Multiscale investigation of CO2 behavior in subsurface

1
Multiscale investigation of CO2 behavior in
subsurface

• A. Tartakovsky
• S. Kerisit
• A. Marquez
• G. Lin
• A. Ward

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Motivation Why Multi-Scale Modeling?
Field Scale
Our present-day knowledge of the soil physical
system is still rather fragmentary. Hence, we
continue to search and re-search for answers to
the newly arising questions. Hillel,
1980. It is important to have a reliable
physically based tool that can provide plausible
estimates of macroscopic properties. Any
theoretical or numerical approach to this problem
not only needs a detailed understanding of the
multiphase displacement mechanisms at the pore
level but also an accurate and realistic
characterization of the structure of the porous
medium. Piri and Blunt, Phys. Rev. E,
026310, 2005
Mesoscale
Pore Scale
Microscopic Scale
Molecular Scale
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Motivation for multiscale approach
Darcy (continuum) scale
Field Scale
Darcy Law
Advection Dispersion Equation
Conductivity, K, and dispersion coefficient, D,
describe properties of the porous media, fluids
and fluid-solid interactions.
Mesoscale
Pore scale
Pore Scale
Microscopic Scale
N-S Equation
Diffusion Equation
Molecular Scale
Empirical boundary conditions are used to
describe surface reactions and fluid-fluid-solid
interactions.
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Project objectives

• Develop a pore-scale model for CO2 behavior in
  subsurface. Main components - multiphase flow
  reactive transport and mineral precipitation/disso
  lution
• Develop a hybrid pore-scale/MD scale model for
  surface chemistry.
• Validate the models using laboratory
  micro-fluidic experiments (M. Oostrom). Use the
  models in experimental design.
• Use the pore-scale models to obtain effective
  transport properties of porous media
• Goals
• Improve the fundamental understanding of the
  geochemical processes involved in the CO2
  sequestration
• Estimate constitutive relationships for the
  reservoir-scale simulations through numerical
  experimentation.

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Research Accomplishments
Pore-scale simulations
CO2 injection
CO2 dissolution
Darcy scale
Hybrid Darcy-pore-scale simulation of calcite
precipitation
Unstable downward migration of dissolved CO2
(Rayleigh-Taylor instability)
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Research Accomplishments
Pore-scale simulation of CO2 injection

• New model for multicomponent multiphase flows
• New model for dissolution of sc CO2
• Stochastic analysis of density driven flows
• Multiscale KMC/diffusion equation model for
  calcite precipitation

CO2 is being injected at the top of the domain
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Research Accomplishments
Pore-scale simulation of CO2 injection

• New model for multicomponent multiphase flows
• New model for dissolution of sc CO2
• Stochastic analysis of density driven flows
• Multiscale KMC/diffusion equation model for
  calcite precipitation

CO2 is being injected at the bottom of the domain
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Research Accomplishments

• New model for multicomponent multiphase flows
• New model for dissolution of sc CO2
• Stochastic analysis of density driven flows
• Multiscale KMC/diffusion equation model for
  calcite precipitation

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Research Accomplishments

• New model for multicomponent multiphase flows
• New model for dissolution of sc CO2
• Darcy-scale stochastic analysis of density driven
  flows
• Multiscale KMC/diffusion equation model for
  calcite precipitation

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Darcy scale Monte Carlo Simulation of
Rayleigh-Taylor Instability
Front position (depth of penetration) vs. time
in homogeneous and heterogeneous media
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Darcy scale Monte Carlo Simulation of
Rayleigh-Taylor Instability
Frequency distribution of front position
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Research Accomplishments

• New model for multicomponent multiphase flows
• New model for dissolution of sc CO2
• Stochastic analysis of density driven flows
• Multiscale KMC/diffusion equation model for
  calcite precipitation

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Calcite growth/dissolution model
Kinetic Monte Carlo model
liquid
Mineral is treated as 3D cubic lattice
Bond strengths and number of nearest neighbors
are used to determine probability of
attachment/detachment
solid
Acute steps
Obtuse steps
(10.4) calcite surface
KMC model takes into account the anisotropy of
the strength of the in-plane bonds
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Calcite growth/dissolution model (cont.)
Dissolution in pure water
RT
Bond strengths can be calibrated to reproduce the
known pH dependence of the rate
Dissolution in sc-CO2/H2O mixtures
pH effect
solubility effect
KMC model allows us to quantify the interplay
between the pH and solubility effects on the rate
of dissolution.
Solubility data from Fein and Walther GCA 1987
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In progress Coupling of KMC model with continuum
treatment of diffusion
Dissolution rate vs. diffusion coefficient
Fully atomistic
Dissolved species concentration
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• Publications
• Tartakovsky et al., Lagrangian particle
  model for multiphase flows, Computer Physics
  Communications, doi10.1016/j.cpc.2009.06.002
  (2009).
• Tartakovsky, Lagrangian simulations of
  unstable gravity driven flow of fluids with
  variable density in randomly heterogeneous porous
  media, submitted to Stochastic Environmental
  Research and Risk Assessment.
• Publication plans (papers in preparation)
• 1. Hybrid KMC continuum (diffusion-reaction
  equations) model of calcite precipitation from
  water CO2 mixture
• 2. Pore-scale model of supercritical CO2 in
  porous and fractured media
• 3. Parallelization of SPH codes on GPUs

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• Proposals
• 1. Developing a Component-Based Framework
  for Hybrid Multiscale Groundwater Modeling (PI
  B. Palmer) SciDAC, DOE, 480K
• 2. Hybrid Model for Ice Sheet Dynamics,
  SciDAC, DOE, 900K
• 3. Multiscale Investigation of Carbon
  Dioxide in Subsurface under Extreme Conditions,
  white paper to BES/DOE, 120K/year
• 4. (Rejected) DOE Energy Frontiers Research
  Center to Improve Long-Term Storage of CO2 and
  Radioactive Waste through Integrated
  Characterization, Monitoring, and Modeling of
  Hydrogeologic Systems, BES/DOE, 300K/year (lead
  institution University of Arizona).
• Conferences
• Special session on Pore-Scale Reactive
  Transport at the 2010 Computational Method in
  Water Resources conference, Barcelona 2010.

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Staffing Distribution in FY09 (through June)
Proposed for FY10
Project Title Multiscale investigations of CO2
behavior in subsurface
Staff Member FY09 FTE FY10 FTE
Tartakovsky, Alexandre M .21 0.16
Kerisit,Sebastien N .07 0.07
Lin,Guang .04 0.04
Marquez,Andres .07 0.07
Ward, Andy .00 0.05
Broyda,Svetlana V .14 0
Misc. FTE .01 0.01
Postdoc 0 0.5
Total .51 0.9
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Smoothed Particle Hydrodynamics Fluids are
discretized with a set of points (particles).
SPH interpolation scheme allows any function to
be expressed in terms of its values at a set of
disordered particles/points Spatial
derivatives can be found
by
analytical differentiation of the kernel SPH
does not require computational mesh.
a
h
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Derivation of SPH interpolation expression
Approximation of Dirac delta function, d,
with kernel function, W, leads to the smoothed
approximation of A
For A known in discrete locations ri, integral
can be approximated as
SPH interpolation does not preserve values of
interpolated functions at the interpolation
points.
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SPH equations
Mass conservation (continuity equation)
Equation of state
(monotonic function)
Momentum conservation (Navier-Stokes equation)
j
i
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SPH equations
Mass conservation (continuity equation)
Momentum conservation (Navier-Stokes equation)
Equation of state
(monotonic function)
Advection-Dispersion Equation
Equation of motion
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Multi-phase multi-component component flow

• solid phase
• wetting liquid
• non-wetting liquid

Pressure is a monotonically increasing function
of density
Continuum surface force formulation (Brackbill,
1992 Morris, 2000)

• surface tension k curvature
• d surface delta function n vector normal
  to the surface

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Van der Waals SPH model for gas-liquid flow with
surface tension (Nugent and Posch, 2000)
van der Waals equation of state
or,
- pair-wise
attractive force
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SPH multiphase multicomponent model

• van der Waals equation of state

is an
attractive force between i and j particles

• Creates surface tension c2,ij is grater for
  interaction between particles representing the
  same fluid than for interaction between particles
  of different fluids

Attractive forces are also used to model wetting
behavior of fluids at fluid-fluid-solid
interfaces.
A.M. Tartakovsky, K.F. Ferris, and P.
Meakin, Computer Physics Communications (in
press).
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Phase separation with vdW SPH model (after Nugent
and Posch, 2000)
T 0.6
T 0.2
T 1.05
T 0.87
Drop configurations for different temperature.
Critical temperature 1.19.
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Van der Waals SPH model for flow of two liquids
with surface tension
Color function


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Van der Waals SPH model for flow of two liquids
with surface tension
Color function


Surface tension as function of k
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Pore-scale model. Two-sided problem
(mineral phase is permeable)
Chemical reaction A(l)B(l)?C(s)
i l ( liquid phase), m (mineral phase)
Boundary conditions at the liquid mineral
interface
Surface reaction
Surface growth
fl 1, fm ltlt1 irreducible porosity of mineral
precipitates dlA,B molecular diffusion
coefficient of solute A or B dmA,B effective
diffusion coefficient of solute A or B inside
mineral phase k reaction rate, Ksp
solubility product
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Sub-pore-scale model. One-sided problem. Mineral
crystal is impermeable.
Mineral phase is modeled as individual crystalsgt
fm dmA,B 0
Boundary conditions at the liquid crystal
interface
Surface reaction
Surface growth
fl 1, fm ltlt1 irreducible porosity of mineral
precipitates dlA,B molecular diffusion
coefficient of solute A or B dmA,B effective
diffusion coefficient of solute A or B inside
precipitate k reaction rate, Ksp
solubility product.
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Governing flow equations
Continuity equation Momentum conservation
Boundary condition at fluid solid interface
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Smoothed Particle Hydrodynamics
diffused-interface model for reactive systems
1. Use the continuum surface force formulation
(Brackbill, 1992) to replace the free boundary
problem with the simpler problem
W
hr is a compact support of W
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SPH discretization of NS and advection-diffusion
equations can be found in (Tartakovsky et al.,
JCP 2007).
SPH discretization of reactive term
For one-sided problems (equations are defined
on one side of interface only), mass balance
considerations yield
Condition on hr (the width of diffused interface)
L characteristic length, Da Damkohler number.
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Validation of the SPH model for diffusion
equation subject to a mixed boundary condition.
Comparison of analytical and SPH solutions. SPH
solutions are obtained with two different
resolutions neq.
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Effect of the width of the diffused interface on
the solution
L characteristic length, Da Damkohler number.
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SPH implementation of crystal growth
Initial mass of the solid phase
Mineral mass in Wl changes according to
Update of the color functions according to
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Surface growth due to precipitation/dissolution
reactions (isotropic reaction rate coefficient)
Large Damkohler numbers RL/D gt unstable
dendrite-like growth Small Damkohler number gt
stable compact growth
Tartakovsky et al, JCP 2007