Inverting for creep strain parameters of uncemented reservoir sands using arbitrary stress-strain data

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 stressstrain 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 stressor 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. In general, linear phenomenological (spring and dashpot) models fail to capture the behavior of viscoelastic materials [13]. However, previous studies on uncemented sands suggest that creep strain can be described using an analytical power law function [11,12]. Furthermore, Chang et al [19] found that creep strain data collected under hydrostatic, triaxial, and uniaxial strain boundary conditions could all be accurately modeled using the same function, implying that a full 3-D representation using tensors is not necessary. So, for the purpose of this study, we will assume that the deformation of uncemented sands can be described using 1-D, phenomenological, linear viscoelastic models. 3. INVERTING FOR CREEP FUNCTION PARAMETERS – THEORY In the context of this study, only two key aspects of linear viscoelasticity theory are needed: linearity, which requires the strain at a given time to be a linear function of stress, and superposition, which requires the strain response of a compound loading history to equal the sum of the strain responses to the individual components of that loading history. In combination, this means that the strain response to any time-series of stresses can be found if the impulse or step function response is known. Recall that creep strain accumulates as a function of time in response to a step increase in stress, ε(t) = σ0 J(t), where ε(t) is creep strain, σ0 is the stress magnitude of a Heaviside step function (H(t)), and J(t) is the creep compliance function. Once J(t) is known, the strain response to any arbitrary loading history can be found using superposition. Figure 1 shows a simple creep strain example composed of a positive step in stress followed by a larger negative step, such that the sum of the stresses is zero. The stress history can be written as the superposition of a step up followed by a step down, as shown in Figure 2, σ(t) = σ 0 Η(t) − Η(t − t1) [ ] (1) and the resulting strain follows from Boltzmann superposition theory, as shown in Figure 3, ε(t) = σ 0 J (t) − J(t − t1) [ ]. (2) To generalize the proceeding example to any arbitrary stress history, the single value of t1 in Equations 1 and 2 may simply be replaced with a variable, τ. The example shown in Figure 1 can now be written as follows: σ(t) = σ (τ ) Η(t − τ) − Η(t − τ + ∆τ) [ ]. (3) By once again exploiting the superposition principle, the associated strain history as a function of time is given by Equation 4, and shown in Figure 4, [ ]. ) ( ) ( ) ( ) ( τ τ τ τ σ ε ∆ + − − − = t J t J t (4) Figure 1: Creep strain and recovery response of a linear viscoelastic material caused by a boxcar stress function. The strain response can be constructed using Boltzmann superposition. Unmodified from [13]. Figure 2: Graphical representation of the boxcar stress function described by Equation 1. + Figure 3: Graphical representation of the linear viscoelastic strain response to the boxcar stress function shown in Figure 2, as described by Equation 2. Figure 4: Graphical representation of Equation 4, showing how an arbitrary stress history can be constructed using a series of boxcar functions. Taking the limit as ∆t goes to zero gives the strain response to an impulse function of stress: dε(t) = −σ(τ) dJ (t − τ ) dτ . (5) Any stress history can be broken up into a series of impulse functions, and the corresponding strain at time t can be found by summing or integrating the effects of each preceding stress pulse. Integrating Equation 5 by parts, to shift the derivative from the creep function to the stress function, results in the following relationship between stress and strain as a function of time: ε(t) = J(t − τ) dσ (τ ) dτ dτ, 0 t ∫ (6) where the only requirement is that the stress history needs to be a piecewise continuous and differentiable function of time. Common experimental protocols call for either step increases in stress followed by creep holds, or continuous loading at a constant rate. Using the Boltzmann superposition principle and the Equations from the previous paragraph, deriving the strain response to a constant stress rate is fairly straightforward. Assuming that the material is subjected to a constant stress rate (S) starting a time zero, and introducing T as an integration variable, results in the following relationship between stress rate and strain: ε(t) = J(t − τ) dσ (τ ) dτ dτ = 0 t ∫ J (t − τ )Sdτ = −S J(T)dT. 0 t ∫ 0 t ∫ (7) By applying Liebnitz’ rule [13], Equation 7 can be simplified to relate stress rate and strain rate: