User's guide of Proceessing Modflow 5.0

Processing Modflow 175
where
z
3 -1
Q x1, Q x2, Q y1, Q y2, Q z1 and Q z2 [L T ] are volume flow rates across the six cell faces.
)x, )y and )z [L] are the dimensions of the cell in the respective coordinate directions;
3 -1 W [L T ] is flow to internal sources or sinks within the cell; and
)h [L] is the change in hydraulic head over a time interval of length )t [T].
y
v sx
x
v sy
vsz+ Mz
M vsz v sz
vsy+ My
M vsy vsx+ Mx
M vsx Fig. 4.2: Flow through a unit volume of a
porous medium
v x1 ' Q x1 / (n @ )y @ )z)
v x2 ' Q x2 / (n @ )y @%)z)
v y1 ' Q y1 / (n @ )x @ )z)
v y2 ' Q y2 / (n @ )x @ )z)
v z1 ' Q z1 / (n @ )x @ )y)
v z2 ' Q z2 / (n @ )x @ )y)
z
Q x1
y
(x1, y1, z1)
x
Q y1
)x
Q z2
)z
Q z1
)y
Q y2
(x2, y2, z2)
Q x2
Fig. 4.3: Finite-difference approach
Eq. 4.2 is the mass balance equation for a finite-difference cell. The left hand side of eq. 4.2
represents the net mass rate of outflow per unit volume of the porous medium, and the right hand
side is the mass rate production per unit volume due to internal sources/sinks and mass storage.
Substitution of Darcy’s law for each flow term in eq. 4.2, ie, Q=)h@K@A/)x, yields an equation
expressed in terms of (unknown) heads at the center of the cell itself and adjacent cells. An
equation of this form is written for every cell in the mesh in which head is free to vary with time.
Once the system of equations is solved and the heads are obtained, the volume flow rates across
the cell faces can be computed from Darcy’s law. The average pore velocity components across
each cell face are
(4.3a)
(4.3b)
(4.3c)
(4.3d)
(4.3e)
(4.3f)
4.1 The Semi-analytical Particle Tracking Method