Summary In this work, we examine the behavior of pressure transient data for single and multiphase flow in heterogeneous reservoirs. In order to illustrate multiphase flow behavior in these systems, we focus on heterogeneous gas condensate reservoirs, however, we also consider other multiphase flow problems. It is well known that in some instances, e. g., water injection/falloff in homogeneous reservoirs, pressure transient data from buildup (or falloff) tests cannot be obtained by superposition of drawdown (injection) pressure responses. In fact, drawdown and buildup reflect properties in different regions of the reservoir. This behavior is common to most occurrences of multiphase reservoir flow, and is exaggerated in the presence of radial heterogeneity. This theoretical work describes the information contained in transient pressure derivative data, and explains the fundamental difference in behavior between multiphase drawdown and buildup pressure transient data in radially heterogeneous reservoirs. We show that whereas multiphase buildup data may be treated like single-phase buildup data, drawdown data is most indicative of properties in that region of the reservoir where mobility is changing most rapidly with time. Theory Isothermal multiphase flow in radially heterogeneous reservoirs is described by a system of partial differential equations that describe conservation of mass of oil, gas and water components in black oil systems and conservation of individual chemical species in compositional systems. When the mass conservation equations are satisfied, Darcy's law, which describes pressure losses in each flowing phase is also satisfied. Thus, for Black Oil systems, we can write at any radial location, (1) where, t is the total mobility, i.e., (2) and, C1 is a constant given by, (3) In writing Eq. (1), we have assumed negligible capillary pressure effects, and that all phase rates are in reservoir bbl/day. Equation (1) obviously applies equally to bounded or to infinite-acting reservoirs. If we consider infinite-acting reservoirs, we can separate variables in Eq. (1) and integrate over radius to obtain (4) Carrying out the indicated integration, noting that (5) we obtain the following equation for the wellbore pressure drop:
Summary This work considers the analysis of pressure data, both drawdown and buildup, obtained at a well producing a reservoir in which the absolute permeability varies with position. A new inverse-solution algorithm is presented that can be applied to estimate the reservoir permeability distribution as a function of distance from the well. Introduction The emergence of reservoir characterization has stimulated efforts to obtain improved information on reservoir heterogeneities. This work considers single-phase flow to a well in a reservoir where permeability varies with distance from the well. We consider methods for estimating the permeability distribution from well-test pressure data. The methods considered were obtained by modifying and extending elegant seminal works of Oliver and Yeh and Agarwal. Oliver used a perturbation theory technique to obtain the wellbore pressure drawdown solution at a single well in an infinite-acting reservoir where absolute permeability varies with position. His solution assumes 2D flow in an (r,) coordinate system and that permeability is a function of r and i.e., k = k (r,). As presented, his solution assumes that permeability varies slightly about a reference, base, or "average" value, kref. In Ref. 3, Oliver used the Backus-Gilbert method to approximate the permeability distribution under the assumption that a reference permeability value can be determined from a semilog plot of pressure vs. time. He applied the method to a three-zone, composite, infinite-acting reservoir where the permeability in the inner and outer zones is k = 2,000 md and in the middle zone is k = 1,500 md. In Refs. 1 and 3, Oliver considers only the analysis of pressure-drawdown data. In this work, we remove Oliver's restrictions and consider the analysis of both drawdown and buildup data obtained at a well with an arbitrary variation in absolute permeability in the radial direction. Most importantly, we derive an inverse-solution algorithm to estimate this permeability distribution directly from well-test pressure data. Unlike Oliver's application of the Backus-Gilbert procedure, our inverse-solution algorithm does not assume that we can compute a reference or base permeability value from a semilog plot of pressure vs. time. In fact, we show that the base permeability value controls only the shifting of the time scale used to evaluate the kernel weighting function in Oliver's solution. Our inverse-solution algorithm, which is recursive but stable, can be applied for large variations in permeabilities and in cases where pressure data exhibit no semilog straight lines. Rosa and Horne examined the same problem as Oliver. While they noted that the pressure response for a multirate test was more sensitive to reservoir heterogeneities, like Oliver, they concluded that the inverse problem (i.e., the determination of permeability distributions) does not have a unique solution. Ref. 5 indicated that, for a multicomposite reservoir, the permeability distribution could be determined (by nonlinear regression analysis) only if the inner and outer radii of each zone were known. Kamal et al. also used a multicomposite model consisting of a few zones to analyze data from a damaged well. They matched pressure-buildup data with the model using nonlinear regression analysis to determine estimates of permeability in each zone.
Significant mud losses often occur while drilling in fractured formations. Severe problems arise when drilling fluids invade high permeability conductive fractures that the well path intercepts. The industry has recently started to monitor mud loses in order to identify the fractures and characterize them. Methodologies are available to characterize the geometry and conductivity of the fractures by quantitative analysis of field measurements. These methodologies are mainly based on mathematical models that describe the physical phenomenon and the mechanism under which the flow within the fractures takes place. Once the models are provided one can look for causes of the problem and methods to minimize it.This paper presents a new model for mud loss of non-Newtonian drilling fluids into naturally fractured formations. Flow of Yield-Power-Law (Herschel-Bulkley) fluids has been coupled with Newtonian reservoir fluid in a single fracture. The governing equations are derived based on principles of conservation of mass and linear momentum for drilling fluid and pressure diffusion for reservoir fluid. Results are obtained based on semi-numerical solutions and plotted in terms of mud loss volumes versus time. The results demonstrate how rheology of the drilling fluid and formation fluid properties can influence mud losses. The relative contributions of both drilling and reservoir fluids is determined and compared.This model allows for predicting fluid losses for a given drilling fluid, formation fluid and operational conditions. Conversely, one can evaluate hydraulic aperture of the fractures by continuously monitoring mud losses and finding the best fit of field measurements of mud loss to the model. Field data are used to demonstrate the practical application of the proposed technique.The proposed model is valuable for drilling operations because it can help in minimizing the loss of expensive drilling fluids through optimization of drilling fluid rheological properties and selecting appropriate lost circulation materials. The model also benefits production operations by minimizing formation damage. In addition, well completion schemes can be optimized based on the improved knowledge of the near-wellbore fracture characteristics.
Summary This work considers the use of Duhamel's principle to analyze pressure drawdown and buildup data when both bottomhole pressures (BHP's) pressure drawdown and buildup data when both bottomhole pressures (BHP's) and sandface flow rates are available. This analysis procedure uses Duhamel's principle to convert pressure data obtained when the sandface rate is variable to the equivalent pressure data that would be obtained for a constant sandface rate of production. The equivalent constant-rate pressure data then can be analyzed with standard procedures (semilog analysis or type-curve matching). Introduction Duhamel's principle was introduced to the petroleum engineering literature in the classic paper of van Everdingen and Hurst. They used Duhamel's principle to obtain the dimensionless wellbore pressure-drop solution for a continuously varying flow rate. Their solution is presented in terms of a convolution integral. The well-known Odeh-Jones method and the more recent methods of Soliman and Stewart et al. can be derived directly from the van Everdingen and Hurst solution by appropriate numerical integration procedures. The methods of Refs. 2, 3, and 4 are restricted because their theoretical basis rests on the assumption that the dimensionless pressure drop term that appears in the convolution integral is given by the semi-log equation or the exponential integral--i.e., the methods assume that if production were at a constant sandface rate, the dimensionless wellbore pressure drop would be given by the sum of the line source solution and the skin factor. Thus, at least theoretically, their methods do not apply to fractured-well problems or to heterogeneous problems such as naturally fractured reservoirs. The methods we consider are general and are not restricted to plane radial flow problems. The methods of Refs. 2, 3, and 4 rely on redefining the time scale. The methods used in this work rely on Duhamel's principle to convert variable-sandface-rate pressure data to the equivalent pressure data that would have pressure data to the equivalent pressure data that would have been obtained if production had been at a constant sandface rate. Our methods are more general than those presented by others but the basic idea is not novel and presented by others but the basic idea is not novel and has been considered previously in Refs. 5 through 8. In fact, our Method 1 (discussed in Appendix A) includes as a special case the method of Kucuk and Ayestaran, and our Method 2 (also discussed in Appendix A) includes as a special case the method of Bostic et al. To the best of our knowledge, Method 3 (discussed in Appendix B) has not been considered previously in any form. Method 3 has advantages over the other two methods when the late-time pressure data are influenced by boundary effects. For example, Method 3 is preferable to other methods when reservoir limit testing is conducted under variable-flow-rate conditions. McEdwards has also presented an analysis procedure based on theoretical concepts similar to those used in Refs. 1 through 8 and in this work. However, his procedure is radically different because he used the convolution integral to calculate the theoretical pressure drops that should occur according to the observed flow rate history. This calculation requires an initial estimate of reservoir properties. The estimates of reservoir properties are then refined by an iterative procedure that minimizes the sum of the squares of the relative differences between the observed and calculated pressure changes. As in Refs. 2 through 4, McEdwards states that the pressure-drop term that appears in the convolution integral is given by the line-source solution. The derivations of Methods 1 and 2 and some numerical analysis considerations pertinent to their application have been presented previously. Brief details on Method 3 are given in Appendix B. Further theoretical details on all three methods can be found in Ref. This paper (1) illustrates the applicability of our techniques to a wide variety of problems, (2) presents a detailed and careful comparison between our analysis techniques and the rate-normalized procedure, (3) presents a new procedure (Method 3) and illustrates its presents a new procedure (Method 3) and illustrates its applicability to reservoir limit testing, and (4) discusses how to apply our method to actual data (Appendices A and B) and considers the analysis of a field test. In this work, we generally use well-known analytical solutions for the theoretical problem considered when these analytical solutions are available. The analytical solutions are available in Laplace space and are inverted with the Stehfest algorithm. For unfractured wells producing layered reservoirs, we generate the necessary producing layered reservoirs, we generate the necessary solutions using a finite-difference model that has been discussed elsewhere. For fractured wells producing commingled layered reservoirs, we generate the pertinent solutions using a finite-difference model that is discussed in Refs. 18 and 19. SPEFE P. 453
Summary Significant fluid loss while drilling through fractured formations is a major problem for drilling operations. From field experience, we know that the type and rheological parameters of the drilling fluid have a strong impact upon the rate and volume of losses. A mathematical model for Herschel-Bulkley [yield-power-law (YPL)] drilling-fluid losses in naturally fractured formations is presented. As a result, the effect of rheological properties of drilling fluid such as yield stress and shear-thinning/-thickening effect (flow-behavior index) on mud losses in fractured formations is investigated. We found that the yield stress can control the ultimate volume of losses while the shear-thinning effect can tremendously decrease the rate of losses. Therefore, mud losses in fractures can be minimized by optimizing the rheology of the drilling fluid properly. The model also allows for quantitative analysis of losses that take into account fluid rheology to characterize the fractures. Hydraulic aperture of conductive fractures can be obtained by continuously monitoring mud losses and fitting field records of mud losses to the model. The proposed model is very useful not only for drilling applications but also for well-completion design and fractured-reservoir-characterization purposes. To examine the validity of the model, a practical application of the proposed technique is demonstrated through a field example of mud-loss measurements in a fractured well in the Gulf of Mexico.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.