1. Richards Eq: General Form
(A) Apply mass conservation principle to REV
Mass Inflow Rate − Mass Outflow Rate
= Change in Mass Storage with Time For a REV with volume mass inflow
rate through face ABCD is
1. Richards Eq: General Form
• Similarly, net inflow rate thru face DCGH is
• and net inflow rate thru face ADHE is
1. Richards Eq: General Form
• The total net inflow rate through all faces is then
• The change in mass storage is
θ: volumetric water content [L3 L−3]
1. Richards Eq: General Form
• Equating net inflow rate and time rate of change in mass storage, and dividing both sides by leads to
• In fact, (1) can be obtained directly from
= (1)
1. Richards Eq: General Form
=
(physics or Lecture 12 notes)
is metric potential, h is total potential
(3)
(B) Apply Darcy’s law to Eq 2
1. Richards Eq: General Form
$ Eq 4 is the 3-d Richards equation—the basic theoretical framework for unsaturated flow in a homogeneous, isotropic porous medium
$ Eq 4 is not applicable to macropore flows
$ Eq 4 (Darcy’s law for unsaturated flow) does not address hysteresis effects
$ Both K and ψ are a function of θ, making Richards equation non-linear and hard to solve
2. Simplified Cases(A) If z (gravity gradient) negligible
compared to the strong matric potential , (4) becomes
tzyyxx
z
ψK
ψK
ψK
(5)
2. Simplified Cases(B) 2-d horizontal flow: (4) becomes
For 1-d horizontal flow (6) becomes
xK
xt
ψ)ψ(
θ
tyyxx
ψ
Kψ
K
(7)
(6)
2. Simplified Cases
(C) Vertical flow: if lateral flow elements negligible, (4) becomes
Rewrite it as
)ψ
1)(ψ(θ
zK
zt
tz
K
zK
z
ψ
(8)
2. Simplified Cases
• Use chain rule of differentiation on term
• Define as water capacity, we have
zz
θ
θ
ψψz
ψ
ψ
θ
C
zCz
θ1ψ (9)
2. Simplified Cases
• Substitute the equations into Richards’ Eq, we have
zC
K
zz
K
zCK
zt
θ)θ()θ(
)θ1
1)(θ(θ
2. Simplified Cases
• Define as soil water diffusivity, we have Richards’ Eq in water content form
zD
zz
K
t
θ)θ(
)θ(θ(10)
C
KD
)θ(
2. Simplified Cases• Now, use the chain rule of differentiation to
the relationship of water content and water potential
• We obtain Richards’ Eq in water potential form
t
ψ
t
ψ
ψ
θθ
Ct
)ψ
1)(ψ(ψ
zK
ztC (11)
2. Simplified Cases
• Generally, if flow is neither vertical nor horizontal, we have
• Where is the angle between flow direction and vertical axis; =90o, horizontal flow, Eq 6; =0o, vertical flow, Eq 8
)ψ
α)(cosψ(θ
xK
xt(12)
2. Simplified Cases
• Consider sink/source terms (e.g. plant uptake), we have
• Where S is the sink/source term (e.g. used to represent plant uptake in HYDRUS 1D/2D)
Sx
Kxt
)ψ
α)(cosψ(θ
(13)