Investigating shear-thinning fluids in porous media with yield stress using a Herschel model PERM Group Imperial College London Taha Sochi & Martin J. Blunt

 Shear stress is proportional to shear rate Constant of proportionality, , is the constant viscosity Newtonian Fluids

Previous condition is not satisfied Three groups of behaviour: 1. Time-independent: shear rate solely depends on instantaneous stress. 2. Time-dependent: shear rate is function of both magnitude and duration of shear. 3. Viscoelastic: shows partial elastic recovery on removal of deforming stress. Non-Newtonian Fluids

We deal with a sub-class of the first group using a Herschel-Bulkley model:    C  n  Shear stress    Yield stress C Consistency factor  Shear rate n Flow behaviour index Current Research

b c a c

For Herschel fluid, the volumetric flow rate in cylindrical tube is:    C n Herschel parameters L Tube length  P Pressure difference  w  PR/2L Where R is the tube radius

Analytical Checks Newtonian:    = 0 n = 1 Power law:    = 0 n  1 Bingham:     0 n = 1

Non-Newtonian Flow Summary Newtonian & non-Newtonian defined. The result verified analytically. Three broad groups of non-Newtonian found. Herschel have six classes. Expression for Q found using two methods.

Network Modelling Obtain 3-dimensional image of the pore space. Build a topologically-equivalent network with pore sizes, shapes & connectivity reflecting the real network. Pores & throats modelled as having triangular, square or circular cross-section.

Most network elements (>99%) are not circular. Account for non-circularity, when calculating Q from Herschel expression for cylinder, by using equivalent radius: where conductance, G, found empirically from numerical simulation. (from Poiseuille)

and hence solve the pressure field across the entire network. Start with initial guess for effective viscosity, , in each network element. Simulating the Flow As pressure drop in each network element is not known, iterative method is used: Invoke conservation of volume applying the relation:

Obtain total flow rate & apparent viscosity. Knowing pressure drop, update effective viscosity of each element using Herschel expression with pseudo-Poiseuille definition. Re-compute pressure using updated viscosities. Iterate until convergence is achieved when specified tolerance error in total Q between two consecutive iteration cycles is reached.

Iteration & Convergence Usually converges quickly (<10 iterations). Algebraic multi-grid solver is used. Could fail to converge due to non-linearity. Convergence failure is usually in the form of oscillation between 2 values. Sometimes, it is slow convergence rather than failure, e.g. convergence observed after several hundred iterations.

To help convergence: 1. Increase the number of iterations. 2. Initialise viscosity vector with single value. 3. Scan fine pressure-line. 4. Adjust the size of solver arrays.

Testing the Code 1. Newtonian & Bingham quantitatively verified. 3. All results are qualitatively reasonable: 2. Comparison with previous code gives similar results.

Initial Results 3. Lack of experimental data. Data is very rare especially for oil. Difficulties with oil: 1. As oil is not a single well-defined species, bulk & in-situ rheologies for the same sample should be available. 2. No correlation could be established to find generic bulk rheology (unlike Xanthan where correlations found from concentration).

Al-Fariss varied permeability on case-by-case basis to fit experimental data. Al-Fariss/Pinder paper SPE 13840: 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. Some initial results:

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. Current code passed the initial tests & could simulate all Herschel classes. Al-Fariss initial results are encouraging. More experimental data need to be obtained & tested.

Plan for Future Work Analysing network flow behaviour at transition between total blocking & partial flow. 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.

Finally… Special thanks to Martin & Schlumberger & Happy New Year to you all!