ARCHIVED REPORTS_XR0011649 - Laserfiche WebLink

Home Browse Search

The model is formuLited as a boundary value y = C(x=0, z=0, y) ' problem that approximates the spreading of a i Z , X contaminant plume in the vertical and horizontal directions pependictilarso_1he prevailing flow Co erf [ 2(DTY/Vy)" ] erf [ 4(DTy/`�y ) ] (19) r path In essence, we have a continuous source where erf W(—W) = --erf (W) has been utilized contaminated parcel moving at a steady one-dimen- In this model„Vy corresponds to V� of sional velocity subject to transverse spreading pro- equation (1), Co is the initial maximum concentra- tresses Afathemaucally, this may be described b the dispersion-convection equation y tion as measured in the vicinity of the solid waste i boundary�Z is the vertical extent of the measure- 32C a2CaC ment zone where the maximum concentration has ' DT{ a x2 + a z2 ) — VY a Y _ 0 (17) been determined in the L icinity of the solid waste boundary,lX is the lateral extent of the plume at where C(x,z,y),is the concentration�of the con- the solid waste boundary, simply taken as the 6 taminant as a function of position, Tt is the Iength of the solid waste facility contributing ' transverse dispersion coefficientry)represents a contaminants to the ground-water flow (see figure Spatial coordinate colinear with the velocity of the 4)CY)is the distance from the solid waste boundary contaminant VY, and x and z represent the hori- measurement to the alternate boundary, and ert s ' I zontal and vertical spatial coordinates perpendicular an error function, which is well tabulated and is to the flowi The problem is thus viewed as a two- presented in Figure 5 The first part of the right- dimensional semi-infinite medium bounded at the hand side of equation (19) (the part involing Z) l top, z = 0 by a zero flux boundary, aC/az = 0 at is for vertical spreading whereas the second part z = 0 Physically this represents z = 0 as the top (the part involving X) is for horizontal spreading of the saturated zone in the aquifer The boundary Consider-the following example Measure- ' condition (y = 0) is determined through measure- men_ts at a site indicate that maximum concentra- iments at the proposed waste boundary as tions extend over a zone that is about 3 meters described previously For simplicity we conserva- (about 10 feet) thick This is Z The waste tively assign the maximum concentration of the boundary contributing to the contaminant flow contaminant, Co, over a specified region of the (X) is about 30 m long (about 100 feet) The plume as measured at the solid waste boundary parameter ratio DT/Vc is about one meter (3 fret) Thus at y = 0 and an alternate boundary is taken as 150 m 0<z<Z (about 500 feet) Equation (19) becomes C. for (shaded region 3 30 ' —X/2<x<X/2 in Figure 4) Cy = Co erf [ 2(1 x elf [150)'= ] -}(1 ,( 150}'1 ] C(x,z,y=0) or 0 otherwise Cy = Co erf (0 12) x erf (0 62) The solution to equation (17) for boundary condi- tions above is C(x,z,y) = Top Of Saturated ' co �crf z + Z z - Z x (O,O) Zone [ ] — erf4 2(DT Y/Vy)i'� 2(DTy/Vy)7” Z i + {erf [ x + X/2 ] _ erf [ x X/2 ] __X ( 2(DT Y/Vy)`I' 2(DTY/Vy)" (18) This is a two-dimensional version of a well known solution presented by Morgenau and Murphy {� } (1956, p 238) The maximum concentration E occurs at the point x = 0, z = 0 The concentration Z I at this point from equarion (18) is Fig 4 Contaminant conditions at waste boundary 309