Peyman Mostaghimi, Martin Blunt, Branko Bijeljic11th January 2010, Pore-scale project meeting

Direct Numerical Simulation of Transport Phenomena on Pore-space Images

1

Flow at pore scale

In petroleum science and engineering, scales of interest may vary from molecular level to a mega level.

The pore scale is of the order of a typical pore which is in the range few microns. Modelling fluid flow at the pore scale can provide a predictive tool for estimating rock and flow properties at larger scales.

One of the most used ways to capture the

morphology of a porous medium as the

main input for pore scale modelling is

micro-CT imaging.

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.

However…….the extraction of networks involves ambiguities and there are some cases where the method does not work so well.

Now have direct three-dimensional imaging of pore spaces.

Why not simulate multiphase flow directly on these images?

Micro-CT imaging AND DIRECT SIMULATION

Post processing Micro-CT images, a matrix can be generated for a core which shows whether there is a solid inside the voxel or a pore.

Zero means that voxel is a pore and one means it is a solid phase.

Two methods to simulate fluid flow in porous media directly without the need for simplified geometries:

- the lattice Boltzmann method (Edo)

- conventional computational fluid dynamics algorithms based on the relevant flow and conservation equations.

Governing equations

Conservation of mass:

Navier-Stokes equation:

Steady-state and incompressible flow:

0

vt

vvvv 2.

Pt

0 v

vvv 2. P

Dimensionless analysis

0

~v

vv

00

0

/

~

lv

PPP

0

~l

220

2~ l00Re

vl

vPvv ~~~~)~~

.~Re( 2

110/.10

/10/1010Re 5

23

3365

msN

mkgsmm

The dimensionless steady-state Navier-Stokes equation:

Reynolds number for flow in porous media:

v20 PStokes Equation :

FORMULATION

0

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

z

w

y

v

x

u

z

P

z

w

y

w

x

w

y

P

z

v

y

v

x

v

x

P

z

u

y

u

x

u

u Equation(u: velocity in x-direction)

v Equation(v: velocity in y-direction)

w Equation(w: velocity in z-direction)

p Equation

Gridding

Marker-and-cell grid:

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

Discretized Form

x

PP

z

uuu

y

uuu

x

uuu KJIKJIKJiKJiKJiKJiKJiKJiKJiKJiKJi

,,1,,

2

1,,,,1,,

2

,1,,,,1,

2

,,1,,,,1 1

)(

2

)(

2

)(

2

y

PP

z

vvv

y

vvv

x

vvv KJIKJIKjIKjIKjIKjIKjIKjIKjIKjIKjI

,1,,,

2

1,,,,1,,

2

,1,,,,1,

2

,,1,,,,1 1

)(

2

)(

2

)(

2

z

PP

z

www

y

www

x

www KJIKJIkJIkJIkJIkJIkJIkJIkJIkJIkJI

1,,,,

2

1,,,,1,,

2

,1,,,,1,

2

,,1,,,,1 1

)(

2

)(

2

)(

2

0,,1,,,,,1,,,,,1

z

ww

y

vv

x

uu kJIkJIKjIKjIKJiKJi

iKJIKJInbnbKJiKJi Appuaua ,,,,1,,,, jKJIKJInbnbKjIKjI Appvava ,,,1,,,,, kKJIKJInbnbkJIkJI Appwawa ,,1,,,,,,

The momentum equations can be rewritten as:

SIMPLE algorithm

• 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* by algebraic

mutigrid solver 3. Solve the p’ equation (The pressure-correction equation) by algebraic

mutigrid solver 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.

Storing matrices in CRS and AMG for solving all linear systems of equations.

Boundary Condition

0pu 0Nu iKJIKJInbnbKJiKJi Appuaua ,,,,1,,,, jKJIKJInbnbKjIKjI Appvava ,,,1,,,,, kKJIKJInbnbkJIkJI Appwawa ,,1,,,,,,

. . . Boundary Condition

)(8

)(

23

2

2

2

yOy

y

uy

y

uuu pn

)(2

)( 32

2

2

yOy

y

uy

y

uuu ps

22

2

)(3

44

3

8

y

uuu

y

u spn

COMPARISON OF THE Three METHODS FOR BC FOR FLOW BETWEEN TWO INFINITE PARALEL PLATES

first method second method third method

VELOCITY PROFILE FOR FLOW BETWEEN TWO INFINITE PARALEL PLATES

We see non-zero velocity even for one block within the channel and for more than one we see agreement to within machine accuracy with the analytical solution.

Lid-driven Cavity

DISPERSION MODELLING

When a miscible fluid is injected in a flowing fluid in a saturated porous media, it will spread by various mechanisms including advection and diffusion.

In brief, dispersion is the spread or mixing of flowing fluids due to all these mechanisms.

To model advection term we use stream tracing algorithm and for diffusion we apply random walk method.

DiffusionAdvection XXtxttx )()(

STREAMLINE TRACING

Interpolation to estimate the velocity vectors at a point within the grid block

KJiiKJiKJi uxx

x

uuu ,,

,,,,1 )(

KjIjKjIKjI vyy

y

vvv ,,

,,,1, )(

kJIkkJIkJI wzz

z

www ,,

,,1,, )(

)ln(,,,,1 i

e

KJiKJix u

u

uu

x

)ln(,,,1, i

e

KjIKjIy v

v

vv

y

)ln(1,,1,, i

e

kJIkJIz w

w

ww

z

),,min( zyx

)1.( ,,,,1

,,,,10

KJiKJi uu

x

iKJiKJi

euuu

xxx

)1.( ,1,,1,

,1,,1,0

KjIKjI vv

y

iKjIKjI

evvv

yyy

)1.( 1,,1,,

1,,1,,0

kJIkJI ww

z

ikJIkJI

ewww

zzz

The time of flight:

The coordinates of exit location:

Diffusion

tDm 2

CosSinXX ..0

SinSinYY ..0

CosZZ .0

Random walking method for both advection and diffusion:

tDtxVtxttx m 2)()()(

Random walking method just for diffusion part of flow :

advection diffusion

tDXtxttx madvection 2)()(

. . . Diffusion

Gridding:

Resolution:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.51

1.52

2.53

3.54

4.55

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.51

1.52

2.53

3.54

4.55

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

m851.8

5050

m 4255.4 m 702.170 3.6Pe 4.0Pe

m

avg

D

LuPe

tDm 2

. . . Particle tracking

Gridding:

Resolution:

Sandpack LV60B

338338m851.8

. . . Particle tracking

Gridding:

Resolution:

Sandpack LV60B

338338m851.8

. . . Particle tracking

Gridding:

Resolution:

Sandpack LV60B

338338m851.8

DISPERSION COEFFIEICENT

The average of positions of particles:

Variance of X can be calculated:

And the longitudinal dispersion coefficient:

For showing the importance of diffusion, dispersion is modelled for a range Peclet number:

N

tXtXX

N

ii

i

1

)()(

N

tYtYY

N

ii

i

1

)()(

N

tZtZZ

N

ii

i

1

)()(

N

XtXXtX

N

ii

i

1

2

22

))(())((

dt

dDL

2

m

avg

D

LuPe

(Bijeljic et al. 2004)

Multiphase flow at the pore scale

Having the interface at different saturations, the flow of each phase can be modelled by the Stokes solver and the relative permeability can be predicted.

Also for reactive transport (Branko), the code can be used to simulate the flow at each time step.

Courtesy of Masa Prodanovic

Many thanks for your attention