Residual graph of a graph with bidirectional edges?
The residual graph is not a graph modeling method. It represents how we can change the flow on edges of a graph $G$ in order to increase the total flow when we compute the maximum flow.

The residual capacity used when you construct the residual graph is defined as 
$$
  c_f(u,v) =
  \begin{cases}
     c(u,v) - f(u,v)  &amp; \text{if $ (u,v) \in E$} \\
     f(v,u) &amp; \text{if $ (v,u) \in E$} \\
     0 &amp; \text{otherwise}
  \end{cases}\
$$

So, we cannot have both $(u,v)$ and $(v,u)$ in $E$ even though the antiparallel edges do not contradict the main network flow properties. 

Your graph contains antiparallel edges which you should get rid of before you run a maximum flow algorithm on that graph, e.g., Ford-Fulkerson algorithm. 

You could transform this graph into equivalent one with no antiparallel edges as following. You choose one antiparallel edge and "split" it into two edges. For example take $AC$ and introduce a new vertex $F$ and two new edges $AF$ and $FC$ with weights equal to $5$, i.e., $w(AF)=w(FC) = 5$. Similarly for every pair of antiparallel edges in the graph.