This talk investigates the upscaling methods to the following
parabolic equation
$$
\partial_t c+\nabla\cdot(\mathbf{u}
c)-\nabla\cdot(\mathbf{D}\nabla c)=f(x,t)
$$
which stems from the application of solute transport in porous
media. Because of the high oscillating permeability of the porous
media, the Darcy velocity $\mathbf{u}$ hence the dispersion tensor
$\mathbf{D}$ has many scales with high contrast. Thus, how to
calculate the macro-scale equivalent coefficients of the above
equation becomes the target of this talk. Two kinds of upscaling
formulations are discussed in this work. The two different
equivalent coefficients computing formulations are based on the
solutions of two different cell (local) problems, which one utilizes
the elliptic operator with terms of all orders while the other only
uses the second order term. Error estimates between the equivalent
coefficients and the homogenized coefficients are given under the
assumption that the oscillating coefficients are periodic (which is
not required by our methods). Numerical experiments are carried out
for the periodic coefficients to demonstrate the accuracy of the
proposed methods. Moreover, we apply the two upscaling methods to
solve the solute transport in a porous medium with a random
log-normal relative permeability. The results show the efficiency
and accuracy of the proposed methods.