for trains of 1 and 10
lamellae. K = 1.
The most striking feature of the Ap curves
for a train of 10 lamellae is the apparent lack
of action except near the pistons. Lamellae
numbers 4 to 9 are at virtually the same Ap
at all times. Even near the pistons there are
only three events where Ap jumps discontinuously. The time-average Ap for the train of
10 lamellae is 0.79, not 10 times but 18 times
that for the single-lamella case; and the variance over time of total Ap is nearly zero for
the train of 10 bubbles, while it is substantial
for the single-lamella case (Fig. 8 ).
Figure 10 makes plain that autocorrelated
lamella jumps cause the higher Ap for the 10bubble train in Fig. 9. This figure illustrates
the position of each lamella within its pore as
well as the relentless movement of the pistons.
Although plotted as though the lamellae are
in adjacent pores, bubbles in highly compressible foams are, in general, many pores long
(Eq. [ 8 ] ). Nine out of ten lamellae in this train
jump at the same time; eight out of ten lamellae reside exclusively in pore throats; and
FLOWING
FOAMS
Piston _ _ _ _ _ - - - - - - - - - - - ~
10 Lamellae
lmella NO. 10
9
8
7
6
5
4
3
aF
7
1j
p~stOe_----------~----
'L--"
T
1
T
1
=1
.=
FIG. 10. L a m e l l a p o s i t i o n s o v e r t i m e for trains o f 1 a n d
10 lamellae. K = 1.
seven out of ten reside always at virtually the
same 2xp within the throats (Fig. 9). The autocorrelation of lamella jumps in long trains
allows lamellae to spend more time in pore
throats, behaving in effect as though K were
higher.
IN POROUS
MEDIA,
II
31
Figure 11 shows how (2xp) avg varies with K
for trains of one lamella and 10, 100, and 300
lamellae (cf. Fig. 3 ). Because the set of initial
bubble volumes is a random variable and affects (2xp) avg, (2xp) avgitself varies from case to
case for trains of many lamellae: in fact, by as
much as 0.12p~ for medium-length trains (3
to 30 lamellae) with intermediate values of K
(0.03 to 0.3). In Fig. 11, we plot the expected
value of (Ap) avg averaged over enough realizations that its uncertainty is less than
0.02p e.
Figure 11 shows that at low, finite compressibility, longer trains behave as though
bubbles were more compressible. (All train
lengths have the same (2xp) avgfor K = 0, however.) For given K, as train length n increases,
(Ap) avg approaches a limit below (Ap) m a x i.e., infinitely long trains do not behave as
though they were infinitely incompressible. As
K increases, behavior approaches the asymptotic, long-train (2xp) avgat smaller values of n.
The behavior of infinite-length trains is academic, however, because in real bubble trains
bubble separation and snap-off acts periodically to truncate train length, especially at low
capillary pressures (2). When bubble separation or snap-off occurs, a liquid-filled throat
,
I (Ap)ma x
_-- I . . . .
1.0
(ZXp)avg
P~
0.5
0
0.01
i
I
i
0.1
1.
10.
K
FIG. 11. Effect o f b u b b l e - t r a i n length n a n d c o m p r e s s i b i l i t y f a c t o r K o n (2xp) avg for c o n i c a l p o r e s w i t h Pb
= 0.5, 0t = 0.1.
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
32
W I L L I A M R. R O S S E N
temporarily isolates the movement of the lamellae upstream from those downstream, severing the autocorrelation of movement of these
lamellae. (Bubble separation and snap-off do
increase (Ap)avg by other means (2).) We expect train length in real foams to be from a
few to a few tens of bubbles (2). Figure 11
shows that for some values of K these train
lengths could more than double (Ap) avg.
DISCUSSION
Compressibility significantly increases the
minimum pressure gradient for foam flow,
( ~ p ) rain if either gas compressibility or bubble
size is large (Eq. [6]). As compressibility increases, the random-walk contribution (2) to
(Vp) min decreases because a, the standard deviation of the Ap between bubbles, decreases
(Fig. 3). In addition, trains of compressible
bubbles have yet-smaller fluctuations in pressure because lamella movement is correlated
(Fig. 9). However, for the model pore shape
examined (conical pore, Pb = 0.5, Ot = 0.1 ),
the increase in the average ~p between bubbles
(Ap) ~vg more than compensates for the drop
in ~. As compressibility factor K increases
from zero to infinity (Fig. 3), (~p)avg increases
by a factor of 10, to a value five times that for
incompressible bubbles including a reasonable
estimate of the random-walk effect (2).
Estimates for Real Foams
One can estimate K for real foams from
surface tension, bubble and pore sizes, and
pressure.
For instance, for steam foams, we take 3/
= 30 d y n / c m , rt = 15 # m , p = 345 kPa (50
psi) (12), P b : 0 . 5 , P t = 0 . 1 , /)O/l)p - - 30, and
nL 2.5 cm -1. This bubble size corresponds to
bubbles about or 400 #m in diameter in bulk,
roughly the size we see exiting Berea core in
nitrogen-foam-propagation experiments in
our laboratory. If steam were an ideal gas, its
compressibility at this pressure would be 2.9
X 10 -7 ( d y n / c m 2 ) -1. However, the compressibility of pure steam in equilibrium with water
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
is infinite; a reasonable estimate with noncondensible gas present (Appendix 1 ) would be
20 times the ideal gas figure, 5.8 × 10 -6 ( d y n /
cm 2)-1. This gives a value o f K = 7, near the
high-compressibility limit in Fig. 3. Autocorrelation oflamella jumps, swelling, and drainage of liquid films and "breathing" of trapped
bubbles would further increase (•p)aVg.
Therefore, we estimate (~xp)"Vg/p~ ~ 1.25 and
(~Tp)rain ~ 1.2 M P a / m (55 psi/ft) (Eq. [1]).
Finer-textured foams would have higher
(~7p) min. Clearly, steam foams cannot flow
from the near-wellbore region unless coalescence coarsens foam texture.
For CO2 foams, we use again rt = 15 t~m
Pb = 0.5, Pt = 0.1, vO/vo ~-- 30, and n~ = 2.5
cm -~. Surface tension is lower, 3' = 5 d y n / c m
(18), however, and at typical reservoir conditions of 66°C and 13.8 MPa (2000 psi),
k ° is much smaller, at 9.7 × 10 -9 ( d y n /
cm2) -1 ((17) and Appendix 1 ). As a result,
K is roughly 0.0023, at the incompressible
limit on Fig. 3.
For nearly incompressible foams like CO2
foams, however, compressibility-like effects
are important, and these effects depend on
capillary pressure and bubble size as well as
on gas compressibility. For low capillary pressures and v°/vp = 30, Eqs. [27] and [30] indicate that the combined effects of swelling
and draining of liquid raises Keerto the range
of 1 to 3; a reasonable estimate of(Ap) avg(Fig.
3) is then 0.6p~, and (Vp) min ~ 100 k P a / m
(4.4 psi/ft). Smaller bubble sizes, e.g., v°/Vp
= 5, reduce Kerr toward the incompressible
limit, but with the finer foam texture (n~
= 15 cm -1) even the incompressible-limit
value (XTp)avg = 0.126p~ gives ( A p ) min - - 126
k P a / m (5.6 psi/ft) (Eq. [1]). Bubble separation at low capillary pressure would further
increase (XTp)min.
At high capillary pressures, the effect of liquid-filled throats is unimportant, because all
but the tightest throats have been occupied by
gas. Equation [27] and Fig. 6 indicate, however, that even at fairly high capillary pressures
(Pci = 2p~), liquid films give Kerr = 0.03 for
33
FLOWING FOAMS IN POROUS MEDIA, II
medium-textured foams (v°/Vp = 30); for
finer textured foams ( vOb/vp = 5 ) Keff is lower
(0.05), and at extremely high capillary pressures, the effect is insignificant. As shown
above, "breathing" could raise Keg for CO2
foams by a factor of about 10, to 0.02 from
the value of 0.002 based on intrinsic compressibility. All of these effects are too small
an increase to raise (Ap)"vg by themselves (Fig.
3), and moreover, they are not additive, because the higher effective compressibility due
to films reduces the "breathing" effect (Eqs.
[ 16 ] and [ 18 ]). However, in this range of values of K around 0.01, autocorrelation of lamella jumps in long trains, expected at high
capillary pressures, could raise (Ap) avg by a
factor of 1 to 2 or more above the incompressible limit (Fig. 11). For v°/vp = 30,
(Ap) avg- 0.126 to 0.25p e gives (Vp) min ~ 2142 k P a / m (0.9 to 1.8 psi/ft); for v ° / % = 5,
(XTp)rain - 125-250 k P a / m (5.6 to 11 psi/ft)
due to finer texture (Eq. [ 1]).
Our predictions for both steam and CO2
foams reflect the pore geometry assumed, i.e.,
a conical pore with Pb = 0 . 5 , Pt = 0 . 1 . A n o t h e r
reasonable model is a conical pore with Pb
= 0.5, Pt = 0.25. If one holds r~ constant between the two models, however, (At))avg is only
about 30% lower for Pt = 0.25. Our value of
rt, 15 t~m, agrees with threshold pressures
measured on a variety of Berea sandstones
with air permeabilities of 0.3 to 0.5 #m 2 (300-
1.0
500 md) (25). Unconsolidated sands presumably would have lower ( • p ) m i n due to
larger rt, and tighter rocks, higher (Vo)rain.
Fit of Model to Data of Falls et al.
For the atmospheric-pressure beadpack experiments of Falls et al. (3), K ranged from
0.055 to 0.43 (Eq. [ 6 ] ). "Breathing," however,
both predicted by theory and observed in these
experiments, would raise the effective K b y orders of magnitude to the infinite-compressibility limit and raise (Ap) avg to (Ap) max.
For highly compressible foams, only the
pore throat matters to (Ap) avg. Here we approximate the pore throat of a beadpack as a
torus with the same ratio of the radius of the
solid matrix rg to the radius o f the pore throat
rt: rt/rg = 0.155. For such a torus,
( A p ) m a x / p e c = 0.54. We replot the data from
Table 2 of Fails et al. in Fig. 12 with this new
estimate of theory, accounting for compressibility and "breathing" of bubbles alongside
the train. The new estimate of theory falls near
the middle of the widely scattered data. Thus,
our model fits these beadpack data well when
compressibility and "breathing" are taken into
account.
CONCLUSIONS
1. The compressibility of the gas phase in
foams can dramatically increase the minimum
~o
o
o
o o
(Ap) avg
o
o
__Q_ . . . .
a o
0.5
o
o~oa
J
o
0~)
Theory
IYield Pressure Dropl
| o Sustain Flow
/ aStart Flow
~
110
1'5
20
Bubble Length (cm)
FIG. 12. Fit of model for (Ap) a~g for compressible foams to data of Falls et al. (Ref. (3)).
Journal of Colloid and Interface Science, Vo]. 136,No. l, April 1990
34
WILLIAM R. ROSSEN
pressure gradient required to maintain flow of
these foams through porous media, ( ~ p ) m i n .
When gas compressibility is high, foam lamellae spend most of their time in pore
throats, where capillary resistance to flow is
greatest, and j u m p over positions where capillary forces promote foam flow.
2. Steam foams behave as highly compressible foams: low pressure and vapor-liquid
equilibrium give the gas phase high compressibility. At low capillary pressures, CO2 foams
also behave as though highly compressible due
to the swelling and drainage of liquid films,
despite the low intrinsic compressibility of
CO2. At high capillary pressures CO2 foams
behave as though slightly to moderately compressible. Reasonable parameter estimates give
values of(Vp) rain of 1.2 M P a / m , 125 k P a / m ,
and 20-40 k P a / m (55, 6, and 1-2 psi/if) for
medium-textured stream, low-capillary-pressure CO2, and high-capillary-pressure CO2
foams, respectively, in rock. Finer textured
CO2 foams have a more substantial (Ap) min.
Clearly, limited coalescence is essential for
deep foam penetration into the formation.
3. The effect of compressibility depends on
both bubble size and gas compressibility.
Large, moderately compressible bubbles, occupying many pores, behave like smaller,
highly compressible bubbles.
4. The contribution of bubble separation
t o ( ~ p ) m i n decreases for highly compressible
foams because the magnitude of the pressure
fluctuations along the train of bubbles decreases. However, the net effect of compressibility is to raise (Vp)min significantly.
5. ( V p ) rain for highly compressible foams
depends only on pore-throat geometry. In porous media of widely varied throat sizes,
(~p) minfor highly compressible foams depends
only on throats represented in the "threshold
pressure" (10) of the porous medium as a
whole.
6. Other phenomena increase the effective
compressibility of foams:
(a) Compression and decompression of
trapped-gas bubbles alongside the train can
Journal of Colloid and Interface Science,
Vol. 136,No. 1, April 1990
raise the effective compressibility of a foam by
orders of magnitude in beadpacks. The effect
on flow of foam through rock can be significant in some cases as well.
(b) At low capillary pressures, the effect
of swelling and drainage of liquid films and
liquid-filled throats can establish a substantial
effective compressibility even for foams of an
incompressible gas. At high capillary pressures,
these effects are negligible, however.
(c) In long trains of compressible bubbles, lamellae tend to stop in pore throats simultaneously, stay there longer, and jump together. This can increase (Ap) avgsubstantially
for long trains.
7. The theory of (Vp) rain of compressible
foams fits the beadpack data of Falls et al. (3)
well within the scatter of the data.
APPENDIX 1: COMPRESSIBILITY OF STEAM
AND CO2 FOAMS
The compressibility of gases in equilibrium
with liquid is complicated by the exchange of
mass with the liquid. For instance, the isothermal compressibility of pure steam in
equilibrium with pure water is infinite: changes
in system volume merely cause condensation
of steam or evaporation of water, with no
change in pressure (26). For steam foams
compressibility is limited by the concentration
of noncondensible gas (27); for CO2 foams it
is limited by the solubility of CO2 in water.
For steam foams we make assumptions
similar to those in Ref. (27). Specifically, we
assume equilibrium between an ideal gas mixture of an insoluble, noncondensible gas and
water vapor and an aqueous phase whose
properties are constant. With these assumptions, one can easily show that the compressibility of steam foam increases proportionately to the inverse of the mole fraction of the
noncondensible gas. As injected, the vapor
phase of steam foams may contain from a few
hundredths of a percent to a few percent of
noncondensible gas (27). Therefore, the mole
fraction of noncondensible gas in the steam-
FLOWING FOAMS IN POROUS MEDIA, II
swept zone is expected to be low, e.g., only a
few percent, and the increase in compressibility great.
For CO2 foams, the effect of dissolution of
CO2 depends on the mass of water in which
it can dissolve, i.e., on the rate at which CO2
can diffuse away from the bubble. Here we
simplify the rate issue by assuming, generously, that CO2 in a bubble can equilibrate
with five times the bubble volume of water
during the period of interest. By interpolating
from data in Ref. (28), we determine that at
66°C and 13.8 MPa (2000 psi), approximately
7 × 10 -6 additional g of CO2 dissolve in 1 g
water due to a pressure increase of 6.9 KPa ( 1
psi). Therefore, the contribution of dissolution
of C O 2 to kT is
k s O. =
l=
ms
z
rnl =
n
z
/7 L =
nsi =
Pl,
P2 =
kT ~ 5(7 × 10-6)/Pco2
- 7.5 × 10 -5 psi -l,
[A1]
where pco2 is the density of CO2, 0.45 g / c m 3.
This result is an order of magnitude less than
the contribution due to the compressibility of
the CO2 itself (17).
APPENDIX 2: NOMENCLATURE
K ~
Ke¢ =
Kfz
Ks=
k
z
ko=
compressibility factor in foam flow
(Eq. [6]).
effective value of K, increased by
"breathing" of side bubbles (Eq.
[ 18 ]) and by drainage of liquid
films (Eq. [25]) and of liquid-filled
throats (Eq. [ 29 ] ).
compressibility factor for effect of
liquid films (Eq. [ 26 ] ).
compressibility factor for effect of
"breathing" of side bubbles (Eq.
[161).
linear coefficient of compressibility
(Eq. [41).
conventional coefficient of compressibility of gas phase, ( - 0 In v~
Op)T, at the average pressure of the
foam.
p~=
Pci
=
e
z
Pcs0
p~ =
pi =
Psij =
Ps =
ap=
( A p ) max =
(ap)av~
=
An/avg
~/~ t tot
=
35
linear coefficient of compressibility
of trapped bubbles alongside train.
length of a pore.
total volume of side bubbles relative to volume of bubbles in train
(Eqs. [181).
average number of liquid-filled
throats adjacent to each pore in the
bubble train.
number of lamellae in a bubble
train.
foam texture: number of lamellae
per unit length in the direction of
flow.
number of lamellae in throats
alongside bubble i in the train.
for a pair of bubbles, the pressure
of the rearward and forward bubbles, respectively.
capillary entry pressure of pore
throats in the bubble train (Eq.
[71).
capillary pressure of bubble i in a
train.
capillary entry pressure of throat ij
alongside bubble train.
representative average value of
pecsij (Eq. [18]).
the pressure of bubble i in the bubble train.
pressure in trapped bubble ij
alongside bubble train.
representative average value of Ps0
(Eq. [181).
difference in pressure between two
adjacent bubbles in a train; positive
if the forward bubble is at lower
pressure (Eq. [ 3 ] ).
maximum value of £xp--occurs in
pore throat.
average capillary Ap per lamella,
not accounting for lower bound in
capillary pressure (Eq. [ 2 ] ).
average capillary Ap per lamella,
accounting for lower bound in
capillary pressure ) Eq. [ 1] ).
Journal of Colloid and Interface Science, Vol. 136, No. 1, April 1990
36
WILLIAM R. ROSSEN
(~7p) min = minimum
R
z
rb =
rf =
r1
rt =
r s ij =
rts =
rts0. z
I) 1 =
v°=
VOs =
/)p =
AVf =
Avsij =
mrs
A/)t =
X z
-y=
#=
pressure gradient to
maintain foam flow.
bead radius in a beadpack.
pore-body radius.
radius of curvature of liquid film
in pore.
radius of curvature of lamella;
positive if convex toward rearward
bubble.
pore-throat radius.
radius of the side lamella ij.
average throat radius of liquidfilled throats alongside the bubble
train.
radius of throat ij alongside bubble
train.
volume occupied by bubble 1 in
the pore of interest.
average bubble volume in bubble
train.
average volume of a trapped bubble alongside the bubble train.
volume of a pore.
the change in volume of water
films in one pore.
change in volume of bubble i due
to movement of lamella ij alongside train.
representative average value of
Avsij (Eq. [181).
change in bubble volume due to
gas intrusion into a liquid-filled
throat.
axial position of lamella attachment to the pore throat.
gas/liquid surface tension.
the mean of Ap(r); equivalent to
( A p ) avg
p b = dimensionless pore-body radius,
rb/l.
p f = dimensionless radius of curvature
of liquid film in pore,
rf/I.
p t ~- dimensionless pore-throat radius,
rt/l.
Pts = dimensionless throat radius of liq-
uid-filled throats alongside
bubble train, rts/l.
Journal of Colloidand InterfaceScience, Vol. 136,No. 1, April1990
the
a = the standard deviation of Ap(z) for
the entire train.
~-= dimensionless time: fraction of
pore transit time.
ACKNOWLEDGMENTS
We thank P. A. Gauglitz and F. Friedmann
for fruitful discussions. L. A. Young helped
accelerate the execution of the algorithm used
to prepare Fig. 11.
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