1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #3: Hydraulic Head and Fluid Potential. p o. p o + p

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1 1.7, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #3: Hydraulc Head and Flud Potental What makes water flow? Consder ressure Water Level o A Water Level C o o + B Pressure at A atmosherc ( o ) Pressure at B > atmosherc ( o + ) Pressure at C atmosherc ( o ) But flow s not from A to B to C. Flow s not down ressure gradent. Hubbert (1940) Potental A hyscal quantty caable of measurement at every ont n a flow system, whose roertes are such that flow always occurs from regons n whch the quantty has less hgher values to those n whch t has lower values regardless of the drecton n sace. Examles: Heat conducts from hgh temerature to low temerature Temerature s a otental Electrcty flows from hgh voltage to low voltage Voltage s a otental Flud otental and hydraulc head Fluds flow from hgh to low flud otental Flow drecton s away from locaton where mechancal energy er unt mass of flud s hgh to where t s low. How does ths relate to measurable quantty? Prof. Charles Harvey Page 1 of 9

2 Groundwater flow s a mechancal rocess forces drvng flud must overcome frctonal forces between orous meda and flud. (generates thermal energy) Work mechancal energy er unt mass requred to move a flud from ont z to ont z. Elevaton: z Pressure: Velocty: v Densty: ρ Volume: V Elevaton: z0 Pressure: o Velocty: v o Densty: ρ o Volume: V o z z Flud otental s mechancal energy er unt mass work to move unt mass Prof. Charles Harvey Page of 9

3 Whch way does water flow (functon of Z and P)? h 1 h 1 h Q h Q z z 1 z 1 z h 1 h 1 h h Q Q z 1 z z 1 z Flud otental s the mechancal energy er unt mass of flud otental at z flud otental at datum + work from z to z. The work to move a unt mass of water has three comonents: 1) Work to lft the mass (where z 0) w mgz' ) Work to accelerate flud from v0 to v mv w 3) Work to rase flud ressure from o to V w 3 Vd m d m d m o o o Prof. Charles Harvey Page 3 of 9

4 Note that a unt mass of flud occues a volume V 1/ρ The Flud Potental (the mechancal energy er unt mass, m1) φ gz + v + d + o φ gz + v + 0 for ncomressble flud (ρ s constant) Ths term s almost always unmortant n groundwater flow, wth the ossble exceton of where the flow s very fast, and Darcy s Law begns to break down. How does otental relate to the level n a e? At a measurement ont ressure s descrbed by: P ρ g(deth) + o ρ gϕ + o ρ g(h-z) + o ϕ z h Return to flud otental equaton 0 φ gz + v + Neglect velocty (knetc) term, and substtute for So, ρ g ( h z ) φ gz + ρ φ gh or h φ / g Thus, head h s a flud otental. Prof. Charles Harvey Page 4 of 9

5 Flow s always from hgh h to low h. H s energy er unt weght. H s drectly measurable, the heght of water above some ont. h z + ϕ Thermal Potental ϕ t Thermal Potental Temerature can be an mortant drvng force for groundwater and sol mosture. Volcanc regons Dee groundwater Nuclear waste dsosal Can cause heat flow and also drve water. Chemcal Potental Adsorton Potental Total Potental s the sum of these, but for saturated condtons for our ntal cases we wll have: ϕ ϕ g + ϕ q * L 1 h T L l l L 3 c l Dervaton of the Groundwater Flow Equaton Darcy s Law n 3D Homogeneous vs. Heterogeneous Isotroc vs. Ansotroc Isotroy Havng the same value n all drectons. s a scalar. h h q x q y q z x y Ansotroc havng drectonal roertes. s really a tensor n 3D Gradent Flow The value that converts one vector to another vector s a tensor. Prof. Charles Harvey Page 5 of 9

6 q x xx x xy y xz q y yx x yy y yz x y qz zx zy z zz The frst ndex s the drecton of flow The second ndex s the gradent drecton Interretaton - xx s a coeffcent along the x-drecton that contrbutes a comonent of flux along the x-axs due to the coeffcent along the z-drecton that contrbutes a comonent of flux along the z-axs due to the comonent of the gradent n the y-drecton. The conductvty ellse (ansotroc vs. sotroc) y yy Flow y yy Flow xx xx Gradent Gradent x x If yy xx then the meda s sotroc and ellse s a crcle. It s convenent to descrbe Darcy s law as: r q h Where s called del and s a gradent oerator, so h s the gradent n all three drectons (n 3D). s a matrx. Prof. Charles Harvey Page 6 of 9

7 q x q y q z xx xy xz yx yy yz zx zy zz x y The magntudes of n conductvtes. the rncal drectons are known as the rncal If the coordnate axes are algned wth the rncal drectons of the conductvty tensor then the cross-terms dro out gvng: q x xx x q y yy y q z zz Prof. Charles Harvey Page 7 of 9

8 Effectve Hydraulc Conductvty H tot Q 1 Q tot Q L tot Q 3 Q 4 Q tot Q 1 + Q + Q 3 + Q 4 H L 1 1 H n H L tot eff H H H + L + L L 4 L n eff n L 4 L Prof. Charles Harvey Page 8 of 9

9 Q H 1 H H tot L tot H 3 H 4 H tot H 1 + H + H 3 + H 4 H tot H 1 H H q 3 eff 1 3 H 4 4 Ltot L 1 L L 3 L 4 n ql tot ql L eff eff n L n Prof. Charles Harvey Page 9 of 9