Upscaling and Inverse Modeling of Groundwater Flow and Mass ...

180 APPENDIX A. FLOWXYZ3D - A THREE-DIMENSIONAL . . .
i-1,j-1,k-1
i-1,j-1,k
i-1,j-1,k+1
i-1,j,k-1
i-1,j+1,k-1
i-1,j,k
i-1,j+1,k
i-1,j,k+1 i-1,j+1,k+1
i,j-1,k-1
i,j-1,k
i,j-1,k+1
i,j,k-1
i,j,k
i,j,k+1
i,j+1,k-1
i,j+1,k
i,j+1,k+1
i+1,j+1,k-1
i+1,j,k-1
i+1,j-1,k-1
i+1,j-1,k
i+1,j,k
i+1,j+1,k
i+1,j-1,k+1 i+1,j,k+1i+1,j+1,k+1
Figure A.1: Schematic illustration of the 3D finite-difference spatial discretization
1
∆x|i,j,k
1
∆y|i,j,k
1
∆z|i,j,k
[ (
(
∂h ∂h ∂h
)�
��i+1/2,j,k
+ Kxy + Kxz
−
∂x ∂y ∂z
∂h ∂h ∂h
)� ]
��i−1/2,j,k
+ Kxy + Kxz
+
∂x ∂y ∂z
∂h ∂h ∂h
)�
��i,j+1/2,k
+ Kyy + Kyz
−
∂x ∂y ∂z
∂h ∂h ∂h
)� ]
��i,j−1/2,k
+ Kyy + Kyz
+
∂x ∂y ∂z
∂h ∂h ∂h
)�
��i,j,k+1/2
+ Kyz + Kzz
−
∂x ∂y ∂z
∂h ∂h ∂h
)� ]
��i,j,k−1/2
+ Kyz + Kzz
= 0
∂x ∂y ∂z
Kxx
Kxx
[ (
(
Kxy
Kxy
[ (
(
Kxz
Kxz
(A.2)
The hydraulic gradients at the interfaces are approximated by central differences
from the heads at the nineteen blocks surrounding (i, j, k), That is,