destination site of type S
m+1 . The loss, return, and conveyance factors are estimated
from available information, including county and municipal water and sewer
planning documents.
Denoting the n m sites of node type, S
m , as S
m
1 , S
m
2 , . . . , S
m
n m
at locations
x m1 , x m2 , . . . , x mn m , where the x may be points, or extended locational data types,
that is, lines or polygons, then, for each node type, S
m , consumptive losses and
return flows at each of the sites S
m
i m
have time-dependent values,
L
m
i m
t
ð Þ ¼ consumptive loss fraction at site S
m
i m
at time, t, i m ¼ 1, 2, . . . , n m
R
m
i m
t
ð Þ ¼ return flow fraction at site S
m
i m
at time, t, i m ¼ 1, 2, . . . , n m
Each of the three conveyance types, C
m m+1 , can have up to (n m  n m + 1 )
conveyances. For example, conveyance type C
2 3 comprises the systems conveying
treated water from the treatment plants to users. It would be possible, in the case of
a highly interconnected system, for each of the n 3 users to obtain some portion of
their water from all of the n 2 treatment plants. Finally, the portion of the water from
site S
m
i m
that is conveyed to site S
mþ1
i mþ1
at time, t, after losses and return flows at site S
m
i m
,
is given by a conveyance factor, c, where c
m mþ1
i m i mþ1
t
ð Þ is the fraction of water leaving
site S
m
i m
that is conveyed to site S
mþ1
i mþ1
at time, t, with i m ¼ 1, 2, . . . , n m and
i m+1 ¼ 1, 2, . . . , n m+1 and where the condition is imposed that
X n mþ1
i mþ1 ¼1
c
m mþ1
i m i mþ1
t
ð Þ ¼ 1
ð4Þ
that is, that 100 % of the water leaving site S
m
i m
, after losses and return flows at the
site are subtracted, is accounted for.
W i 1 ¼ W i 1 t
ð Þ be the value of the withdrawal at withdrawal site S
1
i 1
at time, t,
where i 1 = 1, 2, . . ., n 1 . Several withdrawal sites may contribute to water at a
subsequent site, S
m
i m
, where m = 2, 3, or 4, and water from each W i 1 may have
traveled through multiple conveyance paths before arriving at a subsequent site.
The total amount of water arriving at S
m
i m
, where m = 2, 3, or 4, is the product of all
conveyance factors, c, times the product of all factors representing water remaining
after losses and return flows at sites or conveyances, summed over all potential
withdrawal sites and all potential paths. Then the total loss and total return flows at
site S
m
i m
, for m = 1 to 4, are given by
Loss at S
m
i m
¼
X n 1
i 1
. . .
X n mÀ1
i mÀ1
W i 1 Â
Y
mÀ1
p¼1
1 À L
p
i p
À R
p
i p
 C
p pþ1
i p i pþ1
h
i
 L
m
i m
ð5Þ
156
J. Ducnuigeen et al.
m+1 . The loss, return, and conveyance factors are estimated
from available information, including county and municipal water and sewer
planning documents.
Denoting the n m sites of node type, S
m , as S
m
1 , S
m
2 , . . . , S
m
n m
at locations
x m1 , x m2 , . . . , x mn m , where the x may be points, or extended locational data types,
that is, lines or polygons, then, for each node type, S
m , consumptive losses and
return flows at each of the sites S
m
i m
have time-dependent values,
L
m
i m
t
ð Þ ¼ consumptive loss fraction at site S
m
i m
at time, t, i m ¼ 1, 2, . . . , n m
R
m
i m
t
ð Þ ¼ return flow fraction at site S
m
i m
at time, t, i m ¼ 1, 2, . . . , n m
Each of the three conveyance types, C
m m+1 , can have up to (n m  n m + 1 )
conveyances. For example, conveyance type C
2 3 comprises the systems conveying
treated water from the treatment plants to users. It would be possible, in the case of
a highly interconnected system, for each of the n 3 users to obtain some portion of
their water from all of the n 2 treatment plants. Finally, the portion of the water from
site S
m
i m
that is conveyed to site S
mþ1
i mþ1
at time, t, after losses and return flows at site S
m
i m
,
is given by a conveyance factor, c, where c
m mþ1
i m i mþ1
t
ð Þ is the fraction of water leaving
site S
m
i m
that is conveyed to site S
mþ1
i mþ1
at time, t, with i m ¼ 1, 2, . . . , n m and
i m+1 ¼ 1, 2, . . . , n m+1 and where the condition is imposed that
X n mþ1
i mþ1 ¼1
c
m mþ1
i m i mþ1
t
ð Þ ¼ 1
ð4Þ
that is, that 100 % of the water leaving site S
m
i m
, after losses and return flows at the
site are subtracted, is accounted for.
W i 1 ¼ W i 1 t
ð Þ be the value of the withdrawal at withdrawal site S
1
i 1
at time, t,
where i 1 = 1, 2, . . ., n 1 . Several withdrawal sites may contribute to water at a
subsequent site, S
m
i m
, where m = 2, 3, or 4, and water from each W i 1 may have
traveled through multiple conveyance paths before arriving at a subsequent site.
The total amount of water arriving at S
m
i m
, where m = 2, 3, or 4, is the product of all
conveyance factors, c, times the product of all factors representing water remaining
after losses and return flows at sites or conveyances, summed over all potential
withdrawal sites and all potential paths. Then the total loss and total return flows at
site S
m
i m
, for m = 1 to 4, are given by
Loss at S
m
i m
¼
X n 1
i 1
. . .
X n mÀ1
i mÀ1
W i 1 Â
Y
mÀ1
p¼1
1 À L
p
i p
À R
p
i p
 C
p pþ1
i p i pþ1
h
i
 L
m
i m
ð5Þ
156
J. Ducnuigeen et al.