Stochastic Analysis of Groundwater Flow Processes

CWR 6536Stochastic Subsurface Hydrology

Methods for deriving moments for groundwater flow processes

• Exact analytic solutions– possible only if analytical solution to governing equation

available. Not very realistic• Monte Carlo simulations

– Delhomme, 1979– Smith and Freeze,1979

• Approximate Analytical/Numerical Solutions– Gelhar, 1993; Hoeksema and Kitanidis, 1984;Dagan,

1989; McLaughlin and Wood,1988; James and Graham 1998.

3-D saturated groundwater flow

• K=K(x,y,z;) random hydraulic conductivity field (assumed geologically isotropic)• (x,y,z;) random head field • would like to derive pdfs/moments of random head

field given pdfs/moments of random hydraulic conductivity field

dzK

zdyK

ydxK

x0

Monte Carlo Simulation• 1-D Flow Problem Smith and Freeze, 1979a

• Domain discretized in x-direction• K(x) generated for each block in x-direction for multiple

realizations (200)• (x) solved for each realization • Statistics of (x) calculated at each x over 200 realizations

0)( ;)0(

)(0

LLdxxK

x

1-D Monte Carlo Simulation Results• Mean head field uniform• Head variance increases with Ln K variance • Head variance increases with Ln K correlation scale• Head variance increases with ratio of Ln K correlation

scale to length of domain• Normality of head field only approached in interior of

domain and interval of normality decreases with head variance

• Flux through system random due to finite domain and selection of BCs

2-D Monte Carlo Simulation Results• Uniform flow field

– same trends as for 1-D case– anisotropy of Ln K field affects head variance– deterministic layering and proximinty to

boundaries affects head variance– 2-D head variance reduced from 1-D head

variance by ~ 50%– region of normality larger for 2-D than for 1-D

2-D Monte Carlo Simulation Results• Non-uniform flow field

– head variance increases with mean head gradient– head variance is greater than for uniform flow

case– Uncertainty of model predictions dpeends on

both hydraulic conductivity and flow configuration (governed by dimensionality and BCs)

Concept of Effective Hydraulic Conductivity

• Uniform Keff reproduces the ensemble mean head when inserted into the deterministic model everywhere in domain

• The Darcy flux calculated using the uniform Keff in the deterministic model reproduces the ensemble mean Darcy flux

• For 2-D steady unidirectional flow in an unbounded domain, effective conductivity is the geometric mean

• For bounded non-uniform gradient fields effective conductivity not easily defined

)(x

L

xx

Kq lhG

and i.e.