Appendix
Often, in addition to being constrained by the total cycle volume of a planned oscillator,
another major constraint on field investigations may be the desire to have maximum head
changes at the pumping location stay below some predefined threshold. There are practical
reasons for imposing this constraint: 1) Excessive drawdown during the extraction phase of
oscillation may cause the pumping interval to become dry or experience undesirably high
pressure changes (e.g., may generate suspended sediment and/or rearrange sediments in
the formation adjacent to the well pumping interval), and 2) Excessively high head
decreases or increases near the stimulation location can make extraction / injection much
more energy intensive, which may be undesirable in remote field sites (where power
generation is limited) and in cases where oscillating testing would be planned for long-term
monitoring (where power requirements add to long-term operational costs). The
computations required for optimizing signal propagation distances, under both total
oscillator volume and pumping location head change constraints, are slightly more complex
and require testing of several possible local minima. However, we provide an algorithm
below for optimizing signal period and flow rate amplitude under these constraints, and
include a MATLAB implementation of this algorithm as supplemental material to this paper.
Optimizing the oscillator amplitude and frequency to maximize signal amplitude at
some distance – in the presence of both total cycle volume and pumping location head
change constraints – is a non-linear optimization with nonlinear constraints. The general
constrained optimization problem is:
max M ( P,Q)
Q,P
C1 : c1 ( P,Q) £ 0
C2 : c2 ( P,Q) £ 0
C3 : P ³ Pmin
C4 : P £ Pmax
C5 :Q ³ Qmin
where
(0.1)
M represents the objective function of signal magnitude at a distance under a
particular choice of
P and Q, C1 will represent a constraint on near-pumping-location
head changes, and C2 will represent the total cycle volume constraint. Lastly, constraints
C3 and C4 represent minimum and maximum plausible periods possible for the oscillator
(with both Pmax and Pmin positive), as may be required by specific hardware utilized, and
C5 sets a minimum value (which must be greater than 0) for the flow rate amplitude Q.
Point Source Case
In the case of a point-source oscillator, the objective function M ptsrc ( P,Q) is the amplitude
of the head change expected at the distance of the observation robs by using the
homogeneous, confined solution of Black and Kipp (1981),
1/2
æ
Q
æp Ss æ æ
M ptsrc ( P,Q) =
exp æ-robs æ
æ PK æ
æ æ
4p Krobs
æ
æ
(0.2)
C1 constrains near-pumping-location head change, and uses a common approximation with
rnear as a nearby location [ L ] (e.g., the distance from the pumping well to the observation
well) and hnear a maximum amplitude of head change [ L ] allowed at that location, with
c1, ptsrc ( P,Q) =
1/2
æ
Q
æp S æ æ
exp æ-rnear æ s æ æ - hnear
æ PK æ æ
4p Krnear
æ
C2 constrains the total cycle volume V , with
(0.3)
c2, ptsrc ( P,Q) = Q All symbols other than
pV
P
(0.4)
P and Q are user-defined constants for a particular aquifer
scenario investigated.
In analyzing this nonlinearly-constrained nonlinear optimization problem, we first
note that the objective function Mptsrc is a continuously differentiable monotonic function of
both
P and Q. It is easily shown that the condition ÑM ptsrc = 0 is never met within the
feasible region (there are no local maxima, minima, or saddle points that meet all
constraints), which implies that the optimal values of
P and Q must occur on the
boundaries of the feasible region (i.e., where one or more of the constraint equalities is/are
met).
If C1 equality holds (i.e., c1, ptsrc ( P,Q) = 0 ), this implies
æ æp S æ1/2 æ
Q = 4p hnear Krnear exp ærnear æ s æ æ
æ æ PK æ æ
(0.5)
or, plugging this equality into the objective function,
1/2
æ
hnear rnear
æp Ss æ æ
M ptsrc ( P) =
exp æ( rnear - robs ) æ
æ PK æ
æ æ
robs
æ
æ
" P,Q where c1, ptsrc ( P,Q) = 0
rnear < robs , by definition, so along boundary c1, ptsrc , M ptsrc is a monotonically increasing
function of P, meaning that if the optimum of the full optimization problem (0.1) occurs on
the boundary c1, ptsrc = 0 , it must occur where other constraint equalities are met (i.e., a
point at the intersection of two feasible region boundaries).
A similar analysis for C2 equality shows that along this boundary
(0.6)
M ptsrc ( P) =
1/2
æ
V
æp S æ æ
exp æ-robs æ s æ æ
æ PK æ æ
4PKrobs
æ
(0.7)
" P,Q where c2, ptsrc ( P,Q) = 0
2
As shown in the main manuscript, this function has a local maximum at P = p Ssrobs
/ 4K ,
2
, and thus may represent the location of the global maximum for M ptsrc , if
Q = 4VK / Ssrobs
this particular point meets all other constraints.
It is easily shown that M ptsrc is monotonically increasing or decreasing along the
boundaries where constraint C3 , C4 , or C5 equalities are met. Thus, the following
algorithm can be used to determine the global optimum that maximizes M ptsrc :
2
2
1. Check whether the point P = p Ssrobs
(which is along c2, ptsrc = 0 )
/ 4K , Q = 4VK / Ssrobs
falls within the feasible region. That is, verify this point meets constraints C1 , C3 , C4 ,
and C5 . If so, evaluate M ptsrc at this location.
2. Evaluate M ptsrc at all other intersection points of constraint equalities that define the
boundaries of the feasible region.
3. The global optimum must be, amongst those points found in Steps 1 and 2, the point
with the highest M ptsrc value.
Fully-Penetrating Case
In a similar fashion, the objective function and constraints for the problem with a fullypenetrating oscillator are:
ææ2p r 2 Sæ1/2
æ
Q
M linesrc ( P,Q) =
K 0 ææ obs æ exp ( ip / 4 )æ
2p T ææ PT æ
æ
(0.8)
ææ2p r 2 Sæ1/2
æ
Q
c1,linesrc ( P,Q) =
K 0 ææ near æ exp ( ip / 4 )æ - hnear
2p T ææ PT æ
æ
c2,linesrc ( P,Q) = Q -
pV
P
(0.9)
(0.10)
If C1 equality holds,
æ ææ2p r 2 Sæ1/2
ææ
Q = ( 2p Thnear ) / æK 0 ææ near æ exp ( ip / 4 )ææ " P,Q where c1 ( P,Q) = 0
æ PT æ
æ
ææ
æ æ
æ
which means that along this boundary
M linesrc ( P,Q) = hnear
(
(
)
)
1/2
2
æ æ 2p robs
æ
S
K
exp ( ip / 4 )ææ
PT
0
æ
æ
æ
æ
æ
" P,Q where c1,linesrc ( P,Q) = 0
1/2
2
æ æ 2p rnear
æ
S
æ
K0
exp ( ip / 4 ) æ
PT
æ
ææ
æ æ
(0.12)
This function is monotonic and so, as before, if the global optimum occurs along the
boundary of C1 equality, it must occur at the intersection with another constraint
boundary. As shown earlier, along the boundary c2,linesrc ( P,Q) = 0 , a local minimum may
exist. As before, in the point source case, it is easily verifiable that the objective function is
also monotonic along the constraint boundary equalities for C3 , C4 , and C5 . Thus, similar
to the point source case, an optimum period and frequency under total cycle volume and
near-pumping-location head change constraints can be found as follows:
2
2
1. Check whether the point P = 2p robs
S/ l̂T , Q = l̂TV / 2robs
S (which is along c2 = 0 )
falls within the feasible region. That is, verify this point meets constraints C1 , C3 , C4 and
C5 . If so, evaluate M linesrc at this location.
(0.11)
2. Evaluate M linesrc at all other intersection points of constraint equalities that define the
boundaries of the feasible region.
3. The global optimum must be, amongst those points found in Steps 1 and 2, the point with
the highest M linesrc value.