We consider the standard first passage percolation model in the rescaled graph Z d /n for d ≥ 2, and a domain Ω of boundary Γ in R d. Let Γ 1 and Γ 2 be two disjoint open subsets of Γ, representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behaviour of the flow φ n through a discrete version Ω n of Ω between the corresponding discrete sets Γ 1 n and Γ 2 n. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of φ n /n d−1 below a certain constant are of surface order.