A smooth k-web is a family of k distinct smooth foliations whose leaves intersect generically. Objects of this kind have been thoroughly studied in the setting of algebraic and symplectic geometry. During the talk we will focus on the less-traveled route: webs in the geometry of volume-preserving maps.These unimodular webs possess nontrivial local structure, which manifests itself in the existence of curvature - a symmetric 2-tensor covariant with respect to volume-preserving web-equivalences, named after its classical counterpart defined by Thomsen and Blaschke in the 1920s. This structure can also be described by means of a natural affine connection determined uniquely by the structure of a given web and a choice of the volume form on the ambient space. Our main goal is to construct this connection, and to relate its curvature to the curvature of the web in order to establish another criterion for its triviality.