## 2.7.3. Soil Water[¶](#soil-water "Permalink to this headline")
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Soil water is predicted from a multi-layer model, in which the vertical soil moisture transport is governed by infiltration, surface and sub-surface runoff, gradient diffusion, gravity, and canopy transpiration through root extraction ([Figure 2.7.1](#figure-hydrologic-processes)).

For one-dimensional vertical water flow in soils, the conservation of mass is stated as

(2.7.41)[¶](#equation-7-79 "Permalink to this equation")\\\[\\frac{\\partial \\theta }{\\partial t} =-\\frac{\\partial q}{\\partial z} - e\\\]

where \\(\\theta\\) is the volumetric soil water content (mm3 of water / mm\-3 of soil), \\(t\\) is time (s), \\(z\\) is height above some datum in the soil column (mm) (positive upwards), \\(q\\) is soil water flux (kg m\-2 s\-1 or mm s\-1) (positive upwards), and \\(e\\) is a soil moisture sink term (mm of water mm\-1 of soil s\-1) (ET loss). This equation is solved numerically by dividing the soil column into multiple layers in the vertical and integrating downward over each layer with an upper boundary condition of the infiltration flux into the top soil layer \\(q\_{infl}\\) and a zero-flux lower boundary condition at the bottom of the soil column (sub-surface runoff is removed later in the timestep, section [2.7.5](#lateral-sub-surface-runoff)).

The soil water flux \\(q\\) in equation [(2.7.41)](#equation-7-79) can be described by Darcy’s law [(Dingman 2002)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dingman2002)

(2.7.42)[¶](#equation-7-80 "Permalink to this equation")\\\[q = -k \\frac{\\partial \\psi \_{h} }{\\partial z}\\\]

where \\(k\\) is the hydraulic conductivity (mm s\-1), and \\(\\psi \_{h}\\) is the hydraulic potential (mm). The hydraulic potential is

(2.7.43)[¶](#equation-7-81 "Permalink to this equation")\\\[\\psi \_{h} =\\psi \_{m} +\\psi \_{z}\\\]

where \\(\\psi \_{m}\\) is the soil matric potential (mm) (which is related to the adsorptive and capillary forces within the soil matrix), and \\(\\psi \_{z}\\) is the gravitational potential (mm) (the vertical distance from an arbitrary reference elevation to a point in the soil). If the reference elevation is the soil surface, then \\(\\psi \_{z} =z\\). Letting \\(\\psi =\\psi \_{m}\\), Darcy’s law becomes

(2.7.44)[¶](#equation-7-82 "Permalink to this equation")\\\[q = -k \\left\[\\frac{\\partial \\left(\\psi +z\\right)}{\\partial z} \\right\].\\\]

Equation [(2.7.44)](#equation-7-82) can be further manipulated to yield

(2.7.45)[¶](#equation-7-83 "Permalink to this equation")\\\[q = -k \\left\[\\frac{\\partial \\left(\\psi +z\\right)}{\\partial z} \\right\] = -k \\left(\\frac{\\partial \\psi }{\\partial z} + 1 \\right) \\ .\\\]

Substitution of this equation into equation [(2.7.41)](#equation-7-79), with \\(e = 0\\), yields the Richards equation [(Dingman 2002)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dingman2002)

(2.7.46)[¶](#equation-7-84 "Permalink to this equation")\\\[\\frac{\\partial \\theta }{\\partial t} = \\frac{\\partial }{\\partial z} \\left\[k\\left(\\frac{\\partial \\psi }{\\partial z} + 1 \\right)\\right\].\\\]

In practice (Section [2.7.3.2](#numerical-solution-hydrology)), changes in soil water content are predicted from [(2.7.41)](#equation-7-79) using finite-difference approximations for [(2.7.46)](#equation-7-84).

