Bernabé, Y.

Earth-prints Repository

  
Oscillating flow of a compressible fluid through 
deformable pipes and pipe networks: wave propagation 
phenomena. 
 
Yves Bernabé,  
Earth, Atmospheric and Planetary Sciences Department, MIT 
 
 
 
 
 
 
 
 
 
 
 
Acknowledgements:  This work was partially funded by the US Department of Energy 
under grant DE-FG01-97ER14760. 
 
 Introduction 
Time-dependent flow of interstitial fluid can potentially provide more information about 
the structure of porous media than steady-state flow.  For example, transient and 
oscillating flow methods have been used in the laboratory to measure permeability and 
storativity of rock samples simultaneously [e.g., Hsieh, et al., 1981; Neuzil, et al., 1981; 
Fischer, 1992; Fischer and Paterson, 1992].  However, these methods rely on the 
assumption that the medium investigated is homogeneous.  As a consequence, they fail to 
properly explain certain features observed in heterogeneous media.  For instance, AC 
flow through a strongly heterogeneous medium depends on frequency [e.g., Bernabé, et 
al., 2004; Song and Renner, 2006].  Even in homogeneous media, frequency-dependence 
arises from a transition from viscosity-controlled flow at low frequencies to inertia-
controlled flow at high frequencies [e.g., Johnson, et al., 1987].  In most work on 
frequency-dependent, AC permeability, the interstitial fluid is supposed to be 
incompressible and the pores rigid.  One important consequence is that fluid flow waves 
cannot propagate in the porous medium under these conditions.  To clarify this statement, 
let’s consider a semi-infinite, rigid pipe saturated with an incompressible fluid.  A fluid 
pressure oscillation at the pipe entrance produces an oscillating flow along the pipe, 
which, because of mass conservation and complete absence of storage, is independent on 
distance from the pipe entrance.  In other words, a pressure disturbance and the 
associated flow are instantaneously transmitted from the pipe entrance to infinity.  In 
contrast, studies of blood circulation through elastic arteries [e.g., Zamir, 2000] show that 
the entire body of fluid does not move in unison as in a rigid pipe.  Fluid flow waves 
travel along the elastic arteries with a finite velocity.  These waves are dispersive and 
attenuated.  Their existence has far-reaching consequences since wave reflections and 
interferences greatly affect blood flow in the artery tree [e.g., Duan and Zamir, 1995; 
Zamir, 1998, 2000; Wang and Parker, 2004].  Logically, fluid flow waves must also 
occur in saturated pipe networks and, by extension, in porous media, provided the 
interstitial fluid is compressible and/or the porous medium is deformable.  The purpose of 
this work is to derive a flow wave propagation model for a single pipe and use it to 
investigate flow waves through pipe networks.   
AC flow of compressible fluid through a single infinitely long pipe 
 
Notations:  r, z,     cylindrical coordinates 
  u(r,z,t)     axial v(r,z,t)     radial velocity in fluid 
  p(r,z,t)     pressure variation with respect to constant back pressure 
  x(r,z,t)     axial  y(r,z,t)     radial displacement in solid 
   
Main assumption:   Long-wave approximation (wavelength  >> pipe radius R, and/or, 
wave-speed c >> average flow velocity).  
 
1.  Rigid pipe 
Continuity equation:  p
t
+
u
z
+
v
r
+
v
r
= 0       (with  = 1


p
) 
Navier-Stokes equations:   u
t
= p
z
+ 
2u
r2
+
1
r
u
r
 
 
	 
 
 

  
     v
t
= p
r
+ 
2v
r2
+
1
r
v
r
 v
r2
 
 
	 
 
 

  (with  viscosity). 
       
Solutions are axially traveling fluid flow waves [e.g., Zamir, 2000]: 
u(r,z,t) = U(r) e
-i(t -z/c) 
v(r,z,t) = V(r) e
-i(t -z/c)        
p(r,z,t) = P(r) e
-i(t -z/c)  (with c complex wave-speed). 
 
   P r( )  B          
   U r( ) =
B
c
1 J0 r( )
J0 R( )
 
 
 
 
 
	    
  V r( ) =
rB 2
2c 2J0 R( )
J1 r( )
r
 J1 R( )
R
 
 
	 
 
 

       
 (with B an integration constant,  = i

 and c0 = 1/ the fluid sound velocity) 
 Dispersion equation:   c 2 = c0
2 1 2J1 R( )
RJ0 R( )
 
 
 
 
 
    
 
  
Re(c/co)
Im(c/co)
log10(ω)
log10(ω)0 1 2 3 4 5
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
-0.4
-0.3
-0.2
-0.1
Rigid Pipe
Elastic Pipe
   
Axial volumetric flux q(z,t) = Q e
-i(t -z/c): 
Q =
R2B
c
2J1 R( )
RJ0 R( )
1
 
 
 
	 

 
 = R
2P
 2
2J1 R( )
RJ0 R( )
1
 
 
 
	 

 
   (with P = iB/c )
2. Elastic pipe 
 
Elastic potentials in solid: 
 x r,z,t( ) =

z
+

r
+

r
, and, y r,z,t( ) =

r
 
z
  
Force balance equations:  
  
 2
r2
+
1
r

r
+
 2
z2
=
1
VP
2
 2
t 2
    
  
 2
r2
+
1
r

r
 
r2
+
 2
z2
=
1
VS
2
 2
t 2
   
 
Solutions: 
(r,z,t) = C K0(lr) e-ik(ct –z)    (with k = /c and l2 = k2 (1-c2/VP2) ) 
  (r,z,t) = D K1(mr) e-ik(ct –z) (with m2 = k2 (1-c2/VS2) ) 
 
Dispersion equation: 
 c 4  c 2 2J1 R( )
RJ0 R( )

S
c0
2 + c0
2 + 2VS
2
 
 
 
 
 
	 + 2VS2c02 1
2J1 R( )
RJ0 R( )
 
 
 
 
 
	 = 0    
 
Fluid velocity: 
  U r( ) =
B
c
1 RJ0 r( )
2J1 R( )
1 c
2
c0
2 +
c 2
VS
2

S
 
 
 
 
 
 
 
	 
 

 
 
      
 
V r( ) =
iB
2
R
J1 r( )
J1 R( )
1
c 2
 1
c0
2 +
1
VS
2

S
 
 
 
 
 
 
r
R
1
c 2
 1
c0
2
 
 
 
 
 
 
	 

 
 
 
 
  
 
Fluid volumetric flux:  
  Q = R
2B
c
c02
SVS2
1
 
 
 
 
 
	 
c 2
c0
2
      
 
 
AC flow of compressible fluid through a single pipe of finite length L 
 
Flow wave reflections and interferences 
 
The pressure inside the pipe is sum of two waves traveling in opposite directions:  
 p z,t( ) = B+ei tz / c( ) + Be i t+z / c( )  
 
B+ =
PD  PUeiL / c
eiL / c  e iL / c
 
 B =
PUe
iL / c  PD
eiL / c  e iL / c
  (with PU and PD upstream and downstream amplitudes)  
Volumetric flux q(z,t) also sum of two traveling waves:   
 q z,t( ) =Q+ei tz / c( ) +Qe i t+z / c( )       
 QU =Q
+
+Q , and, QD =Q
+eiL / c +Qe-iL / c   
 
 QU =
iR2
c
PUcos L /c( )  PD
sin L /c( )
 
 

 
 
	 
 
2J1 R( )
RJ0 R( )
1
 
 

 
 
	 
    
 QD =
iR2
c
PU  PDcos L /c( )
sin L /c( )
 
 

 
 
	 
 
2J1 R( )
RJ0 R( )
1
 
 

 
 
	 
    
QU and QD generally different owing to storage inside the pipe.   
Identical only in the limit of an infinitesimally small pipe length.   
 
Apparent upstream and downstream hydraulic conductivities 
 
Steady-state flow:  KSS = -Q/P = R4/8,  
Incompressible AC flow: KAC =
R2
 2
2J1 R( )
RJ0 R( )
1
 
 
 
 
 
	 [Johnson, et al., 1987] 
 
 
 
 
  
 
Re(KAC/KSS)
log10(ω)
log10(ω)
-1 1 2 3 4 5
0.2
0.4
0.6
0.8
1
-1 1 2 3 4 5
0.2
0.4
0.6
0.8
1
Im(KAC/KSS)
 
 KAC/KSS for R = 100 μm.  
 
Sharp transition between low- and high frequency regimes defines transition frequency 
V,I, which scales as 1/R2 [e.g., Johnson, et al., 1987; Charlaix, et al., 1988; Bernabé, 
1997].   
 
Compressible AC flow  (flow wave propagation): 
 
Different upstream and downstream hydraulic conductivities, KU and KD:  
 KU =
LQU
PD  PU
    and  KD =
LQD
PD  PU
    
 
 
 KU = KAC
L
c
PU cos
L
c
 
  
 
  
 PD
sin
L
c
 
  
 
  
PU  PD( )
    
KD = KAC
L
c
PU  PD cos
L
c
 
  
 
  
sin
L
c
 
  
 
  
PU  PD( )
   
 
For an infinitesimally short pipe, KU  KD  KAC at all frequencies.   
For finite pipe length, three regions:  
      (1) at low-frequencies, KU  KD  KAC. 
      (2) at intermediate frequencies, wide irregular oscillations.  
      (3) at high-frequencies, Re(KU/KAC) ~ 1/2, Im(KU/KAC) ~ - and KD/KAC ~ 0.  
Transition frequency 1,2 separating (1) and (2) scales roughly with 1/L.   
 
One important feature, Re(KU/KSS) approaches a finite positive value K/KSS at high-
frequencies (easily explained by remembering that Re(KAC) ~ -3/2 and Im(KAC) ~ -1).  
Most striking, K tends to increase linearly with L and eventually becomes larger 
(possibly orders of magnitude larger) than the steady-state value KSS.   
To the contrary, KD/KSS vanishes in the high-frequency limit.   
This very strong difference in high-frequency behavior between KU and KD is a 
consequence of fluid storage inside the pipe and obviously must increase linearly with the 
pipe length.   
 
 log10(ω)
Im(KU/KSS)
Real and imaginary parts of the normalized apparent conductivity KU/KSS for 
pipes with different lengths L (thick grey line, L = 0.2 m; thin grey line, L = 1 m; thin black
line, L = 5 m; the pipe radius is equal to 100 μm).  The frequencies are given in Hertz.   
log10(ω)
Re(KU/KSS)
5 m
1 m
0.2 m
-1 1 2 3 4 5
0.5
1
1.5
2
2.5
-1 1 2 3 4 5
-1
-0.5
0.5
 log10(ω)
Im(KD/KSS)
Re(KD/KSS)
log10(ω)
Real and imaginary parts of the normalized apparent conductivity KD/KSS for 
pipes with different lengths L (thin black line, L = 5 m; thin grey line, L = 1 m; thick grey
line, L = 0.2 m; the pipe radius is equal to 100 μm). 
5 m
1 m
0.2 m
-1 1 2 3 4 5
-1
-0.5
0.5
1
-1 1 2 3 4 5
-1
-0.5
0.5
1
AC flow through a two-dimensional network of pipes 
 
Standard method for solving a stead-state flow problem on a pipe network can be used 
for an oscillating flow with a given frequency, except that a complex-valued matrix has 
to be inverted [Bernabé, 1997].   
Goal was not to make realistic simulations of rocks but merely to illustrate the effect of 
wave reflections and interferences in relatively complex pipe systems.  Hence, study is 
limited 10x10, 20x20 and 40x40 2D square networks, with semi-periodic boundary 
conditions and narrow pipe radius distribution (low level of heterogeneity).  
Finally, global upstream and downstream conductivities (averaged over 10 realizations) 
calculated using previous definitions, in which QU and QD are the amplitudes of the total 
volumetric flux into/out of the upstream and downstream network sides, PD–PU = -1 is 
the global pressure amplitude difference, and, L refers to the total length of the network 
in the flow direction.   
 
 
Next Figures show real and imaginary parts of KU/KSS as a function of frequency for 
40x40 and 10x10 networks with different total length L.   
Very similar to single pipe curves.  
Transition frequency monotonically increasing function of the network length L and, 
therefore, likely related to wave reflection from the upstream and downstream ends of the 
network).  
Internal reflections (i.e., from the ends of individual bonds) visible at high frequencies 
(but frequency resolution insufficient to accurately delineate the fluctuations of KU/KSS).   
 
 
 
 
 
 
 
 log10(ω)
Im(KU/KSS)
Re(KU/KSS)
log10(ω)
Real and imaginary parts of the normalized apparent permeability KU/KSS for 
40x40 networks with different total lengths (thin, black line, L = 4 m; thicker, dark grey line, 
L = 1 m; thick, light grey line, L = 0.4 m; the mean radius of the pipes is 100 μm).  Note that 
the diagrams show results of discrete calculations with an increment of 0.1 in log10(ω).  The 
discrete data points are joined by straight lines for visibility.  
4 m
1 m
0.4 m
-1 1 2 3 4 5
1
2
3
-1 1 2 3 4 5
-1.5
-1
-0.5
0.5
1
 
 
 
 
 
 
 
 
log10(ω)
Im(KU/KSS)
Re(KU/KSS)
log10(ω)
Real and imaginary parts of the normalized apparent permeability KU/KSS for 
10x10 networks with different total lengths (thin, black line, L = 2.5 m; thicker, dark grey line, 
L = 1 m; thicker, light grey line, L = 0.25 m; the mean radius of the pipes is 100 μm).  
-1 1 2 3 4 5
1
2
3
-1 1 2 3 5
-1.5
-1
-0.5
0.5
1
2.5 m
1 m
0.25 m
 
 
 
 
 
 
 
 
 
 
 
 
 Confirmation of role of wave reflections from the upstream and downstream ends:  
40x40 and 10x10 networks with an identical total length.  
20x20 networks containing a sharp discontinuity in the pipe radius distribution at 
mid-length in the flow direction.   
 
log10(ω)
Im(KU/KSS)
Re(KU/KSS)
log10(ω)
Real and imaginary parts of the normalized apparent permeability KU/KSS for 
networks with same total length (L = 1 m) and same mean radius (R = 100 μm) but different 
number of nodes (thin, black line, 40x40; thin, dark grey line, 10x10).  The thick, light grey 
line represents  the response of 20x20 networks, in which the mean radius was suddenly 
increased by a factor of 3 at mid-length in the flow direction. The black and dark grey curves 
are nearly coincident  up to the point labeled A and the light grey curve is fairly similar.  Above 
point A the grey curves become nearly coincident while the black one is quite different (see text 
for more details).  
40x40
-1 1 2 3 4 5
0.2
0.4
0.6
0.8
1
1.2
1.4
10x10
20x20
A
-1 1 2 4 5
-1
-0.5
0.5
1
A
 
 
 
 
 
Conclusions 
 
1) AC flow of a compressible fluid through a deformable pipe, pipe network, and, by 
extension porous rock propagates as a wave.  
 
2) Flow waves are interface wave traveling along the fluid-solid interface.  They have 
significant dispersion and attenuation.  They can be identified with Biot slow 
compressional wave.   
 
3) The flow field can be greatly affected by wave reflections and interferences.   
 
4) The characteristic length is the distance L between reflectors.  In the Earth, L could 
represent the thickness of geological formations or the distance between fracture 
intersections, and, therefore, could range from tens of centimeters to hundreds of meters.  
Thus, they could have a significant effect on data recorded at seismic frequencies.  


Oscillating flow of a compressible fluid through deformable pipes and pipe networks: wave propagation phenomena

http://www.earth-prints.org/bitstream/2122/2719/1/33_Bernabe.pdf

Oscillating flow of a compressible fluid through deformable pipes and pipe networks: wave propagation phenomena

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