Unsaturated Flow Governing Equations

—Richards’ Equation

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

• Mass outflow rate through face EFGH is

• The net inflow rate is thus

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

• If ρw varies neither spatially nor temporally, (1) becomes

= (2)

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

• Substituting the above equations into Eq 3 leads to

= (4)

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)