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7. Mathematical Models of Groundwater Quality
parameters into the equation. Using first-order approximation, we can also
derive the following expression for the covariance matrix of solution
(Wagner and Gorelick, 1987):
C~ ~ [J]C .. [JY,
(7.4.16)
where C .. is defined by Eq. (7.4.14) and [J] = [o;op] is the Jacobian.
Equations (7.4.15) and (7.4.16) based on the first-order approximation are
simple but too rough.
Tang and Pinder (1979) used the perturbation method to study the uncertainty of the solutions of advection-dispersion equations. A one-dimensional
hydrodynamic dispersion equation can be written in the form of a linear
operator:
LC=j,
(7.4.17)
where operator
L=~+ V~-~(D~),
ot
ox ox ox
(7.4.18)
and hydrodynamic dispersion coefficient D = IX V. Let D, V, and IX be represented by
D = jj + D',
V= V+ V',
IX = Ci + IX',
(7.4.19a)
(7.4.19b)
(7.4.19c)
where the overbar indicates the expected values and the prime indicates the
perturbations from the expected values. By inserting these definitions into
Eq. (7.4.18), operator L will be separated into two parts:
where
o - 0 0 (- 0)
L o = ot + V ox - OX D OX '
L1 = V'~-~(D'~). OX OX OX
Equation (7.4.17) can then be expressed as
LoC + L 1 C =f
(7.4.20)
(7.4.21)
(7.4.22)
(7.4.23)
When using FDM or FEM to discretize this equation, we will use the same
notation as in the continuous form for convenience. After discretization, we
regard L o as the coefficient matrix which depends on V, Ci, Ax, and At; L 1 as
the coefficient matrix varying with V', IX', Ax and At; f as a vector containing
initial and boundary conditions; and C as the unknown solution vector.
Then, Eq. (7.4.23) can also express a set of stochastic algebraic equations.