1Darcy’s Law• Last time– Groundwater flow is inresponse to gradients ofmechanical energy– Three types• Potential• Kinetic– Kinetic energy is usually notimportant in groundwater• Elastic (compressional)– Fluid Potential, !– Energy per unit mass– Hydraulic Head, h– Energy per unit weight– Composed of» Pressure head» Elevation head• Today– Darcy’s Law– Hydraulic Conductivity– Specific Discharge– Seepage Velocity• Effective porosityDarcy’s LawHenry Darcy, a Frenchhydraulic engineerinterested in purifyingwater supplies usingsand filters, conductedexperiments to determinethe flow rate of waterthrough the filters.Published in 1856, his conclusionshave served as the basis for allmodern analysis of ground waterflowA FEW CAREER HIGHLIGHTS:• In 1828, Darcy was assigned to a deep welldrilling project that found water for the city ofDijon, in France, but could not provide anadequate supply for the town. However, underhis own initiative, Henry set out to provide aclean, dependable water supply to the cityfrom more conventional spring water sources.That effort eventually produced a system thatdelivered 8 m3/min from the Rosoir Springthrough 12.7 km of covered aqueduct.• In 1848 he became Chief Director for Waterand Pavements, Paris. In Paris he carried outsignificant research on the flow and frictionlosses in pipes, which forms the basis for theDarcy-Weisbach equation for pipe flow.• He retired to Dijon and, in 1855 and 1856, heconducted the column experiments thatestablished Darcy's law for flow in sands.http://biosystems.okstate.edu/darcy/index.htmFreeze, R. Allen. "Henry Darcy and the Fountains of Dijon." Ground Water 32, no.1(1994): 23–30.2Darcy’s Lawh1"lh2##dhDatumSand-filled columnwith bulk cross-sectional area AQ - Rate ofdischarge [L3/T]Constant head tanks at each endCartoon of a Darcy experiment:In this experiment,water moves due to gravity.What is the relationship betweendischarge (flux) Q andother variables?A plot of Darcy’s actual data:http://biosystems.okstate.edu/darcy/index.htm3What is the relationship betweendischarge (flux) Q andother variables?Darcy found that:1) If he doubled the head difference (dh = h2 – h1),then the discharge Q doubled.2) If he doubled column length (dl),Q halved3) If he doubled column area (A),Q doubled.Conclusions:Q is proportional to dhQ is proportional to 1/dl Q is proportional to Ah1"lh2##"hDatumSand-filled columnwith cross-sectional area AQ - Rate ofdischarge [L3/T]Constant head tanks at each endDarcy’s LawDarcy also found that if he used different kindsof sands in the column, discharge Q changed,but for a particular sand, regardless of Q:dldhKAQ != constant a =dhAdlQThis “proportionality constant” is usually called“hydraulic conductivity” and often is assigned thesymbol, K. Leads to Darcy’s Law:=conductivity x bulk area x ‘gradient’4Why is there a minus sign in Darcy’s Law?(Bradley and Smith, 1995)Darcy’s LawdldhKAQ != “a little vector calculus”Darcy’s Law!"#$%&'=lddhAQK1 Invert Darcy’s Law to express conductivity in terms of discharge, area, and gradient:This is how we measure conductivity:Measure area A; that can be easy.Measure gradient, -dh/dl; can be harder*.Measure discharge, Q; can also be hard.*Imagine trying to measure gradient in a complex geologywith three-dimensional flow and few observation points.5Darcy’s Law!"#$%&=!"#$%&='='TLLLLTLdhAdQK))(())(( 213lWhat are the dimensions of K? Dimensional analysis:= the dimensions of speed or velocitye.g, m/s, meters per second, in SI unitsDarcy’s Law!"#$%&='==TLdldhKAQq discharge specific This expresses Darcy’s Law in terms of discharge.We can also express it in terms of “Darcy velocity”or “specific discharge”, that is, discharge per unitbulk area, A:dldhKAQ != This ‘flux density’ expression is the more common and convenient way to express Darcy’s law. For example, it applies even when area is varying, andwith a little generalization also when gradient or conductivity is varying.6Darcy’s LawdldhKAQ != We will look at three majortopics important to Darcy’sLaw:• Hydraulic HeadGradient• Bulk cross-sectional area of flow• Hydraulic Conductivity(next time)Hydraulic Head• Head is a measure of the total mechanicalenergy per unit weight.• If K, Q and A don’t change with distance,then– hydraulic head varies linearly with distancell ddhdKAQdh !"#"$=##In this experiment,with constant K and A, the headdrops linearly with distance, andthe specific discharge is constant.constant =!=dldhKq7Hydraulic Head• In Darcy’s experiment, do the drops falling from theconstant head tanks have constant velocity?h1"lh2##"hDatumSand-filled columnwith cross-sectional area AQ - Rate ofdischarge [L3/T]Constant head tanks at each endHydraulic Head• In Darcy’s experiment, do the drops falling from theconstant head tanks have constant velocity?• No! Water droplets accelerate at 9.8 m/s2.• Assuming no air resistance, the falling water drop has itspotential energy converted to kinetic energy as it falls:field.energy veconservatia its is,that constant,energy total212==+ mvmgzBut water through our column has constant velocity—why?8Hydraulic HeadBut water through our column has constant velocity—why?##head levelIn this experiment,with constant K, Q and A, the headdrops linearly with distance, andthe specific discharge is constant.constant =!=dldhKqIn a porous medium, the tendency of the fluid to accelerate is opposedby friction against the grains.constant energy total212!=++ mvpVmgzIt’s a dissipative energy field. Where does the energy go?Friction ! Heat; Mechanical Energy ! Thermal Energy~0Darcy’s Law• Could we use Darcy’s Law tomodel the falling drops?• Darcy’s Law works because the driving forces (gravityand pressure) in the fluid are balanced by the viscousresistance of the medium.– Head drop with distance is therefore linear in our simple system.– If inertial forces become important, the head drop is no longerlinear.• In short, if the driving forces of gravity & pressure are notbalanced by viscous forces, Darcy’s Law does not apply.9Darcy’s Law• What happens if the head gradient is too steep?– The fluid will have enough energy to accelerate inspite of the resistance of the grains, and inertial forcesbecome important.– In this case potential energy (head) is not dissipatedlinearly with distance and Darcy’s Law does not apply.• How can we