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Published byHilary Taylor Modified over 6 years ago

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Unsaturated Flow Governing Equations —Richards’ Equation

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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

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1. Richards Eq: General Form Mass outflow rate through face EFGH is The net inflow rate is thus

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1. Richards Eq: General Form Similarly, net inflow rate thru face DCGH is and net inflow rate thru face ADHE is

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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 ]

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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)

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1. Richards Eq: General Form If ρ w varies neither spatially nor temporally, (1) becomes = (2)

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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

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1. Richards Eq: General Form Substituting the above equations into Eq 3 leads to = (4)

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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

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2. Simplified Cases (A) If L z (gravity gradient) negligible compared to the strong matric potential L, (4) becomes (5)

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

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

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

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

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

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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)

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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)

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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)

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