Modelling the Flow of non-Newtonian Fluids in Porous MediaPore Scale Modelling Consortium Imperial College LondonTaha Sochi & Martin Blunt

Definitionof Newtonian & Non-Newtonian Fluids

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.

RheologyOfNon-Newtonian Fluids

Time-Independent

Time-Dependent

Viscoelastic

Thixotropic vs. ViscoelasticTime-dependency of viscoelastic arises because response is not instantaneous.Time-dependent behaviour of thixotropic arises because of change in structure.

Network Modelling OfTime-Independent Fluids

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. Network Modelling Strategy

This is a general time-independent modelt Stressto Yield stressC Consistency factor g Strain raten Flow behaviour index Herschel-Bulkley

This is a shear-thinning modelt Stressmo Zero-shear viscosity g Strain ratet1/2 Stress at mo / 2a Indicial parameter Ellis

Network Modelling OfTime-Dependent Fluids

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

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

3. Flow of strongly shear-dependent fluid in very homogeneous porous medium:Network Modelling Strategya. 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.

GodfreyThis is suggested as a thixotropic modelm Viscosityt Time of shearingmi Initial-time viscosityDm & Dm Viscosity deficits associated with time constants l & l

Stretched Exponential ModelThis is a general time-dependent model m Viscosityt Time of shearingmi Initial-time viscositymin Infinite-time viscosityls Time constant

Network Modelling OfViscoelastic Fluids

There are mainly two effects to model:Network Modelling Strategy1. Time dependency:Apply the same strategy as in the case of time-dependent fluid.

Network Modelling Strategy2. 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 MaxwellThis is the simplest and most popular modelt Stress tensorl1 Relaxation timemo Low-shear viscosityg Rate-of-strain tensor

Oldroyd-Bt Stress tensorl1 Relaxation timel2 Retardation timemo Low-shear viscosityg Rate-of-strain tensorThis is the second in simplicity and popularity

Future WorkImplementation of time-dependent strategyPossible implementation of viscoelastic effects.

Thank YouQuestions?

********The important one, from a causal standpoint, is that, while the time-dependent behaviour ofviscoelastic fluids arises because the response of stresses and strains in the fluid to changesin imposed strains and stresses respectively is not instantaneous, in a thixotropic fluid suchresponse is instantaneous and the time-dependent behaviour arises purely because ofchanges in the structure of the fluid as a result of shear.**********If injection from reservoir, assume large medium to avoid edge effect*********