investigating shear-thinning fluids in porous media with yield stress using a herschel model perm...
TRANSCRIPT
Investigating shear-thinning fluids in
porous media with yield stress using
a Herschel model
PERM Group Imperial College London
Taha Sochi & Martin J. Blunt
Shear stress is proportional to shear rate
Constant of proportionality, , is the constant viscosity
Newtonian Fluids
Previous condition is not satisfied
Three groups of behaviour:
1. Time-independent: shear rate solely depends on instantaneous stress.
2. Time-dependent: shear rate is function of both magnitude and duration of shear.
3. Viscoelastic: shows partial elastic recovery on removal of deforming stress.
Non-Newtonian Fluids
We deal with a sub-class of the first group
using a Herschel-Bulkley model:
Cn
Shear stressYield stressC Consistency factorShear raten Flow behaviour index
Current Research
bb
cc
aa
cc
For Herschel fluid, the volumetric flow rate in cylindrical tube is:
C n Herschel parameters
L Tube length
P Pressure difference
w PR/2L Where R is the tube radius
nnnPL
CQ oowoow
own
n
/11/12)(2
/13)(8
223
/1
11
Analytical ChecksNewtonian: = 0 n = 1
Power law: = 0 n ≠ 1
LC
PRQ
8
. 4
Bingham: ≠ 0 n = 1
nn
PLC
RQ
L
R
n
nn
111
213
4/1
4
8.
44
3
1
3
41
8
.
w
o
w
o
LC
PRQ
Non-Newtonian Flow Summary
Newtonian & non-Newtonian defined.
The result verified analytically.
Three broad groups of non-Newtonian found.
Herschel have six classes.
Expression for Q found using two methods.
Network Modelling
Obtain 3-dimensional image of the pore space.
Build a topologically-equivalent network with pore sizes, shapes & connectivity reflecting the real network.
Pores & throats modelled as having triangular, square or circular cross-section.
Most network elements (>99%) are not circular.
Account for non-circularity, when calculating Q from Herschel expression for cylinder, by using equivalent radius:
4/1
8
G
Req
where conductance, G, found empirically from numerical simulation.
(from Poiseuille)
and hence solve the pressure field across the entire network.
Start with initial guess for effective viscosity, , in each network element.
Simulating the FlowAs pressure drop in each network element is not known, iterative method is used:
Invoke conservation of volume applying the relation:
ii
ii
i L
PGQ
Obtain total flow rate & apparent viscosity.
Knowing pressure drop, update effective viscosity of each element using Herschel expression with pseudo-Poiseuille definition.
Re-compute pressure using updated viscosities.
Iterate until convergence is achieved when specified tolerance error in total Q between two consecutive iteration cycles is reached.
Iteration & Convergence
Usually converges quickly (<10 iterations).
Algebraic multi-grid solver is used.
Could fail to converge due to non-linearity.
Convergence failure is usually in the form of oscillation between 2 values.
Sometimes, it is slow convergence rather than failure, e.g. convergence observed after several hundred iterations.
To help convergence:
1. Increase the number of iterations.
2. Initialise viscosity vector with single value.
3. Scan fine pressure-line.
4. Adjust the size of solver arrays.
Testing the Code
1. Newtonian & Bingham quantitatively verified.
3. All results are qualitatively reasonable:
2. Comparison with previous code gives
similar results.
Initial Results
3. Lack of experimental data.
Data is very rare especially for oil.
Difficulties with oil:
1. As oil is not a single well-defined species, bulk & in-situ rheologies for the same sample should be available.
2. No correlation could be established to find generic bulk rheology (unlike Xanthan where correlations found from concentration).
Al-Fariss varied permeability on case-by-case basis to fit experimental data.
Al-Fariss/Pinder paper SPE 13840:
16 complete sets of data for waxy & crude oils in 2 packed beds of sand.
Simulation run with scaled sand pack network to match permeability.
We did not use any arbitrary factor.
Some initial results:
Discussion & ConclusionsHerschel is a simple & realistic model for wide range of fluids.
Network modelling approach is powerful tool for studying flow in porous media.
Current code passed the initial tests & could simulate all Herschel classes.
Al-Fariss initial results are encouraging.
More experimental data need to be obtained & tested.
Plan for Future WorkAnalysing network flow behaviour at transition between total blocking & partial flow.
Including more physics in the model such as wall- exclusion & adsorption.
Modelling viscoelasticity.
Possibility of studying time-dependent fluids.
Modelling 2-phase flow in porous-media for two non-Newtonian fluids.
Finally…
Special thanks to Martin & Schlumberger
&
Happy New Year to you all!