3.3 Stability condition
The high-resolution Lax–Friedrichs scheme adopted in the present study is an explicit scheme. Therefore, the computa-
tional time step, Dt, is chosen using the CFL condition.
C
n
¼ Dt
Dx g
h þ
gh
p h
i 1:0
17 in which C
n
¼ Courant number. Dt is chosen dynamically
in the numerical model such that eqn 17 is satisfied at all the nodes i ¼ 1,2
… N.
3.4 Subsurface flow
In order to determine the infiltration rate I in the continuity equation for the surface flow, the subsurface flow equations
have to be solved along with an appropriate boundary con- dition at the ground surface. In the present study, a recently
developed strongly implicit finite-difference scheme
20
for the mixed based formulation of the Richards equation is
used to simulate the unsaturated subsurface flow conditions. This scheme ensures mass balance in its solution regardless
of time step size and nodal spacings, and has no limitations when applied to field problems. It is also easy to incorporate
different types of boundary conditions in this scheme. In the present study, two different models 1DS2DSS and
1DS1DSS are developed for simulating the overland flow. In the model 1DS2DSS one-dimensional surface flow with
two-dimensional subsurface flow, the infiltration rate, I, is determined by solving the two-dimensional Richards equa-
tion. In the model 1DS1DSS one-dimensional surface flow with one-dimensional subsurface flow, the subsurface flow
is assumed to occur only in the vertical direction. Fig. 2 shows the definition sketch for the model 1DS1DSS. In
this model, the infiltration rate at any distance x is deter- mined using the one-dimensional Richards equation with
the surface flow depth at that point as the top boundary condition. Model 1DS1DSS results in significant savings
of CPU time. Numerical solution of the Richards equation is described in the following section.
The subsurface flow domain is divided into a number of rectangular blocks. The moisture content, v, and the
pressure head, w, are specified at the center of the block the node, while the velocities are specified at the inter-
block faces. The subscript i refers to the block number in the x direction and the subscript j refers to the block number
in the z direction. The superscripts n and n þ 1 refer to the known and the unknown time levels, respectively. The
finite-difference form of the eqn 5 is
v
n þ 1 i
, j
¹ v
n i
, j
Dt þ
¯ V
i þ 1=2 ,
j
¹ ¯ V
i ¹ 1=2 ,
j
ÿ Dx
þ ¯
V
i ,
j þ 1=2
¹ ¯ V
i ,
j ¹ 1=2
ÿ Dz
¼ ð
18Þ where the bar is used to denote the time-averaged value of
the velocity. Dx and Dz are the nodal spacings in the x and z directions, respectively. The time-averaged velocities are
determined by
¯ V ¼ wV
n þ 1
þ 1 ¹ w
V
n
19 in which, w ¼ time weighting factor. The velocity at any
interblock face is determined using the pressure heads at the neighbouring cell centers. For example;
V
i ,
j þ 1=2
¼ ¹ K
i ,
j þ 1=2
[ w
i ,
j þ 1
¹ w
i ,
j
¹ Dz
] =
Dz 20
in which, K
i,jþ12
is the unsaturated hydraulic conductivity evaluated at the interblock faces between the nodes i,j þ 1
and i,j. Substitution of eqns 19 and 20 in eqn 18 yields
wDt Dx
2
¹ K
n þ 1 i þ 1=2
, j
w
n þ 1 i þ 1
, j
¹ w
n þ 1 i
, j
ÿ þ
K
n þ 1 i ¹ 1=2
, j
w
n þ 1 i
, j
¹ w
n þ 1 i ¹ 1
, j
ÿ þ
wDt Dz
2
¹ K
n þ 1 i
, j þ 1=2
w
n þ 1 i
, j þ 1
¹ w
n þ 1 i
, j
¹ Dz
ÿ þ
K
n þ 1 i
, j ¹ 1=2
w
n þ 1 i
, j
¹ w
n þ 1 i
, j ¹ 1
¹ Dz
ÿ þ v
n þ 1 i
, j
¹ v
n i
, j
¹ 1 ¹ w
Dt Dx
V
n i þ 1=2
, j
¹ V
n i ¹ 1=2
, j
ÿ ¹
1 ¹ w Dt
Dz V
n i
, j þ 1=2
¹ V
n i
, j ¹ 1=2
ÿ ¼
ð 21Þ
The unsaturated hydraulic conductivity at an interblock face is estimated using the pressure heads at the neighbour-
ing cell centers. Cooley
10
, Haverkamp and Vauclin
18
and Narasimhan and Witherspoon
26
have suggested various ways for this purpose. Haverkamp and Vauclin
18
state that the geometric mean is the best choice for estimating
the interblock conductivities. However, Hong et al.
20
reported also experienced during the development of the present model that the iterative solution of eqn 21 fails to
converge if the above procedure is adopted for estimating the K value. This is especially true for infiltration into
initially very dry soils. The geometric mean is strongly weighted towards the lower value and, therefore, water
cannot drain easily if the soil is initially dry. This results in a non-physical build up of pressure. In this study, the
interblock hydraulic conductivity is estimated by the
Fig. 2. Definition sketch for model 1DS1DSS.
Conjunctive surface–subsurface modeling of overland flow 571
weighted arithmetic mean. For example, K
i ,
j þ 1=2
¼ gK
w
i ,
j
þ 1 ¹ g
K w
i ,
j þ 1
22 in which, g is the weight coefficient. Hong et al.
20
suggest a value of 0.5 for g. Eqn 21 is written for all the blocks in
the flow domain and this results in a set of simultaneous algebraic equations in the unknowns w
i,j nþ1
. These simulta- neous equations are highly non-linear since v
nþ1
and K
nþ1
are non-linear functions of w
nþ1
. In the present study, they are solved by using the Newton–Raphson technique.
3.5 Boundary conditions
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