  For $k = 1, 2,...,n-1$ let $V_k = V(\lambda_k)$ be the Weyl module for the
special orthogonal group $G = \mathrm{SO}(2n+1,\F)$ with respect to the $k$-th
fundamental dominant weight $\lambda_k$ of the root system of type $B_n$ and
put $V_n = V(2\lambda_n)$. It is well known that all of these modules are
irreducible when $\mathrm{char}(\F) \neq 2$ while when $\mathrm{char}(\F) = 2$
they admit many proper submodules. In this paper, assuming that
$\mathrm{char}(\F) = 2$, we prove that $V_k$ admits a chain of submodules $V_k
= M_k \supset M_{k-1}\supset ... \supset M_1\supset M_0 \supset M_{-1} = 0$
where $M_i \cong V_i$ for $1,..., k-1$ and $M_0$ is the trivial 1-dimensional
module. We also show that for $i = 1, 2,..., k$ the quotient $M_i/M_{i-2}$ is
isomorphic to the so called $i$-th Grassmann module for $G$. Resting on this
fact we can give a geometric description of $M_{i-1}/M_{i-2}$ as a submodule of
the $i$-th Grassmann module. When $\F$ is perfect $G\cong \mathrm{Sp}(2n,\F)$
and $M_i/M_{i-1}$ is isomorphic to the Weyl module for $\mathrm{Sp}(2n,\F)$
relative to the $i$-th fundamental dominant weight of the root system of type
$C_n$. All irreducible sections of the latter modules are known. Thus, when
$\F$ is perfect, all irreducible sections of $V_k$ are known as well.
