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

Newtonian & Non-Newtonian Fluids Definition of Newtonian & Non-Newtonian Fluids

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.

Rheology Of Non-Newtonian Fluids

Time-Independent

Time-Dependent

Viscoelastic

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.

Time-Independent Fluids Network Modelling Of Time-Independent Fluids

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.

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

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

Park

Time-Dependent Fluids Network Modelling Of Time-Dependent Fluids

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.

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.

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.

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’’

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

Network Modelling Of Viscoelastic Fluids

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.

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.

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

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

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

Thank You Questions?