Numerical Porous MediaKAUST SRI Center

Modeling and simulation of multiscale problems

N Ahmed, VM Calo, Y Efendiev, H Fayed,

O Iliev, Z.Lakdawala, K.Leonard, G.Printsypar

Modeling and simulation of multiscale problems

1. About NumPor2. DRP approach3. Pore scale simulation of non-Newtonian flow 4. Pore scale simulation of reactive flow5. Multiscale algorithms and model reduction6. Summary

DRP Workshop, KFUPM, April 8-9, 2015

Numerical Porous Media

Yalchin Efendiev (Director)

Victor Calo (Co-Director), Craig Douglas, Oleg Iliev, Peter Markowitch

(Associate Directors)

http://numpor.kaust.edu.sa

NumPor: Common solution strategies for diverse applications

NumPor Collaborators

Multiscale Techniques and Applications

• Bridging pore-scale information to field scale with robust multiscale methods and uncertainty quantification tools

• Examples: non-Newtonian, reactive flow or multiphase flow in porous media, geomechanics of fractured reservoirs, etc.

geological structures Intrinsic heterogeneitiesGeological structures

distinct heterogeneities

boundary layer

single pores

field scale macro scale local scale micro/pore scalemacro scale minimum continuum

Length scale

DRP approach

DRP simulation process

Fluid Flow Simulation

Transport and reactionsimulation

Voxelised Geometry

Navier—Stokes equations

Advection—diffusion equation with reactive boundary conditions

MD Simulations

Reaction & Diffusion

coefficients

Imaging, segmentation, characterization

Segmentationof Palatinate Sandstone

• Porosity 25.7 %• Downscaled to 512³ voxels

www.geodict.com

Pore Size Distribution (Sandstone)

Pore-scale simulation of non-Newtonian flows

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Computation with Carreau model in 3D CT image of sandstone

incompressibility (mass

conservation)

momentum conservation𝒯=2𝜇 �̇�𝑖𝑗 𝑠𝑡𝑟𝑒𝑠𝑠𝑡𝑒𝑛𝑠𝑜𝑟

strain tensor

Computation with Carreau model in 3D CT image of sandstone - Viscosity

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Computation with Carreau model in 3D CT image of sandstone - Velocity

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Macroscopic pressure and apparent viscosity

Apparent average viscosity

Pressure drop versus velocity

Pore-scale modeling and simulation of reactive flow

© Fraunhofer ITWM 16

Mathematical model: transport and reaction

Once the velocity, , has been obtained, we solve

with the reactive boundary conditions:

Here the function describes the kinetics of the reaction.

Several different isotherms exist depending on the particular application

under consideration and assumptions made.

2fluid, , 0,

cc D c t

t

v x

Species concentrationAdsorbed species concentrationDiffusion coefficient

(number/m3)(number/m2)(m/s2)

( , , 0) , .f c mm

D c tt

n x

© Fraunhofer ITWM 17

Reaction Isotherms: Henry

• The Henry isotherm is the simplest, and assumes linear adsorption and desorption

• Advantages: Linear – simple(r) to solve both analytically and numerically.

• Easier to perform homogenization.

• Disadvantages : Adsorption rate is unbounded, and independent on adsorbed concentration.

It is only accurate at low concentrations of

ads des( , ) , , 0.f c m c mk tk x

Adsorption coefficientDesorption coefficient

(m/s)(1/s)

© Fraunhofer ITWM 18

Reaction Isotherms: Langmuir

• The first isotherm to be derived mathematically was the Langmuir Isotherm:

• Advantages: Adsorption saturates as increases.

Mathemtaically derived.

The quanttity can often be experimentally evaluated.

• Disdvantages : Assumes non—ionic, non—interacting ideal gas particles

adsorbing at a 2D solid interface.

ads des( , ) 1 , , 0.km

f c m c m tkm

x

Adsorption coefficientDesorption coefficientMaximal possible surface concentration

(m/s)(1/s)(number/m2)

© Fraunhofer ITWM 19

Reaction Isotherms: Frumkin

• Further isotherms exist. For example the Frumkin isotherm:

• Advantages: More accurate for describing interacting molecules

• Disdvantages : Highly nonlinear – complex to solve numerically and

analytically.

The quantity is difficult to determine.

ads des

2( , ) 1 exp , , 0.

m mf c m c m t

kTk k

m

x

T

Adsorption coefficientDesorption coefficientMaximal possible surface concentrationInteraction Boltzmann constantTemperature

(m/s)(1/s)(number/m2)(m4 Kg/(s2 number))(m2 Kg/(s2 K) )K (Kelvin)

© Fraunhofer ITWM 20

Mathematical model: efficiency

II IIads des

ˆˆ ˆ ˆ ˆ1 ,ˆ

Da Dam

c c mm

n

aIads

Id

ds

deses

Da

Da

,

,

k

V

k L

V

I Iads des

ˆ ˆˆ ˆ1 ,

ˆ ˆDa Da

m mc m

t m

ˆ ˆˆ 0.t x

Reactive Robin Boundary condition:

Iads

I

ads

2de

I

des

Iads

I sIdes

Da

D

Da Pe,

PDa e,a

k L

D

k L

D

Ratio reaction rate : advection rate

Damköhler number First type

Ratio reaction rate : diffusion rate

Damköhler number Second typeAdsorption

Desorption

© Fraunhofer ITWM 21

Parameter Set 1:

Pe = 10

DaIads = DaI

des = 10

Parameter Set 2:

Pe = 0.1

DaIads = DaI

des = 10

t = 5 x 10-4

t = 10 x 10-4 t = 15 x 10-4 t = 20 x 10-4

© Fraunhofer ITWM 22

Numerical results: total adsorbed mass

( ) ( , )M t m t dS

x

© Fraunhofer ITWM 23

Parameter Set 3:

Pe = 10

DaIads = DaI

des = 0.1

Parameter Set 4:

Pe = 0.1

DaIads = DaI

des = 0.1

t = 1 x 10-2

t = 5 x 10-2 t = 1 x 10-1 t = 2 x 10-1 seconds

© Fraunhofer ITWM 24

Numerical results: total adsorbed mass

( ) ( , )M t m t dS

x

Multiscale Modeling and Simulation

Multiscale model reduction

• Representing fine-scale features of flow and transport solution via multiscale basis functions.

• Generalizes upscaling techniques and allows systematically increasing the degrees of freedom in each coarse computational grid

• Flexible coarse gridding• Can use single-phase solutions to improve the accuracy• Allows incorporating the information across scales and uncertainties

Illustration of multiscale basis functions

Two-phase flow and transport

reference MS with adaptive grid

Solution comparisons

Adaptive coarse gridding

Approx.Outputs

POD Reduced model Lower complexity

InputsOutputs/snapshot

s

Fine model large complexity

Reservoir Workflow

Reservoir Simulator

timeSVD

POD

Resrvoir Simulator

Darcy-scale Reactive Flow in Porous Media:Wormhole simulation

𝐮=−𝐤𝜇∙𝛁𝑝

𝜕𝜀𝜕𝑡

+𝛁 ∙𝐮=0

𝜕𝜀𝐶 𝑓

𝜕𝑡+𝛁 ∙𝐮𝐶 𝑓=−𝑘𝑐𝑎𝑣 (𝐶 𝑓 −𝐶𝑠 )

=

𝜕𝜀𝜕𝑡

=𝑐𝑅 (𝐶𝑠)

=

: concentration of the bulk fluid

: concentration at the surface of the PM

: Darcy velocity

: mass transfer coefficient

: Chemical reaction rate

: porosity

: Permeability

: viscosity of the fluid

: pressure

: surface reaction rate constant

Kxx

Porosity

VELOCITY (LEFT) AND PRESSURE (RIGHT) DISTRIBUTIONS IN A SLICE OF THE COMPUTATION DOMAIN.

Surface mass at t=0, 1200, 2400 and 3600s at a selected cross section of the domain.

CoRheoS - Simulations

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