Characterization of geologic heterogeneity at a geological carbon storage (GCS) system is crucial for cost-effective carbon injection planning and reliable carbon storage. With recent advances in computational power and sensor technology, large-scale fine-resolution simulations of multiphase flow and reactive transport processes have been available. However, traditional large-scale inversion approaches have limited utility for sites with complex subsurface structures such as faults and microfractures within the host rock matrix. In this work, we present a Bayesian inversion method with deep generative priors tailored for the computationally efficient and accurate characterization of GCS sites. Self-attention generative adversarial network (SAGAN) is used to learn the approximate subsurface property (e.g., permeability and porosity) distribution from discrete fracture network models as a prior and accelerated stochastic inversion is performed on the low-dimensional latent space in a Bayesian framework. Numerical examples with a synthetic fracture field with pressure and heat tracer data sets are presented to test the accuracy, speed, and uncertainty quantification capability of our proposed joint data inversion method.