Transcript Enhanced Mixing in Heterogeneous Buckley Leverett Flow due to Temporal Fluctuations.ppt (706.5Kb)

Enhanced Mixing in Heterogeneous Buckley
Leverett Flow due to Temporal Fluctuations
D Bolster, M Dentz & J Carrera
Contact Info: [email protected]
Introduction
Results
Mathematical Model
Effective Dispersion Coefficient – Ensemble Average

  dxdx'  dt '  q1 ' ( x, t ) q1 ' ( x ' , t ' )  G ( x, x ' , t , t ' )
 x

S
D
dxS

It is well known that heterogeneities in a porous medium have an impact on
the large scale transport. For single phase flow this effect can often be
upscaled using the concept of effective dispersion. Here we use a similar
concept for two phase displacement flow . Beyond spatial heterogeneities
we also study the influence of a temporally fluctuating field as temporal
fluctuations are known to enhance this effective dispersion.
Three Contributions
Heterogeneous Medium
“Homogeneous” Equivalent
Fig 2:Idealised Displacement Problem (Buckley Leverett)
Assumptions:
D  D ( x )  D (t )  D ( x, t )
•Each phase has constant density
•Constant Porosity
Spatial
Heterogeneity
‘Mixing’
•Neglect Buoyancy Effects (Horizontal Plane)


2
qq
Q
Temporal Mixing Term:
Real /Uncertainty?

 
2
qq
Q
2
tt
Temporally
Enhanced
‘Mixing’
Fractional Flow Model
Fig 1:Representing Transport by an Effective Homogeneous Medium
Motivation
S F S F S
 

0
t S x S z
 k1 k2 

  
1
2
F  qT 

k
2


 2 
Temporal Term = Measure of Uncertainty of Location of Front
.qT  0
Fig 3:Uncertainty due to Temporal
Heterogeneity & Temporal Fluctuations
Example 1 – CO2 Sequestration
In the case of carbon sequestration CO2 is injected into a water /brine
filled aquifer. The effective dispersion is particularly useful as it gives
a measure of the ‘contact zone’, which plays a very important role
regarding reactions between the fluids, dissolution and trapping. It
may often desirable to enhance this ‘contact zone’ – or, enhance the
effective dispersion coefficient.
Example 2– Enhanced Oil Recovery
As with sequestration the contact zone plays an important role.
However, here it is typically desirable to minimise spreading. The
study here presents physical insight into how this might be done.
Total Flow Rate
Q  Q  q(t )  q( x)  q( x, t )
Mean
Upscaling
Temporal
Spatial
Spatio Temporal
S
 Q  ( S )   D  S
t
Effective Dispersion
Captures fluctuations
1
 


d

  '   d 2


2





d




D (t )  tt  Q  
0

 2 


1
D ( x )   qq   

d

0




Ratio
Kuo Number (Ratio
Variances
of Timescales)
Fig 4:Fluid Number for
vaious viscosity ratios
Fluid Number (Depends only on ratio of
viscsoties of fluids)
Conclusions
Perturbation Approach
q  Q
 Injected
 Displaced
S  S0  S1   S2
2
2
•As in single phase contaminant transport spatial heterogeneity increases
‘mixing’ (spreading)
•Similarly temporal fluctuations can enhance ‘mixing’
•Temporal fluctuations add an additional level of uncertainty, which can appear
like false ‘mixing’