This paper describes the development of a finite element-based computer model for the determination of stress fields and energy release rates around a delamination in a layered composite. The model is fully three-dimensional. Each layer of a composite material is modelled as homogeneous and orthotropic. The basic finite elements are isoparametric 20-noded bricks with collapsed, prismatic quarter-point elements around the delamination front. Contact theory is used to prevent interpenetration between the faces of a delamination. The modified crack closure method is used to calculate the energy release rates around a delamination. Preliminary experiments are performed to study the growth of a delamination between the layers of a two-layer laminate-one epoxy and the other graphite/epoxy. Initial results from the experiments and numerical simulations are shown to be consistent.
Summary While the effect of 2D proppant transport during hydraulic fracturing has been studied extensively and debated frequently, relatively little attention has been given to acid-fracturing treatments. In addition to fluid-density differences and gravity-driven segregation, spatial variation of the acid temperature and of the rock chemical properties results in more complicated physical phenomena for acid fracturing. Acid diffusion in the fracture has been estimated empirically with no considerations made to model correctly the acid flow across the fracture width. The etched-width profile resulting from 3D flow could differ significantly from that predicted by the 1D, piston-like displacement normally included in acid-fracture simulators. An important consequence of this variation in the etched pattern is a substantially different prediction of the fracture conductivity, and hence the post-stimulation hydrocarbon production. Acid flow in the fracture can affect the outcome of reservoir stimulation and must be considered when designing acid-fracturing treatments. The equations governing fluid flow are developed initially for a 2D pressure profile. These equations are coupled with the formulations for acid wall reaction, heat transfer, and diffusion within the framework of a P3D hydraulic fracturing simulator. The theoretical formulation for acid transport across the fracture width is presented then and coupled with the equations for 2D acid flow. The equation of acid diffusion is solved across the fracture width, hence solving the equations of fluid flow for three dimensions in the fracture. The resulting fracture geometry and conductivity are used subsequently with a reservoir simulator to illustrate the consequences of the 3D acid formulation on hydrocarbon production. Simulated and field examples are presented to illustrate the effects of 3D acid flow on the etched-width distribution and post-stimulation production. Introduction Hydraulic fracturing with acid is an alternative to propped fractures in acid-soluble formations such as dolomite and limestone. In such cases, fracture conductivity is obtained by etching the fracture faces instead of using proppant to prevent the fracture from closing. The acid treatment, however, is more complicated to design because of the difficulty in controlling both the fracture length and conductivity. The former is governed by the chemical reaction between the rock and the fracturing fluid, and the latter by etching patterns formed by the reacting acid in the formation. To design and evaluate such acid treatments, numerical simulators have been developed based on several physical mechanisms. The first mechanism is the acid reaction at the wall that creates the etched width and decreases the acid concentration at the wall. An acid concentration profile is created across the fracture width, with the acid diffusing from the center of the fracture to the fracture wall because of the concentration differences and leakoff velocity (second mechanism). Finally, because of fluid flow, the acid is transported along the fracture. The two limiting cases in modeling these mechanisms are the "diffusion-limited" case and the "reaction-limited" case. The diffusion-limited case occurs when the reaction is very fast, so that the rate of diffusion is limited by the rate at which acid can be transported to the surface and the acid concentration at the wall is apparently zero. The other case is obtained when the acid can be brought to the face as rapidly as it is consumed by reaction. This is called reaction-limited and implies that the wall concentration is approximately equal to the average acid concentration. Several simulators for treatment design and evaluation have been developed1–3 during the past decade based on acid-reaction models and 1D fluid flow. The basic assumptions of these models are the following:Fluid flow in the fracture is assumed to be plug flow. The fluid in the fracture has vertical fronts; i.e., the effects of the fracture width profile and leakoff variation along the fracture height are neglected.Acid diffusion from the center of the fracture to the fracture wall is calculated based on empirical formulations that assume a constant fracture width, infinite reaction rate, and no entrance effects. A new acid model is proposed to eliminate such limitations by solving the fluid-flow equation in three dimensions. This simulator has been developed based on a P3D hydraulic fracturing simulator that has been tested widely and validated.4,5 Two fundamentally different but intrinsically coupled principles are involved in developing a 3D acid fracturing simulator. The first is a 2D fluid transport along the fracture length and height. This formulation requires calculating the pressure gradients and velocities along the fracture length and height. The second concept involves a rigorous calculation for acid diffusion across the fracture width. This is achieved by superimposing a 3D finite difference mesh within the fracture width and tracking the acid concentrations at every cell across the fracture. Finally, the standard principles of acid reaction and fracture-height growth are coupled within this framework to develop the implicit 3D acid simulation. Various physical principles implemented in the development of the simulator that are central to acid fracturing are presented. Secondary considerations in hydraulic fracturing are briefly discussed in the Appendix. The equations describing the acid-reaction model are presented first, followed by the algorithm for 2D acid transport. The 3D formulation that solves for the acid concentration across the width is presented next. Results obtained by applying this simulator to different examples and field cases are given; simple benchmark problems are presented to validate these results against known solutions. Finally, field examples are shown to illustrate its application for real life situations.
Current pressure evaluation techniques that assess fracture behavior from calibration treatments are limited to predicting reservoir parameters that govern fracture growth. In addition, significant variability during the evaluation process can result in subjectivity and hence reduce the effectiveness of such tests. The recently proposed "after-closure" analysis attempts to overcome these limitations by assessing the reservoir- dominated pressure response, following mechanical closure of a hydraulic fracture. After closure linear flow provides an independent assessment of the fracture length induced during the calibration treatment. This prediction constrains the fracture length predicted, either by the conventional pressure decline analysis or a numerical fracture simulator, and hence provides an accuracy check for the pressure evaluation. In addition, the dependency of the pressure response during the linear flow regime on the closure time permits an additional independent assessment for fracture closure pressure. Consequently, the linear flow diagnostic reduces uncertainty in determining the fracture closure pressure that is commonly encountered during the evaluation process. Finally, the reservoir pressure and transmissibility are estimated from the radial flow behavior in a manner similar to conventional well testing. The proposed evaluation methodology combines this after-closure analysis with conventional calibration treatment analysis to predict a cross-validated ensemble of all parameters that govern proppant placement and post stimulation hydrocarbon production. This evaluation methodology thus provides all information required for an optimized design of fracture treatments. The methodology of this analysis, as well as comprehensive field examples illustrating its application and validity, are presented in this paper. P. 375
Summary The most comprehensive hydraulic-fracturing data including the first objective measurements of fracture height, length, and width are acquired from the Gas Research Inst. (GRI)/Dept. of Energy (DOE) Multiwell Site (M-Site) tests. In spite of the availability of extensive and reliable fracturing data, significant deviation between predicted and microseismic-determined fracture geometry was reported. The purpose of this study is to provide a consistent analysis of B-sand experiments by applying a systematic methodology for fracture-treatment evaluation. For this analysis, fracture parameters are estimated initially from laboratory data, well logs, and calibration tests. These parameters are refined by matching simulated pressures to field-measured fracturing pressures recorded during the first linear gel injection. These fracture parameters then are used to compare predicted- and measured-fracture pressures on all subsequent injections. Although general agreement for the fracturing pressures was obtained, a discrepancy was noticed between zone stresses estimated by evaluation and their variation as indicated on published stress logs. Stress data were reinterpreted and an acceptable pressure match was established. Fracture parameters resulting from this study are in agreement with independently inferred estimates. In addition, an apparent difference between closure pressure and microfrac stress is resolved. Finally, good agreement between predicted fracture geometry and microseismic readings is observed for each injection test considered in this study. This study shows that fracture pressures and geometry can be predicted consistently with good accuracy using elementary analysis techniques, without a reliance on ad hoc physical explanations. Background Over the past decade, a series of hydraulic-fracturing experiments, jointly conducted by the GRI and DOE at the M-Site, has provided the most comprehensive data available for hydraulic-fracture treatments. The initial objective of these experiments was to establish the character of gas production from lenticular, low-permeability formations common in the western United States.1 Through the course of the experiments, the focus has evolved toward developing methodologies to increase the accuracy for measurement of field-scale hydraulic fractures.2 The primary effort in this direction has been the successful use of subsurface triaxial accelerometers to locate microseismic events along the extent of a propagating hydraulic fracture. This objective measure of fracture dimensions and other supporting fracturing data provide critical constraint for evaluating fracture models and thus provide an excellent example for comprehensive fracture evaluation. In spite of the availability of such exhaustive and reliable fracturing data, widely used fracture simulators failed to explain comprehensively the observed fracture response for this important data set. This discrepancy for B-sand experiments was reported3 when using both cell-based3 and lumped fracture simulators.4 Although net pressures were matched for calibration treatments,3 disagreement was noticed between the simulated fracture geometry and the geometry outlined by microseismic measurements. Disparity in fracture geometry was particularly pronounced on propped treatment for which not even a satisfactory net pressure match was achieved.3 An undesirable feature of this lenticular formation is the complex geological environment that is prone to inefficient hydraulic fracturing. Nolte discussed a comprehensive list of factors responsible for abnormal fracture behavior.5 A majority of these characteristics are applicable to the in-situ conditions at the M-Site, leading to its classification as the "worst-case scenario."5 Indeed, evaluation of prior tests at this site using both fracture simulations6 and far-field core samples7 refer to these complexities to explain differences between expected and observed behavior. The effect of such complexities can be assessed using a factor, Fc, defined5 asEquation 1 where ?pw=net pressure at the wellbore and seV=effective vertical stress. A low value of Fc is desirable for successful fracture placement. Applying Eq. 1 to the B-sand tests predicted an Fc value of 0.48. A reservoir pressure of 1,950 psi inferred on prior GRI experiments8 was used and an overburden stress based on a gradient of 1.07 psi/ft at 4,530 ft was assumed. The value of ?pw is based on bottomhole fracturing pressure of 4,800 psi recorded at the end of injection during propped treatment and closure pressure of 3,500 psi (estimated later in this paper). Although still higher than the suggested threshold value5 of Fc˜0.35 for the beginning of complex behavior in homogeneous reservoirs, this estimate is significantly lower than an Fc value of 1.02 encountered during fracture tests in the lower Paludal interval at this site.5 A lower Fc value suggests the likelihood of less complicated behavior that is amenable to routine fracture evaluation. This paper presents an evaluation study of the B-sand fracture experiments. Although these tests sought to address numerous other issues, such as proppant encapsulation and fracture conductivity, this analysis focuses on comparing simulator-predicted fracturing pressures and geometry with field measurements. Table 1 lists six injection tests performed during B-sand experiments. However, fluid was flowed back at the end of the first three KCl injections. Because shut-in pressures are essential for assessing fracture behavior, only pumping pressures are considered during these injections. Comprehensive evaluation, however, was performed for linear gel tests and propped treatment listed in Table 1. The fracture simulator used to evaluate these experiments is described in Appendix A. The first linear gel injection in the B-sand is analyzed initially using a systematic evaluation methodology to determine fracture parameters. Fracturing pressures predicted using these parameter estimates then are compared with field measurements on subsequent injections. Simulated geometries are compared with microseismic measurements in each case. The paper finally reconciles discrepancies between independent assessments of these experiments and results arrived at in this study. Primary results are provided in the main body of the paper and additional details are given in the appendices.
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