Let s(k) be the smallest s such that if there exists a k-nowhere zero flow in a regular matroid M, then M can be covered by circuits, the total length of which is at most sM. A recursive formula for the evaluation of s(k) is given: s(kt) ≤ (s(k)(kt − t) + s(t)(kt − k))/(kt − 1). By means of this formula s(k) is found for k = 2, 3, 4, 6, 7, 8. “Natural” proofs for graph theoretical results are obtained.