  The structure of the effective potential $V$ describing causal geodesics near
the throat of an arbitrary spherical wormhole is analyzed. Einstein's equations
relative to a set of regular coordinates covering a vicinity of the throat
imply that any spherical wormhole can be constructed from solutions of an
effective initial value problem with the throat serving as an initial value
surface. The initial data involve matter variables, the area A(0) of the throat
and the gradient $\Lambda(0)$ of the red shift factor on the throat. Whenever
$\Lambda(0)=0$, the effective potential $V$ has a critical point on the throat.
Conditions upon the data are derived ensuring that the critical point is a
local minimum (resp. maximum). For particular families of Quasi-Schwarzschild
wormholes, $V$ exhibits a local minimum on the throat independently upon the
energy $E$ and angular momentum $L^2 $ of the test particles and thus such
wormholes admit stable circular timelike and null geodesics on the throat. For
families of Chaplygin wormholes, we show that such geodesics are unstable.
Based on a suitable power series representation of the metric, properties of
$V$ away from the throat are obtained that are useful for the analysis of
accretion disks and radiation processes near the throat of any spherical
wormhole.
