Creep strain experiments on uncemented reservoir sands suggest that the time-dependent component of deformation can be modeled using linear viscoelasticity theory. The standard approach to solving for the values of the appropriate model parameters is to fit creep strain data as a function of time. However, by writing the creep compliance function in terms of strain-rate rather than strain, it is possible to solve for the values of the model parameters using arbitrary time-histories of stress-strain data. Rewriting the creep compliance function as the conjugate stress relaxation function allows constant loading-rate or step-hold loading data to be used to constrain the model. Complex loading histories can be divided into branches of approximately constant stress- or strain-rate and solved piecewise. After deriving the necessary equations, we show that the method successfully reproduces the known creep compliance function of an example uncemented reservoir sand.
Producing oil and gas from uncemented sand reservoirs is often accompanied by a host of interesting challenges. As these formations can be weak and prone to sand-production [e.g. 1], unique approaches to drilling and completions may be required to minimize the risk of excessively damaging wells. Uncemented sand reservoirs are also associated with significant time-dependent compaction, as observed in numerous fields around the world using 4-D seismic monitoring [2-5], and documented in the laboratory [6-8]. Reservoir compaction can be beneficial if it provides pressure support by mechanically squeezing reservoir fluids as porosity is being lost (so-called compaction drive). On the other hand, the compaction-induced loss of porosity can also cause a dramatic loss in formation permeability [9, 10]. Given the potential complications associated with production in uncemented sands, a model capable of predicting the compaction associated with changes in reservoir pressure is desirable. In previous work, we showed that linear viscoelasticy theory could be used to model the time-dependent compaction of uncemented sand reservoirs from California and the Gulf of Mexico [11]. In the laboratory, time-dependent deformation is most often observed and characterized by conducting creep-strain experiments [e.g. 12]. However, creep-strain data are not routinely collected or reported in the literature, so it would be useful to have a means for obtaining creep function parameters from whatever data is available. In this paper we construct a relatively simple method for deriving creep function parameters from arbitrary data sets using linear viscoelasticity as a framework.
The mathematical framework for linear viscoelasticity theory has been extensively written about in the mechanical and biomedical engineering literature. Lakes [13] provides an excellent overview of both theoretical and experimental methods; an exhaustive treatment of linear viscoelastic constitutive models can be found in Tschoegl [14]. A robust and generic inverse method to solve for creep function parameters using dynamic frequency-domain measurements was introduced by Baumgaertel and Winter [15]. Their methodology has since been extended to take into account time-domain data [16], simplified experimental procedures [17], and different stress paths [18]. Park et al [18] derive a 3-D viscoelastic constitutive law for asphalt that includes the effects of damage.
