Viscous deformation of unconsolidated reservoir sands—Part 2: Linear viscoelastic models

Abstract

Laboratory creep experiments show that dry unconsolidated reservoir sands follow a power law function of time (at constant stress), and cyclic loading tests (at quasi‐static frequencies of 10−6 to 10−2 Hz) show that the bulk modulus increases by a factor of two with increasing frequency while attenuation remains constant. In this paper, we attempt to model these observations using linear viscoelasticity theory by considering several simple phenomenological models. We investigated two classes of models: spring‐dashpot models, which are represented by exponential functions with a single relaxation time, and power law models. Although almost all of the models considered were capable of fitting the creep data with time, they result in very different predictions of attenuation and bulk modulus dispersion. We used the model parameters derived from fitting the creep strain to predict the bulk modulus dispersion and attenuation as a function of frequency, to find a single phenomenological model (and model parameters) that could explain the material's creep response with time as well as its dispersion and attenuation characteristics. Spring‐dashpot models, such as the Burgers and standard linear solid models, produce reasonable fits to the creep strain and bulk modulus dispersion data, but do not reproduce the attenuation data. We find that a combined power law–Maxwell creep model adequately fits all of the data. Extrapolating the power law–Maxwell creep model out to 30 years (to simulate the lifetime of a reservoir) predicts that the static bulk modulus is only 25% of the dynamic modulus. Including the instantaneous component of deformation into the previous prediction results in 2% total vertical strain at the wellbore, in good agreement with field observations.