Alternatives to the Power-Law Fluid Model for Crosslinked Fluids

Summary Measuring the rheological properties of crosslinked fracturing fluids is difficult but important. Fluid properties play a key role in the determination of the final geometry of the created fracture and in the distribution of proppant within the fracture; therefore, an accurate knowledge of these parameters is necessary for optimum treatment design. The first paper1 in this series described a method to measure accurately and reproducibly the rheological properties of crosslinked fracturing fluids. The technique is the first that applies long-accepted mathematical methods to correct the measurements for the deviations in shear rate caused by the non-Newtonian nature of the fluids. This, in turn, allows the rigorous examination of mathematical fluid models to determine which, if any, best describes the flow properties of the fluids. Introduction The problems of characterizing crosslinked fracturing fluids were outlined in the first paper1 in this series. These problems made the application of accepted mathematical techniques to correct measurements for deviations caused by the non-Newtonian character of these fluids difficult to justify. As a result, not making corrections has often led to the wrong choice of fluid models when the mathematical description of the fluid flow is attempted. The technique1 that was used to gather data for this study has been described previously. Dynamic mechanical testing provides a quantity - called the complex viscosity (µ*) - that has been shown by Cox and Merz2 to equal the apparent viscosity (µa) determined in steady-shear measurements. Yasuela et al.3 recently confirmed this relationship with a wide variety of instruments. Use of this relationship, coupled with the increased sensitivity and reproducibility of the mechanical spectrometer, allows an examination of the data analysis techniques currently used in the industry. The API4 currently specifies that the data gathered on fracturing fluids be reported as n' and k', which have been derived from apparent Newtonian shear rates. This promotes consistency in the presentation of data but can lead to the misinterpretation of the results of an experiment. When necessary, model-independent shear-rate conversions were applied before analysis to all the input data in this study to avoid misinterpretation of the results. Background: Analysis of Laboratory Rheology Data The procedure for determining fluid-flow characteristics from laboratory data may be expressed generally as occurring in three distinct, but not independent, steps:data acquisition,analysis and data reduction, andscale-up with the fundamental equations of fluid mechanics or some generalized method, such as that of Metzner and Reed,5 that is based on those relationships. Only the first and second steps are discussed here; a complete discussion of the third step is beyond the scope of this study. Data Acquisition Data for scale-up are normally acquired in the laboratory with capillary-, tube- and extrusional-type rheometers or parallel-plate, cone-and-plate, and concentric-cylinder rotational-type rheometers. When crosslinked gels are measured, each measurement technique suffers from the effects of the viscoelastic nature1 of the gels. Slip at the wall in capillary- and tube-type rheometers makes data obtained with this type of measurement difficult to reproduce. Slip at the wall and the Weissenberg effect complicate the interpretation of data derived from the steady-shear mode of rotational-type viscometers. The method of dynamic testing1 avoids many of those problems and provides reproducible data for the next step in the scale-up process. Analysis and Data Reduction The first step in the data analysis process is the conversion of the experimental measurements - i.e., pressure drop and pump rate or torque and angular velocity - into estimates of shear stress and shear rate. Three methods of conversion can be used:equivalent (apparent) Newtonian shear rate or viscosity,model-dependent conversions, andmodel-independent conversions. Method 1 is specified by API as the method of reporting fluid data. The shear rate, computed as if the fluid were a Newtonian liquid, is used to estimate parameters for non-Newtonian fluid models. It can be shown that this technique is adequate for certain two-parameter models, provided that restrictions are applied to the range of scale-up shear rates and that the rheological parameters are used without modification in generalized methods of scale-up. This method is inadequate, however, if the object of the experiment is both fluid-model optimization and fluid-flow scale-up. The assumptions inherent to this technique will introduce a bias toward three-parameter models that will be carried through the scale-up process, if not isolated and minimized during error determination. Data Acquisition Data for scale-up are normally acquired in the laboratory with capillary-, tube- and extrusional-type rheometers or parallel-plate, cone-and-plate, and concentric-cylinder rotational-type rheometers. When crosslinked gels are measured, each measurement technique suffers from the effects of the viscoelastic nature1 of the gels. Slip at the wall in capillary- and tube-type rheometers makes data obtained with this type of measurement difficult to reproduce. Slip at the wall and the Weissenberg effect complicate the interpretation of data derived from the steady-shear mode of rotational-type viscometers. The method of dynamic testing1 avoids many of those problems and provides reproducible data for the next step in the scale-up process. Analysis and Data Reduction The first step in the data analysis process is the conversion of the experimental measurements - i.e., pressure drop and pump rate or torque and angular velocity - into estimates of shear stress and shear rate. Three methods of conversion can be used:equivalent (apparent) Newtonian shear rate or viscosity,model-dependent conversions, andmodel-independent conversions. Method 1 is specified by API as the method of reporting fluid data. The shear rate, computed as if the fluid were a Newtonian liquid, is used to estimate parameters for non-Newtonian fluid models. It can be shown that this technique is adequate for certain two-parameter models, provided that restrictions are applied to the range of scale-up shear rates and that the rheological parameters are used without modification in generalized methods of scale-up. This method is inadequate, however, if the object of the experiment is both fluid-model optimization and fluid-flow scale-up. The assumptions inherent to this technique will introduce a bias toward three-parameter models that will be carried through the scale-up process, if not isolated and minimized during error determination.