Modelling the Flow of non-Newtonian Fluids in Porous Media

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

2. Definition of Newtonian & Non-Newtonian Fluids

3. 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. Time-dependency of viscoelastic arises because response is not instantaneous.

9. 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 toYield stress CConsistency factor gStrain rate n Flow behaviour index

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

14. Park

15. 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. Possible problems: edge effects in case of injection from reservoir & long CPU time.

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

20. This is a general time-dependent model mViscosity t Time of shearing miInitial-time viscosity minInfinite-time viscosity lsTime constant Stretched Exponential Model

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. This is the simplest and most popular model Upper Convected Maxwell t Stress tensor l1Relaxation time moLow-shear viscosity g Rate-of-strain tensor

25. This is the second in simplicity and popularity t Stress tensor l1Relaxation time l2Retardation time moLow-shear viscosity g Rate-of-strain tensor Oldroyd-B

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

27. Questions? Thank You