Continuum Equation and Basic Equation of Water Flow in Soils January 28, 2002

Elementary Volume - 1  Create a volume with imaginary boundaries within a pool of water (our fluid system)  Call it “elementary volume”

Elementary Volume - 2  What is the scale of elementary volume ?  H 2 O

Elementary Volume - 3  On molecular level, there are molecules and voids. Pick a point in the molecular volume, and your sample is H, O or void  If we take a larger volume, chance is better that we get a sample of water “as a fluid”  Each point in our Representative Elementary Volume (REV) should give us the same properties

Representative Elementary Volume  Volume large enough to be representative of the fluid (same properties everywhere)  Small compared to the fluid system as a whole  Can have any shape

REV  Assume simple shape: The Cube

The Cube  Imagine X-Y-Z axis Z X Y xx xx yy yy zz zz

 Describe volume of water flowing INTO cube xx xx yy yy zz zz Q = q * A Q x = q x *  y *  z

 Same for Q y and Q z inflow Q x = q x *  y *  z Q y = q y *  x *  z Q z = q z *  x *  y xx xx yy yy zz zz

 Describe volume of water flowing OUT of the cube xx xx yy yy zz zz Q = q * A + Change in flow Q x = q x *  y *  z + ( *  x )*  y *  z

Q y = q y *  x *  z + ( *  y ) *  x *  z Q z = q z *  x *  y + ( *  z ) *  x *  y Outflow in 3 directions gives:

Mass Balance  All that flows in must flow out, except for the storage within the volume  Or:

Mass Balance Assumptions  Water is incompressible No compression of water and storage in our “elemental volume”  No sources or sinks in our “elemental volume”  Steady State (no changes over time) Water flowing in equals water flowing out

Thus:

 All Inflow: Q x = q x *  y *  z Q y = q y *  x *  z Q z = q z *  x *  y (q x *  y *  z) + (q y *  x *  z) + (q z *  x *  y)

q z *  x *  y + ( *  z ) *  x *  yq x *  y *  z + ( *  x ) *  y *  z + q y *  x *  z + ( *  y ) *  x *  z +

(q x *  y *  z) + (q y *  x *  z) + (q z *  x *  y) - q x *  y *  z + ( *  x ) *  y *  z q y *  x *  z + ( *  y ) *  x *  z q z *  x *  y + ( *  z ) *  x *  y - -

- ( *  x ) *  y *  z - ( *  y ) *  x *  z - ( *  z ) *  x *  y = 0 OR

Now consider when  S  0  For example, our REV is a cube of soil where the change in volumetric water content (  during time (t) is  Rate of gain (or loss) of water by our REV of soil is the rate of change in volumetric water content multiplied by the volume of our REV:

Thus: Becomes:

“Continuity Equation of water” - ( *  x ) *  y *  z - ( *  y ) *  x *  z - ( *  z ) *  x *  y = Proceeding as before we obtain:

3-D form of Continuity Equation of water is : Where: is the change in volumetric water content with time; q x, q y and q z are fluxes in the x, y and z directions, respectively. In shorthand mathematical notation: Where the symbol  (del) is the Vector differential operator, representing the 3-D gradient in space. OR Where div is the scalar product of the del operator and a vector function called the divergence.

Now apply Darcy’s law and substitute : Into the Continuity Equation, we get : Basic Equation for Water Flow in Soils

Food for Thought:  Now that we have an expression for water flow involving hydraulic conductivity (K) and hydraulic head gradient (H), ….  What about case with constant hydraulic conductivity, K? Flow in Saturated Zone!  What about when K and H is a function of  and matric suction head ? Flow in Unsaturated Zone!

Food for Thought:  An expression exists to define q in steady state…