2019-ApproimateAnalyticalNumericalSolutionstothe：的近似analyticalnumerical解-文檔資料

1、Approximate Analytical/NumericalSolutions to the Groundwater TransportProblemCWR 6536Stochastic Subsurface Hydrology3-D Saturated Groundwater Transport Vj(xy z) spatial random velocity field c(x,y,z, t) spatiotemporal random concentration field No analytic solution exists to this problem 3-D Monte C

2、arlo very CPU intensive Look for approximate analytical/numerical solutions to the 1st and 2nd ensemble moments of the cone fieldSystem of Approximate Moment EqnsQ0雲(yún)口心“+丐口 gs a(_各叮心)+嚴(yán)后,3) a Useas best estimate of c(x,t) Use ac2(x,t)=Pcc(x,t;x,t) as measure of uncertainty Use Pcv(xt;x)and Pcc(x?t;xf

3、) to optimally estimate c or v based on field observationsSolution Techniques Fourier Transform Techniques (Gelhar et al) Finite Difference/Finite Element Techniques (Graham and McLaughlin) Greens Function or Impulse Response Techniques (Neuman et al, Cushman et al, Li and Graham)Fourier Transform T

4、echniques(Gelhar et al) Require an infinite domain Require coefficients in pdes for Pcvi and Pcc to be constant Require input covariance function to be stationary Convert pdes for covariance functions Pcvi and Pcc into algebraic expressions for Scvi and SgSpectral Solution for Steady-State Macrodisp

5、ersive Flux (Pcvi(x,x) 訥伙)-市/(和2 + 購(gòu)應(yīng)2)2 + &3)2” (£)+s(£)旦=0 k 1 dx；alk22 +% (S) = % (0) = I Sg (£)exp( ik 0)dF = f S” c Wk k 1Pg(0)幾陽(yáng)（k）Spectral Solution for Steady-StateMacrodispersive Flux (Pcvi(x,x) Therefore mean equation looks like Ajj determined from spectral relationship b

6、etween Svic and Syivj Svivj is the inverse Fourier transform of Pvivj determined from Pff? flow equation and Darcy's law.Results Assuming 3-D isotropic negative exponential for Pff, and oi/, oc/ «Z (Gelhar and Axness, 1981): Longitudinal macrodispersivity increases with variance and correla

7、tion scale of log conductivity Transverse macrodispersivity increase with variance of log conductivity, independent of correlation scale and depends on local dispersivity Fickian relationship emerges as a result of constant cone gradient assumptionSpectral Solution for Steady-StateConcentration Vari

8、ance*_2叩QSqC伙）一卩岡（i）+ oc2 + 陰)2)hM)+ScM)警"ISc“ (k)(k22 + k32)+ikiSv.v. (k)dxj dxj 1 J_22v ak +*/»dc0 dc0(k 22 + k32)+ikt比二Pg (x,x) = JJJSqC (R)exp(ik 0)d斤=仆)(k)dkOXj »Results Assuming hole-type isotropic negative exponential for Pff, and Ge” ocz «X (Vomvoris and Gelhar, 1990): Si

9、mpler negative exponential spectrum gives infinite concentration variance (caused by small wave number energy, i.e. high wave length variations at a scale large than plume scale) Concentration variance increases with increasing log hydraulic conductivity mean and variance, increasing mean concentration gradient, and decreasing local dispersivityNumerical Solution Solve coupled pdes using finite element or finite difference technique Does not require an infinite domain Does not require coefficients in pdes for Pcvi and Pcc to be constant Does not requi