Investigating the Flow of Yield Stress Investigating the Flow of Yield Stress

Fluids in Porous MediaFluids in Porous Media

Imperial College London & Schlumberger Research CentreImperial College London & Schlumberger Research Centre

Taha SochiTaha Sochi

What is Yield Stress ?What is Yield Stress ?

The stress at which the substance starts flowing.

The substance is solid below its yield stress and fluid above.

DifficultiesDifficulties

The yield stress value is usually obtained by extrapolation and this limits the accuracy.

Before yield, the pressure is not well-defined. To overcome this, the substance is assumed a fluid with very high viscosity.

DifficultiesDifficulties, continued, continued

The yield is highly dependent on the actual shape of the pore space and its fine details. This is compromised by modelling the throats with regular cylindrical ducts.

While in the case of bulk and tube flow the yield stress is a property of the fluid, in the case of porous media it may depend on the porous media as well.

Our Network ModelOur Network ModelNo element yields unless it is part of a spanning path bridging the inlet to the outlet.

Complete experimental datasets in the literature are very scarce.

Experimental Validation Experimental Validation

Sample of the datasets found in the literature with our network simulation results:

Network Yield: Network Yield: Is it a Percolation Phenomenon?Is it a Percolation Phenomenon?

The conventional percolation applies only to homogeneous elements, i.e. elements having equal intrinsic property.The network elements cannot yield independently as a spanning path bridging the inlet to the outlet is a necessary condition for yield.

The pure percolation approach ignores the dynamic aspects of the pressure field.

Predicting Network Threshold Yield PressurePredicting Network Threshold Yield Pressure

1. Invasion Percolation with Memory (IPM):Find the path of minimum yield pressure connecting the inlet to the outlet by increasing the yield pressure continuously. The threshold yield pressure is the value at which the outlet is first reached.

Assumptions: 1. The yield pressure of a number of serially-connected bonds is the sum of their yield pressures.2. Backtracking is allowed.

Predicting Network Threshold Yield PressurePredicting Network Threshold Yield Pressure2. Path of Minimum Pressure (PMP):Find the path of minimum yield pressure connecting the inlet to the outlet by finding the minimum yield pressure needed to reach each node. The threshold yield pressure is the minimum of the values obtained for the nodes at outlet.Assumptions: 1. The yield pressure of a number of serially-connected bonds is the sum of their yield pressures.2. Backtracking is not allowed.

ResultsResults

In most cases, IPM and PMP agree. When they disagree, PMP gives higher and more realistic values.Reason: backtracking is allowed in IPM but not in PMP.

PMP is more efficient in terms of CPU time and memory.

IPM is the method of choice if backtracking should be allowed and vice versa.

Results, continuedResults, continuedBoth IPM and PMP give lower values than the network model, e.g. for sand pack:

Boundaries Threshold Yield Pressure (Pa)

Lower Upper Actual IPM PMP

0.0 1.0 80.94 53.81 54.92

0.0 0.9 71.25 49.85 51.13

0.0 0.8 61.14 43.96 44.08

0.0 0.7 56.34 38.47 38.74

0.0 0.6 51.76 32.93 33.77

0.0 0.5 29.06 21.52 21.52

Results, continuedResults, continued

The main reason is the assumption that The yield pressure of an ensemble of serially-connected bonds is the sum of their yield pressures.

There are two main issues to be addressed:1. Bottleneck.2. Tortuosity.

Bottleneck Bottleneck The bottleneck of a linear ensemble of serially-connected bonds is the element with the smallest radius.

The threshold yield gradient of the ensemble is the threshold yield gradient of the bottleneck, assuming constant pressure gradient.

Therefore, the threshold yield pressure of a linear ensemble of serially-connected bonds is

Pthr, ens = Pthr, bn L

Pthr, bn is the magnitude of the threshold yield gradient of the bottleneck& L is the length of the linear ensemble.

Bottleneck, continued Bottleneck, continued

where,

The magnitude of the external threshold pressure gradient is

Pthr, ext = Pthr, bn / cos

Where is the angle between the ensemble axis & the pressure field direction.

It is straight forward to prove that this yield pressure is greater than or equal the sum of the yield pressures of the single elements, with the equality holding only if the elements have equal radii.

Bottleneck, continued Bottleneck, continued

Tortuosity Tortuosity The effect of tortuosity is a possible increase in the external threshold pressure gradient and a possible change in the bottleneck.

The bottleneck is the element that requires the greatest external gradient for it to reach its threshold yield gradient considering its alignment and assuming constant external gradient.

Tortuosity, continued Tortuosity, continued Therefore, the magnitude of the external threshold yield gradient is

Pthr, ext = MAX{Pthr, i / cosi} i = 1,2…N

Pthr, i is the magnitude of the threshold yield gradient of element i i is the angle between the axis of element i & the external gradient & N is the number of bonds.

where,

Tortuosity, continued Tortuosity, continued So, the threshold yield pressure of the tortuous ensemble is

Pthr, ens = Pthr, ext d

where d is the linear distance from the inlet to the outlet of the tortuous ensemble in the direction of the external gradient.

This yield pressure could be greater than the yield pressure of the corresponding linear ensemble.

Tortuosity, continued Tortuosity, continued

2. The ensemble can communicate with the external pressure field through all nodes and not only through the inlet and outlet, otherwise the tortuosity has no effect. This assumption is reasonable because the ensemble is part of a network with the substance assumed initially to be a fluid with very high viscosity.

This argument is based on two assumptions:1. The external pressure gradient is constant in magnitude and direction, which is a reasonable assumption for the network before yield.

Conclusion Conclusion Even when the bottleneck and tortuosity effects are taken into account, the network threshold yield pressure might not be predicted precisely because of the dynamic aspects of the network pressure field, i.e. the flowing network elements are not just an ensemble of serially-connected bonds but are part of a larger network which requires a stable pressure configuration, especially when we notice that the substance before yield is assumed to be a fluid with very high viscosity and not a solid.

AcknowledgementsAcknowledgements

Pore Scale Modelling Consortium Pore Scale Modelling Consortium

Schlumberger Research Centre Schlumberger Research Centre

Thank YouThank You

Questions?Questions?