1. Modelling the Flow of non-Newtonian Fluids in Porous Media
Pore Scale Modelling Consortium
Imperial College London
Modelling the Flow of non-Newtonian Fluids in Porous Media
Taha Sochi & Martin Blunt

2. Newtonian & Non-Newtonian Fluids
Definition of
Newtonian & Non-Newtonian Fluids

3. Three groups of behaviour:
Newtonian: stress is proportional to strain rate: t  g
Non-Newtonian: this condition is not satisfied.
Three groups of behaviour:
1. Time-independent: strain rate solely depends on instantaneous stress.
2. Time-dependent: strain rate is function of both magnitude and duration of stress.
3. Viscoelastic: shows partial elastic recovery on removal of deforming stress.

4. Rheology Of
Non-Newtonian Fluids

5. Time-Independent

6. Time-Dependent

7. Viscoelastic

8. Thixotropic vs. Viscoelastic
Time-dependent behaviour of thixotropic arises because of change in structure.
The important one, from a causal standpoint, is that, while the time-dependent behaviour of
viscoelastic fluids arises because the response of stresses and strains in the fluid to changes
in imposed strains and stresses respectively is not instantaneous, in a thixotropic fluid such
response is instantaneous and the time-dependent behaviour arises purely because of
changes in the structure of the fluid as a result of shear.
Time-dependency of viscoelastic arises because response is not instantaneous.

9. Time-Independent Fluids
Network Modelling Of
Time-Independent Fluids

10. Network Modelling Strategy
Combine the pore space description of the medium with the bulk rheology of the fluid.
The bulk rheology is used to derive analytical expression for the flow in simplified pore geometry.
Examples: Herschel-Bulkley & Ellis models.

11. Herschel-Bulkley This is a general time-independent model t Stress
to Yield stress
C Consistency factor
g Strain rate
n Flow behaviour index

13. Ellis This is a shear-thinning model t Stress mo Zero-shear viscosity
g Strain rate
t1/2 Stress at mo / 2
a Indicial parameter

14. Park

15. Time-Dependent Fluids
Network Modelling Of
Time-Dependent Fluids

16. Network Modelling Strategy
There are three major cases:
1. Flow of strongly shear-dependent fluid in
medium which is not very homogeneous:
Very difficult to model because:
a. Difficult to track fluid elements in pores and
determine their shear history.
b. Mixing of fluid elements with various shear
history in individual pores.

17. Network Modelling Strategy
2. Flow of shear-independent or weakly shear-
dependent fluid in porous medium:
Apply single time-dependent viscosity function to all pores at each instant of time and hence simulate time development.

18. Network Modelling Strategy
3. Flow of strongly shear-dependent fluid in very
homogeneous porous medium:
a. Define effective pore shear rate.
b. Use very small time step to find viscosity in
the next instant assuming constant shear.
c. Find change in shear and hence make
correction to viscosity.
If injection from reservoir, assume large medium to avoid edge effect
Possible problems: edge effects in case of injection from reservoir & long CPU time.

19. Godfrey This is suggested as a thixotropic model m Viscosity
t Time of shearing
mi Initial-time viscosity
Dm’ & Dm’’ Viscosity deficits associated
with time constants l’ & l’’

20. Stretched Exponential Model
This is a general time-dependent model
m Viscosity
t Time of shearing
mi Initial-time viscosity
min Infinite-time viscosity
ls Time constant

21. Network Modelling Of
Viscoelastic Fluids

22. Network Modelling Strategy
There are mainly two effects to model:
1. Time dependency:
Apply the same strategy as in the case of time-dependent fluid.

23. Network Modelling Strategy
2. Thickening at high flow rate:
As the flow in porous media is mixed shear-extension flow due mainly to convergence-divergence, with the contribution of each component being unquantified and highly dependent on pores actual shape, it is difficult to predict the share of each especially when the pore space description is approximate.
One possibility is to use average behaviour, depending on porous medium, to find the contribution of each as a function of flow rate.

24. Upper Convected Maxwell
This is the simplest and most popular model
t Stress tensor
l1 Relaxation time
mo Low-shear viscosity
g Rate-of-strain tensor

25. Oldroyd-B This is the second in simplicity and popularity
t Stress tensor
l1 Relaxation time
l2 Retardation time
mo Low-shear viscosity
g Rate-of-strain tensor

26. Future Work Implementation of time-dependent strategy
Possible implementation of viscoelastic effects.

27. Thank You
Questions?