January 31, 2011

R ChrisFirst KAUST Study Group in Mathematics for Industry

23rd – 26th January 2011

King Abdullah University of Science and Technology (KAUST)In association with the Oxford Centre for Collaborative Applied Mathematics (OCCAM)

Schlumberger:Filtercake forming mechanisms at fracture and cavity openings

Industrial representative

Frank F. ChangMustapha Abbad

Increasing the productivity of oil wells isachieved by fracturing the well. This is doneby injecting a fracturing fluid (fracfluid) intothe bore, which opens up a fracture. Thenacid is injected into the flow to ensure thecrack remains open. After that, a diverter isinjected, so particles and fibres block thefracture. Then the process is repeated.

The objectives are:

I. to use the particles and fibres in thefluid to block the highest permeabilitychannels

II. to make the blockage that forms tough,but thin;

III. to ensure that the blockage must beable to withstand the next input offracfluid.

The design parameters should be: particlesize and shape, fibre length and flexibility,stickiness of particles and fibres (resincoating), and input concentration andvelocity. Some of the dimensions of theproblem are depicted in the diagram below.

The input fluid velocity, U=0.1m/s,borewidth, L=16cm, fluid densityρ =1000kg/m3, and fluid viscosity μ =10-3-1Pas s which gives a Reynolds numberRe= ρ UL/ μ =10-104. The group focused onthe canonical problem of a pipe with onefracture. The flow model was the steadyNavier-Stokes equations, with no slip orpenetration conditions (impermeability ofbore rock), assuming a fully developedparabolic profile incoming flow and prescribeda zero pressure at outlet.

They first solved for the flow in the absenceof particles. The diagram of streamlines in the

flow for realistic parameters ( ) isgiven below.

A key non-dimensional quantity fordetermining the motion of the suspensionparticles is the Stokes number for a particle,given by St=ρpdp

2/18μ, where ρp and dp arethe particle density and diameterrespectively. To find the particletrajectories, the group modelled them asspheres subject to “Khan-Richardson forces”assuming that the particles do not influencethe flow, and that particle-particleinteractions in the flow are negligible far from

the origin (i.e. suspension is suitably dispersewithin bulk flow – Taylor dispersion).

The behaviour for radius 1mm and 0.1mmparticles are found to be quite different asshown by the figures below (Figure for 1mmfollowed by 0.1mm), indicating that particleshave a critical radius above which they getentrained by a vortex and held in the mainchannel.

The group also studied the effect of the endof the bore hole and found that the flow inthe region of interest was qualitativelysimilar.

Regarding the role of the particle masses,the group noted that reducing the particlemass increases the particles’ capture, butthen more small particles would be requiredto block the hole. This leads to an optimalparticle size for crack clogging.

To maximise the chance of blocking, thegroup recommends maximising the particlesize entering the crack. This leads to anoptimisation problem since smaller particlesare more likely to enter the crack. The rate ofentry of particles should be maximal (c.f.experiment of particles in a chute) - thissuggests injecting particles as a localisedbolus of high concentration.

Spherical particles optimise the likelihoodof clogging for a given total mass of particles.In practice, rigid fibres which are slow at re-orientating are injected with the flow. Theseenhance the likelihood of clogging.

It should be noted that particle and fibrestickiness will enhance clogging in thechannel, but adhesiveness should becontrolled within the bore (to minimisecoagulation).

The group recommends undertaking anexperiment with neutrally buoyant particlesto validate the particle-capture modelling,and another experiment to quantify the effectof Taylor dispersion of particles in the bore.