Solute mixing in porous media results from the interplay between molecular diffusion and the deformation of fluid parcels, which elongates solute elements and steepens concentration gradients in the pore space. In three-dimensional porous media, fluid deformation is asymptotically governed by exponential stretching, arising from the chaotic stretching and folding of fluid elements. However, early time deformation may be dominated by shear, induced by no-slip boundary conditions at grain surfaces, leading to linear elongation. Yet, the persistence and relative contributions of linear shear and exponential stretching to fluid deformation and solute mixing remain poorly understood. Here, we address this question using numerical simulations of fluid deformation and mixing in body-centered cubic bead packs, where the rate of chaotic stretching can be varied with the direction of flow, while maintaining the same average shear rate. We quantify the mean deformation dynamics of elementary surfaces as a function of their initial orientation with respect to streamlines, and their consequences on mixing through Lagrangian methods. We show that surface elements with certain initial orientations relative to the streamline direction experience a transient phase of shear-dominated deformation at early times, leading to persistently larger deformation compared to those with initial orientations that yield purely exponential growth. By expressing the deformation components in streamline coordinates, we derive approximate analytical expressions linking the different components of fluid deformation to shear, helicity and chaotic stretching. We discuss consequences for the mixing of solute sheets and blobs, highlighting generic behaviors as well as differences between these two representations.