us army corps of engineers ® engineer research and development center reactive transport (3): solve...
TRANSCRIPT
US Army Corpsof Engineers®
Engineer Research and Development Center
Reactive Transport (3):Solve Biogeochemistry with The
Primitive Approach
Reactive Transport (3):Solve Biogeochemistry with The
Primitive Approach
Pearce Cheng (x3699)
Last Update: October 8, 2008
US Army Corpsof Engineers®
Engineer Research and Development Center
OutlineOutline
• A Generic Biogeochemical System• Primitive Approach• Advantages and Disadvantages with The Primitive Approach• Solve Biogeochemistry with The Primitive Approach
US Army Corpsof Engineers®
Engineer Research and Development Center
A Generic Biogeochemical SystemA Generic Biogeochemical System
• No. of species = M• No. of reactions = Nr
– Fast reversible (equilibrium) reactions– Slow reversible/irreversible (kinetic) reactions– Fast irreversible (one way instantaneous) reactions
11RνG
U dt
d
where
U
G
1ν ik ik
1R
= the (MxM) identity matrix; = the vector representing the M species concentrations [M/L3];
= the (MxNr) reaction stoichiometry matrix with & as components;
= the vector representing the Nr reaction rates [M/(tL3)].
US Army Corpsof Engineers®
Engineer Research and Development Center
Example for DemonstrationExample for Demonstration 6
7
rN
M
HNTANTAH 1R
NTACoCoNTA 2R
HNTACoHCoNTA
BHHNTA
HNTACoNTANTACoH 2 5R
(R1) a fast reversible (equilibrium) reaction
(R2)
(R3) a slow reversible (kinetic) reaction
(R4) a slow irreversible (kinetic) reaction
(R5)
NTA: sodium nitrilotriacetate; C6H9NO6; a chelating agent used in detergents to improve their cleaning ability
a fast reversible (equilibrium) reaction
a fast reversible (equilibrium) reaction
PHB (R6) a fast irreversible (one way instantaneous) reaction
6R
US Army Corpsof Engineers®
Engineer Research and Development Center
5
6
rN
M
65431
][RRRRR
dt
Hd 521 2
][RRR
dt
NTAd
5431
][RRRR
dt
HNTAd 532
][RRR
dt
Cod
532
][RRR
dt
CoNTAd 64
][RR
dt
Bd
6
][R
dt
Pd
6
7
rN
M
Primitive ApproachPrimitive Approach
US Army Corpsof Engineers®
Engineer Research and Development Center
Advantages with The Primitive Approach
Advantages with The Primitive Approach
• Straightforward in terms of formulation and solution strategy
• Easy to be incorporated into reactive transport equations
US Army Corpsof Engineers®
Engineer Research and Development Center
Disadvantages with The Primitive Approach (1/2)
Disadvantages with The Primitive Approach (1/2)
• The time step size must be sufficiently (or infinitely) small to resolve fast/equilibrium reactions DAE approach
• The integration of the primitive rate equations, due to numerical errors, does not guarantee the conservation of the total mass of component chemical/species, which is necessary. DAE approach
• There is no way to define the subtraction or addition of infinity if more than one equilibrium reactions are involved DAE approach
65431
][RRRRR
dt
Hd
US Army Corpsof Engineers®
Engineer Research and Development Center
Disadvantages with The Primitive Approach (2/2)
Disadvantages with The Primitive Approach (2/2)
• Even if all reactions are slow/kinetic reactions, their rates are coupled via the concentration-versus-time curves of all species. They cannot be formulated and parameterized one reaction by one reaction independently of each other (most difficult issue) Identify the kinetic variables associated with the given experimental concentration-versus-time curves
Conc.
T im e
[Co]532
][RRR
dt
Cod
[P]6
][R
dt
Pd
US Army Corpsof Engineers®
Engineer Research and Development Center
The Primitive Approach Can Be Used When …
The Primitive Approach Can Be Used When …
• Fast reversible (equilibrium) reactions do not exist.– Linear equilibrium assumption may be used for
limited cases (e.g., RT3D strategy)• Reaction networks are adequately constructed.• Time steps used are sufficiently small to resolve all slow
reactions.• Special treatments to account for fast irreversible
reactions are incorporated when necessary.
US Army Corpsof Engineers®
Engineer Research and Development Center
Solve Biogeochemistry with The Primitive Approach
Solve Biogeochemistry with The Primitive Approach
• Solve Reaction Network with the Newton Method – Compute numerical Jacobian (generic approach) – Compute analytical Jacobian (when all reactions can be
represented in analytical form)• Compute adaptive time steps to avoid negative
concentrations• Employ special treatments to handle fast irreversible
reactions
US Army Corpsof Engineers®
Engineer Research and Development Center
Compute Adaptive Time StepsCompute Adaptive Time Steps
• For each chemical species, check the contribution from all slow reactions: – Scenario 1:
– Scenario 2:
• Examine and exclude unnecessary reactions:– If ( is a small positive number, e.g., 10-6), then
(a) Remove the reaction that yields the greatest value of (b) Re-calculate by excluding the reaction just removed(c) If still exists, repeat (a) and (b) until
0 1
nrx
iiijij R tt j
0 1
nrx
iiijij R
nrx
iiijij
oldj
j
R
CtMinimumt
1
,
jttt j
iijij R jt
tt j tt j
• Set ncsreaction tttMinimumt ,...,, 21
NoteNote: A simplified approach is actually : A simplified approach is actually adopted. adopted.
US Army Corpsof Engineers®
Engineer Research and Development Center
Special Treatments to Handle Fast Irreversible Reactions
Special Treatments to Handle Fast Irreversible Reactions
• Fast irreversible reactions are achieved immediately when compared to the slow reactions determine available reactants for fast irreversible reactions to occur
• The resultant chemical distribution after accounting for fast irreversible reactions is used to compute biogeochemistry where only slow reactions are taken into account within each nonlinear iteration.
US Army Corpsof Engineers®
Engineer Research and Development Center
Computation Flow Chart
Computation Flow Chart
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Account for Fast
Irreversible Reactions
Compute Adaptive Time
Steps
US Army Corpsof Engineers®
Engineer Research and Development Center
Demonstration Example: (10 species & 4 reactions)
Demonstration Example: (10 species & 4 reactions)
CkBAkR bf1
211
EDkR f22
FCkR f33
NkMHkR bf444
5R 355 PHkR f
(R1) a slow reversible reaction
(R2) a slow irreversible reaction
(R3) a fast irreversible reaction
(R4) a slow reversible reaction
(R5)
CBA 2
FED 2
HGFC 3R
NMH
QPH 3 a fast irreversible reaction
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Set Working Concentration
before Adjustment
Set Working Concentration
before Adjustment
pAA 00
pBB 00
pCC 00
…
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Adjust Species Concentrations Associated with Fast Irreversible
Reactions
Adjust Species Concentrations Associated with Fast Irreversible
Reactions
US Army Corpsof Engineers®
Engineer Research and Development Center
Adjust Species Concentrations Associated with Fast Irreversible Reactions
Adjust Species Concentrations Associated with Fast Irreversible Reactions
00
00
00
10 , FCMinimumCC
00
00
00
10 , FCMinimumFF
0
000
00
10 , FCMinimumGG
00
00
00
10 , FCMinimumHH
3
001
010
20
P, HMinimumHH
3 3
001
00
010
P, HMinimumPP
3
001
000
10
P, HMinimumQQ
(R3) a fast irreversible reactionHGFC 3R
5R(R5) QPH 3 a fast irreversible reaction
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Set New Initial Concentration
before Nonlinear Iterations
Set New Initial Concentration
before Nonlinear Iterations
10 CC p 10 FFp 10 GGp
20 HH p
10 PPp
10 QQp
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Set Working Concentrations
for Nonlinear Iterations
Set Working Concentrations
for Nonlinear Iterations
pnew AA
pold AA
pwork AA
…
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Compute Reaction Rates
for Slow Reactions
Compute Reaction Rates
for Slow Reactions
US Army Corpsof Engineers®
Engineer Research and Development Center
Compute Reaction Rates for Slow ReactionsCompute Reaction Rates for Slow Reactions
CkBAkR bf1
211
EDkR f22
NkMHkR bf444
(R1) a slow reversible reaction
(R2) a slow irreversible reaction
(R4) a slow reversible reaction
CBA 2
FED 2
NMH
NoteNote: The reaction rates are computed based on the working : The reaction rates are computed based on the working concentration: concentration:
NoteNote: The reaction rates can also be computed with the given : The reaction rates can also be computed with the given formulae when necessary.formulae when necessary.
,...,, workworkwork CBA
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Time Step AdaptionTime Step Adaption
US Army Corpsof Engineers®
Engineer Research and Development Center
• Check the overall reaction rate for each species, e.g., for the i-th species
• Adjustment is necessary only when the overall reaction rate is found negative:
Time Step Adaption (A Simplified Approach)Time Step Adaption (A Simplified Approach)
overalliR
overalli
iprxmip
R
CtMinimumtCC ,
, , If
Otherwise, temporarily de-activate the reactions that Otherwise, temporarily de-activate the reactions that would consume the i-th species for the time being.would consume the i-th species for the time being.
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Special Treatment for
Fast Irreversible Reactions
Special Treatment for
Fast Irreversible Reactions
US Army Corpsof Engineers®
Engineer Research and Development Center
Governing EquationsGoverning Equations(R1)(R2)(R3)(R4)(R5)
slow reversible slow irreversible
fast irreversible slow reversible
CBA 2FED 2
HGFC NMH QPH 3 fast irreversible
rxApnew tSourceRAA 1
rxBpnew tSourceRBB 12
31 RrxCpnew nConsumptiotSourceRCC
rxDpnew tSourceRDD 2
rxEpnew tSourceREE 2
322 RrxFpnew nConsumptiotSourceRFF
rxGRpnew tSourcenConsumptioGG 3
rxHRRpnew tSourceRnConsumptionConsumptioHH 453
rxMpnew tSourceRMM 4
rxNpnew tSourceRNN 4
rxPRpnew tSourcenConsumptioPP 53
rxQRpnew tSourcenConsumptioQQ 5
US Army Corpsof Engineers®
Engineer Research and Development Center
Special Treatment to Account for Fast Irreversible Reactions
Special Treatment to Account for Fast Irreversible Reactions
(R3) a fast irreversible reactionHGFC 3R
5R(R5) QPH 3 a fast irreversible reaction
(When (When RR11 > 0) > 0)
rxCrxp tSourcetRCCAvailable 1
rxCp tSourceC (When (When RR11 0) 0)
rxFrxp tSourcetRFFAvailable 22
FAvailableCAvaiableMinimumnConsumptio R 3 ,
rxHrxRp tSourcetRnConsumptioHHAvailable 43
rxHRp tSourcenConsumptioH 3
(When (When RR44 < < 0)0)(When (When RR44 0) 0)
rxPp tSourcePPAvailable
3
5
PAvailableHAvaiableMinimumnConsumptio R ,
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Compute ResidualsCompute Residuals
US Army Corpsof Engineers®
Engineer Research and Development Center
Compute ResidualsCompute Residuals(R1)(R2)(R3)(R4)(R5)
slow reversible slow irreversible
fast irreversible slow reversible
CBA 2FED 2
HGFC NMH QPH 3 fast irreversible
rxArxpold tSourcetRAA 1AResidual
rxBrxpold tSourcetRBB 1B 2Residual
rxCRrxpold tSourcenConsumptiotRCC 31CResidual
rxDrxpold tSourcetRDD 2DResidual
rxErxpold tSourcetREE 2EResidual
rxFRrxpold tSourcenConsumptiotRFF 32F 2Residual
rxGRpold tSourcenConsumptioGG 3GResidual
rxHRrxRpold tSourcenConsumptiotRnConsumptioHH 543HResidual
rxMrxpold tSourcetRMM 4MResidual
rxNrxpold tSourcetRNN 4NResidual
rxPRpold tSourcenConsumptioPP 5P 3Residual
rxQRpold tSourcenConsumptioQQ 5QResidual
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Compute JacobiansCompute Jacobians
US Army Corpsof Engineers®
Engineer Research and Development Center
Compute JacobiansCompute Jacobians
j
jjj
j C
CCCCCCC
C
,....,,,....,, 21i21ii
Residual ResidualResidual
• Compute Numerical Jacobians
jj CC (When (When CCjj > > CCmm))
mC (When (When CCjj CCmm))
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Full-Pivoting Direct SolverFull-Pivoting Direct Solver
ReferenceReference::Numerical Recipes, Numerical Recipes, 22ndnd Edition, Edition,bybyW.H. Press, W.H. Press, S.A. Teukolsky,S.A. Teukolsky,
W.T. Vetterling,W.T. Vetterling,B.P. FlanneryB.P. Flannery
US Army Corpsof Engineers®
Engineer Research and Development Center
Input Data
Adjust the concentration of species associatedwith fast irreversib le reactions as necessary
Com pute reaction rates for s low reactions
Adjust tim e step s ize based on the com puted s low reaction rates
Account for fast irreversib le reactions w ith specia l treatm ent
Com pute residuals
Com pute jacobians
Solve reaction equations w / fu ll-p ivoting d irect solver
Convergence?
Next tim e step?
Yes
NoYes
No End of tim e loop
Start of tim e loop
Start of nonlinear iteration loop
Check Convergence
Check Convergence
relativerelativej
jEMaximum
Convergence Convergence reached ifreached if
oldj
oldj
newjrelative
jC
CCE
(When (When CCjjoldold > >
CCmm))
wherewhere
0relativejE
(When (When CCjjoldold
CCmm))