1. Investigating shear-thinning fluids in
PERM Group Imperial College London
Investigating shear-thinning fluids in
porous media with yield stress using
a Herschel model
Taha Sochi & Martin J. Blunt

2. t = mg Newtonian Fluids Shear stress is proportional to shear rate
Constant of proportionality, m, is the constant viscosity
Tau is shear stress, gama is shear rate, i.e. rate of change of shear strain-the ratio of deformation to original dimensions. In the case of shear strain, though, it's the amount of deformation perpendicular to a given line rather than parallel to it. The ratio turns out to be tan A, where A is the angle the sheared line makes with its original orientation. (deformation in the direction of motion to the original length perpendicular to the motion).
Newtonian: constant viscosity with shear rate.

3. Previous condition is not satisfied
Non-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.
i.e. shear stress is not proportional to shear rate.
Depends possibly on time lapse between consecutive applications of stress.
Materials possess properties of both fluids and elastic solids. Jelly-like
3. Viscoelastic: shows partial elastic recovery on removal of deforming stress.

4. t = to + Cgn Current Research
We deal with a sub-class of the first group using a Herschel-Bulkley model:
t = to + Cgn
t Shear stress
to Yield stress
C Consistency factor
g Shear rate
n Flow behaviour index
Consistency factor is the viscosity only for Newtonian and Bingham as special cases of Herschel.

5. a b c
6 classes
Graphically the fluid types we deal with (i.e. current code) are shown. No yield stress: n=0, n>0, n<0. Yield stress: n=0, n>0, n<0. c

6. For Herschel fluid, the volumetric flow rate in cylindrical tube is:
to C n Herschel parameters
L Tube length
DP Pressure difference
tw DPR/2L Where R is the tube radius
-Two different derivations. tau-_w is half pressure gradient times radius (used to simplify the expression)

7. Analytical Checks Newtonian: to = 0 n = 1 Power law: to = 0 n ≠ 1
-Each derived independently and obtained as a special case for Herschel.
Bingham: to ≠ n = 1

8. Non-Newtonian Flow Summary
Newtonian & non-Newtonian defined.
Three broad groups of non-Newtonian found.
Herschel have six classes.
Expression for Q found using two methods.
-Q for cylindrical tube
The result verified analytically.

9. Network Modelling Obtain 3-dimensional image of the pore space.
Build a topologically-equivalent network with pore sizes, shapes & connectivity reflecting the real network.
-usually from voxel images. In our case generated by simulating the geological process. This approach used by Bryant and Oren.
S shape factor, A cross sectional area, P perimeter.
Wetting phase reside in the corners. Important for 2-phase flow
Pores & throats modelled as having triangular, square or circular cross-section.

10. Most network elements (>99%) are not circular.
Account for non-circularity, when calculating Q from Herschel expression for cylinder, by using equivalent radius:
(from Poiseuille)
-Most are triangular and some are squares. After Oren.
where conductance, G, found empirically from numerical simulation.

11. Simulating the Flow
As pressure drop in each network element is not known, iterative method is used:
Start with initial guess for effective viscosity, m, in each network element.
Invoke conservation of volume applying the relation:
-Sum is zero (flow in & flow out add to zero). So we have two unknowns delP and Mu.
and hence solve the pressure field across the entire network.

12. Re-compute pressure using updated viscosities.
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.
- Iterate the last two steps
Obtain total flow rate & apparent viscosity.

13. Iteration & Convergence
Algebraic multi-grid solver is used.
Usually converges quickly (<10 iterations).
Could fail to converge due to non-linearity.
Convergence failure is usually in the form of oscillation between 2 values.
- Distinguished by very fast convergence (reduction in computer time) compared with other solvers
-Drawback: require large memory. Not in our case
Sometimes, it is slow convergence rather than failure, e.g. convergence observed after several hundred iterations.

14. 1. Increase the number of iterations.
To help convergence:
1. Increase the number of iterations.
2. Initialise viscosity vector with single value.
3. Scan fine pressure-line.
- Recommended 50. trade-off between convergence and computer time.
- Failure to converge in some cases if previous value used.
- For some reason it might converge at some point but not at its neighbours, e.g. 67.8Pa not 67.5 or 68
- Fail to converge if some vectors are very large.
4. Adjust the size of solver arrays.

15. Testing the Code 1. Newtonian & Bingham quantitatively verified.
2. Comparison with previous code gives
similar results.
- non-Newtonian solver flow results compared with Newtonian plus constant viscosity
-High-Pressure plateau for Bingham.
- Shape of curve, curvature, blocking & yield values
3. All results are qualitatively reasonable:

16. - Simulations with different networks (Berea, scaled and non-scaled SP with different permeabilities)

17. -What seems blocking is due to poor resolution.

22. Initial Results 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.
- Xavier correlations.
2. No correlation could be established to find generic bulk rheology (unlike Xanthan where correlations found from concentration).
3. Lack of experimental data.

23. 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.
Al-Fariss varied permeability on case-by-case basis to fit experimental data.
Two sand columns 0.91m, 315D, 0.36 porosity & 1.0m, 1380 D, 0.44 porosity.
Multiplied by root (Kexp/Knet)
Results: good, bad & between
We did not use any arbitrary factor.
Some initial results:

24. Good
Notice q/PG not Q/PG because we are comparing two different cases. In the past no difference between Q & q apart from scaling. Q used in the past for simplicity.

25. Bad

26. Between.
Discussion of possible source of failure, e.g. sensitivity of some Herschel parameters (n?) and possible experimental errors.
More physics involved & Herschel is a crude & simple model for complex fluids.
Work is still going on. These are initial results.
Mention to other papers on oil and aqueous fluids

27. Discussion & Conclusions
Herschel 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.
-Mention to the papers waiting processing
Al-Fariss initial results are encouraging.
More experimental data need to be obtained & tested.

28. Plan for Future Work
Analysing network flow behaviour at transition between total blocking & partial flow.
Including more physics in the model such as wall- exclusion & adsorption.
Modelling 2-phase flow in porous-media for two non-Newtonian fluids.
-e.g. Streaking behaviour or percolating path.
Wall exclusion where appropriate
Modelling viscoelasticity.
Possibility of studying time-dependent fluids.

29. Finally… Special thanks to Martin & Schlumberger &
Happy New Year to you all!
Thank you all