Peyman Mostaghimi, Prof. Martin Blunt, Dr. Branko Bijeljic 16 January 2009, Imperial College Consortium on Pore-Scale Modelling The level set method and its application to pore-scale modelling 1

Motivation Network modelling – the representation of the pore space by an equivalent representation of pores and throats – has been successful: we now understand trends in recovery with wettability and can predict single and multi-phase properties. BUT…….the extraction of networks involves ambiguities and there are some cases where the method does not work so well – trapping in unconsolidated media, three-phase flow…. Now have direct three-dimensional imaging of pore spaces. Why not simulate multiphase flow directly on these images? Images from PhD dissertation of M.Piri

What I plan to do Micro CT images contains millions of pixels which are either pore or solid. Use a Cartesian grid solver to solve for flow. For multiphase flow the level set method will be applied to predict the position of the interface between different phases. Predict relative permeability

Governing Equations for Flow Navier-Stokes Equation reduces to Stokes Equation with creeping flow assumption:

Formulation u Equation v Equation w Equation p Equation

Numerical Method Finite difference method accompanied by finite volume formulation applied to the partial differential equations. In Stokes equations, pressure and velocity are coupled, SIMPLE (Patankar) method is applied to decouple them in our numerical procedure.

Gridding Staggered grid: Existence of solid phase in each grid causes six velocity components be zero in the 3 dimensional models.

Boundary Conditions -Inlet and Outlet Constant Pressure ( x direction ) -Walls No Flow -Neighbours Rocks No Flow

Semi-Implicit Method for pressure-Linked Equations The SIMPLE (Semi-Implicit Method for pressure-Linked Equations) Algorithm: 1. Guess the pressure field p* 2. Solve the momentum equations to obtain u*,v*,w* 3. Solve the p’ equation (The pressure-correction equation) 4. p=p*+p’ 5. Calculate u, v, w from their starred values using the velocity-correction equations 6. Solve the discretization equation for other variables, such as temperature, concentration, and turbulence quantities. 7. Treat the corrected pressure p as a new guessed pressure p*, return to step 2, and repeat the whole procedure until a converged solution is obtained.

3D results Obtained Velocity Profile for flow in a duct:

What Next -Since a Micro CT image contains at least 10millions of cells, our solver should be as efficient as possible. -We are about to use more efficient solver in our simulator ( AMG solver ) to reduce the run time as much as possible. - Computing Relative-Permeability in multi-phase flow using results obtained by the level set method (in cooperation with Dr.M.Prodanovic in University of Texas at Austin).

Level Set Image from Wikipedia free encyclopaedia

Level Set Method ( Theory ) -LSM firstly introduced by S.Osher LSM is a method for interface moving problems. -In this method, a new dimension is introduced to the case and define the interface as a level set of function φ( x, y). -Make the initial position as a zero level of a higher-dimensional function φ -Track evolution of φ and determine the zero level set. -The method works for any dimension and handles topology changes naturally which has resulted in a vast number of applications including two-phase compressible flow, grid generation, computer vision, image restoration, minimal surfaces and surfaces of prescribed curvature

Φ(x,y,t=0) Φ=0 Φ(x,y,t=1) Φ(x,y,t=2)

Advantage works in any dimension no special treatment needed for topological changes keep track of contour if it self-intersects during its evolution

Level Set Formulation Constraint: level set value of a point on the contour with motion x(t) must always be 0  (x(t), t) = 0 By the chain rule  t +  (x(t), t) · x(t) = 0 Since F supplies the speed in the outward normal direction x(t) · n = F, where n =  / |  | Hence evolution equation for  is  t + F|  | = 0

Speed Function - In the propagation equation, F is speed of motion of interface in normal direction. -In general F is a function of position. The physics of the phenomenon of interest enters the method via F. - Normal velocity F in our application is obtained from a balance of pressure and surface tension forces and is defined only at the interface 19

Interface between two phases

Drainage and Imbibition M. Prodanovic and S. Bryant. Critical curvatures for drainage and imbibition via level set methods. J. Colloid Interface Sci., This slide is from Dr.M.Prodanovic ‘s Lecture at Imperial College

Curvature – saturation curve pore imbibition events  saturation jumps Irreversibility of events  hysteresis This slide is from Dr.M.Prodanovic ‘s Lecture at Imperial College

Level Set Method ( Application ) Image from the paper by M.Pradanovic -accurate description of pore level displacement of immiscible fluids where capillary forces dominate. -irregular porous media geometry further complicates the problem accounting for the interface topological changes M. Prodanovic and S. Bryant. Critical curvatures for drainage and imbibition via level set methods. J. Colloid Interface Sci., This slide is from Dr.M.Prodanovic ‘s Lecture at Imperial College

What Next ? Estimating Absolute and Relative Permeabilities of a core via its MicroCT image.