Viscoplastic Deformation In Unconsolidated Reservoir Sands (Part 1): Laboratory Observations And Time-Dependent End Cap Models

Laboratory studies of deforming unconsolidated reservoir sands from the Wilmington Field, CA and the Gulf of Mexico indicate that a significant portion of the deformation is both time-dependent and permanent. Furthermore, a threshold viscous compaction pressure has been identified in these sands, marking the transition from elastic to viscoplastic behavior, and which in general can be approximated by the maximum in situ effective pressure experienced by the sand at depth. Because the viscous component of deformation is significant, a standard elastic-plastic end cap model is not sufficient, and a model that includes viscoplasticity must be used. An appropriate model for unconsolidated sands can be developed by incorporating Perzyna viscoplasticity theory into the modified Cambridge clay cap model. Perzyna viscoplasticity theory simply states that pressure (and the location of the end cap) should follow a power law function of strain rate when a material is deforming viscoplastically. Hydrostatic compression tests were conducted at volumetric strain rates of 10, 10, and 10 per second in order to find values for the required model parameters, namely the threshold viscous compaction pressure as a function of strain rate. As a result, by using an end cap model and Perzyna viscoplasticity theory, changes in porosity in both the elastic and viscoplastic regimes can be predicted as a function of both stress path and strain rate. models provide a means for mapping out the elasticplastic transition in materials under variety of boundary conditions including shearing, uniaxial stress, and hydrostatic compression. Finally, end cap models have been used to successfully describe a wide range of materials, from unconsolidated soils and clays to cemented sandstones [7]. However, end cap models can be simplistic when used to describe certain materials. For example, materials which fail to deform according to the assumptions of associated plasticity (in associated plasticity the direction of maximum strain is parallel to the direction of maximum stress) cannot be accurately described. Also, since instantaneous time-independent deformation is assumed, end cap models fail to describe timedependent materials. Because unconsolidated reservoir sands and shales exhibit significant timedependent deformation [8,9,10,11], existing end cap models must be modified if they are to be used to describe the constitutive behavior of such materials. This has been done previously, by Adachi and Oka for example [12], who succeeded in describing laboratory data on clays by expanding the original Cambridge clay model to include rate-dependent viscoplastic deformation. Given that end cap models are generally successful in predicting changes in porosity associated with plastic deformation, but breakdown when applied to materials such as unconsolidated sands that exhibit time-dependent deformation, it seems logical to try to expand end cap models such that porosity changes can be accurately predicted even in time-dependent materials. Since porosity changes are directly linked to changes in fluid flow properties in reservoirs, and production associated porosity loss in unconsolidated sand reservoirs can be large (10-15% at Wilmington), accurate prediction of porosity change is crucial to reservoir management. In this paper we expand the modified Cambridge clay model used by Chan and Zoback [2] to include time-dependent deformation by incorporating Perzyna viscoplasticity theory. When assuming that the Cam clay parameter ‘M’ is constant, the new time-dependent model predicts that the end cap expands with increasing strain-rate. Specifically, the elastic-viscoplastic transition pressure is predicted to follow a power law function of volumetric strain-rate. Laboratory tests conducted under constant strain-rate boundary conditions on unconsolidated reservoir sands will be used to verify the model and constrain the necessary parameters. 2. THEORY PART 1–MODIFIED CAMBRIDGE CLAY MODEL The modified Cambridge clay model can be thought of as a simplified case in the general class of “end cap” models based on the concept of critical state. Critical state, or critical void-ratio, of an unconsolidated material is defined as a deformed state in which the individual particles are arranged such that no volume change takes place during shearing [13,14]. For example, when a loose soil sample is sheared, it deforms until finally reaching a state of collapse, at which point (in terms of stress) the sample is said to have reached the plastic yield surface. If loaded beyond this initial yield surface, plastic deformation continues to occur until the soil reaches the critical void ratio (or state), at which point the void ratio remains constant, and all additional deformation occurs as shear. In contrast, when a dense (or cemented) soil or rock sample is sheared, stress increases to some peak value, then decreases to some residual level, at which point it remains basically constant with continuing shear. Initially the dense material compacts, until the peak stress is reached, and then it dilates until the volumetric strain remains constant (at the residual stress), at which point the material is said to have reached the critical state, and shear can continue indefinitely without further volume changes. While end cap models are firmly mathematically grounded in plasticity theory, they were initially developed to describe laboratory observations of yielding soils. Hvorslev [15] observed that the constant void ratio contours on the triaxial plane (σ1 vs. √2σ3 = √2σ2) were identical during both undrained and drained conditions. From this observation, Roscoe and the Cambridge group proposed models for the yielding of soils and clays based on the balance of internal and external work. For the original Cam clay model, it was assumed that only shear strain contributed to dissipated work, and therefore the only component of strain that contributed to the plastic deformation. However, for the modified Cam clay model, both the volumetric and shear strains contribute to the total plastic strain and dissipate work. Other assumptions built into the Cam clay model include the shape of the yield surface, associated plastic flow, time-independent deformation, and constant values for model parameters (independent of pressure and strain) [4,14,16]. In terms of a conventional axisymmetric triaxial (σ2 = σ 3) laboratory apparatus, the work done on a right cylindrical sample is,