7.2. Model Calibration and Parameter Estimation
209
into consideration. This ca se occurs when the natural velo city is so low that
a significant amount of water containing tracer must be added to speed up
the test.
(c) The velocity imposed by injection is dominant and the natural velocity
can be neglected. This is the case when fresh water is injected to form a
relatively steady radial flow between the injection weIl and the observation
wells.
In the last case, when the flow has not reached a steady state, the single
weIl injection model mentioned before should be used. If the flow approaches
a steady state, the following simpler model may be used:
oe IXLA o2e A oe
ot
-r- or2 - r 8,'
C(r,O) = 0;
r > 0,
C(r o , t) = e o , O:s; t < t o ,
(7.2.22)
C(r o , t) = 0,
t ~ t o ,
C(oo,t) = 0,
where A = Q/2nBn; Q is a constant injection rate; B the thickness of the
aquifer; n the effective porosity; and t o shows the instant before which the
tracer is injected and after which fresh water is injected.
This model can be solved by either FDM or FEM, the solution of which
depends on dispersivity IX L • The value of IX L can be identified by fitting the
model output with the observed concentration values in the weIl.
Let us turn to consider the cases (a) and (b). In fact, it is difficult to form
a radial flow on this scale, because the radial velocity v,. = Air decreases
rapidly with increasing of distance, r, between the wells.
The real velocity is the superposition of the natural velocity and radial
velocity, that is
Vx = Vn,x + v,. cos IX, Vy = v",y + v,. sin IX,
(7.2.23)
where v",x, v",y are the two components of the natural velocity v,,; and IX is
the angle between the x axis and the radial vector.
In this case, we have to construct a numerical model to simulate the
experiment. Both longitudinal dispersivity IX L and transverse dispersivity IX T
can be determined by fitting the output of the numerical model with the
concentration records obtained from observation wells.
Global Scale 11 (Average Distance of Propagation Is 20 to 100 m)
The method of multi-weIl tests mentioned above can also be used for this
scale. However, radial flow cannot be formed by injection because of the long
distance of propagation. Therefore, tracer transportation mainly relies on the
natural flow, which prolongs the test procedure. Moreover, the flow velocity
209
into consideration. This ca se occurs when the natural velo city is so low that
a significant amount of water containing tracer must be added to speed up
the test.
(c) The velocity imposed by injection is dominant and the natural velocity
can be neglected. This is the case when fresh water is injected to form a
relatively steady radial flow between the injection weIl and the observation
wells.
In the last case, when the flow has not reached a steady state, the single
weIl injection model mentioned before should be used. If the flow approaches
a steady state, the following simpler model may be used:
oe IXLA o2e A oe
ot
-r- or2 - r 8,'
C(r,O) = 0;
r > 0,
C(r o , t) = e o , O:s; t < t o ,
(7.2.22)
C(r o , t) = 0,
t ~ t o ,
C(oo,t) = 0,
where A = Q/2nBn; Q is a constant injection rate; B the thickness of the
aquifer; n the effective porosity; and t o shows the instant before which the
tracer is injected and after which fresh water is injected.
This model can be solved by either FDM or FEM, the solution of which
depends on dispersivity IX L • The value of IX L can be identified by fitting the
model output with the observed concentration values in the weIl.
Let us turn to consider the cases (a) and (b). In fact, it is difficult to form
a radial flow on this scale, because the radial velocity v,. = Air decreases
rapidly with increasing of distance, r, between the wells.
The real velocity is the superposition of the natural velocity and radial
velocity, that is
Vx = Vn,x + v,. cos IX, Vy = v",y + v,. sin IX,
(7.2.23)
where v",x, v",y are the two components of the natural velocity v,,; and IX is
the angle between the x axis and the radial vector.
In this case, we have to construct a numerical model to simulate the
experiment. Both longitudinal dispersivity IX L and transverse dispersivity IX T
can be determined by fitting the output of the numerical model with the
concentration records obtained from observation wells.
Global Scale 11 (Average Distance of Propagation Is 20 to 100 m)
The method of multi-weIl tests mentioned above can also be used for this
scale. However, radial flow cannot be formed by injection because of the long
distance of propagation. Therefore, tracer transportation mainly relies on the
natural flow, which prolongs the test procedure. Moreover, the flow velocity