  In this paper, we study the properties of weak approximation with
Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction
for surfaces with respect to field extensions of number fields. We assume a
conjecture of M. Stoll. For any nontrivial extension of number fields $L/K,$ we
construct two kinds of smooth, projective, and geometrically connected surfaces
defined over $K.$ For the surface of the first kind, it has a $K$-rational
point, and satisfies weak approximation with Brauer-Manin obstruction off
$\infty_K,$ while its base change by $L$ does not so off $\infty_L.$ For the
surface of the second kind, it is a counterexample to the Hasse principle
explained by the Brauer-Manin obstruction, while the failure of the Hasse
principle of its base change by $L$ cannot be so. We illustrate these
constructions with explicit unconditional examples.
