Continuum Equation and Basic Equation of Water Flow

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

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

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

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

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

REV Ü Assume simple shape: The Cube

The Cube Ü Imagine X-Y-Z axis Z z z X Y x y x y

Ü Describe volume of water flowing INTO cube z z x y x Q=q*A Qx = qx * y * z y

Ü Same for Qy and Qz inflow Qx = qx * y * z Qy = qy * x * z Qz = qz * x * y z z x y x y

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

Outflow in 3 directions gives: Qx = qx * y * z + ( * x ) * y * z Qy = qy * x * z + ( * y ) * x * z Qz = qz * x * y + ( * z ) * x * y

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

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

Thus:

Ü All Inflow: Qx = qx * y * z Qy = qy * x * z Qz = qz * x * y (qx * y * z) + (qy * x * z) + (qz * x * y)

Now consider when S 0 Ü For example, our REV is a cube of soil where the change in volumetric water content (q) during time (t) is Ü 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:

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

3 -D form of Continuity Equation of water is : Where: is the change in volumetric water content with time; qx, qy and qz 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: Ü Now that we have an expression for water flow involving hydraulic conductivity (K) and hydraulic head gradient (H), …. Ü What about case with constant hydraulic conductivity, K? Flow in Saturated Zone! about when K and H is a function of q and matric suction head? Ü What Flow in Unsaturated Zone!

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