Hydrology and Climate
Hydrology and Climate ESS 110
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1 Specific Storage The specific storage S5 of a saturated aquifer is defined as the volume of water that a unit volume of aquifer releases from storage under unit decline in hydraulic head S5 BVWVW h We know that a decrease in hydraulic head infers a decrease in fluid pressure 6h Zipy and an increase in effective stress 6h 60392 The water that is released from storage under conditions of decreasing h is produced by two mechanisms 1 the compactions of the aquifer caused by increasing 0392 and 2 the expansion of the water caused by decreasing p The first of these mechanisms is controlled by the aquifer compressibility a and the second by the fluid compressibility 1 Let us first consider the water produced by the compaction of the aquifer The volume of water expelled from the unit volume of aquifer during compaction will be equal to the reduction in volume ofthe unit volume of aquifer The volumetric reduction 6V5 will be negative but the amount of water produced iiiW1 will be positive so that 6V 6V 11 w1 s and from the definition ofthe compressibility of a porous medium a we know 1 WE VS 60 llZ W aVS 602 so substituting for W in Eq11 gives 6VWJaV5602 13 Therefore for a unit volume VS 1 with a unit decline in hydraulic head 6h 1 and knowing the change in effective stress is defined as 602 pw g 6h gives 6 W pw g6ha1pw g1 14l an apw g 2 Now consider the volume of water produced by the expansion of the water The volumetric expansion is equivalent to the amount of water produced 6wa2 6K From the definition ofthe compressibility of water we know EnvEESS 110 Hydrology and Climate i 6VW Vw 0p gt M VW 6P gt M VW 6F 15 The volume ofwater VW in the total unit volume V is blS where is the porosity With VS 1 6h 1 and 6p pw g h Eq 15 becomes 6V Vp gah 1p g 1 ang pw g The specific storage SS is the sum of the two terms given by Eq 14 and Eq 16 16 SS 0Vw10Vwz apwg pwg swgltawgtiiiaiiia 2 Transmissivity and Storativity of a Confined Aquifer For a confined aquifer ofthickness b the transmissivity T is defined as Tst LL 21 and the storativity aka storage coefficient S is defined as 1 L SESSbpWgba zLZ 22 The storativity of a confined aquifer of thickness b is the vertically averaged specific storage value In confined aquifers storativity ranges in value from 0005 to 000005 Large head changes over extensive areas are required to produce substantial water yields from confined aquifers 3 Transmissivity and Specific Yield of a Unconfined Aquifer In an unconfined aquifer the transmissivity is not as well defined as in a confined aquifer but it can be used It is defined by the same equation Eq 21 but b is now the saturated thickness of the aquifer or the height of the water table above the top of the underlying aquitard that bounds the aquifer The specific yield Sy also know as the drainable porosity is defined as the volume of stored ground water released per unit surface area per unit decline of water table Thus for an unconfined aquifer S Sy Typical EnvEESS 110 Hydrology and Climate values of Sy for sedimentary deposits are in the range 005035 The higher values reflect the fact that releases from storage in unconfined aquifers represent an actual dewatering ofthe soil pores whereas releases from storage in confined aquifers represent only the secondary effects of water expansion and aquifer compaction caused by changes in fluid pressure Fig 31 The favorable storage properties of unconfined aquifers make them more efficient for exploitation by wells When compared to confined aquifers the same yield can be realized with smaller head changes over less extensive areas Unit areas Con ned aquifer Unit declines in heads l Volume of water released a from storage s Figure 31 Definitions of storativity of a confined aquifer right and the specific yield of an unconfined aquifer left 4 Governing Equation for Groundwater Flow Continued Let us now consider the case where flow through the control volume is saturated and transient For the next step we must recall the product rule of calculus also called Leibniz s law which governs the differentiation of products of differentiable functions It may be stated as d d d uvu Vv u 41 Applying product rule to the righthand side of the continuity equation gives EnvEESS 110 Hydrology and Climate T T szlpm 42 0pwqxalpw 6pqul p mapw 6x By 62 W 6t it The first term on the right hand side of Eq 42 is the mass rate of water produced by the compaction of the porous medium as reflected by the change in porosity The second term is the mass rate of water produced by an expansion of the water under a change in its density pw The first term is controlled by the compressibility of the fluid and the second term by the compressibility of the aquifer a The mass rate of water produced time rate of change of fluid mass storage is pw SS ll t and Eq 42 becomes 5Pw 1x alpwqy 00w prS 43 6x By 62 it Let us prove that last step ah ah S a pw Sat pw pwg 6t 6h 6h 2 a pwg at ng at Pi g6 p2 g mj 60 at W 6p at pigpwg j ZZJ 4394 If g Jm g pfW6h 6 61 pw at ll at Expand the terms on the left hand side of Eq 43 using the chain rule and recognize that p qxBx are much greater than terms ofthe form qx BpWBx or 6p q 661 6p 661 r wm r 45 6x A 6x q 6x A 6x Equation 43 then becomes EnvEESS 110 Hydrology and Climate 1 Evapotranspiration Evapotranspiration is a collective term for all the processes by which water in liquid or solid phase at or near the earth s land surfaces becomes atmospheric water vapor The term includes both evaporation from rivers lakes bare soil and vegetative surfaces and sublimation from ice or snow fields Globally about 62 of the precipitation that falls on the continents is evapotranspired Of this about 97 is evapotranspiration from land surfaces and 3 openwater evaporation Evapotranspiration exceeds runoff in most of the river basins on earth The following is a mass balance approach for determining the total volume of water that is being evaporated E v01 within a catchment area I OAV W SWmGWm E SW GW AV 11 vol v01 out out Evol VVVOZ SWout GWout where I is the total quantity of water entering the catchment O is the total quantity of water exiting the catchment AV is the change in storage WW is the volume of water entering the catchment through precipitation SW1 is the surface water input GWm is the groundwater input SW0 is the surface water output and GWOm is the groundwater output Evaporation is typically solved for using Eq 11 Evaporation physics includes the boundary layer mass transport theory diffusiondispersive processes and Fickian mathematics j H20 a 3 H20 3 Low concentration of water molecules V a High concentration of water molecules Figure 11 Evaporation over a water surface 2 Water Vapor Evaporative processes are affected by the amount of water vapor in the air near the evaporating surface and there are several ways of expressing that amount By virtue of their molecular motion and collisions each EnvEESS 110 Hydrology and Climate constituent of a mixture of gases exerts a pressure called a partial pressure which is proportional to its concentration The partial pressure of water is called the vapor pressure e Saturated vapor pressure e is the maximum vapor pressure that is thermodynamically stable and is given by eO6lleXp 2 21 T2373 L where e has units of kPa and T is temperature in units of C A phase diagram is given in Fig 21 A UQUH Freezirlg Melting Condensation ICE Vapor Pressure e k 6 061 lexp T2373 39 39 39 39 39 39 39 39 39 39 39 39 39 39 39 3939 39 391 Vaporization VAPOR Sublimation V Freezing point Temperature T Figure 21 Phase diagram of water Absolute humidity aka vapor density pv is the mass concentration of water vapor in a volume of air pv mVV Note that vapor density is lighter or less dense than dry air The water molecule H20 weighs less than a molecule of air N202 The ideal gas law provides the relationship between vapor pressure and absolute humidity and is expressed as 22 2 PVanT PI L T2 where P is the absolute pressure of the gas V is the volume of gas 11 is the number of moles of gas R is the universal gas constant and T is the absolute temperature The ideal gas law for air is written as EnvEESS 110 Hydrology and Climate PHVnaRTa gt PHV i M a RHMHTH gt Pa V maRa Ta 23 P Pa m RHT gt PapaRaTa gt Ra V Tap where Pa is atmospheric pressure in kPa nu is the number of moles of air nu mg Ma Ta is air temperature in K mg is the air mass Mn is the molar mass of air pa is air density in kg m393 pV mVV and RH is the gas constant for air Ra RMa which for the units given has the value Ra 0288 The density of air pa is temperature dependent and may be calculated as 352 24 p Ta2732 l l where pa is in kgm393 and Ta is the air temperature in C For water vapor the ideal gas law is given as m m eVnVRTa gt eVMV RVMVT gt eVmVRVTa gt eVVRVT V 25 epVRVTa gt RV 6 Tap where e is the vapor pressure in kPa nV is the number of moles of water vapor Ta is air temperature in K mV is the water vapor mass MV is the molar mass of water vapor pV is the vapor density in kg m393 and RV is the gas constant for water vapor which for the units given has the value of RV 0463 The molar mass of water vapor is 0622 times the molecular weight of air or M 0622Ma gt 0622 26 Recognizing that RRa Ma and RRVMV leads to RR MV R M M gt 27 a a R v RV Ma l and substituting in Eq 26 gives R R a 0622 gt RV a 28 RV 0622 The ideal gas law for water vapor Eq 25 is then expressed as EnvEESS 110 Hydrology and Climate R e e RT a gt 29 0622 Ta pV pV 0 22 If Ra 0288 Eq 23 then Eq 29 may be reduced to 0288T T i gt i a 210 pV 0622 pV 217 where e is in kPa Ta is in K pV is in kg m4 Let us now rewrite Eq 29 substituting for Ra from Eq 23 P T P i a J a a qz w 211 Tap p 0622 0622 gt p P where q is specific humidity the ratio of water vapor to air including water vapor and dry air in a particular volume Ifthe temperature and vapor pressure of a parcel of air lie below the curve representing saturation Fig 21 the air is unsaturated and its relative humidity Wu is the ratio usually expressed as a percent of its actual vapor pressure at air temperature ea to its saturation vapor pressure W aid 212 a at m a 3 Physics of Evaporation In Fig 32 dry air with temperature of Ta lies above a horizontal water surface with a temperature of T5 The molecules at the surface are attracted to those in the body of the liquid by hydrogen bonds but some of the surface molecules have sufficient energy to sever the bonds and enter the air The number of molecules with this escape energy increases as T5 increases The water molecules entering the air move in random motion and as these molecules accumulate in the layer of air immediately above the surface some will re enter the liquid The rate of evaporation is the rate at which molecules move from the saturated surface layer into the air above and that rate is proportional to the difference between the vapor pressure ofthe surface layer and the vapor pressure ofthe overlying air ea that is E cc 6 ea 31 where E is the rate of evaporation transfer ea is measured at some representative height and the proportionality depends on that height and on the factors controlling the diffusion of water vapor in the air EnvEESS 110 Hydrology and Climate Temperature Vapor pressure Figure 31 Schematic diagram of flux of water molecules over a water surface Depending on the temperature of the surface and the temperature and humidity of the air the difference between the two vapor pressures can be positive zero or negative If e gt ea evaporation is occurring if e lt ea there is water condensing on the surface and if e 2 ea neither condensation or evaporation is occurring Evaporation will occur even if the relative humidity equals 100 eae 1 as long as e gt 6 However under these conditions the evaporating water will normally condense in the overlying air to form a fog or mist The latent heat of vaporization xiv is the quantity of heat energy that must be absorbed to break the hydrogen bonds when evaporation takes place Thus evaporation is always accompanied by a transfer of heat out of the water body Because of this coupling the rates of latentheat and water mass transfer are directly proportional and given by LEWl5ll ll l where LE is the rate of latent heat transfer Note that if E is expressed in dimensions of LT Eq 32 becomes 32 L M T LT 4 Physics of Turbulent Transfer Near the Ground The land surface is the lowest layer of the atmosphere in which the winds are affected by the frictional resistance of the surface The frictional resistance produces turbulent eddies which are irregular and chaotic motions with vertical components Fig 41 These vertical components are the means by which water vapor EnvEESS 110 Hydrology and Climate and its accompanying latent heat sensible heat and momentum are exchanged between the atmosphere and the land surface L Heat amp mass transfer Momentum transfer Turbulent eddies N Wind velocity distribution uz Vegetation topography roughness Figure 41 Vertical distribution of wind velocity over the ground surface Velocity always decreases towards the land surface therefore momentum transfer has a downward direction Heat and mass water vapor transfer is either up or down and dependent on the directions of temperature and vaporpressure gradients We have already seen that water vapor is transferred between the surface and the air whenever there is a difference in the vapor pressure between the surface and the overlying air Eq 31 and that a transfer of latent heat always accompanies the vapor transfer Eq 32 A second mode of nonradiant heat transfer occurs in the form of sensible heat that is heat energy than can be directly sensed via measurement of the temperature Thus sensibleheat transfer occurs whenever there is a temperature difference between the surface and the air39 the relation that is analogous to Eq 31 is H oc TS Ta 41 where H is the rate of sensibleheat transfer from surface to air per unit area of surface E T IL Z and TS and Ta are the surface and air temperatures respectively When TS ltTa H is negative and sensible heat moves from the air to the surface The heat capacity of air ca is the property that relates a air temperature change to a change in its heatenergy content It is defined as the amount of heat energy AH absorbed released by a air mass ma when its temperature is raised lowered by an amount AT i 42 MaAT M6 The amount of heat energy absorbed released by air for a change in temperature is a AHzmacaAT gt AHmcT 7 MM 66E 43 EnvEESS 110 Hydrology and Climate where Ta is the air temperature and Tb is an arbitrary base temperature usually taken as 0 C Dividing both sides of Eq 43 by the air volume Va gives AH ma i M i Z 5 75V 6 AT T Mp 6 Tc Tb lEllMelM 4394 where h is the concentration of sensible heat and pa is the mass density of air The momentum exchange between the atmosphere and the land surface is based on boundary layer theory The classical approach to boundary layer theory is a no slip boundary condition zero velocity at the ground surface The wind speed distribution u is dependent on wind speed conditions For laminar flow conditions Fig 42a slow flow of wind the horizontal shear stress 239 due to differences of wind velocity at adjacent levels may be expressed as w i iim or a dz LT T LT L where u is dynamic viscosity and z is the vertical distance above the ground surface Note that both 239 and u are functions of Z If flow conditions are turbulent Fig 42b higher wind velocities the dynamic viscosity within Eq 45 is replaced by an empirical constant known as the eddy viscosity 5 Note that eddies typically result in significant deviations in the flow direction A Well de ned velocities W Irregular eddies V V V V V V w W K V V Some order a Laminar ow b Turbulent flow Figure 42 Particle paths in a laminar flow and b turbulent flows Because of turbulent eddies the instantaneous horizontal wind velocity at any level Va fluctuates in time and we can separate the instantaneous velocity into a time averaged component Va and a deviation from that average v caused by the eddies v 2 v i H 46 T EnvEESS 110 Hydrology and Climate 8 Time varying vertical air velocity designated wa are similarly separted into a time averaged component vTa and an instantaneous deviation form that average w I wa Wu wa 47 The intensity of turbulence can be characterized by a quantity called the friction velocity 39 39 12 u E va wa 48 where the minus sign is required because simultaneous values of w and v have opposite signs Dimensional analysis shows us that at a walla characteristic velocity friction velocity u can be defined as 12 u E e 49 Therefore the shear stress may be expressed as a function offriction velocity as follows 2 M 5 M 5 1 pau L3T2 LTZ or L2 410 The Log wind profile is a semi empirical relationship used to describe the vertical distribution of horizontal wind speeds above the ground within the atmospheric surface layer The relationship is well described in the planetary boundary layer literature The logarithmic profile of wind speeds is generally limited to the lowest 100 meters ofthe atmosphere ie the surface layer ofthe atmospheric boundary layer The equation to estimate the wind speed Va at height Z above the ground is Va ln forzgtzdz0 411 where k is a dimensionless constant 201 is called the zeroplane displacement 20 is the roughness height Fig 43 Eq 411 is known as the Prandtlvon Karman Universal VelocityDistribution for turbulent flows Experimental data have shown that k 04 and that the values of 201 and 20 are approximately proportional to the average height of the vegetation or other roughness elements covering the ground surface EnvEESS 110 Hydrology and Climate Velocity Figure 43 Vertical distribution of wind velocity over a vegetative surface height Zveg The zeroplane displacement Zd is about 072 and the roughness height Z0 is about 01Zveg veg 5 Diffusion Diffusion is the process by which constituents of a fluid such as its momentum heat content or a dissolved or suspended constituent are transferred from one position to another within the fluid Such transfers occur whenever there are differences in concentrations of the constituent in different parts of the fluid The rate of transfer of a constituent 1 in the direction 2 is directly proportional to the gradient of concentration 1 in the z direction F 139 D dcli dz 51 where EU is the rate oftransfer flux of 1 in direction 2 per unit area per time C1 is the concentration of 1 and D1 is the diffusivity of 1 in the fluid L2 T4 Eq 51 is the mathematical expression of Fick s First Law of Diffusion The diffusion equation is the basis for a quantitative understanding of the process that transfer water vapor and latent and sensible heat between the surface and the overlying atmosphere Since the absolute humidity pv is the concentration of water vapor in the air the diffusion equation for water vapor V is mmeDvel lwlm dz T L L T where FZV is the flux of water vapor in the z direction and DV is the diffusivity of water vapor L2 T71 Recall that the rate of evaporation E is the rate at which water molecules move from the saturated surface EnvEESS 110 Hydrology and Climate 10 layer into the air above Therefore the evaporation rate is equivalent to the flux of water vapor and Eq 52 becomes dz 53 E V E D Likewise since the upward latent heat transfer rate LE is equivalent to the flux of latent heat in the z direction FZLE we can write F LE LE 54 Since latent heat and water vapor are directly coupled Eq 32a the diffusion equation for latent heat LE is EZLE1VEAVEV1V DVif Z d L2 E M E 5395 FZLE DV1Vie dz T M L4 L2 T Recall from Eq 211 that 0622 LM 5 56 p P P and taking the derivative of Eq 56 with respect to z gives dpV d 0622430 e 0622430 de 5 7 dz dz Pa Pa dz I Eq 57 can be used to rewrite Eq 55 as 0622 FZLEDV1V pa g f 58 Pa dz L T The upward latent heat transfer rate LE Eq 32b is equivalent to FZLE therefore LEpwvivE D1 039622p i 1 5399 Pa dz L T The diffusion equation for sensible heat H is dh L2 E E FHDg 510 Zl H dz Tll l ET l l EnvEESS 110 Hydrology and Climate 11 where FZH is the flux of sensible heat in the z direction and DH is the diffusivity ofsensible heat in air and h is the concentration of sensible heat Substituting the concentration of sensible heat from Eq 44 into Eq 510 gives FlH DH39PHCH39TaYL 511 z where ca is the heat capacity Assume that the arbitrary base temperature Tb is 0 C and that the density and heat capacity are essentially constant under prevailing conditions gives dT FZ H D Ufa 512 H p dz The flux of sensible heat is equivalent to the rate of sensible heat transfer therefore dT FZHH gt H DHpacad 513 z Recall that the momentum equals mass times velocity so that the concentration of momentum momentum per unit volume at any level equals the mass density of the air time the velocity pa va The diffusion equation for momentum M is 610 V EweBM 514 where FZM is the flux of momentum in the z direction and DM is the diffusivity of momentum in turbulent air Assuming constant air density gives weMpa111151i The flux of momentum is equivalent to the negative shear stress 139 therefore FZM riLAI 516 From Eq 410 shear stress is directly proportional to the square of the friction velocity F M p uf 517 Combining Eq 515 and Eq 517 gives dv 2 dv 2 u2 a u gt D a m gt D dz pa M dz M dva dz M p 518 EnvEESS 110 Hydrology and Climate 12 To evaluate the denominator of Eq 518 we rely upon the estimate of wind speed given in Eq 411 that is dv d u z zd u d z zd u z0 1 a 1n 1n dz dz k z0 k dz z0 k z zd z0 519 dva u dz kz zd From Eq 519 the friction velocity u may be expressed as dva k I u z zd dz 5 20 Substituting Eq 520 into Eq 518 gives D kz zd dvadz 2 k2 z zd 2 dvadz2 M dvadz dvadz 521 D M k2Z zZ6 il ll ll zlgl 6 Effects of Atmospheric Stability on Heat and Vapor Transfer Rising air experiences a drop in temperature even though no heat is lost to the outside The drop in temperature is a result ofthe decrease in atmospheric pressure at higher altitudes If the pressure ofthe surrounding air is reduced then the rising air parcel will expand The molecules of air are doing work as they expand This will affect the parcel s temperature which is the average kinetic energy of the molecules in the air parcel One of the results of the Laws of Thermodynamics is that there is an inverse relationship between the volume of an air parcel and its temperature During either expansion or compression the total amount of energy in the parcel remains the same none is added or lost The energy can either be used to do the work of expansion or to maintain the temperature of the parcel but it can t be used for both If the total amount of heat in a parcel of air is held constant no heat is added or released then when the parcel expands its temperature drops When the parcel is compressed its temperature rises In the atmosphere if the parcel of air were forced to descend it would warm up again without taking heat from the outside This is called adiabatic heating and cooling and the term adiabatic implies a change in temperature of the parcel of air without gain or loss of heat from outside the air parcel Adiabatic processes are very important in the atmosphere and adiabatic cooling of rising air is the dominant cause of cloud formation For the atmosphere the drop in temperature of rising unsaturated air is about 1 C100m This rate of temperature change of unsaturated air with changing altitude is called the dry adiabatic lapse rate If the air EnvEESS 110 Hydrology and Climate 13 subsides it also changes temperature It warms up and it is warming up at the dry adiabatic lapse rate Make sure you notice that we are talking about moving air rising or subsiding not still air What happens to the relative humidity Eq 212 Wu E ea e of a parcel of air when the temperature decreases The relative humidity increases because the saturated vapor pressuree decreases for a decrease in temperature Fig 21 If the air is rising and cooling at a rate of 1 C100m eventually it s going to cool off enough for the relative humidity to reach 100 and condensation can take place The dew point is the temperature at which the air becomes saturated and condensation takes place The lifting condensation level is the altitude at which condensation begins You can look up at the windward sides of mountains and see where the lifting condensation level is because that is where you will see the bases of clouds that have formed If condensation is taking place latent heat is being released to the surrounding air So you have two opposing trends going on at the same time within this parcel of air It s rising and cooling but it s also condensing and being warmed If water vapor in the air is condensing the adiabatic rate is less The air is only cooling off at a rate of about 05 C100m This is called the wet adiabatic lapse rate Fig 61 shows unstable neutral and stable lapse rates near the ground The lapse rate is the negative of the actual change of temperature with altitude for the stationary atmosphere ie the temperature gradient dTa dz When a parcel of air is transported upward in a turbulent eddy it cools abatically ie without the loss of heat Thus if the actual lapse rate is steeper than adiabatic unstable the air in the eddy is warmer and hence less dense than the surrounding air and will continue to rise due to buoyancy enhancing vertical transport of heat If the actual lapse rate is less than the adiabatic stable the air in the eddy will be cooler and denser than the surroundings and will sink toward the surface reducing vertical transport EnvEESS 110 Hydrology and Climate 14 Height Inversion lt Unstable Surface gtTa Ts AirTemperature Figure 61 Unstable neutral adiabatic and stable lapse rates near the surface Under neutral conditions the diffusivities of water vapor and sensible heat are identical to the diffusivity of momentum ie DVDM 1 and DH DM 1 because the same turbulent eddies are responsible for the transport of all three quantities However under unstable conditions the vertical movement of eddies is enhanced beyond that due to the wind velocity and there can be significant vertical transport of water vapor andor sensible heat but little transport of momentum so DVDM gt1 and DH DM gt1 These conditions typically occur when wind speed is low and the surface is strongly heated by the sun Conversely when the lapse rate near the ground is stable turbulence is suppressed and DVDM lt1 and DH DM lt1 This situation is typical when warm air overlies a cold surface such as a snowpack Let us first examine latent heat and sensible heat transfer under neutral lapserate conditions DV DH DM For the rate of latent heat transfer we can replace DV in Eq 59 with DM from Eq 521 to give LEDM1V k2ZZd2 gv Pa dz dz Pa dz LEk2ZZd2 1 m w 61 EnvEESS 110 Hydrology and Climate 15 Assuming a logarithmic profile for wind velocity and vapor pressure and integrating between two observational heights 21 and z2 for 22 gt 21 Z 201 20 then yields 2 LE d dz 0622k lvpad dv 2 2 201 Pa lily ardmhwviided Z1 Z1 2 2 V2 Z2 2 LE 1 dz 0622k lvpajezzdv z ZZdjzl Pa V 2 2 LE 1nltz zgti 39m j may 2 0622k2 2 vpaezelvlvl 0622 k2 LE1 a V2Vlezel V 2 21 2d 62 where v E Va 21 and q E ea 21 Eq 62 can be further reduced by taking the lower height as that ofthe nominal surface Le 21 zd 20 22 zav1 0 v2 Va el es and 62 ea and becomes 0622 k2 LE 2Tpa ltv ogtlte egt zd2zo zd 6393 LE AO6ZPAk Vaea es Note that Eq 63 requires a wind speed measurement at only one level For the rate of sensible heat transfer we can replace DH in Eq 513 with DM from Eq 521 to give HDM pH 60 klzzd2ipa 063 2 z z 64 A H k2pacaz zd dz dz Integrating between levels 21 and 22 yields EnvEESS 110 Hydrology and Climate 16 2 v1T2 T1 65 H p c v T T 66 Let us now relax the assumption of neutral lapse rate conditions To account for the effects of non neutral lapse rates we rely upon stabilitycorrection factors for momentum water vapor and sensible heat designated QDM GDV and QDH respectively The stability correction factors are incorporated into the general equations for latent and sensible heat transfer For Latent heat and evaporation Eq 62 2 LE lvwk ZVZVIQZQI 67 M V Z Z 1n 2 01 21 201 For Sensible heat Eq 65 c k2 Pa 5 V2V1TZT1 68 The d3 factors are related to the stability condition ofthe atmosphere which is characterized by the dimensionless Richardson number Ri given by Z39g39zz TZI39T2 71 i LTizllLllgl 2 T2 T1 22732v2 v12 6L2T 2 Rz39 69 where g is the acceleration due to gravity The Richardson number expresses the ratio of potential to kinetic energy Neutral conditions exist when Rz39 0 stable conditions when Rz39 lt 0 and unstable conditions when Rz39 gt 0 To determine Q factor values first calculate Ri then use the relations given in Table 61 Table 61 Formulas for stability factors for computing latent and sensibleheat transfer as functions of Richardson number Stability Factor Rz39 lt 003 003 S Rz39 S 0 0 S Rz39 S 019 QM 1 18Rz 1quot 1 18Rz 1 1 52Rz 1 V1 H 131 18Rz 1quot 1 18Rz 1quot 1 52Rz 1 EnvEESS 110 Hydrology and Climate 1 Transient Unsaturated Flow The degree of saturation is equal to moisture content divided by porosity 639 BM Vwamvam therefore 9 639 For flow in an elemental volume that may be only partially saturated the equation of continuity must reveal the time rate of change of moisture content as well as the time rate of change of water expansion and aquifer compaction The pw term in the continuity equation becomes pw 9 Applying the product rule gives 6x By 62 6t 61qu amy 61qu ap a 6639 9 w 9 ax By 62 ll 6t Jr at pw 6t 6 Wq 6W alp zl 3ltpwegt ltpw agt 11 For unsaturated flow the first two terms on the right hand side of Eq 11 are much smaller than the third term 6 6639 13 6639 639 Wltlt and 639 ltlt 12 at pw at pw at AM at Discarding these two terms gives 6pwqx5quy6quz2pby 13 6x By 62 W at 39 Assuming the water density pw is constant a I i 14 6x By 62 6t Recognize that 6639 6 9 66 66 15 6t 6t 6t at so Eq 14 may be expressed as 16 Recall that the unsaturated form of Darcy s law is EnvEESS 110 Hydrology and Climate q K dhK 62 L411 3 dx dx dx q K K 2461 all W 17 y y dy y dy dy dy qz Z KZ dw K 1dy dz dz dz dz where h zz and assuming isotropic conditions Kr Ky KZ K Inserting Darcy s law into Eq 16 and noting that both K and 9 are functions of 1 gives aimwz rigimwz ri rwinw Eq 18 is the pressure head based equation of flow for transient flow through an unsaturated porous medium It is often called Richards equation The Richards equation which was derived by combining Darcy s law for vertical unsaturated flow with the conservation of mass is the basic theoretical equation for describing the pressure head field at any point in a flow filed at any time The solution requires knowledge of the characteristic curves Kz and 911 Because it is non linear there is no closed form analytical solution except for highly simplified Kz and 911 relations and boundary conditions However the Richards equation can be used as a basis for numerical modeling of infiltration exfiltration and redistribution by specifying appropriate boundary and initial condition dividing the soil into thin layers and applying the equation to each layer sequentially at small increments over time 2 The GreenandAmpt Model There have been many attempts to develop approximate analytical solutions to the Richard s equation for specific situations such as infiltration The Green and Ampt model is one ofthese attempts To develop the Green and Ampt model let us first express Richards equation in one dimension with flow limited to vertical downward movement which we designate as the Z direction Fig 21 Darcy s law for unsaturated flow in the z39 direction is dz dz dz dz dz 61 KVKV qz z 262 21 EnvEESS 110 Hydrology and Climate where 626239 6z396Z39 1 Substituting Darcy s law into Richard s equation Eq 16 and assuming one dimensional downward flow gives 5239 a 6239 5239 5 5239 6239 Z 66 a KZIKZV5l 56 519 5 Kg 2 22 Z Recall the hydraulic relations y6 and K09 for partially saturated soils Fig 22 thus allowing y and K209 Therefore 5196 5 awe K2 ay az at 23 8239 6239 Figure 21 Control volume reoriented for downward vertical flow where IZ and OZ are the mass inflow and outflow rates in the downward direction respectively EnvEESS 110 Hydrology and Climate 7 10 105 09 39 08 4 10 06 07 T e quotD A a E 06 e g 103 gt V M 3 K 0 05 3 a E 392 102 04 g d U H 2 7 03 i m wae Airentry tensmn quot quot 39 quot 0 2 101 A 01 1 I l l 1 00 0 01 02 03 04 05 Water content 6 Figure 22 Typical forms of hydraulic relations for unsaturated soils Consider a block of soil that is homogeneous to an indefinite depth ie and Ks are given parameters that are invariant throughout and there is no water table capillary fringe or impermeable layer and has a horizontal surface at which there is no evapotransporation Fig 23 Where evapotransporation is defined as the sum of evaporation and plant transpiration from the earth s land surface to the atmosphere The water content 6 just prior to t O is also invariant at an initial value of 60 Just before water input begins at t 0 there is no pressure head gradient dip8239 O so the downward flux of water qz zt O is given by Eq 21 as qltzzogtKlteogt KlteogtMarea K 900 6239 24 qZ 29 Kz Note that these conditions are not a steadystate situation because the soil is gradually draining but if we assume that 60 ltlt 610 then qz 20 can be considered negligible EnvEESS 110 Hydrology and Climate Beginning at time I 0 liquid water rain or snowmelt begins arriving at the surface 239 O at a specified rate w and continues at this rate for a specified time tw We need to consider two cases 1 w lt KS and 2 wZKS 39 w Rainfall intensityrw 39 39u Figure 23 Wetting front of the GreenandAmpt model EnvEESS 110 Hydrology and Climate 3 WaterInput Rate Less than Saturated Hydraulic Conductivity Consider a thin surface layer of soil where Eq 23 applies at the instant water input begins If we assume that w gt Kt90 water will enter this layer faster than it is leaving This water goes into storage in the layer increasing its water content 9 The increase in water content 9 corresponds to an increase in the hydraulic conductivity K0 and an increase in the pressure head gradient 6116239 so the flux out of the layer qz also increases However as long as qz is less than the water input rate the water content will continue to increase Fig 31 When the water content reaches the value 6W at which qz w the rate of outflow from the layer equals the rate of inflow and there is no further change in water content until water input ceases This process happens successively in each layer as water input continues producing a descending wetting front The water content equals 6W behind above the wetting front and 60 below it This process results in the successive water content profiles shown in Fig 32 As the wetting front descends the importance ofthe pressure head forces decreases because the denominator of the ip6239 term in Eq 21 increases thus the rate of downward flow approaches Kt9w Note that that the infiltration rate at the ground surface ft is expressed using Darcy s law Eq 21 and written as f0KaKZaKTgKTa KTgK my 0 z z zf 0 31 ftsz9sz9W f Zf where the subscripts f and 0 refer to the wetting front and soil surface respectively eg 110 is the pressure head at the soil surface which is considered negligible This analysis confirms the following model w IfwltKS ftw 0lttStW ft0 tgtt 32 EnvEESS 110 Hydrology and Climate K 0 K0ww K 60 0 Figure 31 Changes in hydraulic conductivity with water content during water input EnvEESS 110 Hydrology and Climate 90 51 tz aw 75 t3 I I 5 5 i 5 Z I l E i l 5 E i I i I 5 l E i I a KS r l I f 0 tw b t Figure 32 a Successive watercontent profiles and b infiltration rate versus time EnvEESS 110 Hydrology and Climate 4 WaterInput Rate Greater than Saturated Hydraulic Conductivity When w gt KS the process just described will occur in the early stages of infiltration Water will arrive at each layer faster than it can be transmitted downward and will initially go into storage raising the water content and the hydraulic conductivity Fig 41 However the water content cannot exceed its value at saturation and the hydraulic conductivity cannot increase beyond KS Once the surface layer reaches saturation the wetting front begins to descend with 9 above the wetting front and t9 60 below it As in the previous situation the pressure force decreases as the wetting front descends while gravity force remains constant thus the downward flux decreases approaching qz KS Since w gt KS some rain will continue to infiltrate after the surface reaches saturation but the excess accumulates on the surface as ponding while the wetting front continues to descend as long as the input continues lfthe ground is sloping the excess water moves down slope as overland flow The instant when the surface layer becomes saturated is called the time of ponding designated IF We can develop an equation to compute IF by approximating the wetting front as a perfectly sharp boundary horizontal line which is at a depth 2tp at the time of ponding Ftpwtp 41 where F is the cumulative infiltration Ft multiplied by the unit area eg A 1m2 of land surface gives you the cumulative volume of water that has infiltrated at time I All the water occupies the soil between the surface and ztp so Ftpzl tp39 6039 4392 Combining Eq 41 and Eq 42 yields the time of ponding as Zl tp39 60 W Ztp39 90 gt I w 43 EnvEESS 110 Hydrology and Climate zft5 N a N t gt b Figure 41 a Successive watercontent profiles and b infiltration rate versus time for infiltration Applying Darcy s law Eq 21 between the surface and the depth 2 tp gives I lf O ljf qz 0t 2ft WZKs Ks s s39 I p p Zftp 0 Zftp Here ljf is the pressure head at the wetting front Solve for 2tp in Eq 44 noting that ljf lt 0 szs Ks W gt 40 S f 45 Ztp p w Ks w Ks Substitution of Eq 45 into Eq 43 then yields EnvEESS 110 Hydrology and Climate 10 11 s w 46 Ks 39 Wf39 60 t p ww KS l flllt 6ogt as the equation for the time of ponding As water input continues after the time of ponding infiltration continues at a rate given by Darcy s law Eq 44 substituting for ztp from Eq 42 as ftKS1 lwfl Ksl 39tp F tp t90 Zf I 4397 Wf 39 60 ft KS1 Ft or w fortlttp ft KiHW fortpStStW 48 0 fortgttw this is the Green and Ampt equation Eq 48 allows us to compute the infiltration rate ft as a function of the cumulative amount of water that has infiltrated Ft Unfortunately Eq 48 does not have time as a variable but instead uses F We can write the following somewhat complicated expression for Ft Ftw forFtltFtp forFtltFtp t forFt F t I p 49 l fl39 90Ftp n1 FtFtlzxf 90mWH forFtgtFtp EnvEESS 110 Hydrology and Climate GreenandAmpt Example Problem Using the Green and Ampt model determine the total runoff and infiltration from a 2hr rainfall event with a 05 cm hr39l intensity When does runoff begin The soil s saturated hydraulic conductivity is 0044cmhr391 initial moisture content before infiltration begins is 025 porosity is 050 and the pressure head at the wetting front is 224cm What s the infiltration rate at the end of the storm Solution The given parameters aretw 2hr w 05 cm hr39l KS 0044cmhr391 60 025 050 and 11 224cm First calculate the cumulative amount of water that has infiltrated by the time of ponding Ks39 Wf39 60 w KS 0044cm hr l 224cm050 025 FOP 05 cm hr39l 0044cm hr39l Ftp 054cm Ft F The time of ponding tp is then calculated as Ft 054cm w 050mhf1 IF 108hr IF Next calculate the time t infiltration rate f and the depth ofthe wetting front 2 using the following equations Ftw forFtltFtp I tp forFtFtp tp KLS FI FIpllf39 6039IHM forFtgtFtp iwfi 90F t w fortlttp f KS1W fort StSt FO F W 0 fortgttw EnvEESS 110 Hydrology and Climate FO fortSt 90 W FM 60 fort gttw To solve increment F by small amounts and calculate the corresponding t f and 2 values Results are shown in the following table Cumulative Infiltration Time infiltration Rate Depth ofWetting Front Hemquot thr fcmhr 1 zcm 0000 F ltFtp no tlttp w050 tStW 000 0300 F ltFtp 06 tlttp w050 tStW 120 0540 FFtp 108 rap 050 rsrw 216 0600 F gtFtp L21 rp 9er 045 tStW 240 0700 F gtFtp 144 rp 9er 040 tStW 280 0800 F gtFtp 171 rp 9er 035 tStW 320 0895 F gtFtp 199 rp 9er 032 tStW 358 0900 F gtFtp 201 r gtrw 000 tgttw 360 1000 F gtFtp gt 200 r gtrw 000 tgttw 360 Just pick these values We find that the infiltration rate at the end of the storm I tw 2hr is 032 cm hr39l The following are sample calculations For F03cmFltFtpand tlttp F 036m w 050th fw050mhr391 239 F M1200m f t90 05 025 EnvEESS 110 Hydrology and Climate For F054cm FFtpand tp StStW ttp 108hr frltrpgt1ltsllwfltgt 1 l 224cm050 025 1 f0044cmhr 1 050mhr 054cm Z F 054cm 2il6cm f 9 05 025 0 For F06cm FgtFtpand tgttp ttp Kls Ft Ftpzf 901HMH lwf 90Ft 06cm 054cm 224cm050 025 l 224cm050 025054cm l 224cm 050 02506cm t108hr 0044cmhf1 1n t121hr 6 fszlm0044cmhr11W F 060m f0450mhf1 F 060m z 2400m 6 05 025 0 EnvEESS 110 Hydrology and Climate