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Unit 10 : Advanced Hydrogeology

Unit 10 : Advanced Hydrogeology. Equations of Mass Transport. Mass Conservation. mass inflow rate - mass outflow rate = change in mass storage with time The change of a vector quantity (such as mass flux, J ) in a three-dimensional (x,y,z) system is:

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Unit 10 : Advanced Hydrogeology

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  1. Unit 10 : Advanced Hydrogeology Equations of Mass Transport

  2. Mass Conservation mass inflow rate - mass outflow rate = change in mass storage with time • The change of a vector quantity (such as mass flux, J) in a three-dimensional (x,y,z) system is: -(J/x + J/y + J/z) = S/t • The minus sign indicates that mass moves from high to low concentration so the flux decreases in the positive coordinate directions. • It gets tedious to write out all the partial derivatives so we use the shorter notation: -div(J) = S/t

  3. Diffusion Equation • The mass in storage in a unit volume of a porous medium is the concentration (C) times the porosity (n) so we can write: -div(J) = (Cn)/t • Suppose our transport process is diffusion, then the mass flux J is given by: J = -nDd*grad(C) where grad(C) is short for C/x + C/y + C/z • The diffusion equation is thus: div[nDd*grad(C)] = (Cn)/t or in more concise notation using the  operator: [nDd* (C)] = (Cn)/t

  4. Isotropic Case [nDd* (C)] = (Cn)/t • If n is neither a function of position (x,y,z) or time then: n. [Dd* (C)] = n.(C)/t [Dd* (C)] = (C)/t • Similarly if the diffusion coefficient is independent of position (isotropic) then: Dd*. [(C)] = (C)/t [(C)] = (1/ Dd*).(C)/t 2(C) = (1/ Dd*).(C)/t where the (del-squared) 2 operator stands for div[grad()]

  5. Laplace’s Equation • For steady-state conditions: 2(C) = 0 Mass inflow rate = Mass outflow rate • Concentrations are constant in time and there is no change in the mass stored with time.

  6. Advection Diffusion Equation • Now suppose our transport process is diffusion and advection, then the components of the mass flux vector J are given by: Jx = -nDd*C /x + nCvx Jy = -nDd*C /y + nCvy Jz = -nDd*C /z + nCvz • These equations are summarized by: J = -nDd* (C) + nCv • The advection diffusion equation is thus: [nDd* (C)] - (nCv) = (Cn)/t • Assuming isotropic n and Dd*: Dd*2(C) - (Cv) = C/t

  7. Simplified Advection Diffusion Equation • Expanding the advection term: Dd*2(C) - v(C) – C (v) = C/t • For transport in a steady state flow field (v)=0 because v=(q/n)(h) and 2(h)=0: Dd*2(C) - v(C) = C/t Notice that if v is zero we recover the diffusion equation. • The one-dimensional form of the advection-diffusion equation is: Dd*2C/x2 - vxC/x = C/t Diffusion Term Advection Term

  8. Dimensionless Form • We can define a dimensionless concentration as C+ = C/Ce where Ce is some reference concentration, the spatial variables x+,y+,z+ as x/Le etc where Le is some characteristic length and the time t+ as t/te where te is any characteristic time: • Now we rewrite the equation: (Dd*/ Le2)2(C+) –(v/Le)(C+) = (1/te)C+/t+ • Dividing through by the coefficient of 2(C+) : 2(C+) – [vLe/Dd*](C+) = [Le2/Dd* te]C+/t+ Peclet Number Fourier Number

  9. Peclet Number • The Peclet Number NPE = [vLe/Dd*] is a dimensionless expression corresponding to the ratio of mass transport by bulk fluid motion (advection) to mass transport by diffusion. • It is easy to see that a large Peclet Number is characteristic of an advective system and that as the advective velocity (v) approaches zero the system become diffusion dominated. • The characteristic length Le is arbitrary. Often a characteristic of the porous medium is chosen such as mean pore diameter. Sometimes a numerical grid spacing is selected.

  10. Fourier Number • The Fourier Number NF = [Le2/Dd* te] is a dimensionless quantity that can be used to characterize a diffusion length for a particular time. Le = (Dd* te/NF)1/2 Le  (Dd* te)1/2 • If Dd* = 10-9 m2/s and te = 1 y = 3.16 x 107 s then the diffusion length is about 0.18 m.

  11. Advection Dispersion Equation • Now suppose our transport process includes dispersion, diffusion and advection, then the components of the mass flux vector J are given by: Jx = -n(axvx + Dd*)C /x + nCvx Jy = -n(ayvy + Dd*)C /y + nCvy Jz = -n(azvz + Dd*)C /z + nCvz • These equations are summarized by: J = -n(D’ + Dd*)(C) + nCv • The advection diffusion equation is thus: [n(D’ + Dd*)(C)] - (nCv) = (Cn)/t • Writing D’ + Dd* as the hydrodynamic dispersion coefficient (D) and expanding (nCv) gives: D2(C) - v(C) - C(v) = C/t

  12. Simplified Advection Dispersion Equation D2(C) - v(C) – C (v) = C/t • For transport in a steady state flow field (v)=0 because v=(q/n)(h) and 2(h)=0: D2(C) - v(C) = C/t Notice that if v is zero we recover the diffusion equation because D’ = a.v = 0 also. • The one-dimensional form of the advection-dispersion equation is: D2C/x2 - vxC/x = C/t • This equation is the “workhorse” of modelling studies in groundwater contamination. Dispersion Term Advection Term

  13. Mass Transport with Reaction • When reactions take place, our mass balance statement changes from: mass inflow rate – mass outflow rate = change in mass storage with time • to: mass inflow rate – mass outflow rate + mass production rate – mass removal rate = change in mass storage with time • The one-dimensional form of the advection-dispersion equation is also changed to include a reaction term: Dx2C/x2 - vxC/x + r/n = C/t where r is the net mass produced per unit volume per unit time (moles/L3T).

  14. Reaction Rate Coefficient • An equation of the form: Dx2C/x2 - vxC/x + r/n = C/t is required for every species in the system. • The general form of the r-term is: r = (nC)/t = knC that is, r is the product of concentration, porosity and a reaction rate coefficient. • The equation this becomes: Dx2C/x2 - vxC/x + kC = C/t

  15. Reactive Transport Dimensionless Form • We can define a dimensionless concentration as C+ = C/Ce where Ce is some reference concentration, the spatial variables x,y,z as x/Le etc where Le is some characteristic length and the time at t/te where te is any characteristic time: • Now we rewrite the equation: (Dx/Le2)2C+/x+2 –(vx/Le)C+/x+ + kC+ = (1/te)C+/t+ • Dividing through by the coefficient of 2C+/x+2 : 2C+/x+2– [vxLe/Dx]C+/x++ [kLe2/Dx]C+ = [Le2/Dxte]C+/t+

  16. Dimensionless Numbers • The equation can be written: 2C+/x+2– NPEC+/x++ DaIIC+ = NFC+/t+ where NPE is the Pelcet number[vxLe/Dx] NF is the Fourier number[Le2/Dxte] DaII is the second Damköhler number [kLe2/Dx] • Another dimensionless Damköhler Number (DaI) can be defined by the equation: 2C+/x+2– NPEC+/x++ NPEDaIC+ = NFC+/t+ where DaI is the first Damköhler number [kLe/v]

  17. Damköhler Numbers • The first Damköhler number (DaI) and measures the tendency for reaction relative to the tendency for advective transport. • The second Damköhler number (DaII) and measures the tendency for reaction relative to the tendency for diffusive transport.

  18. First-Order Kinetic Reactions • A very simple kinetic reaction is first-order decay due to radioactive disintegration: r = (nC)/t = -lnC where l is the decay constant. • The advection-dispersion equation is: Dx2C/x2 - vxC/x - lC = C/t

  19. Sorption Reactions • For equilibrium sorption reactions: r = C*/t where C* is the concentration of the solute on the solid phase. • Defining S as the quantity of mass sorbed on the surface of a porous medium with a bulk density (rb) : C*/t = rbS/t • We can also write the equation using the solid density (rs): C*/t = rs(1-n)S/t • Adopting a linear isotherm: S/t = KdC/t

  20. Retardation Factor • Combing the linear isotherm with the definition of r gives: r = C*/t = Kd rs(1-n)C/t • The advection-dispersion equation becomes: Dx2C/x2 - vxC/x – [Kd rs(1-n)/n]C/t = C/t • Collecting the C/t terms: Dx2C/x2 - vxC/x = [1 + Kd rs(1-n)/n] C/t • Writing the bracketed term on the RHS as Rf : Dx2C/x2 - vxC/x = RfC/t • Rf is called the retardation factor.

  21. Retardation Equation • Dividing through by Rf gives: (Dx/Rf)2C/x2 – (vx/Rf)C/x = C/t • Writing Dx/Rf as DRx and vx/Rf as vRx : DRx2C/x2 – vRxC/x = C/t • This equation is called the retardation equation and it is identical to the advection-dispersion equation except for the coefficients of the spatial derivatives. • This means that the form of solutions for C will be identical for both retarded and non-retarded species.

  22. Solid-Solution Reactions • Reactions between solids and solutions involve many steps. • In general, the rate of reaction is controlled by the slowest step in the reaction path. • The reaction may be transport controlled or surface controlled depending on the relative speed of movement of the fluid past the solid surface.

  23. Assume a thin stationary layer fluid (thickness, s) exists between the solid surface and the moving fluid. Assume reactants move to and from the solid surface by diffusion across this layer in a direction normal to the layer C/n = (Ceq* - C)/s flowing fluid C C C C C stationary layer s Ceq* Ceq* Ceq* Ceq* Ceq* solid surface Transport Controlled Reaction Model

  24. Transport Controlled Flux • Surface mass flux J [mols L-3T-3] away from the surface is given by: J = DdS*(Ceq* - C)/s where S* is the surface area [L2] and Dd is the diffusion coefficient [L2T-1] for movement of reactants and products through the stationary layer. • Now define Dd/s as a mass transfer coefficient k with dimensions [LT-1]: J = kS*(Ceq* - C) • Let C* be the concentration of solute on the solid phase, the flux towards the surface is: C*/t = -nkS*(Ceq* - C) = nkS*(C – Ceq*)

  25. Consider a surface controlled reversible sorption-desorption reaction in a porous medium. Surface sorption depends on a constant k1 and the solution concentration C: S*k1C Surface desorption depends on a constant k2 and the surface concentration C*: S*k2C* solution C C C C C sorption k1 desorption k2 C* C* C* C* C* solid surface Surface Controlled Reaction Model

  26. Surface Controlled Flux • Surface mass flux J [mols L-3T-3] is given by: J = S*(k1C – k2C*) where S* is the surface area [L2]. • With C* as the concentration of solute on the solid phase: C*/t = nS*(k1C - k2C*) C*/t = nk1S*(C – (k2/k1)C*)

  27. Reversible Surface Reactions • Both diffusion-control and surface sorption-desorption derivations give similar equations: C*/t = nkS*(C – Ceq*) C*/t = nk1S*(C – (k2/k1)C*) • The equations can be simplified in terms of parameters and written as: C*/t = nK1C – nK2C* • The corresponding advection-dispersion reactive transport equation becomes: Dx2C/x2 - vxC/x – K1C + K2C* = C/t • Instead of a single linear equation in C we have a pair of coupled equations in C and C*. If we assume C*=Ceq*=constant, the equation is linear in C.

  28. Boundary and Initial Conditions • To solve any of the forms of the advection-dispersion equation we need more information. • This information take the form of boundary conditions: • what happens to concentrations on the boundaries of the region of interest? • Type I Fixed Concentration Boundary • Type II Fixed Flux Boundary • Type III Variable Flux Boundary • and initial conditions: • what are the starting concentrations within the region of interest?

  29. Consider a simple 1-D case where C(x,t) is to be determined. The initial condition is zero concentration everywhere, C(x,0)=0 The LHS boundary condition is constant concentration on the boundary after time to, C(0,t)=0, t<=to C(0,t)=Co, t>to The RHS boundary condition (at an infinite distance) is zero concentration for all times C(,t)=0 Type I Boundaries C(x,0)=0 C(0,t)=0, t<=to C(0,t)=Co, t>to C(,t)=0

  30. A constant flux boundary after time to is specified as: dC/dx(0,t)=J, t>to dC/dx(0,t)=0, t<=to A variable flux boundary after time t1 is specified as: D.dC/dx – v.C = J(t), t>t1 D.dC/dx – v.C = 0, t<=t1 Type II and Type III Boundaries C(x,0) = Co

  31. Column with Constant Source • The concentration distribution in a column is a useful solution of the one-dimensional advection-dispersion equation: Dx2C/x2 - vxC/x = C/t • The boundary conditions are C(0,t) = Co (concentration Co applied at x=0 for all times) • The initial conditions are C(x,0) = 0 (concentration zero everywhere)

  32. Ogata-Banks Solution • The solution is provided by Ogata and Banks (1961): C(x,t) = (Co/2).{ erfc[(x-vxt)/2(Dxt)1/2] + exp(vxx/Dx).erfc[(x+vxt)/2(Dxt)1/2] } where erfc(b) is called the complementary error function (1 – erf(b)). • The second term in the solution involving the exponential function is almost always small and can be neglected. • The simplified solution becomes: C(x,t) = (Co/2).erfc[(x-vxt)/2(Dxt)1/2]

  33. Breakthrough Curve • The (x-vxt) term in the solution identifies the position in terms of the advective front (when x-vxt = 0 and erfc(0)=1 so C(x,t)=0.5Co). • For x>>vxt the position is ahead of the front (x-vxt  and erfc()  0 so C(x,t)  0). • For x<<vxt the position is behind the front (x-vxt and erfc(-)  2 so C(x,t) C(x,t)  Co) x <<vxt x < vxt x > vxt x >> vxt 1.0 0.5 0.0 C/Co x x = vxt

  34. Retardation Solution • The simplified Ogata-Banks solution for retarded species becomes: C(x,t) = (Co/2).erfc[(x-{vx/Rf}t)/2({Dx/Rf}t)1/2] which simplifies to: C(x,t) = (Co/2).erfc[(Rfx-vxt)/2(RfDxt)1/2] • The effect of retardation for a constant continuous source is to delay breakthrough at position x from time tb to time Rf.tb 1.0 0.5 0.0 Retarded species Unretarded species C/Co x = vxt/Rf x = vxt

  35. Bear Solution • The one-dimensional form of the advection-dispersion equation for radioactive decay (and biodegradation) is: Dx2C/x2 - vxC/x - lC = C/t • Bear (1979) provides an analytical solution with initial and boundary conditions C(x,0)=0 and C(0,t)=Co: C(x,t) = (Co/2).{exp(vxx/Dx)[1 - b]}.erfc[(x-bvxt/2(Dxt)1/2] where b = (1+4lDx/vx2)1/2 • Note that for l=0, b=1 and the exponential term disappears and the Bear solution becomes Ogata-Banks solution.

  36. Radioactive Decay Solution • The dimensional group form 4lDx/vx2 = 4lax/vx controls b for finite values of l. The second form is readily recognized as the first Damköhler number (DaI) with ax as the characteristic length. b = (1+4lDx/vx2)1/2 • As the half-life of an element gets larger l = ln(2)/t1/2 gets smaller as does DaI. • For long half-lives where l is small, b is near 1, DaI becomes large and the behaviour is similar to the Ogata-Banks solution. The solute mass is transported faster than it decays. • For short half-lives (such as 3H) l is relatively large, DaI is small and with b>1 the negative exponential term reduces the concentration significantly. The solute mass decays faster than it is transported.

  37. Decay Term • When a decay term is included, the species is both attenuated by the mass removal process and somewhat retarded. • As the Damköhler number gets small transport dominates decay and the curve is similar to the Ogata-Banks solution. 1.0 0.5 0.0 Slowly decaying species Unretarded species C/Co Rapidly decaying species x = vxt

  38. Transverse Dispersion • For one-dimensional problems it is only necessary to specify the source concentration. • For multi-dimensional dispersion the source geometry must be specified. • Possible source geometries include: • point source • linear source • planar source

  39. Multi-dimensional Dispersion Plug Flow Transverse Dispersion Longitudinal Dispersion Longitudinal and Transverse Dispersion Constant Source Co Steady C < Co Dispersion Zone < vt Dispersion Zone > vt

  40. x Z Y/2 Y/2 Multi-Dimensional Solution • Domenico and Robbins (1985) provide an approximate analytical solution for a 3D-plume with a planar source YZ normal to the x-direction: C(x,y,z,t) = (Co/8).erfc [(x-vxt)/2(axvxt)1/2] .{ erf [(y+Y/2)/2(ayx)1/2] – erf [(y-Y/2)/ 2(ayx)1/2] } .{ erf [(z+Z)/2(azx)1/2] – erf [(z-Z)/ 2(azx)1/2] } • The solution is useful for screening level models where it provides an estimate of plume dimensions.

  41. Point Source Solutions • The solutions discussed thus far have dealt with continuous sources. • Baetsle (1969) provides a valuable solution for an instantaneous point source (=spill) with decay. • The original concentration and volume are Co and Vo respectively, l is the decay constant and Dx, Dy,and Dz are the respective dispersion coefficients. The average linear velocity in the x-direction is vx.

  42. Baetsle Solution • The Baetsle solution is: C(x,y,z,t) = [CoVo/8(pt)3/2(DxDyDz)1/2] .exp[ -(x-vxt)2/4Dxt – y2/4Dyt – z2/4Dzt – lt] • The maximum concentration after time t is given by: Cmax = CoVoexp(-lt)/[8(pt)3/2(DxDyDz)1/2] • If the source does not decay this reduces to: Cmax = CoVo/[8(pt)3/2(DxDyDz)1/2] • The dimensions of the cloud after time t are given by: 3sx = (2Dxt)1/2; 3sy = (2Dyt)1/2; 3sz = (2Dzt)1/2

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