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Flow of non-newtonian fluid through porous media
S3.00+ .OO 0020-7225/85 8 1985PcrwnonPIESLtd.
Inl.J. EngngSci.Vol.23,No. 5, pp.571485,1985 Print4 in GreatBritain
FLOW
Department
OF NON-NEWTONIAN FLUID POROUS MEDIA
THROUGH
H. PASCAL and F. PASCAL of Physics, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada
Abstract-This article shows the solutions of the nonlinear equations describing the transient flow of power law fluids through a porous medium. These are obtained by means of a generalized Boltzmann transformation approach for several cases of practical interest in interpretation of wellflow tests of short duration, used currently in oil reservoir engineering for obtaining the reservoir properties.A formulationof a movingboundary problem for a short time, appearing during the production of fluids by elastic decompression, is used in deriving the exact solutions in closed form for certain cases. The presence of a decompression front is physically demonstrated and its location in the reservoir at a given time is determined. The limitations associated with the generalized Boltzmann transformation approach in solving the nonlinear equations of power law fluid flow in oil reservoirs are shown and discussed with regard to the interpretation of the wellflow test analysis. 1.
INTRODUCTION
flow of non-Newtonian fluids through porous media has recently received special attention as a result of increasing practical interest in oil reservoir engineering, where the oil displacement efficency may be improved by using nonNewtonian displacing fluids. Another reason which now justifies this interest is the production of heavy crude oils, where the rheological studies indicated that some of them are also non-Newtonian fluids of power law with yield stress. From a rheological point of view, certain displacing fluids currently used in field operations for a better mobility control in the oil displacement mechanism are of power law, in which the apparent viscosity increases with the shear rate decreasing. Obviously, from an oil reservoir engineering point of view, it is essential to have an adequate understanding of the flow behaviour of non-Newtonian fluids, in particular of the pseudo-plastic type, through a porous medium. In this way, an increased recovery efficiency from techniques based on the injection of non-Newtonian displacing fluids could be obtained. A number of studies on this matter have been recently published in the literature. Our previous article [2] on this subject has shown the rheological effects of non-Newtonian displacing fluids of power law with yield stress on the dynamics of a moving interface separating oil from water. Several relevant conclusions, obtained there, indicated the conditions in which the viscous fingering effect in oil displacement could be eliminated and a piston-like displacement may be possible. To adequately describe the flow of non-Newtonian fluids in porous media, a modified Darcy’s law is required. In another of our previous articles [ 11, this basic relationship between the flow rate and the pressure drop for power law fluids with yield stress has been shown. However, a basic requirement for understanding is knowledge of the coefficents occuring in the modified Darcy’s law for this class of non-Newtonian fluids. While for an unrealistic geometrical model, for example the capillary tube model, these coefficients may be determined and expressed in terms of rheological parameters and geometrical properties of the porous medium, in the case of real porous media the appropriate approach, as it was shown in [3], remains the determination of the coefficients from in situ measurements on the pressure or flow rate behaviour in time, recorded in wells during the transient flow of short duration. Knowledge of this behaviour may permit us the obtaining of useful information regarding the permeability of the porous medium, provided the oil is Newtonian, in which case Darcy’s law holds [4 and 51. For example, the pressure variation in time in a well producing at a constant flow rate, or flow rate variation in a well producing at a constant pressure, recorded and interpreted according to the mathematical solutions of the basic equation describing the flow through a porous THE
STEADY
and unsteady
571
572
H. PASCAL
and
F. PASCAL
medium, are two adequate techniques used now in the field for determining the reservoir properties. These well-test analysis techniques are known in oil reservoir engineering as the pressure drawdown test. In waterflooding projects, where an injection of fluids appears, rather than their production, the well test is known as pressure falloff testing. Obviously, in fact, the well-test analysis is based on the mathematical solutions of the inverse problems of flow through porous media. While for Newtonian oils the interpretation of the well-test data is relatively well understood, in the case of non-Newtonian oils this interpretation is, so far, at best uncertain. The main cause appears to be the nonlinearity of the equations describing the transient flow, where their exact solutions are unknown. In order to illustrate the rheological effects on the deviation from Newtonian flow behaviour, a classical problem arising in oil reservoir engineering was studied, in which oil is of power law with yield stress, and an approximate analytical solution has been presented in [I]. The main objective of this article is to show an exact analytical solution for the problems studied in [l] and to show how this may be used to correctly interpret the welltest data recorded during the transient flow of power law fluids. 2.
BASIC
EQUATIONS
The basic equations describing the steady and unsteady flow of power law fluids with yield stress were derived in our previous articles [ 1 and 21. According to the results obtained there, the basic equations for power law fluids in the absence of yield stress may be written as
(> aP
(‘+a*p
z
gT=na
2 ap z
(1)
for one dimensional flow and
(2) for a plane radial flow. The coefficient a2 in (1) and (2) is given by the relation a2 =
+ (7 >“‘(co4 cJ.
Once the pressure distributions are determined from previous equations, the velocity distributions may be obtained from the modified Darcy’s law and expressed as
(4) and
3.
FORMULATION
OF
PROBLEMS
AND
SOLUTIONS
The problems considered in this article will be designed by means of a pair of symbols describing the oil reservoir boundary conditions. The first symbol will describe the condition at the outface flow, i.e. in wells, while the second will describe that at the inlet end. For example, the case where the pressure at both ends of the reservoir is specified will be termed the p,,, - pe problem. The case in which the flow rates are specified at
Flow of non-Newtonian
fluid through porous media
513
both ends will be termed the QW - Q, problem, while the case where the pressure is specified at one end and the flow rate at the other end will be termed the Q,,, - pe problem. The pw - pe problem corresponds to the first boundary value problem, the Q,,, - Qe problem represents the second boundary value problem, and the Q,,, - pe problem is a mixed problem. 3.1 The p,,,-pe problem (one dimensional flow) The flow system to be analyzed is a linear oil reservoir having a closed outer boundary. It is assumed that at the initial moment t = 0 the pressure in the reservoir is constant, say pe, and for t > 0 the reservoir is depleted by a constant pressure of production, say p,,,, at the outface flow. As a result, the flow is initiated and maintained in the reservoir provided that pw < pe. This case corresponds to a reservoir producing by natural depletion, i.e. by elastic decompression, in which transient flow response appears in reservoir for t > 0. It should be pointed out that during the depletion period we have two phases concerning the reservoir behaviour from a fluid mechanics point of view. For example, by reducing the pressure at the outface flow from pe to p,,,, a decompression front will be generated in the reservoir, which requires a certain time to reach the closed boundary of the reservoir. Obviously, this time depends, as we will see further on, on the reservoir permeability, fluid compressibility, and rheological parameters of the reservoir fluids. The time interval required for the front to reach the closed boundary will be denoted by t, and referred to as the duration of the first phase. At any moment 0 < t < t, the decompression front will be located at a certain position denoted in the article by the equation x = l(t). Taking into account these notations, we can state that the pressure distribution p(x, t) in the reservoir will be, for 0 < t < t, and 0 I x I l(t), the solution of the basic eqn (1) satisfying the double inequality p,,, I p(x, t) 5 pe, while for x r l(t) it has a constant value, namely p(x, t) = pe. For t > t,, the pressure at the closed boundary starts to decline in time and after a certain time of production, termed as the depletion time, the pressure tends to equalize its value at the outface flow. This is the second phase and the corresponding pressure distribution for t > t, will be pw < p(x, t) in the range 0 < x < L, where L is the length of the reservoir. The reservoir is considered to be completely depleted when its energy is insufficient to maintain the production at an imposed economical flow rate. From an oil reservoir engineering point of view, the first phase is of particular interest in well-test analysis, since information obtained in this phase, i.e. prior to when the decompression front reaches the closed boundary, may provide the reservoir properties, which are used in predicting the reservoir performance for a long period of time. Consequently, in well-test analysis the knowledge of the flow parameters in wells for a very short period of time, i.e. for t < t,, is required. In these circumstances, the solution of eqns (1) or (2) may be determined by assuming that the reservoir is of infinite extent, in which case the following boundary and initial conditions may be used for t < t, t=O
P(X, 0) = Pe
t>o
P(W, t) = Pe;
t>o
P(0, t) = Pw
o
However, it may be shown that, for t -e t, and n < 1, the pressure distribution p(x, t) corresponding to the conditions (6) will be identical with that obtained when it is determined from consideration of a moving boundary problem, since p(x, t) = pe for x r l(t). On the other hand, the pressure p(x, t) and its derivative $
should be continuous
functions of x and t for any x > 0 and t > 0, but because p(x, t) is constant for x 2 I(t),
574
H. PASCAL
and F. PASCAL
then 2 = 0. As a result, instead of (6), we have, according to Fig. 1, the following ax X-I(I) conditions:
o
P(O,0 = Pw
(7) o
l(0) = 0
;
Specifically, the pressure distribution p(x, t) for 0 < x < l(t) is the solution of eqn (1) satisfying the conditions specified in (7). The nature of these conditions is motivated by the physical reality of the flow mechanism appearing in oil reservoirs producing by natural depletion, i.e. by elastic decompression, where for 1(t) < x < cc the pressure remains constant to its value corresponding to the initial moment t = 0. As a result, no flow occurs in the reservoir behind the decompression front. Using the transformation 7l = xt-“/(“+l)
(8)
the basic eqn (1) is reduced, for 0 -C x < l(t), to the nonlinear equation
ordinary differential
+ n*a* do* n+l
e
(9)
for 0 < T)< ql where fl
4t)
= -t”l(“+‘)
(10)
.
The boundary conditions (7) now become
Phi)
= Pei
0
and
Pm
= Pw-
(11)
These three conditions determine, along with the solution of eqn (9) the front location 1(t) and pressure distribution p(x, t) in the range 0 < x < l(t), as it will be shown further on. For a Newtonian fluid, we have 12 = 1 in (9) and in this particular case eqn (9) becomes
(12)
Flow of non-Newtonian fluid through porous media
515
and its solution may be written as follows:
Pe-P(X,O
(13)
Pe - Pw
where (14)
For n < 1, from (9) one obtains
and
from
dp
= 0, so that dv Ill=ll,
( 11) we have
Cr in (15) is expressed
as C1
,u2 1 - n = -VT. Integration of eqn (15) with condition p(0) = p,,, yields a relation for 2 1+n determining the pressure distribution in the range 0 < x < l(t). This may be written as
(16) where J,(?)
=
1”’
(1
[2)‘d”-“@;
-
O
(17)
To determine qI, one can use the condition ~(7,) = pe. As a result, from (16) we have
Pe
_
Pw
=
[!g ~~‘-‘)n(“““-~)J”(q,)
(18)
in which
s 1
Jn(m)
=
(1
_
(2)““1-$@
(19)
0
does not depend on vl. Using the following notations
s I
AP
=
Pe - Pw
and
L, =
(1 _ [2)““‘-“&,
0
(20)
from (18) one obtains q, =
[$““““[“2”’
: ; ;I”““’
=
constant
(21)
whereas from (8) I(t) is
(22)
H. PASCAL and F. PASCAL
576
Relation (22) determines the front location at any time t, since L, may be obtained from (20) exactly for certain values of 12,as it will be shown further, or approximately for any n, using some numerical integration formulas. As eqn (22) indicates, the front location depends strongly on the rheological parameters n and u*, i.e. on k/pe,-. For the capillary tube model k/pef is related to n and H by relation [3]. (23)
in which the rheological parameter H denotes the consistency index, k denotes the permeability, and 4 denotes the porosity. As is well known, for power law fluids the apparent viscosity is shear rate dependent. Consequently, the rheological effects on the front location will depend on the pressure drop Ap. We again find this dependence by the relation (22). The following relations illustrate how these effects appear on I(t), for some particular cases
c-1 ZAP
1f2t ,/4
7r
n = +;
r(t) =
3Ap 2
‘i3t ,,,
t-1
n = ;;
l(t) =
(-1 a12 2
l/3
35Ap 1i7tj,, (-1 16
3. n = 5,
l(t) =
3a* 317
*
(24)
56 c-1
For a Newtonian fluid, i.e. n = 1, from (13) we have 4
4t) =
--$ ;
a*= f
(cly$ + c/J.
CW
\/
This relation shows that the front location is independent of Ap. Once nI is determined by relation (2 l), the pressure distribution may be obtained from (16) and expressed as
P(d - Pw _ JAV) . Jn(d ’ Pe-Pw
O
(25)
91
in which J”(v), given by relation (17), may be calculated for any n and 9, using an adequate numerical integral formula, but for certain values of n an exact solution in closed form can be obtained. For example, using the transformation t = sin CYin (17) we have
(26)
Flow of non-Newtonian
fluid through porous media
511
whereas one can see that J,,(v) may be determined by means of the formulas
s l+n l-n
for
= 2k
2kn=m,k=
sin(2k-2m+ 2k-2m+
(cos a)2k+‘da = L 22k s for
or
l+n -=2k+ l-n
1,
1
1,2 ,...,
and
(27)
1)cu 1 or
n=-
k k+l’
k=l,2,3
,....
(28)
In order to illustrate the flow deviations from Newtonian behaviour, where n = 1, several illustrative examples of practical interest will be considered, namely n = 4, $, $, :, and $ For these cases, the function Jn(q) defined by (26) may be expressed as (29)
(30)
(31)
(32)
(33) From these relations for TJ= ql, we have the values of L, = Jn(vl) as: 2 L,/2 = - ; 3
LJ,J = gi ;
and
L5,, = 51
32
(34)
and according to (22) the front location. A problem of special interest in oil reservoir engineering is the prediction of the flowrate decline in time at the outface flow, i.e. at x = 0, during the natural depletion mechanism under constant pressure. Since the pressure distribution is known from (16) then the modified Darcy’s law (4) allows the knowledge of the flow-rate variation expressed in terms of variables x and t. This variation may be written as
(35) in which 71~is known from (21). At the outface flow, i.e. at x = 0, the relation (35) determines the flow-rate decline in time. 3.2 The QW - pe problem (one-dimensional flow) As mentioned earlier, in practice, instead of a constant pressure of production at the outface flow, a variable flow rate may appear there. In this case, knowledge of pressure variation in time at the outface flow is of great practical interest. As field observations ES 23:5-F
578
H. PASCAL
and
F. PASCAL
have shown, the flow rate at the outface flow declines as a continuous monotonic function of time. On the other hand, eqn (35) indicates that, for a constant pressure at x = 0, the flow-rate decline is expressed by the relation 0,
t)
=
Q.
=
ct-‘/(‘+“)
(36)
where C is a constant. As an illustrative example, we will consider the case when the flow rate at the outface flow is of the form (36), so that the following boundary conditions occur
PI@),11= Pe
(37)
The conditions (37) for eqn (9) become
P(%>
dp
= Pe;
&
and
0 9=1),
=
dp
= A = constant. G o=O
(38)
From (37), one obtains for the flow rate variation imposed at the outface flow
Q,,
=
Q(O,
t)
=
F
(39)
It should be noted that transformation (8) implies some limitations in obtaining the exact solution for nonlinear eqn (9). These appear as a result that the boundary conditions must also be expressed in terms of variable B only. For example, while for a constant pressure of production specified at the outface flow this condition is satisfied, as can be seen from (1 l), for the QW- pe problem a variable flow rate at the outface is required, as is evident from (37) and (38). Following the same procedure as with the pw - pe problem, one can obtain the solution of the basic eqn (1) satisfying the conditions specified in (37). = A so that C, , in (15), will Using the condition imposed at x = 0, we have dp dv ,,=o be C, = A(‘-“)‘“. As a result, eqn (15) becomes -dp =
A(
Bq2)“l(1-n),
1 -
dq where B
=
A’“-“l”(&
+
c,)
1/n~n( 1 - n) 2(1 +n)’
(41)
= 0, specified Now one may determine the front location Z(t) using the condition dp dv rl=n, in (38), obtaining from (40) vI = +j = constant
Flow of non-Newtonian
519
fluid through porous media
which, taking into account (42), leads to (43) Once I(t) is determined from (43) the pressure distribution integrating (40). Since P(T),) = pe, from (40) we have
P(v) = Pe - A
s ”
(1
p(n) may be known by
Bq2)“/(‘-“)dq.
_
(44)
1)
By means of the transformation nG = [, which from (42) becomes q/vi = 4, the pressure distribution may be written as follows:
0
P(v) = pe - AmJn ;
;
O
in which (46) This integral is similar to (17) so that all the comments with regard to its calculation are also valid here. In order to illustrate the flow deviations from Newtonian behaviour, the following two cases are considered: n = 4 and $. The pressure distributions obtained from (45) and (46) are expressed as: n = f;
n = L. 3,
A Pw)-Pe=$j
[
tix 1 Btix3 t’/3--y--y--~
p(x,t)-p.=$[~arcsin~+~$(l
in which B is determined
2 I
-$y-t]
(464 (46b)
from (41) for n = i and n = 4. 4. RADIAL
FLOW
In this section, the solution of the basic eqn (2) describing the radial flow in an oil reservoir for two cases of practical interest will be shown. These correspond to a centrally located well in the reservoir, considered of infinite extent, and producing at a given pressure or flow rate specified in the wells. As in previous sections, we have in this case a moving boundary problem as well, with the following conditions at the moving boundary: pU(t),
Pel = PC?;
o
l(0) = 0 in which I(t) is the front location at the moment t. For R > l(t) the pressure ap mustbep(R,t)=p,forO
(47) distribution may be seen we have to < l(t) in the
580
H. PASCAL and F. PASCAL
time interval 0 < t < t,,,, where RW denotes the well radius. At R = R, we can have two cases. The first case corresponds to a constant pressure of production, imposed in the well, while the second relates to a variable flow rate of production. As a result, at the well we have P&V, 0 = PW
(48)
or (49) These conditions provide for each case the front location l(t), as will be shown further on. Using the transformation ,, = Rt-“t(“+l)
(50)
eqn (2) may be written as
9 + n dp
dv=
t
dp
na2
-(l+nVG
II ds
(2n-‘)‘n =o
(51)
>
in which q, < TJI< ql, where r], = RJ-*‘(~+‘) and 7, = l(t)t-“‘(“+‘). The conditions (47) expressed in terms of the variable n become
P(n) = Pe
at,
and
=-- l
aR I R=I(f)
=o
&J
dq ?=l),
t”‘(“+‘)
(52)
whereas, for conditions specified in (48) and (49), we have
(53)
By means of the transformation of the Bernoulli type
$
= u, Eq. (5 1) is immediately reduced to an equation
WV0 = 0
which has the general solution u(q) =
clq”-’
-
[
(1 - n)a2 (1 +n)(3-n)“=
and therefore
dp &=y
1 c1
-
(1 - n)a2 3-n (1 + n)(3 - n) q
(54)
n/t 1-N 1 n/Cl-n) 1 *
(55)
(56)
The solving of the partial differential eqn (2) is now reduced to the solving of the differential eqn (56) with the conditions specified in (52) and (53). The last condition in = 0, determines the constant Cr and therefore
dp
1
(1 - da2
dq
q”
(1 + n)(3 - n) 1
-=-
“‘(1-n)(q3-n _ 1
q3-n)n,(,-n)
(57)
Flow of non-Newtonian
fluid through porous media
581
Integrating (57) by taking into account that P(T),) = pe, one obtains p(,,)
=
pe
_
B
s ”
s-“(,,-”
_ ,,3-n)n/(‘-n)d,,
(58)
I)
or using the transformation
C;= 1 11
fi9)
=
pe
_
s’
B,f+n)/(‘-n)
[-“(
1 -
[3-n)n/(‘-ddf
Jt1
(59)
in which (1 9w <
tl <
and
91
B =
_
n)a2
(1 + n)(3 - n)
1 * Ml-n)
(60)
Equation (58) shows that the determination of pressure distribution in the reservoir, expressed by function p(n), requires knowledge of vl. To determine nl, the two cases specified in (53) will be analyzed separately. For the pw - pe problem, the condition at the outface flow p(r),,) = pw yields the determination of ql . Since the well radius R, is very small, we may take in (59), as a first approximation, n&r1 = R,//(t) N 0. In this case, from (59) we have Ap = pe - p,,, = Bq\‘+“)‘(‘-@J,,(O)
(61)
Jn(0) = s,I E-“(1 - [3-“)““‘-“)dc;.
(62)
where
Relation (6 1) determines 9, (1-nM1+n) = constant
(63)
whereas the front location is I(t) = ~,Pfll+“). By a detailed study of the integral appearing in the previous relation, it may be seen that J,(n/a,) cannot, in general, be expressed by elementary functions, and therefore an adequate numerical integral formula should be used. However, for IZ = 4 the integrals (59) and (62) may be calculated exactly, and in this case the pressure distribution p(n) may be expressed in a closed form, as we will show further on. For the QW- pe problem, as already pointed out in the previous section, the use of transformation (50) requires a variable flow rate at the outface flow, in order to obtain for nl a constant value. In the radial how case, one obtains q1 = constant, i.e. dl becomes independent of variables R and t, if the flow rate in the well declines according to the relation Q(R,,
t) =
Q,, =
At(“-‘)/(“+I),
n-c 1.
(64)
The boundary condition at the well becomes
(65) so that from (57) taking into account that &;” may be neglected compared to r]:-“, we have for n1
582
H. PASCAL
711=
L
and F. PASCAL
= constant
B
(66)
whereas for the front location one obtains l(t) =
q,t”‘(“+‘).
(67)
The flow rate variation in a well producing at the constant pressure may be known from (57) with 7 = n,,,, in which nI is determined from (63), while the pressure in a well producing at the variable rate is known from (59) with q = nW, where n1 is determined from (66). For example, the pressure distribution in the case of constant pressure, corresponding to n = 4, is expressed as
P,-NM=
Ap[l -f$($f-)‘“+$~];
where the front location is I(t) =
n=f
W’a)
t’13; Ap = pe - pw.
The pressure distribution in the case of a variable flow rate, obtained from (59) and (66) with n = 4, is pe
_
p(R,
t)
=
1
B,,5
&f 6’5 (k) (&y[;- %(~)“‘(~)“‘”
where the front location is l(t) =
From (67a) one can observe that,
at the well, assuming R, = 0, we have p(0, t) = p,,, = constant, while from (67b) one obtains ~$0, t) = constant. Physically, this means that a constant pressure of production may be maintained in the well, provided the flow rate declines according to the relation (64). 5.
SOME
DISCUSSION
ON
LINEARIZED EQNS
(1)
AND
SOLUTIONS
OF
THE
BASIC
(2)
The basic eqns (1) and (2) describe the unsteady flow of power law fluids through porous media, in which for n < 1 the apparent viscosity increases with the shear rate decreasing. In deriving these equations, an empirical rheological equation must be used to relate the apparent viscosity to the shear rate. This is expressed as Pap = ffci)“-‘;
n
(68)
where in terms of flow velocity in porous medium + is expressed as (69) Since rheological eqns (68) and (69) are empirical, their limitations are quite evident. As the experimental observations, reported from the rheological studies, have shown, there are certain non-Newtonian fluids in which the observed and predicted data cannot be
Flow of non-Newtonian fluid through
porous media
583
adequately fitted by using relations (68) and (69) particularly at very high and very low shear rates. As the rheological effects are flow rate dependent, naturally the question arises whether or not the basic equations (1) and (2) would be valid for any values of the flow rate or pressure drop which appear currently during the flow in reservoirs. Obviously, the limitations of eqns (1) and (2) will depend strongly on the possibility of using the rheological eqns (68) and (69) in a given range of shear rate or flow rate variations expected to appear in the flow in oil reservoirs, in which the rheological behaviour could be known from laboratory measurements. As eqns (1) and (2) are nonlinear, naturally we now ask if these could be linearized in order to obtain analytically simpler relations for interpretation of the flow parameter measurements recorded in wells. The exact solutions in a closed form for eqns (1) and (2), associated with boundary and initial conditions of practical interest, have been shown in previous sections. As a result, a correct appreciation of some approaches of linearization is now possible, comparing the results obtained from the linearized solutions with those previously presented. For example, by means of the modified Darcy’s law (5), eqn (2) may be expressed as j$
+ ; g
= n(27rh)‘_” y(,g
+ c,)W”Q”-‘(R,l)
;
.
(70)
This equation suggests that by replacing Q(R, t) by QO, where Q0 is the flow rate imposed in the well, it becomes a linear equation having the variable coefficient R’-“. In this case eqn (70) may be rewritten as follows: (71) where
“? Q = MC04+ c&7) k
(” )‘-’ =
constant.
Tj--$
(72)
It is of particular interest to show that an equation similar to (7 1) may arise in conditions of flow through a porous medium for a fluid having the variable viscosity, with regard to the radial distance R, expressed as 1-n
(73) in which II,,,is the viscosity corresponding to R,. In this situation, the following equation for pressure distribution appears (74) Comparing eqns (71) and (74) we can see that they have the same form, so that their mathematical solutions will be similar. It should be noted that the radial heat conduction through nonhomogeneous solids, with the thermal properties varying with position R, also leads to equations similar to (71) or (74); see Carslaw and Jaeger [7]. The exact solutions in closed form for equations of the form (7 1) or (74) have been found by these authors and Luikov [6]. For details on this matter the reader is referred to [6 and 71. Consequently, the linearization based on an assumption of constant flow rate, i.e. Q(R, t) = Q. in (70) may permit us the use of the analytical solutions indicated by Carslaw and Jaeger [7], in which coefficients corresponding to eqn (71) must be replaced. However, we should recognize that these solutions are linearized approximations of the nonlinear eqn (70) which have recently been pointed out and discussed in [8] with regard to the
584
H. PASCAL
and
F. PASCAL
interpretation of the transient pressure response in wells, in which a constant flow rate is maintained. Naturally, the problem arises whether or not the linearized solutions, based on a constant flow rate approximation, could correctly include the rheological effects of power law fluids which, as already mentioned, are flow rate dependent according to relation (70). The flow rate Q(R, t) in (70) is a monotonic decreasing function with regard to the radial distance R, so that we have Q(R, t) < Qo for R, < R < l(t). Physically, this means that the rheological effects appearing in the flow through a porous medium will strongly depend on the flow rate profile in the reservoir, i.e. on function Q(R, t) in zone R, -c R < l(t). Obviously, for power law fluids of the pseudo-plastic type, where n < 1, these effects become more significant where flow rate is smaller, particularly at a greater distance from injection or production wells. From the above discussion, it is evident that approximate analytical solutions derived from linearized eqn (71) are questionable in obtaining the conclusive interpretations of the well-flow test for non-Newtonian fluids of power law. From this point of view, the problem remains the finding of the conditions of the validity of the approximate solutions. More specifically, we are interested in accurately knowing the limitations of the analytical solutions derived from linearized eqn (7 I), for various conditions of flow and rheological parameters which currently appear in practice. For example, eqn (57) indicates how the function Q(R, t) in (70) depends on variables R and t, as well as the rheological parameter n. Obviously, for the case when instead of a flow rate imposed at the well, a pressure is imposed there, the linearization of eqn (70) is no longer possible, because in this case Q,-, is unknown. 6.
DISCUSSION
OF
RESULTS
The generalized Boltzman transformation, expressed by relations (8) or (50) may be used in finding the solutions of the nonlinear equations for power law fluids associated with the interpretation of the well-flow test of short duration. The presence of a moving boundary problem occurring in the flow-test analysis of short duration may allow us to obtain the exact analytical solutions in certain cases of practical interest. For example, the pressure distribution and flow-rate decline in a well producing at a constant pressure may be exactly known. However, some limitations associated with the generalized Boltzman transformation approach arise when the boundary condition imposed at the well is expressed in terms of flow rate rather than of a constant pressure. In this case, the solution in the closed form could be obtained only for a certain profile of variable flow rate in the well. This basic requirement appears as a result that 71~must be independent of time, i.e. we should have v1 = constant. While this condition may be satisfied for a boundary condition at the well expressed by a constant pressure, for a constant flow rate the parameter q1 becomes time dependent. In order to obtain q1 time independent for the case when a flow rate is given in the well, this imposed flow rate should have a certain decline in time, as shown in the article. Although the generalized Boltzman transformation fails to solve the case of constant flow rate, it is evident that this approach may permit us to find solutions for a better interpretation of the well-flow test for non-Newtonian fluids, since the need for linearization of the basic equations is no longer required. Another significant advantage of this approach remains the possibility to obtain the solution in closed form for nonlinear equations describing the flow, where the supposition principle cannot be applied. While for a linear equation this principle provides the solution corresponding to the variable rate, provided the solution at the constant rate is known, in the case of nonlinear equations, for example eqns (1) and (2), the Boltzman transformation approach may give an exact solution for certain cases of variable rate. The formulation of a moving boundary problem associated with nonlinear eqns (1) and (2) appears naturally from eqns (15) and (56), where for n < 1 their integration requires a finite interval of variation of r], which leads to the location of a time depending boundary. REFERENCES [I] H. PASCAL, Int. J. Numerical and Analytical Methods in Geomechanics 7(3), 289 (1983). [2] H. PASCAL, Dynamics of moving interface in porous media for power law fluids with yield stress. Int. J. Engng Sci. 22(5), 577 (1984).
Flow of non-Newtonian fluid through porous media
585
[3] H. PASCAL, Some problems related to the quantitative evaluation of physical properties of the porous medium from flow tests with non-Newtonian fluids. Znt. J. Engng Sci. 23(3), 307 (1985). [4] S. C. MATTHEWS and G. D. RUSSEL, Pressure buildup and flow tests in well. Monograph Series Vol. 1, Sot. Petroleum Eng. (1967). [5] R. C. EARLOUGHES, Jr., Advances in well test analysis. Monograph Series, Sot. Petroleum Eng. (1977). [6] A. LUIKOV, Analytical Heat Diffusion Theory. Academic Press, New York (1968). [7] H. S. CARSLAW and J. C. JAEGER, Conduction of Heat in Solids. Oxford University Press, Oxford (1959). [8] C. IKOKU and H. J. RAMEY, J. Energy Resources Tech. 104, 149 (June 1982). (Received 22 May 1984)
APPENDIX In order to justify the relation (73) one can consider the case when a power law fluid is injected in the well at a constant flow rate. From the continuity equation for a steady flow we have Q = 2rhRv
= constant
(A.1)
or
Q
v=2*hR. On the other hand, for a power law fluid the shear rate is expressed in terms of velocity [2] 3n+l
v (A.3)
‘=y= whereas taking into account the relation between the apparent viscosity p, and shear rate i one has Lb=*.
H
(A.4)
The previous relations (A.l-A.4) lead to the relation (73), which indicates that the apparent viscosity for a pseudo-plastic fluid, i.e. n < 1, is an increasing monotonic function with the radius increasing, provided that a steady flow in the reservoir exists.
Inl.J. EngngSci.Vol.23,No. 5, pp.571485,1985 Print4 in GreatBritain
FLOW
Department
OF NON-NEWTONIAN FLUID POROUS MEDIA
THROUGH
H. PASCAL and F. PASCAL of Physics, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada
Abstract-This article shows the solutions of the nonlinear equations describing the transient flow of power law fluids through a porous medium. These are obtained by means of a generalized Boltzmann transformation approach for several cases of practical interest in interpretation of wellflow tests of short duration, used currently in oil reservoir engineering for obtaining the reservoir properties.A formulationof a movingboundary problem for a short time, appearing during the production of fluids by elastic decompression, is used in deriving the exact solutions in closed form for certain cases. The presence of a decompression front is physically demonstrated and its location in the reservoir at a given time is determined. The limitations associated with the generalized Boltzmann transformation approach in solving the nonlinear equations of power law fluid flow in oil reservoirs are shown and discussed with regard to the interpretation of the wellflow test analysis. 1.
INTRODUCTION
flow of non-Newtonian fluids through porous media has recently received special attention as a result of increasing practical interest in oil reservoir engineering, where the oil displacement efficency may be improved by using nonNewtonian displacing fluids. Another reason which now justifies this interest is the production of heavy crude oils, where the rheological studies indicated that some of them are also non-Newtonian fluids of power law with yield stress. From a rheological point of view, certain displacing fluids currently used in field operations for a better mobility control in the oil displacement mechanism are of power law, in which the apparent viscosity increases with the shear rate decreasing. Obviously, from an oil reservoir engineering point of view, it is essential to have an adequate understanding of the flow behaviour of non-Newtonian fluids, in particular of the pseudo-plastic type, through a porous medium. In this way, an increased recovery efficiency from techniques based on the injection of non-Newtonian displacing fluids could be obtained. A number of studies on this matter have been recently published in the literature. Our previous article [2] on this subject has shown the rheological effects of non-Newtonian displacing fluids of power law with yield stress on the dynamics of a moving interface separating oil from water. Several relevant conclusions, obtained there, indicated the conditions in which the viscous fingering effect in oil displacement could be eliminated and a piston-like displacement may be possible. To adequately describe the flow of non-Newtonian fluids in porous media, a modified Darcy’s law is required. In another of our previous articles [ 11, this basic relationship between the flow rate and the pressure drop for power law fluids with yield stress has been shown. However, a basic requirement for understanding is knowledge of the coefficents occuring in the modified Darcy’s law for this class of non-Newtonian fluids. While for an unrealistic geometrical model, for example the capillary tube model, these coefficients may be determined and expressed in terms of rheological parameters and geometrical properties of the porous medium, in the case of real porous media the appropriate approach, as it was shown in [3], remains the determination of the coefficients from in situ measurements on the pressure or flow rate behaviour in time, recorded in wells during the transient flow of short duration. Knowledge of this behaviour may permit us the obtaining of useful information regarding the permeability of the porous medium, provided the oil is Newtonian, in which case Darcy’s law holds [4 and 51. For example, the pressure variation in time in a well producing at a constant flow rate, or flow rate variation in a well producing at a constant pressure, recorded and interpreted according to the mathematical solutions of the basic equation describing the flow through a porous THE
STEADY
and unsteady
571
572
H. PASCAL
and
F. PASCAL
medium, are two adequate techniques used now in the field for determining the reservoir properties. These well-test analysis techniques are known in oil reservoir engineering as the pressure drawdown test. In waterflooding projects, where an injection of fluids appears, rather than their production, the well test is known as pressure falloff testing. Obviously, in fact, the well-test analysis is based on the mathematical solutions of the inverse problems of flow through porous media. While for Newtonian oils the interpretation of the well-test data is relatively well understood, in the case of non-Newtonian oils this interpretation is, so far, at best uncertain. The main cause appears to be the nonlinearity of the equations describing the transient flow, where their exact solutions are unknown. In order to illustrate the rheological effects on the deviation from Newtonian flow behaviour, a classical problem arising in oil reservoir engineering was studied, in which oil is of power law with yield stress, and an approximate analytical solution has been presented in [I]. The main objective of this article is to show an exact analytical solution for the problems studied in [l] and to show how this may be used to correctly interpret the welltest data recorded during the transient flow of power law fluids. 2.
BASIC
EQUATIONS
The basic equations describing the steady and unsteady flow of power law fluids with yield stress were derived in our previous articles [ 1 and 21. According to the results obtained there, the basic equations for power law fluids in the absence of yield stress may be written as
(> aP
(‘+a*p
z
gT=na
2 ap z
(1)
for one dimensional flow and
(2) for a plane radial flow. The coefficient a2 in (1) and (2) is given by the relation a2 =
+ (7 >“‘(co4 cJ.
Once the pressure distributions are determined from previous equations, the velocity distributions may be obtained from the modified Darcy’s law and expressed as
(4) and
3.
FORMULATION
OF
PROBLEMS
AND
SOLUTIONS
The problems considered in this article will be designed by means of a pair of symbols describing the oil reservoir boundary conditions. The first symbol will describe the condition at the outface flow, i.e. in wells, while the second will describe that at the inlet end. For example, the case where the pressure at both ends of the reservoir is specified will be termed the p,,, - pe problem. The case in which the flow rates are specified at
Flow of non-Newtonian
fluid through porous media
513
both ends will be termed the QW - Q, problem, while the case where the pressure is specified at one end and the flow rate at the other end will be termed the Q,,, - pe problem. The pw - pe problem corresponds to the first boundary value problem, the Q,,, - Qe problem represents the second boundary value problem, and the Q,,, - pe problem is a mixed problem. 3.1 The p,,,-pe problem (one dimensional flow) The flow system to be analyzed is a linear oil reservoir having a closed outer boundary. It is assumed that at the initial moment t = 0 the pressure in the reservoir is constant, say pe, and for t > 0 the reservoir is depleted by a constant pressure of production, say p,,,, at the outface flow. As a result, the flow is initiated and maintained in the reservoir provided that pw < pe. This case corresponds to a reservoir producing by natural depletion, i.e. by elastic decompression, in which transient flow response appears in reservoir for t > 0. It should be pointed out that during the depletion period we have two phases concerning the reservoir behaviour from a fluid mechanics point of view. For example, by reducing the pressure at the outface flow from pe to p,,,, a decompression front will be generated in the reservoir, which requires a certain time to reach the closed boundary of the reservoir. Obviously, this time depends, as we will see further on, on the reservoir permeability, fluid compressibility, and rheological parameters of the reservoir fluids. The time interval required for the front to reach the closed boundary will be denoted by t, and referred to as the duration of the first phase. At any moment 0 < t < t, the decompression front will be located at a certain position denoted in the article by the equation x = l(t). Taking into account these notations, we can state that the pressure distribution p(x, t) in the reservoir will be, for 0 < t < t, and 0 I x I l(t), the solution of the basic eqn (1) satisfying the double inequality p,,, I p(x, t) 5 pe, while for x r l(t) it has a constant value, namely p(x, t) = pe. For t > t,, the pressure at the closed boundary starts to decline in time and after a certain time of production, termed as the depletion time, the pressure tends to equalize its value at the outface flow. This is the second phase and the corresponding pressure distribution for t > t, will be pw < p(x, t) in the range 0 < x < L, where L is the length of the reservoir. The reservoir is considered to be completely depleted when its energy is insufficient to maintain the production at an imposed economical flow rate. From an oil reservoir engineering point of view, the first phase is of particular interest in well-test analysis, since information obtained in this phase, i.e. prior to when the decompression front reaches the closed boundary, may provide the reservoir properties, which are used in predicting the reservoir performance for a long period of time. Consequently, in well-test analysis the knowledge of the flow parameters in wells for a very short period of time, i.e. for t < t,, is required. In these circumstances, the solution of eqns (1) or (2) may be determined by assuming that the reservoir is of infinite extent, in which case the following boundary and initial conditions may be used for t < t, t=O
P(X, 0) = Pe
t>o
P(W, t) = Pe;
t>o
P(0, t) = Pw
o
However, it may be shown that, for t -e t, and n < 1, the pressure distribution p(x, t) corresponding to the conditions (6) will be identical with that obtained when it is determined from consideration of a moving boundary problem, since p(x, t) = pe for x r l(t). On the other hand, the pressure p(x, t) and its derivative $
should be continuous
functions of x and t for any x > 0 and t > 0, but because p(x, t) is constant for x 2 I(t),
574
H. PASCAL
and F. PASCAL
then 2 = 0. As a result, instead of (6), we have, according to Fig. 1, the following ax X-I(I) conditions:
o
P(O,0 = Pw
(7) o
l(0) = 0
;
Specifically, the pressure distribution p(x, t) for 0 < x < l(t) is the solution of eqn (1) satisfying the conditions specified in (7). The nature of these conditions is motivated by the physical reality of the flow mechanism appearing in oil reservoirs producing by natural depletion, i.e. by elastic decompression, where for 1(t) < x < cc the pressure remains constant to its value corresponding to the initial moment t = 0. As a result, no flow occurs in the reservoir behind the decompression front. Using the transformation 7l = xt-“/(“+l)
(8)
the basic eqn (1) is reduced, for 0 -C x < l(t), to the nonlinear equation
ordinary differential
+ n*a* do* n+l
e
(9)
for 0 < T)< ql where fl
4t)
= -t”l(“+‘)
(10)
.
The boundary conditions (7) now become
Phi)
= Pei
0
and
Pm
= Pw-
(11)
These three conditions determine, along with the solution of eqn (9) the front location 1(t) and pressure distribution p(x, t) in the range 0 < x < l(t), as it will be shown further on. For a Newtonian fluid, we have 12 = 1 in (9) and in this particular case eqn (9) becomes
(12)
Flow of non-Newtonian fluid through porous media
515
and its solution may be written as follows:
Pe-P(X,O
(13)
Pe - Pw
where (14)
For n < 1, from (9) one obtains
and
from
dp
= 0, so that dv Ill=ll,
( 11) we have
Cr in (15) is expressed
as C1
,u2 1 - n = -VT. Integration of eqn (15) with condition p(0) = p,,, yields a relation for 2 1+n determining the pressure distribution in the range 0 < x < l(t). This may be written as
(16) where J,(?)
=
1”’
(1
[2)‘d”-“@;
-
O
(17)
To determine qI, one can use the condition ~(7,) = pe. As a result, from (16) we have
Pe
_
Pw
=
[!g ~~‘-‘)n(“““-~)J”(q,)
(18)
in which
s 1
Jn(m)
=
(1
_
(2)““1-$@
(19)
0
does not depend on vl. Using the following notations
s I
AP
=
Pe - Pw
and
L, =
(1 _ [2)““‘-“&,
0
(20)
from (18) one obtains q, =
[$““““[“2”’
: ; ;I”““’
=
constant
(21)
whereas from (8) I(t) is
(22)
H. PASCAL and F. PASCAL
576
Relation (22) determines the front location at any time t, since L, may be obtained from (20) exactly for certain values of 12,as it will be shown further, or approximately for any n, using some numerical integration formulas. As eqn (22) indicates, the front location depends strongly on the rheological parameters n and u*, i.e. on k/pe,-. For the capillary tube model k/pef is related to n and H by relation [3]. (23)
in which the rheological parameter H denotes the consistency index, k denotes the permeability, and 4 denotes the porosity. As is well known, for power law fluids the apparent viscosity is shear rate dependent. Consequently, the rheological effects on the front location will depend on the pressure drop Ap. We again find this dependence by the relation (22). The following relations illustrate how these effects appear on I(t), for some particular cases
c-1 ZAP
1f2t ,/4
7r
n = +;
r(t) =
3Ap 2
‘i3t ,,,
t-1
n = ;;
l(t) =
(-1 a12 2
l/3
35Ap 1i7tj,, (-1 16
3. n = 5,
l(t) =
3a* 317
*
(24)
56 c-1
For a Newtonian fluid, i.e. n = 1, from (13) we have 4
4t) =
--$ ;
a*= f
(cly$ + c/J.
CW
\/
This relation shows that the front location is independent of Ap. Once nI is determined by relation (2 l), the pressure distribution may be obtained from (16) and expressed as
P(d - Pw _ JAV) . Jn(d ’ Pe-Pw
O
(25)
91
in which J”(v), given by relation (17), may be calculated for any n and 9, using an adequate numerical integral formula, but for certain values of n an exact solution in closed form can be obtained. For example, using the transformation t = sin CYin (17) we have
(26)
Flow of non-Newtonian
fluid through porous media
511
whereas one can see that J,,(v) may be determined by means of the formulas
s l+n l-n
for
= 2k
2kn=m,k=
sin(2k-2m+ 2k-2m+
(cos a)2k+‘da = L 22k s for
or
l+n -=2k+ l-n
1,
1
1,2 ,...,
and
(27)
1)cu 1 or
n=-
k k+l’
k=l,2,3
,....
(28)
In order to illustrate the flow deviations from Newtonian behaviour, where n = 1, several illustrative examples of practical interest will be considered, namely n = 4, $, $, :, and $ For these cases, the function Jn(q) defined by (26) may be expressed as (29)
(30)
(31)
(32)
(33) From these relations for TJ= ql, we have the values of L, = Jn(vl) as: 2 L,/2 = - ; 3
LJ,J = gi ;
and
L5,, = 51
32
(34)
and according to (22) the front location. A problem of special interest in oil reservoir engineering is the prediction of the flowrate decline in time at the outface flow, i.e. at x = 0, during the natural depletion mechanism under constant pressure. Since the pressure distribution is known from (16) then the modified Darcy’s law (4) allows the knowledge of the flow-rate variation expressed in terms of variables x and t. This variation may be written as
(35) in which 71~is known from (21). At the outface flow, i.e. at x = 0, the relation (35) determines the flow-rate decline in time. 3.2 The QW - pe problem (one-dimensional flow) As mentioned earlier, in practice, instead of a constant pressure of production at the outface flow, a variable flow rate may appear there. In this case, knowledge of pressure variation in time at the outface flow is of great practical interest. As field observations ES 23:5-F
578
H. PASCAL
and
F. PASCAL
have shown, the flow rate at the outface flow declines as a continuous monotonic function of time. On the other hand, eqn (35) indicates that, for a constant pressure at x = 0, the flow-rate decline is expressed by the relation 0,
t)
=
Q.
=
ct-‘/(‘+“)
(36)
where C is a constant. As an illustrative example, we will consider the case when the flow rate at the outface flow is of the form (36), so that the following boundary conditions occur
PI@),11= Pe
(37)
The conditions (37) for eqn (9) become
P(%>
dp
= Pe;
&
and
0 9=1),
=
dp
= A = constant. G o=O
(38)
From (37), one obtains for the flow rate variation imposed at the outface flow
Q,,
=
Q(O,
t)
=
F
(39)
It should be noted that transformation (8) implies some limitations in obtaining the exact solution for nonlinear eqn (9). These appear as a result that the boundary conditions must also be expressed in terms of variable B only. For example, while for a constant pressure of production specified at the outface flow this condition is satisfied, as can be seen from (1 l), for the QW- pe problem a variable flow rate at the outface is required, as is evident from (37) and (38). Following the same procedure as with the pw - pe problem, one can obtain the solution of the basic eqn (1) satisfying the conditions specified in (37). = A so that C, , in (15), will Using the condition imposed at x = 0, we have dp dv ,,=o be C, = A(‘-“)‘“. As a result, eqn (15) becomes -dp =
A(
Bq2)“l(1-n),
1 -
dq where B
=
A’“-“l”(&
+
c,)
1/n~n( 1 - n) 2(1 +n)’
(41)
= 0, specified Now one may determine the front location Z(t) using the condition dp dv rl=n, in (38), obtaining from (40) vI = +j = constant
Flow of non-Newtonian
519
fluid through porous media
which, taking into account (42), leads to (43) Once I(t) is determined from (43) the pressure distribution integrating (40). Since P(T),) = pe, from (40) we have
P(v) = Pe - A
s ”
(1
p(n) may be known by
Bq2)“/(‘-“)dq.
_
(44)
1)
By means of the transformation nG = [, which from (42) becomes q/vi = 4, the pressure distribution may be written as follows:
0
P(v) = pe - AmJn ;
;
O
in which (46) This integral is similar to (17) so that all the comments with regard to its calculation are also valid here. In order to illustrate the flow deviations from Newtonian behaviour, the following two cases are considered: n = 4 and $. The pressure distributions obtained from (45) and (46) are expressed as: n = f;
n = L. 3,
A Pw)-Pe=$j
[
tix 1 Btix3 t’/3--y--y--~
p(x,t)-p.=$[~arcsin~+~$(l
in which B is determined
2 I
-$y-t]
(464 (46b)
from (41) for n = i and n = 4. 4. RADIAL
FLOW
In this section, the solution of the basic eqn (2) describing the radial flow in an oil reservoir for two cases of practical interest will be shown. These correspond to a centrally located well in the reservoir, considered of infinite extent, and producing at a given pressure or flow rate specified in the wells. As in previous sections, we have in this case a moving boundary problem as well, with the following conditions at the moving boundary: pU(t),
Pel = PC?;
o
l(0) = 0 in which I(t) is the front location at the moment t. For R > l(t) the pressure ap mustbep(R,t)=p,forO
(47) distribution may be seen we have to < l(t) in the
580
H. PASCAL and F. PASCAL
time interval 0 < t < t,,,, where RW denotes the well radius. At R = R, we can have two cases. The first case corresponds to a constant pressure of production, imposed in the well, while the second relates to a variable flow rate of production. As a result, at the well we have P&V, 0 = PW
(48)
or (49) These conditions provide for each case the front location l(t), as will be shown further on. Using the transformation ,, = Rt-“t(“+l)
(50)
eqn (2) may be written as
9 + n dp
dv=
t
dp
na2
-(l+nVG
II ds
(2n-‘)‘n =o
(51)
>
in which q, < TJI< ql, where r], = RJ-*‘(~+‘) and 7, = l(t)t-“‘(“+‘). The conditions (47) expressed in terms of the variable n become
P(n) = Pe
at,
and
=-- l
aR I R=I(f)
=o
&J
dq ?=l),
t”‘(“+‘)
(52)
whereas, for conditions specified in (48) and (49), we have
(53)
By means of the transformation of the Bernoulli type
$
= u, Eq. (5 1) is immediately reduced to an equation
WV0 = 0
which has the general solution u(q) =
clq”-’
-
[
(1 - n)a2 (1 +n)(3-n)“=
and therefore
dp &=y
1 c1
-
(1 - n)a2 3-n (1 + n)(3 - n) q
(54)
n/t 1-N 1 n/Cl-n) 1 *
(55)
(56)
The solving of the partial differential eqn (2) is now reduced to the solving of the differential eqn (56) with the conditions specified in (52) and (53). The last condition in = 0, determines the constant Cr and therefore
dp
1
(1 - da2
dq
q”
(1 + n)(3 - n) 1
-=-
“‘(1-n)(q3-n _ 1
q3-n)n,(,-n)
(57)
Flow of non-Newtonian
fluid through porous media
581
Integrating (57) by taking into account that P(T),) = pe, one obtains p(,,)
=
pe
_
B
s ”
s-“(,,-”
_ ,,3-n)n/(‘-n)d,,
(58)
I)
or using the transformation
C;= 1 11
fi9)
=
pe
_
s’
B,f+n)/(‘-n)
[-“(
1 -
[3-n)n/(‘-ddf
Jt1
(59)
in which (1 9w <
tl <
and
91
B =
_
n)a2
(1 + n)(3 - n)
1 * Ml-n)
(60)
Equation (58) shows that the determination of pressure distribution in the reservoir, expressed by function p(n), requires knowledge of vl. To determine nl, the two cases specified in (53) will be analyzed separately. For the pw - pe problem, the condition at the outface flow p(r),,) = pw yields the determination of ql . Since the well radius R, is very small, we may take in (59), as a first approximation, n&r1 = R,//(t) N 0. In this case, from (59) we have Ap = pe - p,,, = Bq\‘+“)‘(‘-@J,,(O)
(61)
Jn(0) = s,I E-“(1 - [3-“)““‘-“)dc;.
(62)
where
Relation (6 1) determines 9, (1-nM1+n) = constant
(63)
whereas the front location is I(t) = ~,Pfll+“). By a detailed study of the integral appearing in the previous relation, it may be seen that J,(n/a,) cannot, in general, be expressed by elementary functions, and therefore an adequate numerical integral formula should be used. However, for IZ = 4 the integrals (59) and (62) may be calculated exactly, and in this case the pressure distribution p(n) may be expressed in a closed form, as we will show further on. For the QW- pe problem, as already pointed out in the previous section, the use of transformation (50) requires a variable flow rate at the outface flow, in order to obtain for nl a constant value. In the radial how case, one obtains q1 = constant, i.e. dl becomes independent of variables R and t, if the flow rate in the well declines according to the relation Q(R,,
t) =
Q,, =
At(“-‘)/(“+I),
n-c 1.
(64)
The boundary condition at the well becomes
(65) so that from (57) taking into account that &;” may be neglected compared to r]:-“, we have for n1
582
H. PASCAL
711=
L
and F. PASCAL
= constant
B
(66)
whereas for the front location one obtains l(t) =
q,t”‘(“+‘).
(67)
The flow rate variation in a well producing at the constant pressure may be known from (57) with 7 = n,,,, in which nI is determined from (63), while the pressure in a well producing at the variable rate is known from (59) with q = nW, where n1 is determined from (66). For example, the pressure distribution in the case of constant pressure, corresponding to n = 4, is expressed as
P,-NM=
Ap[l -f$($f-)‘“+$~];
where the front location is I(t) =
n=f
W’a)
t’13; Ap = pe - pw.
The pressure distribution in the case of a variable flow rate, obtained from (59) and (66) with n = 4, is pe
_
p(R,
t)
=
1
B,,5
&f 6’5 (k) (&y[;- %(~)“‘(~)“‘”
where the front location is l(t) =
From (67a) one can observe that,
at the well, assuming R, = 0, we have p(0, t) = p,,, = constant, while from (67b) one obtains ~$0, t) = constant. Physically, this means that a constant pressure of production may be maintained in the well, provided the flow rate declines according to the relation (64). 5.
SOME
DISCUSSION
ON
LINEARIZED EQNS
(1)
AND
SOLUTIONS
OF
THE
BASIC
(2)
The basic eqns (1) and (2) describe the unsteady flow of power law fluids through porous media, in which for n < 1 the apparent viscosity increases with the shear rate decreasing. In deriving these equations, an empirical rheological equation must be used to relate the apparent viscosity to the shear rate. This is expressed as Pap = ffci)“-‘;
n
(68)
where in terms of flow velocity in porous medium + is expressed as (69) Since rheological eqns (68) and (69) are empirical, their limitations are quite evident. As the experimental observations, reported from the rheological studies, have shown, there are certain non-Newtonian fluids in which the observed and predicted data cannot be
Flow of non-Newtonian fluid through
porous media
583
adequately fitted by using relations (68) and (69) particularly at very high and very low shear rates. As the rheological effects are flow rate dependent, naturally the question arises whether or not the basic equations (1) and (2) would be valid for any values of the flow rate or pressure drop which appear currently during the flow in reservoirs. Obviously, the limitations of eqns (1) and (2) will depend strongly on the possibility of using the rheological eqns (68) and (69) in a given range of shear rate or flow rate variations expected to appear in the flow in oil reservoirs, in which the rheological behaviour could be known from laboratory measurements. As eqns (1) and (2) are nonlinear, naturally we now ask if these could be linearized in order to obtain analytically simpler relations for interpretation of the flow parameter measurements recorded in wells. The exact solutions in a closed form for eqns (1) and (2), associated with boundary and initial conditions of practical interest, have been shown in previous sections. As a result, a correct appreciation of some approaches of linearization is now possible, comparing the results obtained from the linearized solutions with those previously presented. For example, by means of the modified Darcy’s law (5), eqn (2) may be expressed as j$
+ ; g
= n(27rh)‘_” y(,g
+ c,)W”Q”-‘(R,l)
;
.
(70)
This equation suggests that by replacing Q(R, t) by QO, where Q0 is the flow rate imposed in the well, it becomes a linear equation having the variable coefficient R’-“. In this case eqn (70) may be rewritten as follows: (71) where
“? Q = MC04+ c&7) k
(” )‘-’ =
constant.
Tj--$
(72)
It is of particular interest to show that an equation similar to (7 1) may arise in conditions of flow through a porous medium for a fluid having the variable viscosity, with regard to the radial distance R, expressed as 1-n
(73) in which II,,,is the viscosity corresponding to R,. In this situation, the following equation for pressure distribution appears (74) Comparing eqns (71) and (74) we can see that they have the same form, so that their mathematical solutions will be similar. It should be noted that the radial heat conduction through nonhomogeneous solids, with the thermal properties varying with position R, also leads to equations similar to (71) or (74); see Carslaw and Jaeger [7]. The exact solutions in closed form for equations of the form (7 1) or (74) have been found by these authors and Luikov [6]. For details on this matter the reader is referred to [6 and 71. Consequently, the linearization based on an assumption of constant flow rate, i.e. Q(R, t) = Q. in (70) may permit us the use of the analytical solutions indicated by Carslaw and Jaeger [7], in which coefficients corresponding to eqn (71) must be replaced. However, we should recognize that these solutions are linearized approximations of the nonlinear eqn (70) which have recently been pointed out and discussed in [8] with regard to the
584
H. PASCAL
and
F. PASCAL
interpretation of the transient pressure response in wells, in which a constant flow rate is maintained. Naturally, the problem arises whether or not the linearized solutions, based on a constant flow rate approximation, could correctly include the rheological effects of power law fluids which, as already mentioned, are flow rate dependent according to relation (70). The flow rate Q(R, t) in (70) is a monotonic decreasing function with regard to the radial distance R, so that we have Q(R, t) < Qo for R, < R < l(t). Physically, this means that the rheological effects appearing in the flow through a porous medium will strongly depend on the flow rate profile in the reservoir, i.e. on function Q(R, t) in zone R, -c R < l(t). Obviously, for power law fluids of the pseudo-plastic type, where n < 1, these effects become more significant where flow rate is smaller, particularly at a greater distance from injection or production wells. From the above discussion, it is evident that approximate analytical solutions derived from linearized eqn (71) are questionable in obtaining the conclusive interpretations of the well-flow test for non-Newtonian fluids of power law. From this point of view, the problem remains the finding of the conditions of the validity of the approximate solutions. More specifically, we are interested in accurately knowing the limitations of the analytical solutions derived from linearized eqn (7 I), for various conditions of flow and rheological parameters which currently appear in practice. For example, eqn (57) indicates how the function Q(R, t) in (70) depends on variables R and t, as well as the rheological parameter n. Obviously, for the case when instead of a flow rate imposed at the well, a pressure is imposed there, the linearization of eqn (70) is no longer possible, because in this case Q,-, is unknown. 6.
DISCUSSION
OF
RESULTS
The generalized Boltzman transformation, expressed by relations (8) or (50) may be used in finding the solutions of the nonlinear equations for power law fluids associated with the interpretation of the well-flow test of short duration. The presence of a moving boundary problem occurring in the flow-test analysis of short duration may allow us to obtain the exact analytical solutions in certain cases of practical interest. For example, the pressure distribution and flow-rate decline in a well producing at a constant pressure may be exactly known. However, some limitations associated with the generalized Boltzman transformation approach arise when the boundary condition imposed at the well is expressed in terms of flow rate rather than of a constant pressure. In this case, the solution in the closed form could be obtained only for a certain profile of variable flow rate in the well. This basic requirement appears as a result that 71~must be independent of time, i.e. we should have v1 = constant. While this condition may be satisfied for a boundary condition at the well expressed by a constant pressure, for a constant flow rate the parameter q1 becomes time dependent. In order to obtain q1 time independent for the case when a flow rate is given in the well, this imposed flow rate should have a certain decline in time, as shown in the article. Although the generalized Boltzman transformation fails to solve the case of constant flow rate, it is evident that this approach may permit us to find solutions for a better interpretation of the well-flow test for non-Newtonian fluids, since the need for linearization of the basic equations is no longer required. Another significant advantage of this approach remains the possibility to obtain the solution in closed form for nonlinear equations describing the flow, where the supposition principle cannot be applied. While for a linear equation this principle provides the solution corresponding to the variable rate, provided the solution at the constant rate is known, in the case of nonlinear equations, for example eqns (1) and (2), the Boltzman transformation approach may give an exact solution for certain cases of variable rate. The formulation of a moving boundary problem associated with nonlinear eqns (1) and (2) appears naturally from eqns (15) and (56), where for n < 1 their integration requires a finite interval of variation of r], which leads to the location of a time depending boundary. REFERENCES [I] H. PASCAL, Int. J. Numerical and Analytical Methods in Geomechanics 7(3), 289 (1983). [2] H. PASCAL, Dynamics of moving interface in porous media for power law fluids with yield stress. Int. J. Engng Sci. 22(5), 577 (1984).
Flow of non-Newtonian fluid through porous media
585
[3] H. PASCAL, Some problems related to the quantitative evaluation of physical properties of the porous medium from flow tests with non-Newtonian fluids. Znt. J. Engng Sci. 23(3), 307 (1985). [4] S. C. MATTHEWS and G. D. RUSSEL, Pressure buildup and flow tests in well. Monograph Series Vol. 1, Sot. Petroleum Eng. (1967). [5] R. C. EARLOUGHES, Jr., Advances in well test analysis. Monograph Series, Sot. Petroleum Eng. (1977). [6] A. LUIKOV, Analytical Heat Diffusion Theory. Academic Press, New York (1968). [7] H. S. CARSLAW and J. C. JAEGER, Conduction of Heat in Solids. Oxford University Press, Oxford (1959). [8] C. IKOKU and H. J. RAMEY, J. Energy Resources Tech. 104, 149 (June 1982). (Received 22 May 1984)
APPENDIX In order to justify the relation (73) one can consider the case when a power law fluid is injected in the well at a constant flow rate. From the continuity equation for a steady flow we have Q = 2rhRv
= constant
(A.1)
or
Q
v=2*hR. On the other hand, for a power law fluid the shear rate is expressed in terms of velocity [2] 3n+l
v (A.3)
‘=y= whereas taking into account the relation between the apparent viscosity p, and shear rate i one has Lb=*.
H
(A.4)
The previous relations (A.l-A.4) lead to the relation (73), which indicates that the apparent viscosity for a pseudo-plastic fluid, i.e. n < 1, is an increasing monotonic function with the radius increasing, provided that a steady flow in the reservoir exists.