This document summarizes research on modeling the flow of Herschel fluids in porous media. It discusses using the Herschel model to describe non-Newtonian fluids and presents results showing how fluid yield stress affects flow patterns in Berea and sandstone networks. It also compares network and single tube models, validates the network model using experimental data, and discusses incorporating additional physics into future models like viscoelasticity and two-phase flow.

1. Flow of Herschel fluid in porous media
Part II
PERM Group Imperial College London
Taha Sochi & Prof. Martin J. Blunt

2. Introduction
Newtonian & non-Newtonian fluids.
Groups of non-Newtonian.
Herschel model:
τ = το + Cγn
Herschel classes.

3. For Herschel fluid, the volumetric flow rate
in cylindrical tube is:
το C n Herschel parameters
L Tube length
∆P Pressure difference
τw ∆PR/2L Where R is the tube radius
( ) 





+
+
+
−
+
+
−
−





∆
=
+
nnnP
L
C
Q oowoow
own
n
/11/12
)(2
/13
)(8
223
/1
11 ττττττ
ττ
π

5. The Latest Results for Fluid with
Yield Stress
Berea (50): interior elements start flowing
with total blocking.
Berea (70): interior elements plus border
elements start flowing partially.
Berea (300): Large number of elements
flowing.

6. S. Pack (30): interior elements start
flowing with total blocking.
S. Pack (50): interior elements plus border
elements start flowing partially.
S. Pack (300): Large number of elements
flowing.

7. Percolation: flow starts when percolation
threshold pressure gradient is reached.
Streaking: flow starts before percolation
threshold pressure gradient is reached.
Fluid with Yield Stress
Percolation vs. Streaking
Streaking is observed for both Berea and
sand pack networks:

8. το
= 1.0Pa Network actual
threshold pressure
gradient (kPa/m)
Percolation threshold
pressure gradient
(kPa/m)
Tube threshold
pressure gradient
(kPa/m)
Berea (CN = 4.2) 15 93 173
Sand Pack (CN = 5.5) 17 40 41
το
= 10.0Pa Network actual
threshold pressure
gradient (kPa/m)
Percolation threshold
pressure gradient
(kPa/m)
Tube threshold
pressure gradient
(kPa/m)
Berea (CN = 4.2) 150 930 1730
Sand Pack (CN = 5.5) 170 400 410

9. The comparison is made between the
network and a single tube, within a bundle of
tubes, having the same Newtonian
volumetric flow rate & porosity.
Network vs. Single Tube
Comparison & Explanation

11. The bulk rheology, as function of concentration,
is obtained from the actual experimental data
points.
9 complete sets of data for Bingham aqueous
solution of Carbopol 941 in packed column of
glass beads.
Simulation run with scaled sand pack network
with single permeability obtained from the most
Newtonian-like case.
Results:
Experimental Data: Chase et al.

13. Al-Fariss varied permeability on case-by-case
basis to fit experimental data.
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.
Results:
Experimental Data: Al-Fariss et al.

15. Network Consistency Check
To rule out the possibility of erratic behaviour of
the network or gross error, simulation run for
the ‘worst’ case parameters with n varying
smoothly between 0.80 to 1.00.
From the results, a 3D graph, n-P-Q, obtained:
Thus; the network is well-behaved & no gross
error.

16. Analysing Al-Fariss Data
The 16 data sets are divided into 4 groups.
In each group, the common factor between the
data sets is fluid and porous medium, and the
difference is temperature only.
There is an element of inconsistency in the data
as there is no obvious correlation between fluid
properties and temperature, and hence no
correlation between bulk & in-situ rheologies.

17. One possibility is that the wax-oil mixture is not
homogeneous so more complex physical
phenomena; e.g. wax precipitation; took place.
This might be inferred from the more consistent
results of waxy crude oil.

18. 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.

19. Plan for Future Work
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.

20. Finally…
Special thanks to…
Martin
Thank you all…
& Schlumberger