CONSERVATIVE CHARACTERISTIC METHODS
FOR LINEAR TRANSPORT PROBLEMS
Todd Arbogast
Department of Mathematics
and
Center for Subsurface Modeling,
Institute for Computational Engineering and Sciences (ICES)
The University of Texas at Austin
Chieh-Sen (Jason) Huang
Department of Applied Mathematics
and National Center for Theoretical Sciences
National Sun Yat-sen University (Taiwan)
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
Outline
1.
2.
3.
4.
5.
The PDE’s of Transport Problems and Local Conservation Principles
Characteristic Methods for Approximating the solution
Satisfying the Local Volume Conservation Principle Discretely
Numerical Results
Conclusions
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
Transport Problems
and Local Conservation
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
Conservative Fluid Flow
Suppose
ξ
v
ξv
Q
is
is
is
is
a conserved quantity ξ (mass/volume)
the fluid velocity (length/time)
the flux of ξ (mass/area/time)
an external source or sink of fluid (mass/volume/time)
Within a region of space R, the total amount of ξ changes in time by
Z
d
ξ dx
|dt R
{z
}
=−
Change in R
Z
| ∂R
R
ξt dx
{z
Flow across ∂R
=⇒ conservation locally on R
Z
ξ v · ν da(x) +
=−
Z
|R
}
∇ · (ξ v) dx
{z
}
Z
Q dx
| R {z
}
Sources/sinks
+
Z
R
Q dx
Divergence Theorem
This is true for each region R, so in fact
ξt + ∇ · (ξ v) = Q
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A Transport Problem–1
One incompressible fluid (tracer) flowing miscibly in another
incompressible fluid, within an incompressible medium.
Velocity of the bulk fluid. Conservation of bulk fluid mass (ξ = φρ) gives
ξt + ∇ · (ξ v) = Q
u
φ
ρ
q
=⇒
∇·u=q
is the (unknown) bulk fluid velocity (v = u/φ)
is the porosity (constant in time)
is the (constant) density
is the source/sink (wells, Q = ρq)
Simple Tracer Transport. Conservation of tracer mass (ξ = φc) gives
φct + ∇ · (cu) = cI q+ + cq− ≡ qc(c)
c is the (unknown) tracer concentration
cI is the given concentration of injected fluid
q+/q− is q when positive/negative
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A Transport Problem–2
However, transport is not the only process occurring!
Mass flux.
v = cu − D∇c
(Transported plus Diffusive Flux)
where
D is the diffusion/dispersion coefficient
Chemical reactions.
q = qc(c) + R(c)
(Wells plus Reactions)
where
R is the reaction term
Tracer Transport. Conservation of tracer mass gives
φct + ∇ · (cu − D∇c) = qc(c) + R(c)
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Operator Splitting of Transport Equation—1
φct + ∇ · (cu − D∇c) = qc(c) + R(c)
Discretization in time: ∆t > 0 and tn = n∆t.
We want to solve the transport and reactive part of the equation
explicitly and the diffusive part implicitly. Thus, we want
cn+1 − cn
φ
+ ∇ · (cnu) − ∇ · (D∇cn+1) = qc(cn) + R(cn )
∆t
This is equivalent to the three steps
(Reaction)
c̃ − cn
φ
= R(cn )
∆t
(Transport)
φ
ĉ − c̃
+ ∇ · (cnu) = qc(cn )
∆t
cn+1 − ĉ
(Diffusion)
φ
− ∇ · (D∇cn+1) = 0
∆t
with some intermediate c̃ and ĉ.
!
φct = R(c)
!
φct + ∇ · (cu) = qc(c)
!
φct − ∇ · (D∇c) = 0
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Operator Splitting of Transport Equation—2
Nonlinear Ordinary Differential Equation part (Reaction)
φct = R(c)
Linear Hyperbolic part (Transport)
φct + ∇ · (cu) = qc(c)
Linear Parabolic part (Diffusion/dispersion)
φct − ∇ · (D∇c) = 0
We discuss approximations of the transport step only.
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Locally Conservative Methods
A locally conservative method is one for which the approximate solution
satisfies the conservation principle, but only over certain discrete regions.
Normally, one would take the grid elements R and require
Z
R
φct + ∇ · (cu) dx =
Z
R
q dx
but we will need to be more general than this.
Remark: The reactive and diffusive steps must also be solved by locally
conservative methods, or local conservation will break down!
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Characteristic Methods
for Linear Transport
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Characteristic Tracing of Points
The characteristic trace-forward of the point x is denoted x̌ = x̌(x; t).
It satisfies the ordinary differential equation
u(x̌, t)
dx̌
=
,
dt
φ(x̌)
x̌(tn) = x
tn < t ≤ tn+1
In the absence of sources/sinks and diffusion, fluid particles simply travel
along the characteristics of the equation.
Time
tn+1
6
x̌
tn
-
x
Space
The concentration is constant along this space-time path, since
dc(x̌, t)
∂c
dx̌
u
1
=
+ ∇c ·
= ct + ∇c · =
φct + u · ∇c = 0
dt
∂t
dt
φ
φ
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Characteristic Trace-back of Points
The characteristic trace-back of the point x is denoted x̂ = x̂(x; t).
It satisfies the (time backward) ordinary differential equation
u(x̂, t)
dx̂
=
,
dt
φ(x̂)
x̂(tn+1) = x
tn ≤ t < tn+1
In the absence of sources/sinks and diffusion, fluid particles simply travel
along the characteristics of the equation.
Time
tn+1
6
x
tn
-
x̂
Space
Again, the concentration is constant along this space-time path.
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Modified Method of Characteristics (MMOC)
(Douglas and Russell, 1982)
Key idea: Use a finite difference approximation of the characteristic
derivative
dc
u(x, t)
c(x, t + ∆t) − c(x̂, t)
≡ ct (x, t) +
· ∇c(x, t) ≈
dt
φ
∆t
This results in the approximation
tn+1
6
c(x)
c(x, t + ∆t) − c(x̂, t)
φ
= (cI − c)q+
∆t
at each grid point
tn
-
c(x̂)
Problems: Because the method is based on points, it violates local mass
conservation constraints for both the bulk fluid and the tracer.
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Characteristic Trace-back of Regions
To obtain mass conservation, ...
Key idea: Trace regions rather than points!
The particles in a grid element E trace back to a region Ê
Ê = {x̂ ∈ Ω : x̂ = x̂(x; tn) for some x ∈ E}.
In space and time, we actually trace a region E = E(E) given by
E = {(x̂, t) ∈ Ω × [tn, tn+1] : x̂ = x̂(x; t) for some x ∈ E}.
tn+1
6
E
E
tn
-
Ê
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Local Mass Conservation of the Tracer
φct + ∇ · (cu) = qc(c)
Integrate in space-time over E and use the divergence theorem
ZZ
E
cu
φc
∇x,t ·
=
Z
E
!
dx dt =
φcn+1 dx −
ZZ
∂E
Z
Ê
c
!
u
· νx,t dσ
φ
φcn dx +
Z
S
c
!
u
· νx,t dσ
φ
The last
! term is integration on the space-time sides S of E,
u
but
is orthogonal to νx,t there!
φ
tn+1
The local mass constraint:
Z
E
φcn+1 dx =
Z
φcn dx
ÊZZ
+
6
E
qc dx dt −
Z
SB
E
E
cu · ν dσ
tn SB
Ê
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j
νx,t
-
Local Mass Conservation of the Bulk Fluid
A similar local mass constraint holds for the bulk fluid (c ≡ 1)
φt + ∇ · u = ∇ · u = q
Since we are dealing with incompressible fluids, we call this the local
volume constraint.
The local volume constraint:
Z
E
φ dx =
Z
Ê
φ dx +
ZZ
E
q dx dt −
Z
SB
u · ν dσ
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Characteristics Mixed Method (CMM)
(A., Chilakapati, and Wheeler, 1992; A. and Wheeler, 1995)
Use lowest order Raviart-Thomas mixed finite elements. The scalar test
function is a constant on each element in space.
Z
E
φcn+1 dx =
Z
Ê
φcn dx +
ZZ
E
qc dx dt −
Z
SB
cu · ν dσ
y
Remark: Practical implementation requires
that Ê be approximated by Ẽ ≈ Ê, a
polygon. This is equivalent to modifying
the velocity field, so tracer mass is still
conserved locally by the above equation.
y
y
y
Ê ≈ Ẽ
y
Vol(Ê) 6= Vol(Ẽ)
y
y
y
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Volume Error
Problem: The local volume constraint may be violated for this
perturbed velocity! This is because the volume constraint does not
enter into the definition of the method.
This leads to incorrect densities of the tracer, which leads, e.g., to bad
approximation of reaction dynamics.
2.00
0.10
0.01
-0.01
-0.10
Relative volume errors around an injection well.
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Local Volume Conservation
in Characteristic Methods
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Volume Conservation
Key idea: To obtain volume conservation, define Ẽ ≈ Ê to be a simple
shape that satisfies the volume constraint
Z
E
φ dx −
Z
Ẽ
φ dx =
ZZ
E
q dx dt −
Z
Strategy: Suppose that E is a rectangle.
Perturb the vertices and midpoints of Ê
only a little so that we get a polygon Ẽ
with 8 sides such that the above constraint
is satisfied.
y
y
yy
y
y
y
y
Ê ≈ Ẽ
We call this method the Volume Corrected
Characteristics Mixed Method (VCCMM).
Problem: It is easy to introduce systematic
bias into the flow field and thereby produce
unphysical flows. We must do the adjustment very carefully!
SB
u · ν dσ
y
y
Vol(Ê) = Vol(Ẽ)
y
y
yy
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y
y
An Example of Unphysical Flow
10 years
Volume not conserved
60
60
50
50
40
40
30
30
20
20
10
10
10
30 years
Volume conserved
20
30
40
50
60
60
60
50
50
40
40
30
30
20
20
10
10
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
Biased trace-back adjustment has introduced unphysical flow
corresponding to a large, incorrect velocity channel.
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Forward Trace of Injection Wells
Most of the error is near injection wells. Characteristic tracing back in
time traces into the well, which is difficult to approximate.
Key idea 1: Trace the well forward (out of the well). Adjust region to
satisfy the volume constraint (cf. Healy and Russell, 2000).
The characteristic trace-forward of the point x is denoted x̌ = x̌(x; t),
and it satisfies the ordinary differential equation
dx̌
u(x̌, t)
=
,
dt
φ(x̌)
x̌(tn) = x
tn < t ≤ tn+1
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Conservation Near Injection Wells
Well volume constraint: Adjust W̃ so
Z
W̃
W
φ dx +
ZZ
Ef
Z
Z
Vol(E ∩ W̃ )
φ dx =
φ dx +
Vol(W̃ )
Ẽ
E
Transport:
Z
q dx dt
Element volume constraint: Adjust Ẽ so
W̃
E
Ẽ
W
φ dx =
Z
Z
Vol(E ∩ W̃ )
φcn+1 dx =
φcn dx +
Vol(W̃ )
E
Ẽ
ZZ
E
qc dx dt
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ZZ
E
q dx dt
Inflow boundaries
Like injection wells, inflow boundaries trace back “out of the domain.”
Idea: Either trace inflow boundaries forward, or “fold” the time axis
down to the xy-plane to create a ghost region.
y
y
t
tn
tn+1
−u · ν
x
t
x
φ
Ẽ
∂Ω
Ω
Volume constraint: Replace φ by u · ν in the ghost region:
Z
E
φ dx −
Z
Ẽ∩Ω
φ dx +
Z
S̃B
u · ν dσ =
Z
E
φ dx −
Z
Ẽ
φ dx =
ZZ
Mass constraint: Replace φcn by cn
I u · ν in the ghost region:
Z
E
φcn+1 dx −
Z
Ẽ
φcn dx =
ZZ
E
qc dx dt.
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E
q dx dt,
Trace-back Point Adjustment
Key idea 2: Adjust points in the direction of the flow; that is, along the
characteristics in time (cf. Douglas, Huang, and Pereira, 1999).
To define x̃, for τ n ≈ tn, we solve (backwards)
dx̃
u(x̃, t)
=
,
dt
φ(x̃)
x̃(tn+1) = x
τ n < t ≤ tn+1
We convert space error into time error:
tn+1
6
x
τn
-
tn
-
x̂ x̃
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Trace-back (or Forward) Point Adjustment—1
Proceed away from injection wells and inflow boundaries by “layers.”
For each layer, obtain volume conservation in two steps.
1. Volume conservation of the layer. Adjust the exterior contour of the
entire layer along the characteristics until the volume of the layer is
correct (within a small tolerance). That is, in the absence of other
sources, inflow boundaries, and sinks,
Z
X
E in the layer E
w
``
`w
DD
w
Dw
DD
Dw
w
```w
D
D
Dw
w
DD
w`
Dw
`
`w w
w
a
w
w
a
aw
w
!
!
g!
g
C
```
!Cg
!!
×
×
×
×
g
D
D
Dg
a
a
a
g
a
g
a
a
E
E
E
E
E
E
φ dx =
X
Z
E in the layer Ẽ
φ dx
Adjusted point (fixed)
g Points adjusted simultaneously in the
direction of the characteristic (we use
a type of “bisection” algorithm)
× Points adjusted individually transverse
to the flow in Step 2
w
@
Flow
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The University of Texas at Austin, USA
Trace-back (or Forward) Point Adjustment—2
2. Element volume conservation. Within the layer, sequentially adjust
the interior midpoint of each element transverse to the flow until the
volume of the element is correct (within a small tolerance).
>
xi,j+1
Flow
z
z
z
XXX
L
BB
XXX
L
z
B
L
L
B
L
B
B
L
L
B
z
XXX
z
XX
XXX X
z
×
<
×>
xi,j+1/2
z
L
L
L
L
L
L
L
z
z
×
Adjusted point (fixed)
× Points individually
adjusted transverse to
the flow
z
xi,j
Remark: This is an extremely fast direct algorithm.
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The University of Texas at Austin, USA
Numerical Results
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Numerical Results
Darcy’s Law completes the equations:
k
u = − ∇p
µ
p is the fluid pressure
k is the permeability
µ is the fluid viscosity
Measure the variability of k by the dimensionless coefficient of variation
Cv =
1
Mk
Z
1
(k(x) − Mk )2 dx
Vol(Ω) Ω
!1/2
where the mean is
Z
1
Mk =
k(x) dx
Vol(Ω) Ω
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A Nuclear Contamination Problem—1
The permeability is log-normal and fractal.
Mk = 2 × 10−10 cm2 (about 20 md)
Cv = 0.522 (varies over five orders of magnitude).
256
-
-
192
1E-08
1E-09
1E-10
1E-11
1E-12
1E-13
Inflow
128
Y
-
64
Outflow
-
-
-
0
0
64
t
128
192
256
Injection well
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A Nuclear Contamination Problem—2
If we use a small time step of ∆t = 1.5 years, we can trace back into
the injection well.
2.00
0.10
0.01
-0.01
-0.10
CMM volume errors
VCCMM (volume errors 10−9)
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A Nuclear Contamination Problem—3
Concentration at 30 years on a 64 × 64 grid, with ∆t = 1.5 yr ≈ 2.6 CFL
192
192
160
1.1E-05 160
1.0E-05
9.0E-06
8.0E-06
128
7.0E-06
6.0E-06
5.0E-06
4.0E-06 96
128
96
64
32
64
96
128
CMM
160
64
32
1.1E-05
1.0E-05
9.0E-06
8.0E-06
7.0E-06
6.0E-06
5.0E-06
4.0E-06
64
96
128
VCCMM
CMM overshoots the maximum concentration of 1E-5
by 34% up to 1.34E-5.
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160
The Courant-Friedrichs-Lewy (CFL) Condition
Explicit methods in 1-dimension have a time step restriction, known as
the Courant-Friedrichs-Lewy (CFL) time-step, given by
hφ(x)
x∈Ω |u(x)|
∆t ≤ ∆tCFL,1-D = max
where h is the grid spacing.
In 2-dimensions, we should limit ∆t to half this value,
hφ(x)
∆t ≤ ∆tCFL,2-D = max
x∈Ω 2|u(x)|
Godunov’s method is a popular method, that is unstable if the CFL
condition is violated.
In principle, characteristic methods are not subject to this constraint,
and large time steps can be used.
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A Nuclear Contamination Problem—4
Concentration at 30 years on a 64 × 64 grid
192
192
1.0E-05
9.0E-06 160
8.0E-06
7.0E-06
6.0E-06 128
5.0E-06
4.0E-06
96
160
128
96
64
32
64
96
128
160
Godunov ∆t = 0.586 yr = 1 CFL
•
•
•
•
64
32
1.0E-05
9.0E-06
8.0E-06
7.0E-06
6.0E-06
5.0E-06
4.0E-06
64
96
128
160
VCCMM-TF ∆t = 3 yr = 5.1 CFL
We use trace-forwarding near the well.
No overshoot for either method.
Less numerical diffusion for VCCMM-TF.
51 Godunov steps vs. 10 for VCCMM-TF.
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A Nuclear Contamination Problem—5
Concentration at 30 years on a 128 × 128 grid
64
192
1.0E-05
9.0E-06 160
8.0E-06
7.0E-06
6.0E-06 128
5.0E-06
4.0E-06
96
96
128
160
192
32
64
96
128
160
Godunov ∆t = 0.146 yr = 1 CFL
•
•
•
•
64
32
1.0E-05
9.0E-06
8.0E-06
7.0E-06
6.0E-06
5.0E-06
4.0E-06
64
96
128
160
VCCMM-TF ∆t = 1 yr = 6.8 CFL
We use trace-forwarding near the well.
No overshoot for either method.
Less numerical diffusion for VCCMM-TF.
205 Godunov steps vs. 30 for VCCMM-TF.
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A Nuclear Contamination Problem—6
Concentration at 30 years on a 256 × 256 grid
64
192
1.0E-05
9.0E-06 160
8.0E-06
7.0E-06
6.0E-06 128
5.0E-06
4.0E-06
96
96
128
160
192
32
64
96
128
160
64
32
1.0E-05
9.0E-06
8.0E-06
7.0E-06
6.0E-06
5.0E-06
4.0E-06
64
96
128
160
Godunov ∆t = .0366 yr = 1 CFL VCCMM-TF ∆t = 0.5 yr = 13.7 CFL
•
•
•
•
We use trace-forwarding near the well.
No overshoot for either method.
Less numerical diffusion for VCCMM-TF.
820 Godunov steps vs. 60 for VCCMM-TF.
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A Quarter Five-Spot Problem—1
Geostatistically generated permeability.
Mk = 100 md
Cv = 2.58 (varies over four orders of magnitude).
5E-12
1E-12
5E-13
1E-13
5E-14
1E-14
5E-15
1E-15
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A Quarter Five-Spot Problem—2
Concentration at 3.36 years using ∆t = 0.012 yr = 10.68 CFL
1.00
0.85
0.70
0.55
0.40
0.25
0.10
CMM
1.00
0.85
0.70
0.55
0.40
0.25
0.10
VCCMM-TF
• CMM shows both overshoot and undershoot.
• Very large initial volume imbalances throughout the domain.
• If ∆t = 0.0136 yr = 12.10 CFL initially creates degenerate trace-back
regions, which cannot be used.
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A Linear Flood Problem—1
A test with an inflow boundary.
• Permeability field of the quarter five-spot problem
• Linear pressure drop across the domain in the x-direction.
Concentration at 25 years using ∆t = 0.18 year = 3 CFL.
1.000
0.800
0.600
0.400
0.200
0.000
CMM
VCCMM
• CMM has severe overshoots up to c = 1.65.
• Initial volume errors exceed 10% in the interior of the domain.
• If ∆t = 9 CFL, initial relative volume errors are around 25%.
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1.000
0.800
0.600
0.400
0.200
0.000
A Linear Flood Problem—2
Concentration at 25 years using trace-forwarding of the inflow boundary.
1.000
0.800
0.600
0.400
0.200
0.000
VCCMM-TF
∆t = 0.18 yr = 3 CFL
1.000
0.800
0.600
0.400
0.200
0.000
VCCMM-TF
∆t = 0.36 yr = 6 CFL
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Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
A Fluvial Domain Problem—1
•
•
•
•
Domain 600 × 600 feet2 solved on a 60 × 60 grid.
Permeability of 3 values, Mk = 4.056 Darcy and Cv = 1.15. φ = 0.2.
Wells in opposite corners, injecting 1 pore volume every 3 years.
∆t = 0.015 years = 12.84 CFL.
0.1
1.0
10.0
The permeability, in Darcies
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
A Fluvial Domain Problem—2
VCCMM-TF concentration
1.000
0.800
0.600
0.400
0.200
0.000
Time 1.05 years (step 70)
1.000
0.800
0.600
0.400
0.200
0.000
Time 1.65 years (step 110)
Remark: CMM alone produces negative concentrations on a 30 × 30 grid
with ∆t = 0.01 year, indicating that the trace-back regions self intersect.
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
A Fluvial Domain Problem—3
CMM with trace-forwarding of wells only
1.000
0.800
0.600
0.400
0.200
0.000
Time 1.05 years (step 70)
1.000
0.800
0.600
0.400
0.200
0.000
Time 1.65 years (step 110)
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
Comparison with an Analytic Solution
Comparison of VCCMM-TF concentration with an analytic solution for
radial flow from a well in a horizontally infinite, uniform porous medium.
1
1
0.9
0.9
Analytic
0.8
0.7
0.7
VCCMM-TF
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
50
100
Longitudinal dispersion 20 cm
VCCMM-TF
0.6
0.5
0
Analytic
0.8
0
50
100
Longitudinal dispersion 100 cm
The analytic solution is derived by a code from P.A. Hsieh (1986).
VCCMM-TF uses ∆t = 8 CFL.
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA
Conclusions
1. A critical aspect of approximating hyperbolic transport problems is
to conserve the mass of the tracer locally.
2. It is just as critical to the conserve locally the mass of the bulk fluid.
3. When adjusting trace-back points, care must be used to avoid
introducing systematic bias into the transport computation. The key
is to adjust along the characteristics when possible.
4. Trace-forwarding is needed around injection wells.
5. Inflow boundaries can be treated transparently through a space-time
“fold-down” strategy, or with trace-forwarding.
6. In principle, the only restriction on the time step is that the
traceback regions not degenerate or self-intersect (this problem can
be alleviated by tracing more points of the boundary ∂E).
7. VCCMM-TF allows us to take large time steps (such as 14 times the
2-D CFL limited time step) with no overshoots, which results in less
numerical dispersion than Godunov’s method.
Center for Subsurface Modeling
Institute for Computational Engineering and Sciences
The University of Texas at Austin, USA