Groundwater Flow and Contaminant Transport
Groundwater Flow and Contaminant Transport BSysE 595
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Mass Transport Equations 1 Mass Conservation Statement Mass in ow rate Mass out ow rate Net mass in ow rate Change in mass storage with time MT 1 l 2 General Form For a REV let J be the mass ux vector molesLZT the LHS of Eq 1 divJ denoting net mass in ow rate per unit volume If changes in mass in the REV oc changes in concentration the RHS of Eq 1 denoting time rate of change of mass per unit volume Combining LHS and RHS gives aCn 6t divJ 2 3 The Diffusion Equation Fick s law for diffusion in sediments J D ngradC 3 D bulk diffusion coefficient accounting for tortuosity Substituting 3 into 2 leads to divD ngrad C y 4 If n not changing with space amp time and D constant spatially 4 becomes the diffusion equation 1 divgradC 5 at or 2 2 2 6C6C6CZV2C 1 g 6 6x2 6y2 dz2 Dat If the net concentration does not change With time 6 is simplified to Laplace equation V2 C 0 4 The Advection Diffusion Equation Combining the advective and diffusive uxes 6C Jx Dn van 6x 6C Jy Dy na y 1nd 6C Jz Dz Ila z szn For isotropic media J DngradCan 7 J v vectors With x y 2 components Substituting 7 into 2 yields VD nVC Vvcn 8 Expanding the advection term and assuming constant D over space and n over time amp space gives 6 C 9 DV2C V39VC CV39v E For steadystate ow Vv 0 and DV2C vVC 6 10 A noadvection condition reverts 10 to 6 lac WC 1 at A steadyCprofile condition leads to DV2C vVC0 11 Eq 10 11 are called the advection diffusion equations 5 The Advection Dispersion Equation Combining diffusion and mechanical dispersion processes and assuming the hydrodynamic dispersion coefficient D D D constant over space yields DVZC vVC CVv 6C t 12 Eq 12 is the advection dispersion equation If D small 12 reverts to the advection diffusion equation further if no advection it reduces to the diffusion equation Modeling Transverse Dispersion 1 C Characteristics Assuming lateral dispersion only ie no longitudinal dispersion ccx 0 then no solute would travel beyond the advective front Solute however is not restricted to a single tube but can spread in both y and z direction into adjacent tubes Cx t lt C0 occurs because volume occupied by solute increases with x for x lt vt 2 Source Geometry Possible geometrical configurations of the contaminant source are l a point 2 a vertical line 3 a planar area Along the xaXis for y 0 and z 0 the maximum C is given by Cmax C0 ow C max A line source 181221 2v1txocy12 C m 4Q point source 1812b 4vitxoc 05 2 C0 source concentration M L 3 Q volumetric flow rate per unit length for the line source L2 T l volumetric flow rate for the point source L3 T l 3 General Solutions for Planar Sources For spreading in ydirection Cxy 1 erf y Y2 M y Y2 C0 2 2 ayx12 ayx12 18 13 For spreading in z direction m MM Mai C0 2 2oczx 2 2oczx 2 1814 For spreading in both y and z directions the product rule is applied to obtain Cxyz 1 M v Y2 M v Y2 X C0 4 2ayx 1 2 2ayx 1 2 erf M erf 2azx 12 2azx 12 1815 Along the xaXis for y 0 and z 0 the maximum C is given by Cmx Coerf L 1816 4 ocyx12 2 oczx12 WATER AND ENERGY IN THE ATMOSPHERE Appendix D Summarized from Physical Hydrology by SL Dingman 2002 1 Physics of radiant energy All matter at T gt zero K radiates energy 7 form electromagnetic waves 7 speed 30gtlt108ms 1 Emission rate from StefanBoltzmann law Q 6011 Eq Dl Q rate of energy emission EL ZT I 6 emissivity dimensionless O StefanBoltzmann constant EL ZT IQ A T absolute temperature 9 7 A surface with 61 is a black body 7 Most earth materials have 6Z l 7 The spectrum of electromagnetic radiation extends over 21 orders of magnitude only near ultraviolet visible and infrared 02780 pm rays important in balancing earth energy and climate 2 Terms A bsorptance 05 the fraction of the incident energy absorbed by a surface raises temperature of the matter and or causes phase change Re ectance p the fraction of the incident energy re ected by the surface does not affect the matter and continues traveling in a new direction T ransrnittance T the fraction of the incident energy transmitted through the matter does not affect the matter and continues traveling in the original direction 060 t PO t TO 1 Eq D4 Albedo the re ectance integrated over the visible wavelengths 04707 pm 11 Composition and vertical structure ofthe atmosphere 1 Composition Mixture of gases with suspended liquid and solid particles 7 Many are constant in time and space 7 Variable constituents that affect energy balance and the formation of precipitation including water in three phases dust and C02 2 Vertical structure 7 Atmosphere extends from surface to 700 km 7 Divided into T roposphere Stratosphere and Mesosphere based on P and T features 7 Only Troposphere and Stratosphere 0750 km affect climate and hydrologic activity 111 Pressure tem perature relation 1 Ideal gas law PTapa Ra Eq D5 P atmospheric pressure T a air temperature pa mass density of air Ra gas constant for air 2 Adiabatie coolingwarming Changes in T do not involve removal or input of heat but only changes in P and p Lapse rate vertical temperature gradient in the troposphere 7 65C km 1 Inversion temperature gradient near the surface may temporarily reverse direction IV Water vapor Partial pressure Due to their molecular motions and collisions each constituent of a mixture of gases exerts a pressure Vapor pressure e the partial pressure of water vapor Atmospheric pressure formed from the sum of all gases partial pressure Saturation vapor pressure emU the max vapor pressure that is thermodynamically stable at a temperature T 8mm 7 611 expl73TT2373 Eq D7 em in mb T in C Absolute humidityvapor density pv mass concentration of water vapor in a volume of air Relative humidity Wa the ratio of actual vapor pressure to its saturation vapor pressure W 7 new Eq D Io Dew point T de the temperature to which a parcel of air with a given vapor pressure has to be cooled to reach saturation Tdlne 0492600708 7 000421ne Eq D11 V Physics of evaporation 1 Mass transfer equilibrium state at the surface escapereentry and EMU 2 Dalton 3 law Em 24T 7 2 Eq D12 7 25111 7 8 can be positive E occurs zero equilibrium or negative condensation occurs 7 Fog may form when 85111 gt 8 while the air is saturated 3 Latent heat transfer LE7 pAVE Eq D13a LE7 pa AQE Eq D13b LE rate of latent heat transfer EL ZT I p mass density of water AV latent heat of vaporization EM I If latent heat of fusion EM I E the rate of evaporation or condensation LT I VI Physics of precipitation Four processes 1 cooling to the dew point 2 condensation 3 droplet growth 4 importation of water vapor Cooling by other processes may produce fog drizzle Only vertical lift can cause rates of cooling high enough to produce signi cant precipitation 7 Air containing water vapor but with no impurities can have relative humidities up to 800 A stable droplet with diameter of 10 4 mm contain about lOgmolecules The chance to form such droplets by random collision is basically zero 7 Therefore foreign particles as cloud condensation nuclei CCN larger than 10 4 mm to which water molecules are attracted via hydrogen bonds must be present for condensation to occur 7 Droplets must grow to a size such that their fall velocity gt the rate of uplift amp they can also survive evaporation 7 Droplets grow by collision Icecrystal grows at the expense of the liquid droplets as esat ice lt esat water 7 Importation of water vapor is provided by wind VII Physics ofturbulent transfer near the ground 1 F ick s first law of diffusion FZX 7DXdCXdz Eq D 23 FZX rate of transfer of X DX diffusivity of X CX concentration of X 2 Mathematical equations for transfer of momentum latent heat and sensible heat 7 Dingman 2002 7 Other references Modeling Advection and Longitudinal Dispersivity 1 Combining ld Solutions to Form a 3d Solution If three ld solutions are of the form m 2 F1 ax x 1 C0 C CE F2ocyy 0z F3oc z z then an approximate solution to a 3d problem may be given by C 9 9 9t w F10 xsxs t F2 y y F3az z 0 Q Describe the major mass transport processes in various directions implied in this problem Solution to ld Advection and Longitudinal Dispersion problems For the ld advectiondispersion equation 2 D v 181 6x2 6x at with BC and IC CO t C0 continuous source Cx 0 0 given by Ogata and Banks 1961 Cx t Z l erfc x vt C0 2 12 exp 3 erfc th D 2Dt12 182 D dispersion coefficient in xdirection erfc the complementary error function erfB the error function v velocity of the tracer or pore velocity The second term in Eq 182 is often negligible Dt may be written as axvt if ignoring molecular diffusion so Eq 182 is simplified to m erfc M 183 C0 2 2axvt 2 Values for erfc B and erf B are given in Table 181 and Fig 183 Useful relations erfc B 1 erf B erf 13 el fl3 erfc B 1 erfB 3 Physical Meaning of the Argument of erfc When x vt 0 or x vt the argument is zero erfc 0 1 and CC0 05 When x vtgtgt0 or xgtgtvt the argument B is positive as it a 00 erfcoo 0 and CC0 0 practically C s 0 when B 2 When x vtltlt0 or xltltvt the argument B is negative as it 6 oo erfc o 2 and CC0 1 practicallyC z C0 when B 2 The denominator 2ocxvt 2 has the units of length and can be regarded as the longitudinal spreading around the advective front Longitudinal dispersion moves mass ahead of the advective front as the advective front furthers away from source with time the spreading becomes larger Mass Transport Processes 1 Mass in Water Forms of mass in water ions molecules solid particles Mass in water undergoes i transport ii reactions thus solute transport and fate 2 Transport Processes 1 Advection The process by which moving ground water carries with it dissolved solutes The direction and rate of mass transport coincide with groundwater ow mass added to ow tubes Will stay in those tubes direction of mass spreading is defined by ow lines Factors in uencing groundwater ow patterns also control mass transport patterns 1 The velocity of advective mass transport is given by the Darcy s law and v Twith i ne K6h ne 6 2 Hydrodynamic Dispersion A process of uid mixing acting to dilute the solute and lower its concentration it consists of diffusion and mechanical dispersion Dispersion spreads mass beyond the region it normally would occupy due to advection only Re ected by the difference between the source loading curve CC0 vs t at entry and the breakthrough curve CC0 vs t at outlet For ld ow the advective front vXt tL time since loading starts generally corresponds to CC0O5 For 3d ow dispersion occurs in three directions longitudinal transverse and vertical Diffusion dissolved ionic and molecular species move from areas of higher C to areas of lower C Diffusion through water follows Fick s laws J mg Fick s ist law dx J solute ux ML ZT I D diffusion coefficient L2T 1 C solute concentration M L 3 concentration gradient M Lquot x 2 6 C 1 Fick s 2 1 law at 8x 2 Diffusion in aquifer is not as fast as in H20 Why Thus a more suitable form is J D or J DngradC raanl g V V REV volume 1 tortuosity Within REV D bulk diffusion coefficient accounting for tortuosity Mechanical dispersion mixing caused by local variations in velocity around some mean velocity of flow D ocva ld ow Hence the hydrodynamic dispersion is DL DDn ochxDn Id ow 3 Retardation Consider solutes in two classes i Conservative not reacting With porous media and natural ground water not undergoing biological or radioactive decay eg Cl 1 ii Reactive opposite to conservative solutes Retardation a general term for many processes that act to remove solutes in ground water re ected by solute front traveling more slowly than the advecting ground water Darcy s Law for Unsaturated Flow 1 Surface Tension A molecule at the surface of liquid water is subject to a net inward force due to H bonding with the molecules below the water surface So there is a tendency for the surface area of water to be reduced as much as possible Surface tension 0 is the energy required to increase the surface area of a liquid by a unit amount or the tension force divided by the distance over which it acts Examples steel pin oating or water bug sliding on water surface 2 Capillarity Resulting from surface tension Place a tube upright in water If the tube materials attract water then the H20 molecules in contact with and thus the entire mass of water within the tube are drawn upward until the attraction balances the weight of the water in the tube Capillary rise lJ Fu ocos0c 21w Fd1rr211 y 20 cosO Fu Fd q 2 z E quotY quotY Given tube material and liquid 1 depends on r Soil H20 tensionsuction expressed as negative pressure values as PAtmO can change from O V when 6v 7 approaches n to more than 150 m T of head when 6v is very low 2 3 Vadose Zone Fluid Potential v negligible 1 lt O and may dominate g for very dry 8 conditions 4 Darcy s Law for Unsaturated Flow q Ko a z 339 K6Vh q specific discharge LT l K6 unsaturated hydraulic conductivity LT l z elevation head L I negative pressure tension suction head L V h hydraulic gradient For unsaturated ow both K and I are a function of 6 6T 111T 6T KT and HKW 5 Hysteresis During wetting the small pores fill first during drying the large pores empty first causing hysteresis 0r inkbottle effect Challenge in modeling unsaturated flow Warmup Questions What is hydrologic cycle Why is a proper understanding of hydrologic cycle relevant to studying ground water hydrology Draw a diagram showing the hydrologic cycle Hydrologic Cycle I Defined as the global pattern of continuously circulating water between the ocean the atmosphere and land I Its dynamic operation and the interactive processes frame the entire theoretical study of hydrology I Maj or characteristics 1 A dynamic system powered by the solar radiation and embraced by constant ow A closed system to which no new water is added or lost in any significant amount A recycling system which enables water to remain clean A system in balance barring the generally adverse impacts of human activities pumping damming introducing into it contaminants An interactive system signified by water s readily changing states and moving between the atmosphere groundwater aquifers and surface water bodies I Main processes 1 In atmosphere Precipitation Evaporation Transpiration Interception In hydrosphere Overland ow Runoff In lithosphere 60 km crust partial upper mantle Infiltration Inter ow Groundwater ow Case Study Mono Lake Fetter 1994 Mono Lake Committee 2003 Located in California s spectacular Eastern Sierra Mono Lake is the largest natural lake entirely within the state and an oasis in the dry Great Basin and a vital habitat for millions of migratory and nesting birds For over 20 years signi cant efforts have been devoted to protect Mono Lake from destruction and to heal the damage done in the Mono Basin Mono Lake is a terminal lake What is a terminal lake The Mono Lake basin has an area of 180000 ha Inputs to the lake under natural conditions are direct precipitation averaging 02 m annually runoff from the land areas via gaged streams roughly 185gtlt 108 n13 per year and un gaged runoff and ground water ow which is about 456gtlt107 n13 per year The lake evaporation averages 11 m per year Estimate the surface area of the lake under the water balance assumption When first surveyed in 1856 the elevation of Mono Lake was 1953 m amsl What is amsl and what is this elevation in ft In 1941 water was first diverted from four of ve major streams feeding Mono Lake into LA Aqueduct and thence to southern CA Diversions amounted as much as 123 gtlt108 n13 per year The historic low of the lake level 19422 m was reached in 1982 The lake level rose to 1945 m during a very wet period of 198271984 In 1989 the diversions were halted under a temporary court order that prohibited any diversions that would result in a lake level of less than 1944 m However the lake s level still declined even without any diversions Why did this occur so that by the end of 1992 it was 19426 m In 1941 the year that diversions began the surface area of Mono Lake was 21670 ha When the lake level declined by 12 m from 1941 to 1981 the surface area shrank to 16200 ha The annual diversion of 123gtlt108 n13 would cover the 16200 ha lake to a depth of 076 m Really The water level fell because the amount of the diversion plus the natural evaporation from the lake exceeds the amount of the precipitation onto the lake surface plus the remaining surface in ow and ground water ow If the courts were again to permit the unrestrained annual diversion of 123gtlt108 n13 of water from the Mono Lake basin the lake level could fall to as low as 193071934 m resulting in a lake surface area of only 9520711200 ha Roughly sketch the vertical lake pro le through the center of the lake assuming a circular lake surface and symmetric geometry for the crosssection Why would there be a range of the lake level Why wouldn t the lake level continue to drop One consequence of the reduction of the volume of Mono Lake has been an increase in the salinity of the lake Mono Lake contains about 280 million tons of dissolved salts Before water diversions began in 1941 salinity was about 527 g L 1 or 54 dissolved solids byweight What is the density ofsalt At the lowest lake level in 1982 it was 99 g L For comparison the ocean is about 315 g L 1 or 32 dissolved solids by weight The increase in salinity has resulted in a reduction of the brine shrimp and brine y population and in turn the population of migratory birds of the lake If the lake level were allowed to fall to 1931 m the salinity could reach 22 which would eliminate the brine shrimp and ies at all The other consequence was the connection of one of the islands to the main land by a land bridge due to the drop of the lake level The connection allowed coyotes to cross from the mainland to the island severely disrupting the nesting California gulls Today Mono Lake enjoys a greater measure of protection than at any time since water diversions began In 1994 the California State Water Resources Control Board issued its decision D 1631 on Mono Lake which set minimum ows for the streams and limits on water exports designed to allow the lake level to rise and stabilize at an elevation of 19488 m and ordered the LA Department of Water and Power to restore the damaged streams and waterfowl habitat The elevation of Mono Lake in 2002 is 19460 m and the salinity is about 78 g L Once it rises to its stabilization level How long approximately would it take for Mono Lake to rise from its 2002 level to the stabilization level salinity will average 69 g L 1 5 Programming Analytical Solutions to Mass Transport Equations I Importance of Modeling Modeling is one of the activities that must be practiced to develop necessary coding skills the ability to formulate a problem in terms of boundary conditions and other data and the ability to use a computer I Purpose of Programming the Solutions To obtain the solution in a more efficient manner I Major Coding Components 0 Reading values of transport parameter from a file 0 Solving for C at speci ed times and locations 0 Writing the results to a file I General Procedures in Solving for Transport Problems 1 Formulate the problem mathematically and select appropriate solution 2 Organize input data 3 Construct the computer code 4 Run the code using known information 5 Interpret results I Sample Code Statement of Problem An organic contaminant is disposed of continuously in a narrow trench that fully penetrates a shallow semiinfinite aquifer For the aquifer v 231gtlt10 6 ms at 43 m R 30 The contaminant concentration at the source Co remains constant with time at 10 mgL Calculate the contaminant concentration after three years at 10m intervals from the source Analytical Solution For ld mass transport subject to advection dispersion and simple sorption fo V C x t C 2allthRf1 2 BMW 186 bdk 000 100 110 120 PROGRAM RONE FOR Last change 11 Nov 96 1044 CODE FOR EVALUATING ONE DIMENSIONAL DISPERSION WITH RETARDATION DIMENSION C50 READ NECESSARY INPUT DATA OPEN5FILE INDAT OPEN6FILE OUTDAT READ5lOO NX WRITE6lOONX READ5llO COALFXVXRFTYRDELX WRITE6llOCOALFXVXRFTYRDELX TSECTYR36586400 X 0 CALCULATE CONCENTRATIONS DO 200 IlNX XXDELX CICO2ERFCRFXVXTSEC 2SQRTALFXVXTSECRF WRITE RESULTS WRITE6120TYR WRITE6llOCII1NX FORMATI5 FORMAT6F102 FORMATF50 STOP END YEARS CALCULATE ERFC FUNCTION ERFCZ DATA BCDEFGO705230784 0422820123 0092705272 1520143EO4 2765672EO4 430638E05 IFZGE3ANDZLE3GO TO 100 IFZLT3ERFC20 IFZGT3ERFC00 RETURN XABSZ IFXGT30ERFCO56419EXP00XX X5XlXl5X2X25Xl IFXLE30ERFC1lBXCXXDX3 EX4FX5GX6l6 IFZLT00ERFC20ERFC RETURN END Input Data 18 10 43 00000231 3 3 10 Output Results 3 YEARS 099 098 096 091 082 070 055 039 025 014 007 003 001 000 000 000 000 000 I Exercise Read the following statement codes input and output data Provide remarks to replace those question marks Also list the variables for which the values are provided in the input data file and describe the characteristics of the plume as simulated and shown in the output file Statement of Problem Diethyl ether de with a fixed initial concentration 0 10000 pgL enters the ground water from a source offinite size The transport includes dispersion in L 439 quot advection in 439 t39 and retardation due to sorption The initial concentration of de is zero Calculate C for a grid of points on a horizontal plane passing through the middle of the source at Z 0 Note the origin 000 is set at the center of the source area Code PROGRAM Last Chan e 11 No CODE FOR EVALUATING WITH 9 v 96 907 pm DIMENSIONAL TRANSPORT DIMENSION C3030 READ OPEN5FILE IN3DAT OPEN6FILE OUT3DAT READ5lOO NXNY WRITE6100NXNY READ5110 C0ALFXALFYALFZVXRF WRITE6110C0ALFXALFYALFZVXRF READ5110 TYRDELXDELYZ WRITE6110TYRDELXDELYZ READ5llOBZBY WRITE6110BZBY TSECTYR36586400 VXVXRF X00 DO 200 IlNX XXDELX Y DELYFLOATNYl2 DO 200 JlNY YYDELY AACO8ERFCX VXTSEC2ALFXVXTSECO5 BBERFYBY22ALFYXO5 ERFY BY22 ALFYX05 1 CCERFZBZ22ALFZXO5 ERFZ BZ22 ALFZXO5 CIJAABBCC CONTINUE WRITE6120TYR DO 210 IlNX WRITE6130CIJJ1NY FORMAT2I5 FORMAT6F102 FORMATF50 FORMAT7F92 TOP YEARS END FUNCTION ERFCZ DATA BCDEFGO70523078404228201230092705272 l520143EO42765672EO4430638E05 IFZGE3ANDZLE3GO TO 100 IFZLT3ERFC20 IFZGT3ERFC00 RETURN XABSZ IFXGT30ERFCO56419EXP00XX l X5XlXl5X2X25Xl IFXLE30ERFC1lBXCXXDX3EX4 l FX5GX6l6 IFZLT00ERFC20ERFC RETURN END FUNCTION ERFZ IFZLE00ERFERFCZ1 IFZGT00ERF1ERFCZ RETURN END Input Data 10 7 10000 1 01 15 75 15 00 5 25 Output data 15 YEARS 000 00 2486 12 9575 000 595 2757 84 8319 000 3459 2700 07 7137 007 6030 1886 02 4514 000 121 2163 47 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 10E06 mu OOOOOI OrbU39lO OOOOOOOOOO Bernoulli Equation and Darcy s Law for Saturated Flow 1 Bernoulli s Equation In uid mechanics for an ideal uid under steady ow the uid potential q is stated in terms of Bernoulli Equation P 2 I f E gz v7 constant along a streamline P P0 The three terms represent the pressure gravity and velocity potentials respectively The Hubbert s and Bernoulli s representations of uid potential are identical 2 Darcy s Law h h K i 2 Kg KgmdhltmMD q Al Q volumetric ow rate L3T 1 q volumetric ow rate per unit area also called Darcy s velocity or specific discharge LT l K constant of proportionality LT l h1 h2 hydraulic heads at positions 1 and 2 L AZ distance between positions 1 and 2 L hydraulic gradient 3 Darcy s Velocity vs PoreLinear Velocity As water flows through pore openings instead of a completely hollow pipe Darcy s q is not real A new term porelinear velocity v is defined as n effective porosity 4 Physical Interpretation of K Experiments show that Cdng6h a 2 v ltpg vltgtltlvltgtltd2 a v C constant of proportionality Comparing Eq 1 and 2 shows 2 01ng p K K containing properties of both the porous medium and fluid is called hydraulic conductivity Let k Cd 2 then k a function of only the porous medium is called permeability 5 Applicability of Darcy s Law valid for laminar ows and in porous media of reasonable values of permeability k 3 Unsaturated Flow Governing Equations 1 Richards Eq A Apply mass conservation principle to REV Mass In ow Rate Mass Out ow rate Change in Mass Storage with Time For a REV with volume AxAyAz mass in ow rate through face ABCD is pwqx AyAz Mass out ow rate through face EFGH is 6 Ax pwaqx AyAz pwqx The net in ow rate is thus 6pwqx x AxAyAz Similarly net in ow rate thru face ABFE is 6pwqy AxAyAz 6y and net in ow rate thru face BCGF is 6pqu Z AxAyAz The total net in ow rate through all faces is then apwqx apwqy 6pqu ax 6y az AxAyAz The change in mass storage is a 6 AxAyAz at where 6 is the volumetric water content L3 L 3 Equating net in ow rate and time rate of change in mass storage and dividing both sides by AxAyAz leads to 6pw196pwqpapwq Z 6pw0 6x a y dz 6 t 1 In fact 1 can be obtained directly from 6pw3 V39Pw4 d1Vqu T If pw varies neither spatially nor ternporally 1 becomes 66 8t aqx 2 6x 8y dz B Apply Darcy s law to Eq 2 i Kah 1 Kah 1 Kah 6x 6x 6y 6y az az 2 Q at 3 h z 1 Bernoulli eq 39 zai0 aw0 a 1 8x 6x 36y 6y az 62 Substituting the above equations into Eq 3 leads to i fa 1 i fa III i fa q aI 6x 6x 6 y 6 y az az az Q at 4 Eq 4 is the 3d Richards equation the basic theoretical framework for unsaturated flow in a homogeneous isotropic porous medium Eq 4 is not applicable to macropore ows Eq 4 Darcy s law for unsaturated ow does not address hysteresis effects Both K and I are a function of 6 making Richards equation nonlinear and hard to solve 2 Simplified Cases If V2 gravity gradient negligible 4 becomes 1amp1 1Kaw 1Kaw z 8x 8x 6y 6y dz 82 at If lateral ow elements negligible 4 becomes 1 fa w aI 6 6 az dz 87 at Sources of GroundWater Contamination 1 Terms Water quality the quantity of dissolved solids and gases suspended solids hydrogen ions pathogenic organisms and heat in a given quantity Of water Dingmam 1994 Pollutant when the C of a constituent reaches a level that adversely affects the suitability of water for desired use it becomes a pollutant mammalm 1995 Contaminant a substance Whose presence in water is not normally expected can be either dissolved constituents or nonaqueous phase liquids NAPLS added to the water from man s activities Domenico and Schwartz 1990 2 Complexity of Contamination Problems Denserthan water DNAPLs sink through water Lighterthanwater LNAPLs oat on water Transport of NAPLs highly complex as the contaminants can migrate as a separate liquid phase dissolved phase and vapor phase 3 Contamination Sources Classified according to three attributes 1 Degree of localization Point smallscale leaking tanks sanitary landfillls disposal ponds Nonpoint largescale agricultural farming household disposal systems highway de icing salts acid rain 2 Loading history Onetime release pulse spill from a tank Longterm continuous loading NAPLs dissolving at low rates over decades 3 Contaminant type Resulting from industrial agricultural or domestic activities Contaminants occurring in drinking water vs those persisting Within the food chain 2 4 Organization Based on Reaction Type and Mode of Occurrence l Radionuclides Main generator nuclear industry Sources mining and milling of U U enrichment and fuel fabrication power plant operation fuel reprocessing and waste disposal Health hazards cancer genetic defects 2 Trace Metals Main generator industry and agricultural practices Sources mining ef uents industrial waste water urban runoff and wastes agricultural wastes and fertilizers fossil fuels Health hazards bioaccumulation in the food chain leading to metal poisoning 3 Nutrients Main generator agricultural practices Sources land application of fertilizers cattle feeding lots cultivated Virgin soils Health hazards methemoglobinemia 4 Other Inorganic Species Including metals present in nontrace quantities Main generator industrial and domestic activities Sources mine tailings and spoil sanitary landfills industrial waste water saline brine Health hazards less severe than caused by others high C leading to cell or blood chemistry 5 Organic Contaminants Main generator industrial agricultural and domestic activities Sources petroleum auto transformercapacitor and agricultural production Health hazards may cause cancer liver damage impairment of cardiovascular function 6 Biological Contaminants Including pathogenic bacteria viruses and parasites Main generator human and animals Sources human and animal sewage waste water Health hazards typhoid fever cholera polio hepatitis diarrhea Porous Media Porosity Compressibility and Tortuosity 1 Porosity What is a fish net Porosity n is the part of the porous media that is void space VV n VT Vv void volume VT total volume Void ratio 6 B H 5 lg Vs solid volume en relations 12 Primary interstitial porosity vs secondary fracture or solution porosity Total porosity vs effective porosity regarding pore connections 2 Compressibility Rock rnatriX cornpresseseXpands as the grain tograin pressure increasesdecreases Fig 44 In a waterfilled rock rnatriX the pressure in the solid phase by Virtue of points of contact is called inter granular pressureeffective stress 5 the pressure due to the weight of water is pore Water pressuresneutral stresses P o 5 P o total vertical stress For Fig 44 A0 AP at the initial loading and A0 A5 at the end P acts on all sides of the rock rnatriX but does not cause particles to press against each other 5 causes rock cornpression Fig 45 an example of stress transfer 3 Tortuosity The natural porous media render the ow paths sinuous in form Tortuosity actual length of the flow path diVided by the straightline distance between the ends of the ow path Modeling Retardation Due to Linear Sorption For ld advective mass transport with dispersion and linear sorption the equation is 2 2 l 184 Rf 3x2 Rf ax at For BC and IC CO t C0 continuous source Cx 0 0 the approximate solution is Cx t 1 erfc C0 2 fo vt 186 2axvtRf12 v velocity of advecting water Rf 1 pst is the retardation factor n Note if Rf 1 the Ogata Banks solution is exactly recovered The transport velocity of contaminant vc is related to the groundwater velocity v by 187 Eq 187 often referred to as the retardation equation predicts the position of a plume front due to advective transport With linear sorption The ratio vvc describes how faster ground water or nonsorbing species is moving relative to the contaminant being sorbed Note when K 0 vc v For binary exchange reactions the partition K CEC coeff1c1ent K d is replaced by s and the 1 retardation equation is given by l Rf 1 MKSCEC 189 vc m Ks selectivity coefficient unitless 1 total competing cation C in solution rneqrnL CEC cation exchange ratio meq 100 g mass centirnolekg 2 Due to Radioactive Decay For ld advective mass transport With dispersion and radioactive decay the equation is 2 D vE ACJ C 1416 6x2 6x at 0693 1 A decay constant A 1 halflife For BC and IC CO t C0 continuous source Cx 0 0 the solution is Cxtlexp i 114Mx12 C0 2 Zax v f x vt1 4ocxv12 c 2 ocxvt12 1810 v velocity of contaminant equal to velocity of advective water when there is no retardation If A 0 Eq 1810 reduces to the Ogata Banks solution if the argument of exp 6 00 C gt 0 The dimensionless group 4Aaxv is critical if A large exp 6 0 and C gt 0 species decays faster than it is transported if v large taxv 0 exp 6 1 species moves faster than it decays ocx appears in two places in the exp term off setting its effect The advective front is modified by decay as vt1 4 Maxv 2 instead of vt If the observation point is far behind the modified advective front the argument of erfc a 2 and erfc a 2 a steadystate C profile is obtained as L C eXP Max 0 which is independent of t Solute Plumes 1 General Principles Solute plumes serve as a manifestation of transport and reaction processes Advection and dispersion determine the maximum extent of plume spread and geometric character of C distribution Chemical reactions attenuate the plume spread and reduce the contaminated region Advection is by far the most dominant process in shaping the plumes Hydrodynamic dispersion is usually a second order process except in the cases involving fractured rocks Where gradual dispersion may not occur 2 Specific Process Impacts 1 Advection The magnitude and direction of advection are controlled by K distribution Within the ow field configuration of h field water table level or potentiometric surface presence of sources or sinks shape of ow domain Due to the lack of dispersion or retardation solute plumes have a uniform C equal to the source C mass added to stream tubes remain within the tubes Understand Fig 174 a b Understand Fig 174 c e 2 Advection Dispersion Adding dispersion to advective transport can cause significant change in plume shape Increasing the dispersion coefficient will increase the size of a plume and the extent of mixing with uncontaminated ground water While decreasing the maX C Understand Fig 175 a d Understand Fig 176 a c 3 Advection Dispersion Reaction The more or faster a reaction can remove a contaminant from the solution the smaller the plume Will be at a given time Irrespective the specific reaction form and model chemical reactions that reduce solution C are of great importance in controlling plume shapes Understand Fig 177 a c 3 Understand Fig 178 a d 4 Effect of Loading Functions Loading function in uences plume shape adding same quantity of solute over a longer time leads to enlarged closertosource plume with lower maX C Understand Fig 179 a c 5 Nonideal Plumes Often in reality plumes are irregularly shaped with nonsystematic concentration distribution which is particularly true in fractured and karst systems 6 Nonuniqueness of Model Parameters When processes and properties are not well known interpretation of plumes is difficult eg a broad dispersion zone at the front of a plume may be a result of a high XL and constant loading or a low 06L but increasing loading rate Saturated Flow Equation Initial and Boundary Conditions 1 Significance of Boundary B Conditions BC Mathematical ground water flow models include a governing equation BC and initial conditions In steadystate simulations BC determines the ow pattern for transient flow cases BC affects the results if the transient stress reaches the B 2 Types of BC Type 1 specified head or Dirichlet conditions for which head is given eg a river Type 2 specified flow or Neumann conditions for which the derivative of head flux across the B is given eg seepage to underlying bedrock Type 3 headdependent ow or Cauchymixed BC for which ux across the B is calculated based on a specified head on one side of the B and the modelcalculated head on the other side ofthe B 6g For a 3D problem let I be the complete B we may have h Hxay9z9t on S1 1 ah ah ah Kxa nx Kya yny KzG znz gxyzt fx9y9z9th 0 on S2 2 where 71 my and n are the direction cosines of the outward normal n to the surface and g and f are known a prioi then S1 U S2 I1 Eq 1 and 2 are the Dirichlet and Cauchy BC when f 0 Eq 2 reduces to the Neumann BC In groundwater study a special case for Type 1 BC is a constant head BC when H xyzt is constant in time and for Type 2 a no ow BC when a h 0 an 3 Initial Conditions IC In groundwater hydrology IC refer to the entire head distribution at the beginning of the simulation and thus are BC in time Type 1 static steady state h constant in time and space used more in drawdown calculation Type 2 dynamic average steady state h constant in time but not space most often used Type 3 dynamic cyclic conditions h varies in both space and time used when transient effect is cyclic eg pumping around a city Conservation of Fluid Mass 1 Physics of Fluid Flow Conservation Statement Fluid mass conservation Mass In ow Rate Mass Out ow rate Change in Mass Storage with Time This statement is generally valid regardless of the size of the domain of interest Yet when using the statement together with Darcy s law macroscopic a threshold exists for the domain s1ze Representative Elementary Volume REV the smallest volume for point P which includes sufficient number of pores to permit meaningful statistical averaging and below which no single value exists that can represent the porosity at P Mathematical Representation Given a REV with volume AxAyAz mass in ow rate through face ABCD is pw qx AyA z mass out ow rate through face EFGH is 8 A qux pwqquot x AyAz 6x The net in ow rate is thus 6 M AxAyAz ax Similarly net in ow rate through face ABFE is 6pwqy AxAyAz 6y and net in ow rate though face BCGF is 6pqu Z AxAyAz The total net in ow rate through all faces is then apwqx apwqy 6pqu AxA Az ax 6y az y 45 in units of M T l t The change in mass storage is 6 AxAyAz 46