CE/ENVE 320 CE/ENVE 320 ––VadoseVadoseZone Hydrology/Soil PhysicsZone Hydrology/Soil PhysicsSpring 2004Spring 2004Copyright © Markus Tuller and Dani Or 2002Copyright © Markus Tuller and Dani Or 2002--20042004Water Flow in Saturated SoilsDarcy’s Law33 Hillel, pp. 173 - 177Copyright© Markus Tuller and Dani Or2002-2004Flow occurs from locations with high potential energy to locations of lower potential energy in pursuit of equilibrium state.The driving force for flow is called potential (energy) gradient, the difference in potentials between two points in a system separated by a certain distance.Non Equilibrium and FlowPotential Gradient iPotential Gradient iHigh potential energyLow potential energyψ1ψ2L∆ψ1= ψ1- ψ2LLi21ψ∆−=ψ−ψ−=i…… potential gradientψ….. potential energyL…... distance betweenthe locations [L]Copyright© Markus Tuller and Dani Or2002-2004In general, gradients can develop due to differences in:--PressurePressure--Position in a gravity fieldPosition in a gravity field--Chemical concentrationChemical concentration--TemperatureTemperature--Position in an electrical fieldPosition in an electrical fieldleading to spontaneous flow of mass or energy.We will focus on flow due to differences in hydraulic potential in this section (neglecting solute potential).Hydraulic Potentialpmzhψψψψ++==hψHydraulic potential =zψ=mψ=pψGravimetric potential Matrix potentialPressure potentialCopyright© Markus Tuller and Dani Or2002-2004Definition of Liquid ViscosityNewton’s Law of ViscosityNewton’s Law of ViscosityEarly concepts in fluid dynamics are based on perfect fluids that are assumed to be frictionless and incompressible. In a perfect fluid contacting layers can exhibit no tangential forces (shearing stresses) only normal forces (pressures).Perfect fluids do not exist. In the flow of real fluids adjacent layers do transmit tangential stresses (drag), and the existence of intermolecular attraction causes fluid molecules in contact withsolid surfaces to adhere to it rather than to slip over it.The flow of a real fluid is associated with the property of viscosity.Before we discuss flow in soils it is advantageous to introduce some basic concepts related to flow in general.Copyright© Markus Tuller and Dani Or2002-2004Liquid viscosityThe nature of viscosity can be visualized considering fluid motion between two parallel plates; one at rest, the other one moving at constant velocity.Under laminar flow conditions water molecules are moving in adjacent parallel layers. The layers transmit tangential stresses (drag) due to attraction between fluid molecules.Motion of fluid between parallel platesThe existence of intermolecular attraction causes fluid molecules to adhere on the solid walls.Copyright© Markus Tuller and Dani Or2002-2004Newton’s law of viscosityThe velocity distribution in the liquid is linear.Maintaining the relative motion of the plates at constant velocity requires the application of a constant tangential force to overcome the frictional resistance in the fluid.This resistance per unit area of the plate is proportional to the velocity of the upper plate and inversely proportional to the distance between the plates. The shearing stress τ at any point is proportional to the velocity gradient.The viscosity η is the proportionality factor between τ and the velocity gradient dydvAFητ==τ shearing stress (force F acting on an area A) [M L-1 t-2]dv/dy velocity gradient perpendicular to the stressed area (shear rate) [t-1]η viscosity coefficient of the liquid in [Pa s] [M L-1t-1] Viscosity is the property of the fluid to resist the rate of shearing and can be visualized as an internal friction. Newton’s Law of ViscosityNewton’s Law of ViscosityCopyright© Markus Tuller and Dani Or2002-2004Fluid flow in cylindrical tubesLPyyPLy∆=⇒∆=222τππτFluid flows through a cylindrical tube having a diameter of 2R and length L. We assume that the flow is laminar and caused by a pressure gradient ∆P=P2-P1. π2yPFp ∆=LyFfπτ2=Pressure Force:Frictional Resistance Force:We equate the pressure and frictional resistance forces and solve for τCopyright© Markus Tuller and Dani Or2002-2004Flow through cylindrical tubesNow we can introduce Newton's law of viscosity. Substituting the integration constant back into our previous result yields the expression for the velocity profile as a function of distance from the tube axisdyyLPdvLPydydvηη22∆−=⇒∆=C2yL2P)y(vdyyL2Pdv2+η∆−=⇒η∆−=∫∫η∆=⇒=+η∆−4RLPC0C2RL2P22()2222yRL4P4RLP2yL2P)y(v −η∆=η∆+η∆−=The resulting ODE can be solved by integration. Since we know that the velocity at y=R is equal to zero we can solve for the integration constantCopyright© Markus Tuller and Dani Or2002-2004Poiseuille’s law for flow in cylindrical tubesL4RPv2maxη∆=We know that the velocity is maximum at the center of the tube where y=0, and can calculate vmax. If we divide this expression by the tube cross section we receive the average flow velocity as: To calculate the Discharge Rate (volume of water flowing through the tube per unit time) we have to integrate the velocity profile over the cross-sectional tube area. This can be done very simple by calculatingthe volume of a paraboloid of revolution.This relationship is known as Poiseuille’s law. It shows that the volume of flow is proportional to the pressure trop per unit distance and to the fourth power of the tube radius. LP8RQL4RP2RtVQ422∆ηπη∆π=⇒==∆η=LP8Rv2Copyright© Markus Tuller and Dani Or2002-2004Example: laminar flow in tubes What is the average (laminar) flow velocity of water at 20oC through a 50m long tube having a diameter of d=0.1m under a pressure difference of 100 Pa ? Viscosity of water at 20 Viscosity of water at 20 ooC:C:ηη= 0.001 Pa s= 0.001 Pa ssm625.0vmPasPam50100001.0805.005.0v2=××=∆η=LP8Rv2Copyright© Markus Tuller and Dani Or2002-2004Water Flow in SoilsImages of porous media pore space reveal that pores do not resemble uniform and smooth circular tubes that form the basis for Poiseuille’s law. Flow in porous media is generally described by macroscopic or averaging terms that replace microscopic description of individual flow pathways.The first one able to quantitatively describe saturated flow through porous media was HENRY DARCY a French engineer.33 Copyright© Markus Tuller and