Let $p:E\to B$ be a fibration. Suppose $B$ has a simplicial decomposition. For each $n\in\mathbb{Z}_{\ge0}$, let $C_n$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\sigma:\Delta^n\to E$ is a singular simplex, $\tau:\Delta^n\to B$ is a simplicial simplex, and $\tau=p\circ\sigma$. There is a boundary operator $\partial:C_n\to C_{n-1}$ defined in the usual way, making use of the faces of $\Delta^n$. Clearly $\partial\circ\partial=0$, and $(C_*,\partial)$ is a chain complex.

Update: If $B$ is a finite-dimensional simplicial complex, then for $n>\dim B$, $C_n$ is defined inductively as follows. For $n=\dim B+1$, let $C_{\dim B+1}$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\sigma:\Delta^n\to E$ is a singular simplex, $\tau:\Delta^n\to B$ is a singular simplex such that $\tau=p\circ\sigma$ and $\partial(\sigma,\tau)\in C_{\dim B}$, where $\partial(\sigma,\tau)$ is defined in the usual way. Define $C_n$ similarly for all larger $n$.

My question is: Is the homology of $(C_*,\partial)$ isomorphic to the singular homology of $E$?