Modeling Tools for Environmental Engineers and Scientists

6.2.1 FLOW OF WATER THROUGH THE SATURATED ZONEThe flow of water through the saturated zone, commonly referred to asgroundwater flow, is a very well-studied area and is a prerequisite in simulatingthe fate, transport, remediation, and management of contaminants in groundwater.Fluid flow through a porous medium, as in groundwater flow, studied byDarcy in the 1850s, forms the basis of today’s knowledge of groundwater modeling.His results, known as Darcy’s Law, can be stated as follows:u = Q A = –K d h (6.1)dxwhere u is the average (or Darcy) velocity of groundwater flow (LT –1 ), Q isthe volumetric groundwater flow rate (L 3 T –1 ), A is the area normal to thedirection of groundwater flow (L 2 ), K is the hydraulic conductivity (LT –1 ), his the hydraulic head (L), and x is the distance along direction of flow (L).Sometimes, u is referred to as specific discharge or Darcy flux. Note that theactual velocity, known as the pore velocity or seepage velocity, u s , will bemore than the average velocity, u, by a factor of three or more, due to theporosity n (–). The two velocities are related through the following expression:u s = nQA = u n (6.2)By applying a material balance on water across an elemental control volumein the saturated zone, the following general equation can be derived:– ∂( ∂ u)x– ∂( ∂ v)y– ∂( ∂ w)z= ∂( ∂ n)t= n ∂ (∂ )t+ ∂ (n)∂ (6.3)twhere u, v, and w are the velocity components (LT –1 ) in the x, y, and z directionsand ρ is the density of water (ML –3 ). The three terms in the left-hand sideof the above general equation represent the net advective flow across the element;the first term on the right-hand side represents the compressibility of thewater, while the last term represents the compressibility of the soil matrix.Substituting from Darcy’s Law for the velocities, u, v, and w, under steadystate flow conditions, the general equation simplifies to:∂ ∂ x K x ∂ h ∂∂x ∂ Y K y ∂ h ∂∂y ∂ z K z ∂ h∂z= 0 (6.4)and by further simplification, assuming homogenous soil matrix with K x = K y= K z , the above reduces to a simpler form, known as the Laplace equation:222 ∂ h∂x2 ∂ h∂y2 ∂ h∂z2 = 0 (6.5)© 2002 by CRC Press LLC