Numerical Porous Media KAUST SRI Center Modeling and simulation of multiscale problems N Ahmed, VM Calo, Y Efendiev, H Fayed, O Iliev , Z.Lakdawala, K.Leonard, G.Printsypar
Dec 19, 2015
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
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 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
11
Computation with Carreau model in 3D CT image of sandstone
incompressibility (mass
conservation)
momentum conservation𝒯=2𝜇 �̇�𝑖𝑗 𝑠𝑡𝑟𝑒𝑠𝑠𝑡𝑒𝑛𝑠𝑜𝑟
strain tensor
14
Macroscopic pressure and apparent viscosity
Apparent average viscosity
Pressure drop versus velocity
© 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 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
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
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