​We derive computable error estimates for finite element approximations of elliptic partial differential equations (PDE) with rough stochastic conductivities. These estimates have applications to subsurface flow problems in geophysics where the conductivities are assumed to have  lognormal distributions with low regularity. The standard techniques for a posteriori error estimation fail to produce reliable estimates in this setting as they inadequately capture high frequency content present in the solution. The novel estimates presented here are for means of observables of the Galerkin and quadrature errors committed in piecewise linear finite element approximations. Moreover, the estimates, based on local error indicators, can be computed at a relatively low cost and are a first step towards developing adaptive Multilevel Monte Carlo (MLMC) methods for this class of problems. Our theory is supported by numerical experiments on test problems in one and two dimensions.​