Chemical Hydrograph
Separation
1
180
150
Tracer 2
STEP 1 MIXING
DIAGRAMS
Streamflow
120
End-member 1
90
End-member 2
60
End-member 3
30
0
0
20
40
60
80
100
Tracer 1
•Generate all plots for all pair-wise combinations of tracers;
• The simple rule to identify conservative tracers is to see if streamflow samples
can be bound by a polygon formed by potential end-members or scatter around a
line defined by two end-members;
• Be aware of outliers and curvature which may indicate chemical reactions!
2
Assumptions
• Only 2 components in Streamflow
• Mixing is complete
• Tracer signal is distinct for each
component
• No evaporation or exchange with the
atmosphere
• Concentrations of the tracer are constant
over time or changes are known
3
MIXING
MODEL: 3
COMPONENT
S(Using
Specific
Discharge)
• Two
Conservative
Tracers
• Mass Balance
Equations for
Water and Tracers
Simultaneous Equations
Q1  Q2  Q3  Qt
C11Q1  C21Q2  C31Q3  Ct1Qt
C12Q1  C22Q2  C32Q3  Ct2Qt
Solutions
(Ct1  C31 )(C22  C32 )  (C21  C31 )(Ct2  C32 )
Q1  1
Qt
(C1  C31 )(C22  C32 )  (C21  C31 )(C12  C32 )
Ct1  C31
C11  C31
Q2  1
Qt  1
Q1
C2  C31
C2  C31
Q3  Qt  Q1  Q2
Q - Discharge
C - Tracer Concentration
Subscripts - # Components
Superscripts - # Tracers
4
MIXING
MODEL: 3
COMPONENT
S(Using
Discharge
Fractions)
Simultaneous Equations
f1  f 2  f 3  1
C11 f1  C21 f 2  C31 f 3  Ct1
C12 f1  C22 f 2  C32 f 3  Ct2
Solutions
(Ct1  C31 )(C22  C32 )  (C21  C31 )(Ct2  C32 )
f1  1
(C1  C31 )(C22  C32 )  (C21  C31 )(C12  C32 )
• Two
Conservative
Tracers
Ct1  C31 C11  C31
f2  1

f1
C2  C31 C21  C31
• Mass Balance
Equations for
Water and Tracers
f - Discharge Fraction
f 3  1  f1  f 2
C - Tracer Concentration
Subscripts - # Components
Superscripts - # Tracers
5
MIXING MODEL:
Generalization
Using Matrices
• One tracer for 2 components
and two tracers for 3
components
• N tracers for N+1
components? -- Yes
• However, solutions would be
too difficult for more than 3
components
• So, matrix operation is
necessary
Simultaneous Equations
f1  f 2  f 3  1
Cx f x  Ct
C11 f1  C21 f 2  C31 f 3  Ct1
C12 f1  C22 f 2  C32 f 3  Ct2
Where
1
1
1
f1
1
C x  C11
C21
C31
f x  f2
Ct  Ct1
C12
C22
C32
f3
Ct2
Solutions
f x  C x1Ct
Note:
• Cx-1 is the inverse matrix of Cx
• This procedure can be generalized to
N tracers for N+1 components
6
Cook and Herczeg, 2000, Environmental Tracers in Subsurface Hydrology
Stable isotopes are efficient tools for identifying
the history of water…
Microsoft® ClipArt
NO3
d15N = 20‰
d13C = -27‰