MODELING FLOW IN VUGGY MEDIA
Advisor Todd Arbogast - Mathematics, University of Texas at Austin
Diana Battefeld - Physics, Brown University
Cheng Chen - Engineering, Northwestern University
Abdullah Cihan - Engineering, University of Tennessee, Knoxville
Daniel Pringle - Geophysics, University of Alaska, Fairbanks
Masa Prodanovic – Comp. Applied Math, University of Texas at Austin
Danail Vassilev - Computational Mathematics, University of Pittsburgh
Hua Zhang - Agronomy, Louisiana State University
Summer School In Geophysical Porous Media
July 17-28, 2006
Purdue University, West Lafayette, Indiana
Motivation
Environmental: Understanding movement of
contaminants in vuggy subsurface systems, pumping
and protection of water aquifiers.
Oil Industry: Oil recovery - Help oil industries to
manage oil reserves more efficiently.
Geological: System unlike those studied normally by
geologists.
Mathematical: Model fluid flow and transport in vuggy
(dual porosity) media.
What is a Vug?
Vug: an orifice in a porous medium
relatively large compared to pore size
Internal structure of Vug system: X-ray Computed Tomography (CT) Scans
of Pipe Creek Reef sample
Where?
This sample: Pipe Creek, Texas
River basin Area
Limestone matrix with caprinid fossils
4-6 cm diameter Cretaceous caprinid fossil
Goal:
To characterize the vuggy media’s pore space and to
understand flow and tracer transport. We use CT scans,
permeability measurements, and experimental tracer
curves to achieve this goal.
We will proceed as follows:
I. CT Image analysis used to characterize pore space & connectivity
II. Permeability computation from simplified Darcy flow models
III. Tracer analysis what we can infer from the experiments
I
CT IMAGE ANALYSIS
3DMA-Rock software
http://www.ams.sunysb.edu/~lindquis/3dma/3dma_rock/3dma_rock.html
Segmentation (Indicator
Kriging, Oh and Lindquist ’99)
Medial Axis (MA) extraction
(Lee-Kashyap-Chu ’94)
+ some trimming
Throat finding +
pore partitioning
Interphase areas, fluid
blob characterization,
pore scale saturation…
Pore/throat
characterization
(distributions)
Medial Axis
 Centrally located skeleton
 Preserves topology and geometry of a 3D
object
 a powerful search tool
 Obtained by morphological thinning
algorithms (3DMA-Rock implements LeeKashyap-Chu '94. algorithm)
 Branch cluster (node) – digitized version of
a point (graph vertex) where medial axis paths
meet
 Medial axis path (link) – digitized version of
a curve (graph link)
 First applied to (simulated) porous media by
Thovert/Sales/Adler ’93. and to sandstones by
Spanne/Thovert/Jacquin/Lindquist/Jones/Adle
r ’94.
Terminology : Medial axis = skeleton= percolating backbone = deformation retract
MA computation = morphological skeletonization = morphological thinning/erosion
Grass-fire algorithm = burn algorithm = erosion algorithm = distance labeling
Objective: Pore-throat network
Most throat finding algorithms
minimize area A (and not
hydraulic radius R)!
R = A/P = (aL2)/(bL)=(a/b)L = cL
(a,b,c constants reflective of shape)
Throat: R1 > R2 < R3
 Hydraulic radius of fluid R1 = c1L1 & R2 = c2L2, assume c1=c2
pathway cross-section:
for close cross-sections,
R = Area/Perimeter
Then
 Throat: a pathway cross
R1 < R2 iff L1 < L2 iff
section of minimal R
 Pore: pore space opening c1L12 < c2L22 iff A1 < A2
enclosed by grain & throat
surfaces
Throat Finding Approaches
 Delaunay tesselation of sphere packs
 “too many” pores obtained; merging close clusters needed
 Finney '70, Mason '71, Bryant ’93., Willson ’00.
 Morphological thinning based pore partitioning
 Baldwin/Sederman/Mantle/Alexander/Gladden '96.
 widely used in soil sci.; pore space partitioned, no throats
identified
 Maximal ball algorithm
 Silin/Patzek 2003., results in “ball-n-stick diagram”
 Throat finding algorithms
1. Multi-orientation scanning (Zhao/MacDonald/Kwiecien '94.)
2. Dijkstra based throats (Venkatarangan/Lindquist '99.) and
extensions (Prodanovic/Lindquist/Seright ‘05) MA based
3. Planar throats (Liang/Ioannidis/Chatzis '99.) MA based
4. Wedge based throats (Shin/Lindquist '02.) MA based
Sample Analysis
Does upscaling by coarsening work?
 cut central 300x300x100 subset of the segmented image
 original z-direction slices multiplied 3 times to match resolution in x- & ydirections  3003
 the image coarsened by using majority rule in the boxes of appropriate size
3003
Porosity 17.8%
Connectivity 83.3%
1003
Porosity 17.2%
Connectivity 83.0%
503
Porosity 16.9%
Connectivity 84.7%
Connectivity = percentage of pore space in the largest connected component
Vug Space Medial Axis (MA)
3003

1003
503
Small (MA) structures are destroyed by upscaling (above). Main fluid pathways are
preserved: the shortest z-direction pathways are shown below. 503 opens up one
more path!
Medial Axis Path Statistics
1400
350
1200
300
Ntotal = 5500
800
600
150
100
200
50
0
20
40
60
80
 = 0.80
200
400
0
 = 1.60 ( 5 mm)
250
Number
Number
1000
0
-1
100
0
1
Path Lengths [mm]
2
3
4
5
ln(Path Lengths [mm])
2500
7
6
1/r2eff
2
=  (1/r ) along path
ln(Number)
Number
2000
1500
1000
N(r) ~ r -1.58
5
4
3
2
500
0
1
0
2
4
6
8
reff [mm]
10
12
14
0
-1
-0.5
0
0.5
1
1.5
ln(reff [mm])
2
2.5
3
Sample Heterogeneity
 Shortest paths across x-, y- and z- directions for 1003 sample
 Average geometric tortuosity (ratio of actual length and side-side
distance) for the shortest paths is ~1.8 for all directions, and ~2.0
for all possible paths across the sample
Vug-throat network
 3DMA-Rock throat finding algorithms used to identify
individual vugs
vug
surface
MA path
Branch
cluster
A vug from 3003 sample, 9983 mm3
Relative positions of 10 largest
vugs from 3003 sample
Vug-throat Network Statistics
Small structures disappear
in the upscaled image
• If small structures are important for the flow, the simulation in
upscaled images will not produce good results
• Dead-end vugs (not included the above volume stats) can be
identified and occupy 4% of the connected pore space component
II
PERMEABILITY COMPUTATION
Flow and Transport Modeling
Provide link between structure and experiment.
Can system be understood in terms of an effective permeability?
What controls tracer transport?
Matrix (no vugs)
k = 10 mDarcy
Sub-sample (~10 cm)3 k = 100 Darcy
k
v   P
Darcy Flow

c

   ( Dc  cv ) Advection Diffusion Equation
t
Flow and Transport: Approaches
1. Medial axis and statistics
2. CT output structures
Simple pipe flow
‘Darcy code’: modeling
Simple pipe networks
2D Lattice Boltzmann
Medial-axis pipe networks
3. Dual porosity model fitting
Insights from Poiseuille Flow
u 
k

2
r0
k pipe 
8
p
r0
Parallel
k eff
ai
  ki
li
Large k dominate
1 Darcy (D) ~ 10-12 m2
 ai 
1
   ki 
Series
k eff
 li 
Small k dominate
1
Insights from Poiseuille Flow
u 
k

p
2
r0
k pipe 
8
d = 10 cm
1 Darcy (D) ~ 10-12 m2
kmx =10 mD, kpipe = 45 D
n=2: keff = 90 Darcy
r = 1 mm
Small constrictions can strongly control keff
Pipe Network Model
Dead
end v=0
P=1
P=0
INPUT: Medial axis path network
NB: more usual to use pore/throat network, but we had medial axis data
available first!
Pipe Network Formulation
q4
Poiseuille flow
r 4
Pl  Pr 
qi  
8Li
Mass balance
q
in
 k1
 0

 ...

 0
 1

 0
 ...

 0
... 0
... 0
... ...
... k NE
... 0
... 0
... ...
... 1
P6
q6
q1
P7
P1=1
  qout
k1
 k2
...
0
0
0
...
0
P4
q2
1 0
0 1
... ...
0 0
0 0
1 1
... ...
0 0
...
...
...
...
...
...
...
...
P8=0
P2
q3
q8
P3
0   p1   0 
0   p2   0 
...  ...  ...
  

1   p NN   0 

 
0   q1   1 

0   q2   0 

  
...  ...  ...

0   q NE   0 
P5
q7
q5
Conjugate gradient method
on normal equations:
ATAx=ATb
Simulated Permeability
Kzu
Kxu
Kzd
Kxub
Kxd
Kyu
Kyd
Kzu
Kzd
Darcy
50×50×50a
1 x106
2 x106
8 x 106
1 x 106
1x106
1 x106
100×100×10
0
10500
30200
152
439
9630
839
300×300×30
a Image resolutions.
0
1320
3080
Medial axis1110
flow paths985
are used1400
for simulation.
b
Flow directions.
606
Simulated Permeability
Kzu
Kxu
Kzd
Kxub
Kxd
Kyu
Kyd
Kzu
Kzd
Darcy
50×50×50a
1 x106
2 x106
8 x 106
1 x 106
1x106
1 x106
100×100×10
0
10500
30200
152
439
9630
839
300×300×30
a Image resolutions.
0
1320
3080
Medial axis1110
flow paths985
are used1400
for simulation.
b
606
Flow directions.
Upscaling dependence
and Variability
2-scale problem !
Voxel-sized restrictions to flow
Vug scale ~ sample size
Darcy Code: Parssim
Parallel subsurface simulator www.ices.utexas.edu/~arbogast/parssim/
IN:
3D geometry + physical parameters
OUT:(steady-state) velocity field, concentration field
48 x 48 x 48 voxels
(i) Full sample,
Coarse resolution
( L= 16.4 cm)
k = 416 Darcy
(ii) Sub-samples,
max resolution
( L = 2.6 cm)
k(s1) = 1282 Darcy
k(s2) = 170 Darcy
k(vug) = 107 D, k(mx) = 0.01 D,
ΔP = 104 Pa (0.1 atm),
2-D Lattice Boltzmann Modeling
Flow and passive tracer transport
Limited input file size: 100 x 100
voxels
Q: How to project ‘very’ 3D structure
to 2D ?
A: Examine 1-2 ‘connecting’ paths
instead.
Main flow
Dead ends
2D toy model made
by tracing and
projecting 2 paths
Flow Field
t = 0 (sec)
t = 85.3 (sec)
t = 341.2 (sec)
t = 1108.9 (sec)
t = 1706 (sec)
t = 3412 (sec)
t = 6397.5 (sec)
LBM Breakthrough Curve
Reynolds number, Re = 0.10
Inlet
Permeability,
k = 1.1 x 106 Darcy
Outlet
Breakthrough Curves
Numerical Permeability Results
Method
Network Model (300)3
(Medial Axis)
k [Darcy]
1400 ± 800
PARSSIM upscaled
subsample 1
subsample 2
416
1282
170
2D Lattice Boltzmann
1 x 106
(?)
Within 1-2 orders of magnitude of experiments: this was our target!
2 scale problem: really need Lsample > vugs and voxel < mm
To finish: 3D Lattice Boltzmann, pore-throat network system
Darcy code: systematic examination of upscaling and k values.
New numerical scheme in Arbogast and Brunson, submitted 2006.
III
TRACER TRANSPORT
WHAT WE HAVE SEEN SO FAR:
A. VUGS ARE INTERCONNECTED
B. WE CAN EXTRACT AN EFFECTIVE PERMEABILITY
C. APPROXIMATION: DARCY FLOW (only valid at larger
scales)
D. BETTER: POISEUILLE FLOW
NOW LET’S SEE WHAT WE EXPECT FROM EXPERIMENTS
TOY MODEL: 3D cellular vug-network (periodic)
A
HEURISTIC ARGUMENTS USING STOKES FLOW:
causes for marker behaviour
•Early Breakthrough: Tracer finds a pathway through the network
•Multiple Plateaus: Tracer finds other (longer) ways through the network
– result: new plateaus whenever a new way opens.
•Abrupt Drop: Time drop correlated to end of injection. A time delay is
due to cells on the main paths, acting as tracer reservoirs for a small
amount of time.
•Single Plateau: Weakly interconnected vugs are drained slowly. The
large main volumes of vugs act as the reservoir for a longer time, leading
to a plateau of constant height until they are drained.
•Long Tail: Remaining tracer in thin arms is slowly washed out after main
volumes are drained – diffusive process (exponential decay).
Implement model on
computer:
test arguments!
TEST HYPOTHESIS (Experimentally/Numerically, 3D)
•Multiple plateaus: Close one of the main
pathways and see if the early plateaus change.
•Abrupt drop: Increase injection time – drop
should be shifted by same time (the delay does
not change)
•Single Plateau: Inject the fluid for smaller time
interval – no reservoir build up: Plateau should
decrease in height.
•Long tail: increase the diameter of the smallest
arms in the toy model: exp. decay should be
faster.
TEST HYPOTHESIS (Experimentally or Numerically,
3D)
•Multiple plateaus: experimentally- close one of
the main pathways and see if the early plateaus
change.
From CT scan we see multiple pathways
In Poiseuille flow model, plateaus are non-existent (no reservoirs).
Macroscopic Models
Dual Porosity Model
m
im

C

C
m
  im
  D m mC m  v m mC m
t
t

Mobile
Immobile
 im

Dispersion
C im
 a C m  C im
t

Advection

Diffusion between mobile and immobile region
Dual Porosity Model CXTFIT
Diffusion between mobile and immobile regions important in transport.
( i.e. from vugs in/out of porous matrix ).
C/Co
1.0
0.8
Tracer data
0.6
CXTFIT free fit
0.4
0.2
0.0
0
4
8
12
V/Vo
16
20
Stream Tube Model
L1
r1
r4
r2
r3
Li effective flow length
ri effective radius
Correlated random variables with known
distributions
L3
Solute transport in each tube follows ADE
C
 Di C   v i C
t
Hydrodynamic dispersion Di
vi
Di 
i
vi - Poiseuille flow
Lo: Length of the column
Tortuosity ti=(L0/Li)2
Longitudinal dispersivity a=0.01 m
Flux averaged effluent concentration
Parallel 4-Tube Model
1.0
Tracer data
0.8
C/Co
4 tube model
0.6
Generally good fit!
Fit could be optimized..
0.4
No connectivity in model!
0.2
0.0
0
4
8
V/Vo
12
16
20
3D models (outlook)
•Easy: use periodic cell model as depicted, insert fluid, solve flow
numerically (maybe analytic estimate – use cylinders, Pois. flow.)
•Better: Construct fractal Vug-network: bifurcate progressively to smaller
branches (thereafter numerics as above).
•Best: Add Monte Carlo to choose (around mean):
- Length of arms before bifurcation
- Angle of outgoing arms
- Diameter after bifurcation of arms
Goal: find effective permeability, check heuristic arguments, …
Doable in 2-weeks time: Single Parallelpiped Uniform/Non-Uniform DarcyFlow 3D Model.
References
•T. Arbogast and H.L Lehr, Homogenization of a Darcy-Stokes system
modeling vuggy porous media., 2003. (Preprint available
www.ices.utexas.edu/reports/2002.html, TICAM report 02-44)
• H. L. Lehr, Analysis of a Darcy-Stokes system modeling flow through
vuggy porous media, Ph.D Thesis 2004
•W. B. Lindquist, 3DMA-Rock, A Software Package for Automated
Analysis of Rock Pore Structure in 3-D computed Microtomography
Images
http://www.ams.sunysb.edu/~lindquis/3dma/3dma_rock/3dma_rock.html
•S. Succi, Lattice Boltzmann Equations for Fluid Dynamics and Beyond
(Numerical Mathematics and Scientific Computation), 2001.
•M. C. Sukop, D. T. Jr. Thorne, Lattice Boltzmann Modeling: An
introduction for Geoscientists and Engineering, 2005
•D. Zhang and Q. Kang, Pore Scale simulation of solute transport in
fractured porous media., Geophysical Research Letters, 31, 2004