Issues other than obtaining mass conservation when solving Richards’ equation are clearly important as well;
the strong non-linear nature of the Richards’ equation often causes serious problems in the iterative search for a
convergent solution. In extreme situations, some schemes simply fail to find a convergent solution. In addition,
source terms that are non-linear functions of the soil water potential, which normally are present when dealing with
realistic field situations, can have additional negative effects on a procedure’s ability to find a convergent solution.
These problems are addressed in the present study where the performance and ease of implementation for three
procedures for solving Richards’ equation are compared.
The first procedure referred to as P1 is based on the popular implicit and mass-conserving iteration scheme by
Celia et al.,
3,4
and is implemented as described by Ahuja et al.
1
The second procedure P2 is a new procedure, where mass conservation is obtained by applying the approxi-
mation of the specific water content suggested indepen- dently by Cooley
5
and Milly.
14
The third procedure P3 is based on a standard implicit discretization, and for that
reason it does not facilitate mass conservation. The main focus is on the two mass-conserving procedures where the
problems arising from the described non-linearities are solved in significantly different ways.
2 THEORY
All three procedures produce solutions of Richards’ equation where a source term has been included:
] v
] t
¼ ]
] z
K ]
w ]
z ¹
1 þ
S 1
where v is the volumetric soil water content, w is the soil water potential, K is the hydraulic conductivity, t is the
time, z is the depth below the soil surface, assumed to be positive downward, and S is the source term.
The iteration schemes in the three procedures are listed below, where superscripts n and n þ 1 are used for the time
levels at time t and t þ Dt, respectively, and m and m þ 1 denote the iteration levels where values at level m are known.
2.1 Procedure P1
As described by Ahuja et al.,
1
an initial guess in each time step of the desired solution is calculated by a single Euler
step using eqn 2: v
n þ 1 ,
1 j
¼ v
n j
þ Dt
K
n j þ 1=2
0:5Dz
j
Dz
j
þ Dz
j þ 1
w
n j þ 1
¹ w
n j
¹ K
n j ¹ 1=2
0:5Dz
j
Dz
j ¹ 1
þ Dz
j
w
n j
¹ w
n j ¹ 1
¹ K
n j þ 1=2
¹ K
n j ¹ 1=2
Dz
j
ð 2Þ
Based on values of v
n þ 1 ,
1 j
, values of w
n þ 1 ,
1 j
are calculated and the following iteration steps are carried out using
the mass-conserving iteration scheme derived by Celia et al.:
3
ˆ C
n þ 1 ,
m j
Dt d
n þ 1 ,
m þ 1 j
¹ K
n þ 1 ,
m j þ 1=2
0:5Dz
j
Dz
j
þ Dz
j þ 1
3 d
n þ 1 ,
m þ 1 j þ 1
¹ d
n þ 1 ,
m þ 1 j
þ K
n þ 1 ,
m j ¹ 1=2
0:5Dz
j
Dz
j ¹ 1
þ Dz
j
d
n þ 1 ,
m þ 1 j
¹ d
n þ 1 ,
m þ 1 j ¹ 1
¼ K
n þ 1 ,
m j þ 1=2
0:5z
j
Dz
j
þ Dz
j þ 1
w
n þ 1 ,
m j þ 1
¹ w
n þ 1 ,
m j
¹ K
n þ 1 ,
m j ¹ 1=2
0:5Dz
j
Dz
j ¹ 1
þ Dz
j
w
n þ 1 ,
m j
¹ w
n þ 1 ,
m j ¹ 1
¹ K
n þ 1 ,
m j þ 1=2
¹ K
n þ 1 ,
m j ¹ 1=2
Dz
j
¹ v
n þ 1 ,
m j
¹ v
n j
Dt þ
S
n j
ð 3Þ
where ˆ
C
n þ 1 ,
m j
¼ dv
dw
n þ 1 ,
m
4 and
d
n þ 1 ,
m þ 1 j
¼ w
n þ 1 ,
m þ 1 j
¹ w
n þ 1 ,
m j
5 Since all values at iteration level m are known, eqn 3
describes a linear tridiagonal system of equations in d
n þ 1 ,
m þ 1 j
. After the system of equations is solved using the Thomas algorithm,
17
new values of w
n þ 1 ,
m þ 1 j
are cal- culated using eqn 5 and the process is repeated eqns 3–
5 until a convergent solution is found the criterion is described below. If convergence fails the number of itera-
tions exceeds a certain maximum before the criterion for convergence is reached, the time step is cut in half and the
entire process is repeated starting with eqn 2. If conver- gence still fails, the time step is set back again and the
process is repeated. This setting back of the time step will continue until a convergent solution is found or a pre-
defined minimum time step is reached. If a convergent solution is not found using this minimum time step, the
initial guess given by the Euler step is used as the solution. After a solution is found, and if the time step is less than a
predefined maximum, the time step is increased by a given factor for future calculations. It should be noted that
d
n þ 1 ,
m þ 1 j
and the entire left side of eqn 3, goes toward zero when a convergent solution is approached. The source
term is included explicitly as it is evaluated at time level n eqn 3.
2.2 Procedure P2
