Physics of Aquatic Systems

Transcription

1 Pysics of Aquatic Systems Contents of Session 7: Groundwater 7.1 Introduction to ydrogeology 7.2 Groundwater flow 7.3 Transport in groundwater 7. Flow and Transport in Groundwater Werner Aescbac Hertig Introduction to Hydrogeology Definition of Groundwater Groundwater = Water in te saturated zone below te water table Importance of Groundwater as Resource Most important source of drinking water een in umid areas (Vadose zone) Unsaturated zone: p < 0 (Preatic zone) Water table: p = 0 ( = p atm ) Saturated zone: p > 0 Origin of water witdrawals in Germany Statististical Yearbook 2010 Ground and spring water in 2007: Total witdrawals: 5.8 out of 32.3 km 3 18% Public supply: 3.6 out of 5.1 km 3 70% 3 4 Types of Porous Media sediment fractured rock karst Hydraulic Head Total energy at eigt z pores fissures caities Microscopic: Porous media are extremely eterogenous Macroscopic description by aeraging oer a certain olume: Representatie Elementary Volume (REV) Vpore Major geometric property: Porosity n n Vtotal REV sould be sufficiently large tat n constant Effectie porosity n e n: porosity aailable for flow and storage from Fitts, Groundwater Science, Academic Press E = pv+ mgz Energy per weigt E p = z mg = ρg + Hydraulic ead is measured by piezometers (open pipes) also called piezometric ead (Piezometeröe) 5 6 1

2 Hydraulic Head and Types of Aquifers Confined Aquifer in a Basin piezometer soil aquifer permeable unsaturated permeable saturated piezometer z = 0 p ρg z Unconfined Aquifer = water leel in aquifer = water table or preatic surface soil Confined Aquifer aquitard aquifer impermeable screen z = 0 p ρg z > water leel in aquifer = potentiometric surface 7 8 Patterns and Time Scales of Groundwater Flow Lage scale cross section Flow direction Numbers: Residence times (a) Interbasin Groundwater Flow Recarge areas Discarge area 7.2 Groundwater Flow Scale analysis of te Naier Stokes equation for flow in small pores terms in ms 2, L = L = 10 4 m, u = w = 10 5 m/s, f = 10 4 s 1, ν = 10 6 m 2 s 1 Reynolds number: UL 3 Re = = 10 ν Laminar flow! orizontal: u 1 p ( ( ) ) 2 ( w cosϕ sin ) u = x ρ x Ω ϕ +νδ ertical: w 1 p = ( ( ) ) z g + 2Ωucosϕ +νδw ρ z from Fitts, Darcy's Law Generalization of Darcy's Law Darcy in 1 D: q= K d dx column Empirically (Darcy 1856): Q Δ d q= = K = K A Δl dl Darcy elocity ydraulic conductiity ead gradient 3 D: Porous media are usually anisotropic, i.e. te ydraulic conductiity K depends on flow direction. Replace K by conductiity tensor K. For coordinate system along layers: Kxx 0 0 K = 0 K yy Kzz Darcy in 3 D: q= K Typically: K xx = K yy > K zz

3 Properties of Different Porous Media Cone of Depression of a Well Darcyelocities: m/d m/a Pumping draws down te ydraulic ead around a well, producing a cone of depression. Pumping tests: Measuring drawdown at know discarge yields ydraulic conductiity K Drawdown in Different Aquifers Pumping Tests to Determine K Drawdown in unconfined aquifer Drawdown in confined aquifer Pumping wit constant rate Q: Measuring (r) yields K Confined aquifer wit constant tickness m: d Q = A q = AK dr d Q Q Q 1 = = = dr AK 2πrmK 2πmKr Q r () = lnr () + c 2πmK Q r 1 K = ln 2πm( 1 2) r Pumping Tests to Determine K Pumping wit constant rate Q: Measuring (r) yields K Unconfined aquifer wit ariable water table (r): d Q Q Q 1 1 = = = dr AK 2πrK πk 2 r d d 2 Q 1 2 = = dr dr π K r () () r = ln r + c πk 2 Q Q r 1 ln r 2 K = π 2 2 ( 1 2) Water Storage in Aquifers Storatiity S S describes te capacity of an aquifer to release groundwater from storage in response to a decline in ydraulic ead. Te water olume produced from an area A in response to a ead decline of d is gien by: dvw = S A d S = dvw ( A d) Unconfined aquifer: Large storatiity by cange of water table Specific yield: S = S n (drainable porosity) y eff Confined aquifer: Small elastic storage by deformation of medium Specific storage: S 1dVw Ss = = m V d tickness 1dρ 1dV wit compressibility γ = = dp V dp ρ dp= ρgd compressib. of matrix and water S =ρg( γ + nγ ) s m w

4 Equation of Groundwater Flow Flowlines around a Well Continuity Mass balance oer control olume: cange in storage equals sum of in and outflows oer boundaries plus external in/output J w. 1 V SS = diq + w Jw wit Ss = V Combination i wit idarcy q = K ( K ) = Ss J Special cases Stationary flow ( / = 0), no sources: + K omogenous and isotropic (unrealistic): w = 0 ( K ) Δ = 0 Bird's eye iew lines of constant ead flow lines from Fitts, Patterns of Groundwater Flow Hydrological Groundwater Balance Cross section lines of const. ead (equipotential lines) flow lines How muc water can be pumped witout depleting an aquifer? Natural State: R0 D0 ET = dv = 0 dt Wit pumping (extraction E) ( 0 0 irrigation ) R +Δ R +Δ R + GW dv ( D0 + ΔD) ( ET0 + ΔET ) E = dt from Mook, 2001 E =Δ R +ΔR dv ΔD ΔET dt 0 irrigation induced recarge reduced discarge capture depletion Aescbac Hertig, W. and Gleeson, T., submitted to Nature Geoscience Groundwater Sources Increased witdrawal from aquifers as arious "sources": Storage decrease Discarge decrease Recarge increase Example: Reduced Discarge by Eapotranspiration Closed basin in semi arid climate, constant recarge ia riers Extraction rate by pumping = recarge rate Wat is a sustainable use of groundwater? Is constant storage a sufficient condition? Alley et al., Science 296: Bredeoeft, Ground Water, 40, Oer time, discarge by ET goes to zero: Natural egetation dies

5 Groundwater Flow Equation: Head Diffusion = Ss J Flow equation ( K ) Assuming K to be isotropic and uniform, no sources: ( K ) = KΔ = Ss or =αδ t w K wit α= S Tis is a diffusion equation for te ydraulic ead, wit te ydraulic diffusiity α. α is a measure for te speed at wic a pressure pulse will propagate (diffuse) troug te system. α is large for low specific storage, i.e., low compressibility. Hydraulic response time: (for system of dimension L) 2 2 L SL s T* = = α K s Groundwater Flow Modeling Groundwater flow (and transport) models are standard tools to simulate and predict te ydraulics of groundwater systems. Numerically sole flow equations (Darcy + mass conseration) ondiscrete grid Need boundary conditions (flows/eads at model boundaries) Adjust conductiity and fluxes to reproduce eads Boundary and Initial Conditions Flow Pattern in 3 D Conditions needed to sole te flow equations Boundary conditions: Type 1 (Diriclet): Head (t) at te boundary gien special case "constant ead": = const. (e.g. rier, lake) Type 2 (Neumann): Flux (i.e. / r ) at te boundary gien special case "no flow": / r = 0 (impermeable boundary) Type 3 (Caucy): Combination of 1 and 2 Initial conditions: Initial distribution of ead 0 (x,y,z) Mattle et al., J. Hydrol. 242: Problems of Groundwater Flow Modeling Flow models are usually calibrated wit measured eads Bot water fluxes (recarge, boundary fluxes) and ydraulic conductiity (distribution) are unknown No unequiocal solution for tis problem! E.g.: 1 D, ead measured at 2 points: measured Δ 2 1 Δ q = Darcy : = Δx x x Δx K 2 1 unknown Infinite number of solutions, as only te ratio q/k is constrained Tracers can proide additional constraints: elocities, fluxes (q) 7.3 Transport Processes in Groundwater Adection (moement of solutes wit groundwater flow) F = qc = n c ad e Molecular Diffusion (stocastic moement of solutes) n Fdif = Dm c τ: Tortuosity τ Dispersion (spreading due to different flow pats and elocities) F = n D c dis e Dispersion is proportional to and anisotropic: larger parallel to flow (longitudinal, D L ) tan perpendicular to flow (transersal, D T ). D = α D =α α 10α L L T T L T

6 Origin of Dispersion Scale Dependence of Dispersion Mecanical dispersion Macrodispersion Dispersion increases wit te size of te flow domain! Retardation due to Adsorption Many substances tend to adsorb on aquifer matrix. Tis leads to a slow down of te transport (retardation), as in cromatograpy. Adsorption isoterm: Definition of c a : c c = k c a a d m ma m 1 n Vρ a a = = m ( ) Mass balance: ( ) tot d a d m a m indices: a = adsorbed d = dissoled m = matrix m = m + m = nvc + 1 n Vρ c = ( ) 1 n = nvcd + ( 1 n) Vρ mkacd = nvcd 1+ ρmka n Retardation factor: R = mtot md md R m Rm M Concentrations: tot d d ctot = = = nr = nrcd Vtot Vpore n Vpore Equations of Groundwater Transport Mass balance: Canges in mass density of solute in control olume V are due to fluxes across boundaries and sources/sinks σ ( nrc) n = dif +σ= n c + D c + n D c + J + J c τ tot e m e c w in For n = n e = const. and D m << D L, D T (molecular diffusion neglected): c 1 R = ( c) + ( D c) + ( J + c Jwcin) n Jw wit ( c) = c ( ) + c and c( ) = c n c 1 1 Jc J = w c + ( D c) + + ( c in c) R R nr nr Simplified Case of Transport in Groundwater Transport equation for a case witout retardation and sources Adection Dispersion equation: c c = ( c) + ( D c) Adection Dispersion Completely analogous to surface water, only wit dispersion instead of turbulent diffusion! Summary Groundwater is an important water resource Hydraulic ead (potential) is a central quantity Darcy flow = ydraulic conductiity ₓ gradient of ead Determination of K by pumping tests locally possible Storatiity: Large in unconfined, small in confined aquif. Flow equation from continuity & Darcy: Head diffusion Transport: Dispersion instead of turbulent diffusion Transport: Retardation, porosity, and sources play a role