Storing and recovering water, carbon and heat from geologic reservoirs is central to managing resources in a changing climate. We tested the hypothesis that the strain tensor caused by injecting or producing fluids can be measured at shallow depths and interpreted to advance understanding of underlying aquifers or reservoirs. Geodetic-grade strainmeters were deployed at 30m depth overlying the Bartlesville Formation, a 500-m-deep sandstone near Tulsa, OK. The strainmeters are 220m east of injection well 9A completed in a permeable lens at the base of the Bartlesville Formation. Water was injected into well 9A at approximately 1.0 L/s during four tests that ranged in duration from a few hours to a few weeks. The horizontal strain increased (tension) and the circumferential strain was a few times larger than the radial strain. The vertical strain decreased (compression) during injection. Strain rates were approximately 100 n/day during the first few hours, but the rates decreased and were approximately 10 n/day during most of the tests. Four independent methods of poroelastic simulation and inversion predict reservoir properties and geometries that are similar to each other and consistent with independent information about the reservoir. All strain interpretations predict that a boundary to the permeable lens occurs beneath the vicinity of the AVN strainmeters, which is consistent with core data from the site. The boundary of the permeable lens is located by matching the radial and circumferential strains, which demonstrates the value of measuring the strain tensor.