424
Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
Analysis of the flow of non-Newtonian viscoelastic fluids in fractal reservoir with the
fractional derivative
TONG Dengke & WANG Ruihe
Department of Applied Mathematics, Petroleum University, Dongying 257061, China
Correspondence should be addressed to Tong Dengke (email: [email protected])
Received December 27, 2003
Abstract
In this paper, fractional order derivative, fractal dimension and spectral
dimension are introduced into the seepage flow mechanics to establish the relaxation
models of non-Newtonian viscoelastic fluids with the fractional derivative in fractal
reservoirs. A new type integral transform is introduced, and the flow characteristics of
non-Newtonian viscoelastic fluids with the fractional order derivative through a fractal
reservoir are studied by using the integral transform, the discrete Laplace transform of
sequential fractional derivatives and the generalized Mittag-Leffler function. Exact
solutions are obtained for arbitrary fractional order derivative. The long-time and
short-time asymptotic solutions for an infinite formation are also obtained. The pressure
transient behavior of non-Newtonian viscoelastic fluids flow through an infinite fractal
reservoir is studied by using the Stehfest’s inversion method of the numerical Laplace
transform. It is shown that the clearer the viscoelastic characteristics of the fluid, the more
the fluid is sensitive to the order of the fractional derivative. The new type integral
transform provides a new analytical tool for studying the seepage mechanics of fluid in
fractal porous media.
Keywords: fractional calculus, non-Newtonian viscoelastic fluids, porous media, fractal, exact solution.
DOI: 10.1360/ 03yw0208
In both the oil reservoir engineering and seepage flow mechanics, heavy oil with
relaxation property shows non-Newtonian rheological characteristics. The relationship
between shear rate γ and shear stress τ is nonlinear. Because of the relaxation
phenomena of heavy oil flow in porous media, the equation of motion can be written
as[1]
qr + λ v
∂ qr
k ⎛∂p
∂2p ⎞
=− ⎜
+ λp
⎟,
∂t
µ ⎝ ∂r
∂ r∂ t ⎠
Copyright by Science in China Press 2004
(1)
Flow analysis of non-Newtonian viscoelastic fluids
425
where λv and λp are velocity relaxation and pressure retardation times.
For most porous media, the above motion equation (1) can basically reflect the flow
characteristics of non-Newtonian viscoelastic fluids with relaxation property. Ametov[1]
and Luan[2] studied the flow characteristics of non-Newtonian viscoelastic fluids with
relaxation property from theory and production practice by using eq. (1). In recent years,
fractional calculus has served as the dynamic basis for the fractal geometry and fractional dimension. Fractional calculus has been very successful in describing the constitutive relationship of viscoelastic fluid. The starting point of fractional order derivative
model of non-Newtonian fluid is usually a classical differential equation that is modified
by replacing the time derivative of an integer order by the so-called Riemann-Liouville
fractional calculus operators. Thus the description of the fractional order constitutive
relationship of viscoelastic fluid is more general. Friedrich[3], Huang[4], Tan[5] and Xu[6,7]
introduced fractional calculus into the rheology and analyzed various problems. They
indicated that the fractional calculus approach is more appropriate for the viscoelastic
fluid. Park[8] introduced fractional calculus into the flow equation of fluid in fractal reservoir. He deemed that the fractional calculus approach tallies with real fractal reservoir,
in particular, the description of the pressure behavior of the early-time stage is more accurate. Using fractional calculus to the experimental data of the viscoelastic glue fluid,
Jiang[9] obtained a very good fit. Generally, fractional derivative is introduced in the
flow model of viscoelastic fluid through porous media. The equation of motion is written
as
qr + λvα Dα qr = −
∂p⎞
k ⎛∂p
+ λ γp Dγ
⎜
⎟,
µ ⎝ ∂r
∂r ⎠
(2)
where Dα and Dγ are fractional calculus operators of orders α and β with respect to t
respectively and are defined as
Dα [ y (t )] =
1
d t y( z)
dz 0 ≤ α ≤ 1 ,
Γ(1 − α ) dt ∫0 (t − z )α
where Γ(x) is Gamma function. While α =γ =1, it may be simplified into eq. (1).
The aim of this paper is to study the flow of relaxation property of a viscoelastic
fluid with the fractional derivative in fractal reservoir. A new type integral transform is
introduced. While α, γ is arbitrary fraction, the exact solution for the flow equation is
obtained by using the Laplace transform of fractional order derivative and the generalized Mittag-Leffler function. Many previous and classical results can be considered as
special cases of our results. For example when α =γ =1, df =2, θ =0, our results become
Luan’s solution[2] for the flow of non-Newtonian viscoelastic fluid in porous media. The
flow characteristic and pressure sensitive to parameters of the viscoelastic fluids with
fractional derivative through porous media are discussed by using the numerical inversion of Laplace transform and asymptotic solutions. It is shown that the clearer the viswww.scichina.com
426
Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
coelastic characteristics of the fluid, the more sensitive the fluid is to the order of the
fractional derivative. The new type integral transform provides a new analytical tool for
studying the seepage mechanics of fluid in fractal porous media.
1
The flow model of the viscoelastic fluid with fractional derivative
Consider the fractal permeable network embedded in impermeable Euclidean matrix, where the fractal network dimension is df, and the Euclidean matrix dimension is
d(d =1, 2, 3). Fluid flow only occurs in a fractal network and satisfies creeping flow law
given by eq. (2). The approximate form of the continuous equation may be written as[8]
∂ P ∂ Qr
+
= 0,
∂r ∂r
Vs N (r )cf
(3)
where Vs is the volume per site, N(r) is site density, and N (r ) = ar
d f −1
, the parameter a
is related directly to its fracture porosity, cf is fluid compressibility, and Qr is volumetric
flow rate. We consider the flow velocity qr, and
Qr = Br d −1qr ,
(4)
where B describes the appropriate symmetry (e.g. B=A, 2πh, 4π for rectilinear, cylindrical and spherical symmetry, respectively).
k (r ) =
maVs df − d −θ
,
r
B
(5)
where m is a local structural property of the fractal network, and θ is related to the
so-called spectral exponent of the fractal network.
Inserting eqs. (2), (4) and (5) into eq. (3) yields the dimensionless flow equation
∂
1
∂
∂
α
Dα pD ) = d −1
( pD + λvD
[rD β
( pD + λ γpD Dγ pD )],
f
r
r
∂ tD
∂
∂
rD
D
D
pD =
λ pD =
aVs m( p0 − p)
Q µ rw1− β
mλ p
µ c f rwθ + 2
,
tD =
mt
,
µ c f rwθ + 2
rD =
(6)
mλv
r
, λvD =
,
rw
µ c f rwθ + 2
, β = d f − θ − 1,
with the initial condition
pD
the inner boundary condition
Copyright by Science in China Press 2004
tD = 0
= 0,
(7)
Flow analysis of non-Newtonian viscoelastic fluids
427
∂
( p + λ γpD Dγ pD )
∂ rD D
rD =1
= −1,
(8)
and the outer boundary condition
lim pD = 0,
(9)
rD →∞
or
∂ pD
∂ rD
rD = R =
0,
(10)
= 0.
(11)
or
pD
rD = R
2 The exact solution for the flow model of the viscoelastic fluid with fractional
derivative
2.1
The generalized finite integral transform
In order to solve the above initial and boundary value problem, a new integral
transform is introduced. We consider the following eigenvalue problem for a finite formation:
1
ξ
d f −1
d ⎛ β dE ⎞
2
⎜ξ
⎟ = −λ E ,
dξ ⎝
dξ ⎠
(12)
⎛ dE
⎞
+ β1 E ⎟
= 0,
⎜ α1
⎝ dξ
⎠ ξ =1
(13)
⎛ dE
⎞
+ β2 E ⎟
= 0.
⎜α 2
⎝ dξ
⎠ ξ =R
(14)
We obtain the eienvalue λk, k=0, 1, 2, …, where λ0=0 and λk (k=1,2, …) is the kth
positive root of the following equation:
a
a
α1α 2 R a −1λ 2ϕν −1，ν −（
1 b，bR , λ ) + α1 β 2 λϕν −1，（
ν b，bR , λ ) +
a
a
α 2 β1 R a −1λϕν，ν −（
1 b，bR , λ ) + β1 β 2ϕν ，（
ν b，bR , λ ) = 0,
where b =
(15)
2
θ +2
, a=
, ν = 1 − ds / 2 .
θ +2
2
ϕ m, n ( x, y, λ ) = J m ( xλ )Yn ( yλ ) − Ym ( xλ ) J n ( yλ ) ,
where Jn(x) and Yn(x) are the Bessel functions of the first and second kind, respectively.
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428
Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
The eigenfunction of eigenvalue λk may be written as
⎧1, λk = 0,
⎪
E (ξ , λk ) = ⎨ 1− β
⎪⎩ξ 2 [α 2 R a −1λk ϕν ,ν −1 (bξ a , bR a , λk ) + β 2ϕν ,ν (bξ a , bR a , λk )],
λk ≠ 0.
Eigenfunction systems construct complete orthogonal systems with weighting function ξ
d f −1
on [1, R]. The orthogonality of the system can be expressed as
R
∫1
E (ξ , λk ) E (ξ , λn )ξ
d f −1
⎧ H (λk ), n = k ,
dξ = ⎨
n ≠ k,
⎩0,
where
H (λk ) =
A2
− [(b 2 λk2 − ν 2 )α12 + ( β1 b + α1ν ) 2 ]B 2
R2
,
π 2 b3λk2 A2
2[(b 2 R 2 aα 22 λk2 − α 22ν 2 ) + ( β 2 bR + α 2ν ) 2 ]
A = α1λk Jν −1 (bλk ) + β1 Jν (bλk ) , B = α 2 R a −1λk Jν −1 (bλk R a ) + β 2 Jν (bλk R a ).
A finite integral transform is introduced by using the orthogonality of eigenfunction
system.
Definition.
If u(ξ) is piecewise continuous in (1, R), then
R
∫1 u (ξ ) E (ξ , λk )ξ
d f −1
dξ k = 0,1, 2,
(16)
is called the finite integral transform of the function u(ξ), and written as u (λk ) or
F1[u (ξ )] . The inversion transform of u (λk ) is obtained by using the orthogonality of
eigenfunctions
u (ξ ) =
i
π 2 b3
2
λk2 A2 u (λk ) E (ξ , λk )
∞
∑
k =0
[(b 2 R 2 a λk2 − ν 2 )α 22 + ( β 2 bR + α 2ν ) 2 ]
2
A
− [(b 2 λk2 − ν 2 )α12 + ( β1 b + α1ν ) 2 ]B 2
R2
.
(17)
Theorem. If the function u(ξ ) possesses continuous first derivative and piecewise continuous second derivatives in [1, R], we have the following equality:
⎡ 1 d ⎛ β du ⎞ ⎤
F1 ⎢ d −1
⎜ξ
⎟⎥
f
dξ ⎝
dξ ⎠ ⎦⎥
⎣⎢ ξ
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Flow analysis of non-Newtonian viscoelastic fluids
429
⎧ R β E ⎡ du
⎤
⎤
E ⎡ du
α2
+ β 2u ⎥ − ⎢α1
+ β1u ⎥ − λk2u (λk ), (α1 ≠ 0,α 2 ≠ 0)
⎪
⎢
⎦ R α1 ⎣ dξ
⎦1
⎪ α 2 ⎣ dξ
⎪
⎤
E ⎡ du
⎪− R β ∂E
[u ]R − ⎢α1
(α1 ≠ 0,α 2 = 0)
+ β1u ⎥ − λk2u (λk ),
⎪
α1 ⎣ dξ
∂ξ ξ = R
⎦
1
⎪
=⎨ β
⎪ R E ⎡α du + β u ⎤ + ∂E [u ] − λ 2 u (λ ),
(α1 = 0,α 2 ≠ 0)
2 ⎥
1
k
k
⎪ α 2 ⎢⎣ 2 dξ
⎦ R ∂ξ ξ =1
⎪
⎪ β ∂E
∂E
[u ]R +
[u ]1 − λk2u (λk ),
(α1 = 0,α 2 = 0)
⎪− R
∂
∂
ξ
ξ
⎪⎩
ξ =R
ξ =1
Proof.
(18)
⎡ 1 d ⎛ β du ⎞ ⎤
R d ⎛ β du ⎞
F1 ⎢ d −1
ξ
=∫
⎥
⎜
⎟
⎜ξ
⎟E (ξ , λk )dξ
f
dξ ⎠
dξ ⎝
dξ ⎠ ⎦⎥ 1 dξ ⎝
⎣⎢ ξ
R
R
⎡⎛
⎡⎛
R
∂ ⎛ β ∂ E (ξ , λk ) ⎞ ⎤⎥
β du ⎞
β ∂ E (ξ , λk ) ⎞
⎢
⎢
= ⎜ξ
⎟ E (ξ , λk ) − ⎜ u (ξ )ξ
⎟ + ∫1 u (ξ ) ⎜ ξ
⎟dξ ,
∂ξ
∂ξ ⎝
∂ξ
dξ ⎠
⎢⎝
⎢⎝
⎠1
⎠ ⎥⎦
1
⎣
⎣
⎧ R β E ⎡ du
⎤
⎤
E ⎡ du
2
⎪
⎢α 2 dξ + β 2 u ⎥ − α ⎢α1 dξ + β1u ⎥ − λk u (λk ),
α
⎦R
⎦1
1 ⎣
⎪ 2 ⎣
⎪
⎤
E ⎡ du
⎪− R β ∂E
[u ]R − ⎢α1
+ β1u ⎥ − λk2u (λk ),
⎪
α1 ⎣ dξ
∂ξ ξ = R
⎦1
⎪
=⎨ β
⎪ R E ⎡α du + β u ⎤ + ∂E [u ] − λ 2u (λ ),
k
k
2 ⎥
1
⎪ α 2 ⎢⎣ 2 dξ
∂ξ ξ =1
⎦
R
⎪
⎪ β ∂E
∂E
[u ]R +
[u ]1 − λk2u (λk ).
⎪− R
∂ξ ξ = R
∂ξ x =1
⎪⎩
(α1 ≠ 0, α 2 ≠ 0)
(α1 ≠ 0, α 2 = 0)
(α1 = 0, α 2 ≠ 0)
(α1 = 0, α 2 = 0)
While df=2, β =1, it will be simplified into a finite Hankel transform.
2.2 Exact solution for the relaxation model of non-Newtonian viscoelastic fluid flow in
an infinite reservoir
The initial and boundary value problem Ι made up of eqs. (6)—(9) is the case of
constant production rate from an infinite reservoir. We consider the initial value condition of integer order for fractional differential equation. The operator Dα is taken as the
Miller-Ross sequential fractional derivative. Let
∞
pD (rD , s ) = ∫ e − stD pD (rD , t D )dt D .
0
Application of the Laplace transformation to (6)—(9) yields
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Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
α α
s (1 + λvD
s )
γ
γ
(1 + λ pD s )
∂pD
∂rD
1
pD =
d −1
rD f
=−
rD =1
∂ ⎛ β ∂pD ⎞
⎜ rD
⎟,
∂rD ⎝
∂rD ⎠
1
s(1 + λ γpD sγ )
(19)
,
(20)
lim pD = 0.
(21)
rD →∞
Assume
1− β
pD = rD 2 f ( ρ ), ρ = brDa .
(22)
It can be shown that
α α
s (1 + λvD
s )
∂ 2 f 1 ∂ f ν2
f,
+
−
∂ρ 2 ρ ∂ρ ρ 2
(23)
⎛ ∂ f (1 − β ) ⎞
1
f⎟
,
+
=−
⎜
2
s (1 + λ γpD s )
⎝ ∂ρ
⎠ ρ =b
(24)
1 + λ γpD sγ
f =
2d f ⎞
⎛
where ν = 1 − d s / 2 ⎜ d s =
⎟,
θ + 2⎠
⎝
1− β
pD =
rD 2 Kν (b x( s )rDa )
s (1 + λ γpD sγ ) x( s ) Kν −1 (b x( s ))
, x( s ) =
α α
s (1 + λvD
s )
1 + λ γpD sγ
.
(25)
Eq. (25) is reduced to the following form at the wellbore (rD=1):
pD =
Kν (b x( s ))
γ
s (1 + λ pD sγ ) x( s ) Kν −1 (b x( s ))
.
(26)
The Weber transformation is given by[10]
∞
f = W [ f ] = ∫ ρ f ( ρ )ϕ1 ( ρ , λ )d ρ ,
b
where ϕ1 ( ρ , λ ) = Jν (λρ )Yν −1 (bλ ) − Yν (λρ ) Jν −1 (bλ ).
Applying the generalized Weber transformation to eqs. (22) and (23) yields
f =
2
α α
s )]
πλ s[(1 + λ pD s )λ 2 + s (1 + λvD
Copyright by Science in China Press 2004
γ
γ
.
(27)
Flow analysis of non-Newtonian viscoelastic fluids
431
The inverse of Weber transformation can be expressed as[10]
f (ρ ) = ∫
∞
0
λ f ϕ1 ( ρ , λ )d λ
2
Jν −1 (bλ ) + Yν2−1 (bλ )
.
(28)
Substituting (27) and (28) into (22), we have
1− β
pD (rD , s ) = ∫
2rD 2 ϕ1 (brDa , λ )d λ
∞
α α
[ Jν2−1 (bλ ) + Yν2−1 (bλ )]π s [(1 + λ γpD sγ )λ 2 + s (1 + λvD
s )]
0
1− β
=
where A(λ , s ) =
∞
2rD 2
A(λ , s )ϕ1 (brDa , λ )d λ
∫0 [ J 2
π
2
1
α
λvD s
,
ν −1 (bλ ) + Yν −1 (bλ )]π
2 +α
(29)
.
γ
+ s + λ pD
λ 2 s1+γ + λ 2 s
2
Analytical solution for the initial and boundary value problem can be obtained as
long as the inverse of Laplace transform of the function A(λ , s ) is given. We will use
the discrete inverse Laplace transform method to get the analytical solution. Firstly, we
rewrite A(λ , s ) as a Taylor series form
A (λ , s ) =
=
1
α
λ vD s
2 +α
1
+s
2
1+
λ
γ
pD λ
s +
−α
λ vD
s +λ2
α 1+α
λ vD
s +s
−α −2
λ vD
s
α
2 γ
1
1+
−α −1
λ vD
s (λ γpD λ 2 sγ
−α
sα + λ vD
+ λ 2)
−α −2
⎞
s ⎛ λ γpD 2 γ −1
(−1)m λ vD
−α −1
⎜
λ s + λ vD
s ⎟
=∑ α
−α m +1 ⎜ α
⎟
m = 0 ( s + λ vD )
⎝ λ vD
⎠
∞
−α −2
(−1)m λ vD
s
m
γk
⎛ m ⎞ λ pD 2 m γ k − m
=∑ α
λ s
−α m +1 ∑ ⎜ ⎟ mα
m = 0 ( s + λ vD )
k = 0 ⎝ k ⎠ λ vD
∞
=
1
λ αvD
m
⎛ λ2 ⎞
∑ (−1)m ⎜⎜ λ α ⎟⎟
m =0
⎝ vD ⎠
∞
m
m
⎛ m⎞
sγ k − m − 2
∑ ⎜ k ⎟λ γpDk (sα + λ −α )m+1 .
k =0 ⎝
⎠
(30)
vD
Applying the inverse Laplace transform to eq. (30), we obtain
A(λ , t D ) =
1
α
λ vD
(−1)m ⎛ λ 2 ⎞
⎜⎜ α ⎟⎟
∑
m = 0 m ! ⎝ λ vD ⎠
∞
m
m
⎛ m⎞
−α α
t D ),
∑ ⎜ k ⎟λ γpDk tD(1+α )( m+1)−γ k Eα(m,α)+ m+ 2−γ k (−λ vD
k =0 ⎝
⎠
(31)
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Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
in which Eµ ,ν ( z ) denotes Mittag-Leffler function. To solve eq. (31), we use the following property of M-L function[11]:
µ −ν
⎪⎧ n ! s
⎪⎫ µ n +ν −1 ( n )
1µ
=t
L−1 ⎨ µ
Eµ ,ν (± ct µ ) (Re( s ) > c ).
n +1 ⎬
⎩⎪ ( s ∓ c) ⎭⎪
(32)
The exact solution for the flow model of an infinite formation is obtained by using
eq. (31)
1− β
pD (rD , t D ) =
(−1) m +1 1
m ! λ αvDm
m =0
∞
2rD 2
∑
πλ αvD
∞
λ 2 mϕ1 (rDa , λ )d λ
2
ν −1 (λ ) + Yν −1 (λ )]
∫0 [ J 2
(33)
⎛ m ⎞ k (1+α )( m +1) −γ k ( m )
−α α
tD
Eα ,α + m + 2−γ k (−λ vD
t D ).
i ∑ ⎜ ⎟λ γpD
k
k =0 ⎝ ⎠
m
If one sets α =γ =1, formula (31) can be simplified into
⎧⎪
⎫⎪
1
A(λ , t D ) = L−1{ A(λ , s )} = L−1 ⎨
2
2
2 ⎬
⎪⎩ s[λvD s + (λ λ pD + 1) s + λ ] ⎭⎪
1 ⎧⎪1 − e − s1 (λ )tD 1 − e− s2 (λ )tD ⎫⎪
=
+
⎨
⎬,
λvD ⎩⎪ s1 ( s2 − s1 ) s2 ( s1 − s2 ) ⎭⎪
(34)
where −s1 and −s2 satisfy λvDs2+(λ2λpD+1)s+λ2=0.
By using eq. (34), eq. (33) can be simplified into
1− β
pD (rD , t D ) =
2rD 2
πλvD
∞ [ Jν
∫0
(bλ rDa )Yν −1 (bλ ) − Jν −1 (bλ )Yν (bλ rDa )]
Jν2−1 (bλ ) + Yν2−1 (bλ )
⎧⎪1 − e − s1 (λ )tD 1 − e − s2 (λ )tD
i ⎨
+
s2 ( s1 − s2 )
⎩⎪ s1 ( s2 − s1 )
⎫⎪
⎬ d λ.
⎭⎪
(35)
This is the Tong’s solution for the flow of non-Newtonian viscoelastic fluid in fractal reservoir[12]. If one sets α=γ =1, df =2, θ=0, formula (33) can be reduced to the exact
solution for the flow of heavy oil in an infinite homogeneous reservoir[2]. Because the
relaxation of the pressure gradient and velocity of heavy-oil is considered in eqs. (33)
and (35), they are generally valid. For instance, for λvD=λpD=0, (35) can be simplified
into the following solution for Newtonian fluid flow in a fractal reservoir.
1− β
pD =
2rD 2
π
∞
∫0
(1 − e −u
2
tD
Copyright by Science in China Press 2004
)[ Jν (brDa u )Yν −1 (bu ) − Jν −1 (bu )Yν (brDa u )]du
u 2 [ Jν2−1 (bu ) + Yν2−1 (bu )]
.
(36)
Flow analysis of non-Newtonian viscoelastic fluids
2.3
433
Asymptotic solution
1) By using the property of Laplace transform, the following equality holds true for
the final state (s→0):
α α
s (1 + λvD
s )
1 + λ γpD sγ
= s.
Eq. (26) may be simplified into
pwD =
Kν (b s )
3
s2 K
ν −1 (b
.
s)
The long-time approximate solution in real space can be obtained by using the approximate formula of the modified Bessel function
pwD (t D ) =
pwD (t D ) =
(θ + 2)2ν
tνD , v ≠ 0,
(θ + 2 − d f )Γ(d s / 2)
1
θ +2
(37)
[ln t D + 2 ln(θ + 2) − c], v = 0,
(38)
where c is Euler’s constant.
Eq. (37) indicates that the long-time pressure drop in heavy-oil can be used in
computing fractal and formation parameters, as the interference of the rheological parameters λv, λp disappears.
p0 − p (t ) = ∆p = C1tν ,
C1 =
Q (θ + 2)2v µ 1−ν
.
aVs cνf m1−v Γ(1 − v)(θ + 2 − d f )
(39)
Eq. (39) shows that a plot of the pressure transient in logarithmic coordinates yields a
straight line with a slope v and an interception logC1. Through analysis of the drawdown
test, one gets v. It is a pity that df and θ are indeterminate without other information. The
pressure response of viscoelastic fluid flow in a fractal reservoir is smaller at the
early-time and greater at the late-time.
2) At the early-time (s→∞), the following equality is tenable:
⎛ 1 + λ α sα
vD
⎜
⎜ 1 + λ γpD sγ
⎝
α 1+α −γ
⎞
λvD
⎟s =
s 2 .
α
⎟
λ
pD
⎠
Using the approximate formula of the modified Bessel function, one gets
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434
Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
pwD =
1
λα
s (1 + λ pD s ) vD
s
λ γpD
γ
γ
1+α −γ
2
.
As s is very large, sγ λ γpD >> 1 . The solution in real space can be given by
α +γ +1
pwD =
tD 2
.
⎛ 3+α +γ ⎞ α γ
Γ⎜
⎟ λ vD λ pD
2
⎝
⎠
(40)
The short-time asymptotic solution shows that a plot of the pressure transient in
α + γ +1
logarithmic coordinates yields straight lines with slope
. The physical mean2
ing of eq. (40) is that well test data of heavy-oil field reveal not only the effect of fractional derivative but also slowly-moving effect of the relaxation of heavy-oil in a short
time. Then the orders α and γ of fractional derivative and the rheological parameters λv,
λp can be determined.
The pressure distributions are obtained by applying the Stehfest’s method of the
numerical inversion of Laplace transform to eq. (26). The pressure behavior is as follows:
1) The dimensionless pressure solution exhibits two line segments in the lg-lg plot
of the PwD~tD (figs. 1 and 2).
a. The straight line with a slope
α + γ +1
2
reflects the effect of the fractional de-
rivative.
b. The straight line with a slope v=1−ds/2 reflects the effect of spectral dimension
and the interception reveals the effect of the formation parameters.
2) The pressure curves are influenced by the α, γ. The effects of α, γ were mainly in
short-time stages (tD<106). In the initial stage, the larger the value of γ, the smaller the
dimensionless pressure. With increasing time, the pressure curves show a sudden change
and pressure curves intersect at a point. The dimensionless pressure henceforth is
smaller for a smaller value of γ . That is why the pressure at initial stages is larger for the
smaller order of corresponding fractional derivative. The pressure change is slower in
the whole flow process. The pressure at initial stages is smaller for the larger order of
corresponding fractional derivative. With increasing time, the pressure rises so fast that
exceeds the pressure of the smaller order of corresponding fractional derivative; thus the
pressure curves have a sudden change in fig. 1.The pressure curves merge into a single
one in later stages. The effect of α is similar to that of γ, but the effect of α is smaller
Copyright by Science in China Press 2004
Flow analysis of non-Newtonian viscoelastic fluids
435
than that of γ (figs.1 and 2).
Fig. 1. Effect of γ on plot of pressure curve.
Fig. 2. Effect of α on plot of pressure curve.
3) The curve of the pressure derivative has a specific shape. (figs. 3 and 4). The
pressure derivative curves have a similar shape in the lg-lg plot of the dpwD/dlntD~tD, but
the scope of curve change is greater and the time the curves intersect at a point is earlier.
Each dlnpwD/dlntD curve has a much more pronounced specific shape. The distinguishing
characteristics makes it easy to find a unique curve to match field data.
4) The effects of spectral dimension ds are mainly in later stages. The dimensionless
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436
Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
Fig. 3. Effect of γ on plot of pressure derivative curve.
Fig. 4. Effect of γ on plot of dlnpwD/dlntD~tD.
pressure at any time is smaller for a greater value of ds (fig. 5). Fig. 6 shows that the
clearer the fluid viscoelastic characteristic, the more sensitive pressure curve is to the
order of the fractional derivative. That is, the greater the value of λpD, the more sensitive
the pressure curve is to the order of the fractional derivative.
2.4 Exact solution for the relaxation model of non-Newtonian viscoelastic fluid flow in
a finite closed reservoir
The initial and boundary value problem ΙΙ that is made up of eqs. (6)—(8) and (10)
Copyright by Science in China Press 2004
Flow analysis of non-Newtonian viscoelastic fluids
437
Fig. 5. Effect of ds on plot of pressure curve.
Fig. 6. Effects of γ and λpD on plot of pressure curve.
is the case of constant production rate from a finite closed reservoir. Applying Laplace
transformation to the initial and boundary value ΙΙ , one obtains the boundary value
problem made up of eqs. (19), (20) and (41)
∂pD
∂rD
= 0.
(41)
rD = R
The solution in Laplace space for a finite closed formation is given by
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Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
1− β
pD =
rD 2 [ Iν −1 ( x( s ) R a ) Kν ( x( s )rDa ) + Kν −1 ( x( s ) R a ) Iν ( x( s )rDa )]
s (1 + λ γpD sγ ) x( s )[ Iν −1 ( x( s ) R a ) Kν −1 ( x( s )) − Kν −1 ( x( s ) R a ) Iν −1 ( x( s ))]
. (42)
Eq. (42) is reduced to the following form at the wellbore (rD=1):
pwD =
Iν −1 ( x( s ) R a ) Kν ( x( s )) + Kν −1 ( x( s ) R a ) Iν ( x( s ))
s (1 + λ γpD sγ ) x( s )[ Iν −1 ( x( s ) R a ) Kν −1 ( x( s )) − Kν −1 ( x( s ) R a ) Iν −1 ( x( s))]
. (43)
In order to gain the analytical solution of model II, by finite integral transform introduced in sec. 3.1, if one sets α1=α2=1, β1=β2=0, eq. (16) can be simplified into
R d −1
pD (λn , s ) = ∫ rD f
1
pD (rD , s )ϕ 2 (λn , rD )drD .
(44)
The inversion transform of pD (λn , s ) may be written as
∞
pD (rD , s ) = ∑ pD (λn , s )
n =0
ϕ 2 (λn , rD )
,
N 2 (λn )
(45)
where
⎧ 1− β a −1
a
a
⎪ 2
ϕ 2 (λn , rD ) = ⎨rD R [Yν −1 (bλn ) Jν (bλn rD ) − Jν −1 (bλn )Yν (bλn rD )], n = 1, 2,
⎪⎩1,
n = 0.
λn(n=1, 2, …)is the nth positive root of the following equation:
,
λ0=0,
Jν −1 (bλ )Yν −1 (bλ R a ) − Yν −1 (bλ ) Jν −1 (bλ R a ) = 0,
⎧π 2b
λ 2n Jν2−1 (bλn )
,
⎪
2a−2
[ Jν2−1 (bλn ) − Jν2−1 (bλn R a )]
1
⎪ 2 R
=⎨
N 2 (λn ) ⎪ d f
n = 0.
,
⎪ df
⎩ R −1
n = 1, 2,
.
(46)
Applying integral transform (44) to eq. (19), under boundary conditions (11) and
(23), we have
pD (λn , s ) = −
2 R a −1 Jν −1 (bλn R a )
α α
π sbλ n Jν −1 (bλn )[(1 + λ γpD sγ )λ n2 + s (1 + λvD
s )]
.
Substituting eq. (47) into eq. (45), one gets
pD (rD , s ) =
∞
df
R
df
−1
p0 (rD , s ) − ∑ A1 (λn , s )
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n =1
πλn Jν −1 (bλn R a ) Jν −1 (bλn )ϕ 2 (λn , rD )
,
Jν2−1 (bλn ) − Jν2−1 (bλn R a )
(47)
Flow analysis of non-Newtonian viscoelastic fluids
where p0 (rD , s ) =
1
s (1 + λ
2
α
α
vD s
)
439
1
, A1 (λn , s) =
⎡
1
p0 (rD , t D ) = L−1 ⎢ 2
s
λ
(1
+
⎣
α
λ vD s
2 +α
γ
+ s + λ pD
λ n2 s1+γ + λ n2 s
2
⎡
⎢ sα − 2
⎤
−1 ⎡ 1 ⎤
−1 ⎢
L
L
=
−
⎢ s2 ⎥
α α ⎥
⎢ sα + 1
⎣ ⎦
vD s ) ⎦
⎢
λ αvD
⎣
,
⎤
⎥
⎥
⎥
⎥
⎦
(48)
−α α
t D ).
= t D − t D Eα ,2 (−λ vD
A1 (λ n , t D ) =
α
(−1) m ⎛ λ 2
∑ m! ⎜⎜ λ αn
m =0
⎝ vD
∞
1
λ vD
⎞
⎟⎟
⎠
m
m
⎛ m⎞
−α α
t D ). (49)
∑ ⎜ k ⎟λ γpDk tD(1+α )(m+1)−γ k Eα(m,α) + m+ 2−γ k (−λ vD
k =0 ⎝
⎠
The exact solution of the model ΙΙ is obtained using eqs. (48) and (49)
pD (rD , t D ) =
−
1− β
π rD 2
λ αvD
G1 (λn ) =
∞
df
R
df
−1
−α α
[t D − t D Eα ,2 (−λ vD
t D )]
(50)
∞
m m
⎛ ⎞
(−1)m
−α α
(
)
λ
G
t D ),
∑ ∑ m! 1 n ∑ ⎜ k ⎟λ γpDk tD(1+α )( m+1)−γ k Eα( m,α)+ m+ 2−γ k (−λ vD
n =1 m = 0
k =0 ⎝ ⎠
λn Jν −1 (bR a λn ) Jν −1 (bλn )[ Jν (brDa λn )Yν −1 (bR a λn ) − Yν (brDa λn ) Jν −1 (bR a λn )]
.
Jν2−1 (bλn ) − Jν2−1 (bR a λn )
2.5 Exact solution for the relaxation model of non-Newtonian viscoelastic fluid flow in
a finite constant-pressure outer boundary
The initial and boundary value problem III that is made up of eqs. (6)—(8) and (11)
is the case of constant production rate from a finite constant-pressure outer boundary
reservoir. Then applying Laplace transformation to the initial and boundary value III,
one obtains the boundary value problem that is made up of eqs. (19), (20) and (51)
pD
rD = R
= 0.
(51)
The solution in Laplace space for a finite constant-pressure outer boundary formation is
given by
pD =
β
Iν ( x( s ) R ) Kν ( x( s )rD ) − Kν ( x( s ) R) Iν ( x( s )rD )
s (1 + λ pD s β ) x( s )[ Iν ( x( s ) R ) Kν −1 ( x( s )) − Kν ( x( s) R ) Iν −1 ( x( s ))]
.
(52)
Applying finite integral transform introduced in subsec. 2.1 to eq. (19) and a
method similar to that in sec. 2.4, and setting α1=1, α2=0, β1=0, β2=1, one gets the exact
solution of the model III
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440
Science in China Ser. G Physics, Mechanics & Astronomy 2004 Vol.47 No.4 424—441
pD (rD , t D ) =
π
λ αvD
(−1) m +1
G2 (λn )
m!
n =1 m = 0
∞
∞
∑∑
⎛ m⎞
−α α
t D ),
i ∑ ⎜ ⎟λ βpDk t D(1+α )( m +1) − β k Eα( m,α) + m + 2 −γ k (−λ vD
k
k =0 ⎝ ⎠
m
(53)
where
G2 (λn ) =
λn Jν (bR a λn ) Jν −1 (bλn )[ Jν (brDa λn )Yν (bR a λn ) − Yν (brDa λn ) Jν (bR a λn )]
,
Jν2−1 (bλn ) − Jν2−1 (bR a λn )
λn (n=1, 2, … ) is the nth positive root of the equation Jν −1 (bx)Yν (bR a x) −
Yν −1 (bx) Jν (bR a x) = 0.
3
Conclusions
(1) Fractional calculus and fractal dimension are introduced into the flow equation
of non-Newtonian viscoelastic fluids with relaxation property flow in fractal reservoirs.
The flow model has more extensive adaptability than traditional model.
(2) The exact solutions for the fractional derivative flow models of non-Newtonian
viscoelastic fluid in fractal reservoir with both infinite and finite formation are obtained
by using the discrete Laplace transform of sequential fractional derivatives and the generalized Mittag-Leffler function.
(3) A new type integral transform is presented. Hankel transform is its special case.
The new type integral transform provided a new analytical tool for studying the seepage
mechanics of fluid in porous media.
(4) By making λv→0 and λp→0, solutions (33), (53) and (55) are simplified into the
solution for Newtonian fluid flow in fractal reservoir.
(5) While α =β =1, solutions (33), (53) and (55) are simplified into the solution for
flow model of non-Newtonian viscoelastic fluid flow in fractal reservoir.
(6) With changing fractional derivative, the pressure curves have sudden changes
and pressure derivative curves are clear. Fig. 6 shows that the clearer the viscoelastic
characteristic of the fluid is, the more it is sensitive to the order of the fractional derivative.
Acknowledgements This work was supported by Project of China National 973 Program: Basic studies on formation mechanism and economic exploitation of coalbed gas reservoir (2002CB211708) and the Natural Science
Foundation of Shandong Province (Y2003F01).
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