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 [L 3 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 ρ w 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 L z (gravity gradient) negligible compared to the strong matric potential L, (4) becomes (5)

2. Simplified Cases (B) 2-d horizontal flow: (4) becomes For 1-d horizontal flow (6) becomes (7) (6)

2. Simplified Cases (C) Vertical flow: if lateral flow elements negligible, (4) becomes Rewrite it as (8)

2. Simplified Cases Use chain rule of differentiation on term Define as water capacity, we have (9)

2. Simplified Cases Substitute the equations into Richards’ Eq, we have

2. Simplified Cases Define as soil water diffusivity, we have Richards’ Eq in water content form (10)

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 (11)

2. Simplified Cases Generally, if flow is neither vertical nor horizontal, we have Where  is the angle between flow direction and vertical axis;  =90 o, horizontal flow, Eq 6;  =0 o, vertical flow, Eq 8 (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) (13)