modeling co2 flow in porous media
TRANSCRIPT
Modeling CO2 Flow in Porous Media
Ernst A. van Nierop, Antonio C. Baclig
C12 Energy
RECS June 6th, 2012, Birmingham AL.
TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA
CO2 capture
no CO2 emissions clean,
abundant power to
homes and businesses
CO2 storage
enhanced oil recovery
burn convert
power plant
• 13 storage sites in 9 states • Perpetual carbon dioxide storage rights at ~500k private acres
• 10 Gt+ of CO2 storage capacity – equivalent to 30 years of emissions from 15% of the US coal fleet
C12 Energy Project Locations
MADRONE CAPITAL
C12 Energy Investors and Partners
4
Investors Partners
Outline
• Short review of fluid mechanics
– From Navier-Stokes to Darcy’s Law.
• Upscaling multi-phase flow from pores to reservoirs
– Relative permeability, capillary trapping, and so forth.
• Demonstration with a commercial simulator
– Walk through set-up of simulation, play with results.
– Plume evolution with & without geologic structure
– Effect of gridding
– Effect of heterogeneity
– Long term fate of CO2, and “engineering” the reservoir
– Area of Review determination by pressure or CO2 plume size
Short review of fluid mechanics
Acknowledgement: these slides are based on course notes by Prof. H.A. Stone.
Take groups of fluid molecules (the continuum approach)
Apply a stress or shear to them
Ask yourself: How will the parcel of fluid respond?
Will it deform? Expand? Move out of the way?
Five unknowns: density (r),
pressure (p),
velocity (u = [u,v,w]),
Five equations:
mass conservation (“continuity”)
momentum conservation (in [x, y, z])
equation of state (relates r to p, and other “state” variables)
1.
2.
3.
4.
5.
more
Mass conservation (“continuity”)
time rate of change
of the mass in V rate of mass flow
into V
rate of mass flow out of V =
_
@@t(½A¢x) = (½uA)jx¡ (½uA)jx+¢x
@ ½
@t+r¢ (½u) = 0In the limit of small Dx, and for flow in 3-D, this becomes:
flow flow p(x; t)
½(x; t)
u(x; t)
p(x+¢x; t)
½(x+¢x; t)
u(x+¢x; t)¢x
V
A
Momentum conservation (1/2)
time rate of change of momentum in V
net flow of momentum into V
forces acting on
the surface of V = +
@@t(½A¢xu) = (½uAu)jx¡ (½uAu)jx+¢x+(pA)jx¡ (pA)jx+¢x
@ (½u)
@t+
@(½u2)
@x= ¡@p
@x
But something is missing.... what about other forces on the surface of V?
½¡@ u@t
+u @ u@x
¢=¡@p
@x
, which, combined with continuity, gives:
flow flow p(x; t)
½(x; t)
u(x; t)
p(x+¢x; t)
½(x+¢x; t)
u(x+¢x; t)¢x
V
A
Momentum conservation (2/2)
flow flow p(x; t)
½(x; t)
u(x; t)
p(x+¢x; t)
½(x+¢x; t)
u(x+¢x; t)¢x
More general formulation: Cauchy stress equation of motion
½¡@u@t
+u ¢ ru¢= ½ f +r¢T
body forces e.g. gravity, electric fields, etc.
surface forces e.g. pressure, friction (viscosity), etc.
T=¡pe I+ ¿for convenience, we write:
stresses that do NOT
depend on u
stresses that DO
depend on u
0 = ½g¡rp
V
A
(Remember, we had before.) ½¡@ u@t
+u @ u@x
¢=¡@p
@x
For example, fluid statics: u = 0, f = g and = 0; so ¿
Momentum conservation (2/2)
flow flow p(x; t)
½(x; t)
u(x; t)
p(x+¢x; t)
½(x+¢x; t)
u(x+¢x; t)¢x
More general formulation: Cauchy stress equation of motion
½¡@u@t
+u ¢ ru¢= ½ f +r¢T
body forces e.g. gravity, electric fields, etc.
surface forces e.g. pressure, friction (viscosity), etc.
T=¡pe I+ ¿for convenience, we write:
stresses that do NOT
depend on u
stresses that DO
depend on u
• must depend on gradients of fluid velocity, so in its simplest form, • Viscosity is just the coefficient that relates stress to velocity gradients. • Thus, in its most common form, the Navier-Stokes equations are:
@ ½
@t+r¢ (½u) = 0
V
½ (ut + u ¢ ru) = ¡rp+ ¹r2u+ ½g
A
¿ ¿ = ¹ru
Navier-Stokes in words
@ ½
@t+r¢ (½u) = 0
½ (ut + u ¢ ru) = ¡rp+ ¹r2u+ ½g
“fluid particles are neither generated or destroyed, they just move around”
“fluids will move when subjected to pressure gradients and external foces, but viscosity will always resist the fluid motion”
½ (ut +u ¢ ru) =¡rp+¹r2u+ ½g
Simple (but very relevant!) application
why you should watch your cholesterol
@ ½
@t+r¢ (½u) = 0
2. steady flow
4. uni-directional flow
3. incompressible
1. gravity neglected
) dp
dx= ¹
rddr
¡rdudr
¢
Integrate with respect to r : dudr
= r2¹
dp
dx , and again: u =¡R2
4¹
dp
dx
³1¡ r2
R2
´
A 10% reduction in R raises dp/dx (or reduces Q) by 35%!
Q=¡ ¼8¹
dp
dxR4
ut = 0
½= constant; )r¢u = 0
u= u )r¢u= dudx
= 0)u ¢ ru= 0
Integrate over tube cross section to get total flow:
Spot the similarities...
u =¡R2
4¹
dp
dx
³1¡ r2
R2
´Flow through a tube
Flow through a rock
u
L
¹u = ¡ kÁ¹
dp
dx
viscosity m
p1 p2
R
u
L
viscosity m
p1 p2
permeability k porosity f
Darcy’s Law
Another approach: Darcy’s experiment
Outline
• Short review of fluid mechanics
– From Navier-Stokes to Darcy’s Law.
• Upscaling multi-phase flow from pores to reservoirs
– Relative permeability, capillary trapping, and so forth.
• Demonstration with a commercial simulator
– Walk through set-up of simulation, play with results.
– Plume evolution with & without geologic structure
– Effect of gridding
– Effect of heterogeneity
– Long term fate of CO2, and “engineering” the reservoir
– Area of Review determination by pressure or CO2 plume size
Darcy’s law for multi-phase flow (1/2)
water CO2
¾pg; ½g; ¹g pw; ½w; ¹w
Added challenges to incorporate: 1. surface tension (affects capillary pressure, and relative permeability) 2. density differences (affects buoyancy) 3. viscosity differences (affects relative permeability)
4. rock wettability (affects relative permeability)
ug = ¡k kr;g¹gÁ
(rpg +¢½g)
For example, within the gas phase....
¹u = ¡ kÁ¹
dp
dx(For single phase flow, it was: )
Darcy’s law for multi-phase flow (2/2)
uw = ¡k kr;w¹w Á
rPw = ¡k kr;w¹wÁ
(rpw ¡¢½g)
ug = ¡k kr;g¹g Á
rPg = ¡k kr;g¹gÁ
(rpg +¢½g)
¢½ = ½w ¡ ½g
pg ¡ pw = pc ¼ 2 ¾Rpore
water CO2
¾½g; ¹g ½w; ¹w
How do we go from flow at the pore scale, to flow at the reservoir scale?
The (imperfect, yet sort of useful) answer: relative permeability.
It is impractical to know and model all pores and channels perfectly.
Relative Permeability
The more there is of stuff X, the easier it is for stuff X to flow.
Bachu & Bennion, Env. Geo. 54, (2008)
“drainage”
(CO2 injection) “imbibition”
(post-injection)
Capillary Trapping
http://www2.bren.ucsb.edu/~keller/micromodels.htm
Capillary Trapping in Simulation
hysteresis in krelative
0 0.5 1 0
0.5
1 kr,i
Sg
Sw 1 0
kr,w
kr,g connate water
saturation max. residual gas
Capillary Pressure: the way it is presented
Water Saturation, Sw
Capillary Pressure
connate water
saturation
Capillary Pressure: what it actually is
pg ¡ pw = pc ¼ 2 ¾Rpore
water CO2
¾½g; ¹g ½w; ¹w
Bachu & Bennion, Env. Geo. 54, (2008)
Outline
• Short review of fluid mechanics
– From Navier-Stokes to Darcy’s Law.
• Upscaling multi-phase flow from pores to reservoirs
– Relative permeability, capillary trapping, and so forth.
• Demonstration with a commercial simulator
– Walk through set-up of simulation, play with results.
– Plume evolution with & without geologic structure
– Effect of gridding
– Effect of heterogeneity
– Long term fate of CO2, and “engineering” the reservoir
– Area of Review determination by pressure or CO2 plume size
With and Without Structure
• 5 degree dip • 0.5 Mt/yr for 15 years • ~ 1,400 m depth
• T = 40oC • Salinity = 15,000 ppm TDS • Grid block sizes: 100 m x 100 m x 10 m
• Parabolic anticline • 0.5 Mt/yr for 15 years • ~ 1,450 m depth • T = 40oC
• Salinity = 15,000 ppm TDS • Grid block sizes: 100 m x 100 m x 5 m
Without structure, top-view
5 years (post injection)
10 years
50 years 15 years
With structure, top-view
5 years (post-injection)
10 years
50 years 15 years
Effect of Gridding: coarse vs. telescopic
Effect of Gridding: coarse vs. telescopic
50% increase in R
so 125% increase in footprint
Effect of Gridding: coarse
Effect of Gridding: telescopic
Pe
rme
ab
ility
Mo
de
l C
O2 P
lum
e
Homogeneous Permeability
Heterogeneous Permeability Channels
Effect of Heterogeneity
Slope of 8% 100 kt/yr for 5 years “no-flow” boundary conditions
Long term fate of CO2
“mobile”
dissolved
residually trapped
Long term fate of CO2: reservoir engineering
“mobile”
dissolved
residually trapped
“mobile”
dissolved
residually trapped
Calculating Area of Review
• Area of Review is defined as the area within which the CO2
sequestration project could cause endangerment to a USDW.
• Endangerment comes from presence of mobile CO2, and
from increased pressure that can push brine into the nearest
USDW through leakage pathways.
“MESPOP” maximum extent of separate phase or pressure
• Conservatively, guidance recommends assuming that there is
an open borehole somewhere, and calculating the maximum
pressure rise in the injection reservoir that would just prevent
brine from reaching the USDW through that open borehole....
Area of Review: example
surface
USDW
caprock
injection formation
H
rw
rb
injection
well abandoned
open borehole
Pressure rise required to push column of brine up from injection formation to USDW (assuming open frictionless borehole, and initially hydrostatic pressures):
Dp = (rb-rw) g H
or, in normalized form:
Dp/p = ((rb-rw)/ rw) (H/D)
D
Area of Review: two examples
Dp/p = ((rb-rw)/ rw) (H/D)
surface
USDW
caprock
injection formation
H
rw
rb
injection well
abandoned open borehole
D
1. High salinity & large H/D (Illinois)
rb = 1200 kg/m3, H = 2000 m, D = 2050 m
Dp/p = 19.5%
2. Low salinity & small H/D (North Dakota)
rb = 1005 kg/m3, H = 1000 m, D = 1500 m
Dp/p = 0.3%
Area of Review: two examples
1. High salinity & large H/D (Illinois) rb = 1200 kg/m3, H = 2000 m, D = 2050 m
Dp/p = 19.5%
2. Low salinity & small H/D (North Dakota)
rb = 1005 kg/m3, H = 1000 m, D = 1500 m Dp/p = 0.3%
3.5 yrs into injection End of Injection
Points for further thought
• Types of Modeling
– Finite difference
– Streamlines
– Vertical equilibrium
– Invasion percolation
• What to look for as a regulator
– how are uncertainties accounted for? (statistics, ensembles, deterministic worst-case scenarios, etc.)
– technical merits of simulations (numerical settings, gridding,
boundary conditions, etc.)
• Geochemistry, geomechanics, well design, etc.
Questions?
Contact Information:
Ernst A. van Nierop
tel: 617-849-8006
Additional Slides
The continuum assumption
Continuum = averaging. This is OK if the length scale and time scale of the averaging is much smaller than the scales of the motion we are interested in.
If the flow occurs over the “macro” length scale, it will be described well by fluid properties that are averaged on a smaller “ave” scale. This length scale
needs to be larger again than the molecular scale on which stuff actually happens.
1 ¹m3 contains about 3 £ 1010 water molecules or 1010 benzene mols., or 107
gas molecules. Even if it is 100 times smaller on each side (10 nm £ 10 nm £10 nm), there are still 104 molecules in a liquid. Typical collision time scales
are 10¡12s, so the time scale is not a problem in most cases.
Statistical physics tells us that thermal fluctuations scale as dN ≈ N1/2 so if we require the average to work well, i.e. be within 10% of the fluctuations, then we require dN/N ≈ N-1/2 <0.1; or N ≈ 100 particles!
back
Name Position Experience
Kurt Zenz House, Ph.D. President Ph.D., Geoscience, Harvard; post doctoral research, MIT; Bain and Company; National Science Foundation
Justin Dawe Chief Executive Officer Horizon Wind Energy; MBA, Harvard; MS, Stanford
Tony Meggs
Chairman Talisman Energy; BP; Exxon; visiting scientist, MIT
Charles Brankman, Ph.D. Director of Geosciences Ph.D., Earth and Planetary Sciences, Harvard; William Lettis & Associates
Jill Daniel Director of Finance SteelRiver Infrastructure Partners; Babcock & Brown; The Carlyle Group; Bechtel ; MBA, Stanford
Daniel Enderton, Ph.D. Director of External Affairs
MIT Energy Initiative; Ph.D., Climate Physics and Chemistry, MIT
Barclay Rogers Director of Development Chapman Tripp; Sierra Club; MBA, University of Cambridge; JD, Lewis & Clark
Ernst van Nierop, Ph.D. Director of Engineering Ph.D., Engineering, Harvard; MS, Applied Physics, University of Twente, Netherlands; Akzo Nobel
C12 Energy Leadership Team