ABSTRACT:

Sand production onset and mass/rate predictions are difficult to validate in the field because the field sand production data are often unreliable. To overcome this difficulty recent research has concentrated in representing as realistically as possible the field conditions in laboratory experiments and focused on the effects of fluid saturation of the test specimens and the effects of stress anisotropy. The laboratory results were used to calibrate a semi-analytical model and a numerical nonlinear finite element model for developed for sand onset and sand mass/rate prediction studies. The correlations pertain to sand production functions that depend on the fluid saturation and stress anisotropy and account for the observed physical behavior in the laboratory. The models are compared for field case analyses including parametric studies of various completion and operational parameters. They show the importance of anisotropy and fluid saturation during the upscaling from laboratory data to the field.

1. INTRODUCTION

Sand management of sand producing field requires accurate predictions of sand onset and sand mass to optimize the well completion method and the drawdown and depletion during the life of the well. Research in sand production has addressed various issues in this topic by e.g., classifying sandstones according to their sand production potential, by looking at multiphase flow conditions and the effect of water breakthrough and gas flow in gas reservoirs.

Recent research has concentrated in representing as realistically as possible the field conditions in laboratory experiments and focused on the effects of fluid saturation of the test specimens and the effects of stress anisotropy. The laboratory results are used to calibrate both a semi analytical model for sand onset and sand mass prediction and a numerical model with empirical correlations based on field data. Both models are used in the industry for field case analyses and sand onset and mass predictions.

<p>Wormholes are an effective fluid conduit that dominate the flow path in karst aquifers and are artificially induced in geo-energy applications through acid injection. As acidic fluids infiltrate geologic formations, they react with the minerals in the formation. The reaction localizes and forms a dendritic dissolution pattern under certain conditions, known as the reaction infiltration instability. This instability is instigated by material heterogeneities in most computational models. However, studies have demonstrated that injection of water into a homogeneous plaster can initiate and grow wormholes. In this study, we show that material heterogeneities suppress the wormhole growth in carbonate rocks compared with a homogeneous counterpart. Wormholes were numerically simulated through injection of a strong acid (hydrochloric acid) under both homogeneous and heterogeneous permeability fields using a phase-field approach. The phase-field variable represents calcite dissolution in a diffused manner and is coupled with a reactive flow model assuming convective and diffusive acid transport in the liquid phase and significantly high surface reaction rate, which emulate typical high-rate matrix acidizing treatments in carbonate reservoirs. Heterogeneous permeability fields localize the flow in high-permeability domains and enhance the splitting and branching of wormholes. The length of the dominant wormholes can be suppressed as an increasing amount of acid infiltrates into the branched wormholes. Our findings indicate that material heterogeneities should not be treated as a trigger for wormholes in the numerical simulation but as one of the parameters to control their nucleation and growth. </p>

<p>Wellbore instability is one of the most serious drilling problems increasing well cost in well construction processes. It is widely known that many wellbore instability problems are reported in shale formations where water sensitive clay mineral exist. The problems become further complicated when the shale exhibits variation in strength properties along and across bedding planes. In this study, a coupled thermal-hydro-mechanical-chemical (THMC) model was developed for time-dependent anisotropic wellbore stability analysis considering chemical interactions between swelling shale and drilling fluids, thermal effects, and poro-elastoplastic stress-strain behaviors.</p><p>The THMC simulator developed in this work assumes that the shale formation behaves as an ion exchange membrane where swelling depends on chemical potential of drilling fluids invading from the wellbore to the pore spaces. The time-dependent chemical potential changes of water within the shale are evaluated using an analytical diffusion equation resulting in the evolution of swelling strain around the wellbore. On the other hand, the thermal and pressure diffusion equations are evaluated numerically by finite differences. The stress changes associated with thermal, hydro, and chemical effects are coupled to the 3D poroelastoplastic finite element model. The effects of bedding planes are also taken into account in the FEM model through the crack tensor method in which the normal and tangential stiffnesses of the bedding planes have stress dependency. The failure of the formation rock is judged based on the critical plastic strain limit.</p><p>The numerical analysis results indicate that the rock strength anisotropy induced by the existence of bedding planes is the most important factor influencing the stability of the wellbore among various THMC process parameters investigated in this work. The numerical results also reveal that an established theory to orient the wellbore in the direction of the minimum principal stress is not always a favorable option when the effect of the anisotropy of in-situ stresses and the distribution angle of bedding planes cancel out each other. Depending on both the distribution angle of bedding plane and ratio of the vertical to the horizontal stress, the trend of minimum mud pressure showed a great variation as predicted by the yield and failure criterion implemented in the model. Furthermore, the analysis results reveal that the distribution and evolution of plastic strains caused by the THMC processes have the time dependency, which can be controlled by the temperature and salinity of the drilling fluids.</p><p>The numerical wellbore stability analysis model considering shale swelling and bedding plane effects provides an effective tool for designing optimum well trajectories and determining safe mud weight windows for drilling complex shale formations. The time-dependent margins of safe mud weight window of drilling can be fine-tuned when the interaction among various parameters is fully considered as the THMC processes.</p>

Abstract

It has been reported that hydraulic fracturing treatments with smaller cluster spacing and larger fracturing fluids volumes yield better production performance in Permian Basin, Bakken, and Eagle Ford. Degradable diverting agents can play an important role as temporary plugging materials for multiple, tightly-spaced fracturing operations. However, applications of degradable diverting agents are often limited to moderate to high reservoir temperatures. In this study, a new degradable diverting agent is developed for use in low temperature reservoir applications.

Butane-diol vinyl alcohol co-polymer (BVOH) which has controllable water solubility is evaluated as diverting agents for hydraulic fracturing treatments. Using a high pressure-high temperature filtration apparatus, filtration properties of BVOH diverting agents are measured for various powder-to-pellet ratios under a range of temperature conditions. Filter media with 1 to 3 mm width slots, that simulate fracture openings, are used for the filtration test. The filtrate properties are evaluated based on spurt losses and filtration coefficients for quantitative evaluation. An analytical diverting agent model that considers swelling of the polymer in water is also developed for evaluating the filtration process of multimodal particles.

The experimental results presented in this work indicate that the degradable BVOH materials can be used as effective plugging agents for fracture-like narrow slits. Based on spurt losses and leakoff coefficients obtained under different powder-to-pellet ratios and temperature conditions, the performance of the diverting agents is quantitatively evaluated. The optimum powder-to-pellet ratio for BVOH materials are determined to be 80 to 20. The experimental results also reveal that the degree of BVOH crystallinity provides a dominant effect on the solubility of BVOH powder. The test results also indicate that the diverting agent plug properties started degrading under the temperature greater than 140°F as designed. The BVOH diverting agent developed in this work provides effective diversion effects under low to moderate temperature conditions (e.g., 80 to 100°F). The analytical plugging and bridging model developed in this work, which takes into account swelling properties of the polymer, show very good matches to the experimental results.

The degradable diverting agent developed for low temperature applications improve operational efficiency and economics of multistage hydraulic fracturing treatments in shallow reservoirs and operations where immediate fracturing fluid flowback is required. The plugging and bridging model with bimodal particle system developed in this study helps stimulation engineers select and optimize diverting agent material types, particle size distribution, and diverting agent concentration for various well, stimulation, and reservoir conditions.

© 2019, Society of Petroleum Engineers Induced seismicity caused by underground fluid injection occurs because of pore pressure changes that lead to stress changes in the reservoir and the surrounding formations. Despite that noticeable seismic events from fluid injection is very rare, proper assessment of possible seismic events is important. The objective of this study is to develop numerical models that simulate stress changes, fault slips, and ground floor movements induced by underground fluids injection. The numerical analysis process presented in this work consists of three steps. First, stress changes around the reservoir due to fluid injection are analyzed using a FEM-BEM (Finite Element Method - Boundary Element Method) coupled model. Secondly, the stability of faults located near the reservoir is evaluated using the displacement discontinuity method. Thirdly, elastic waves caused by the fault slip is simulated using a FEM model where seismic response on the surface are calculated. A field case study is also presented to demonstrate the applicability of the numerical model developed in this work. The numerical analysis results indicate that stress concentration occurs around a boundary between the basement and sandstone beneath the reservoir. This affects the stability of existing faults in this region. As a result, when the fault is slipped, seismicity may be triggered. It is assumed that the slip is caused by stress changes in the faulted region as well as a pore pressure change in the fault, which is caused by volumetric strain changes of the fluid in the fault. A field case study based on wastewater injection in the Southwestern region of the United States where injection induced seismicity events have been recently reported, is also performed in this work. In this case study, the variation of rock strength is considered one of important factors in induced seismicity events. The novelty of our model is the ability to quantitatively assess the risk of induced seismicity due to wastewater injection, which can be also applied to other applications such as CCS and underground gas storages. Moreover, conducting risk assessment by these numerical models can improve safety of underground fluid injection operations.

© 2019, International Petroleum Technology Conference Massive steam injection during SAGD operation may result in significant changes in pore pressure, temperature, stress and strain in the overlying caprock as well as the injected formations. These changes lead to containment breach of the caprock as reported in the steam release incident at the Joslyn Creek field in 2006. To avoid such a catastrophic event, the integrity of the caprock and risks of steam release must be properly evaluated during planning and operating SAGD wells. In this study, a thermo-poro-mechanical model is developed to evaluate the integrity of the caprock due to temperature and pressure changes observed during SAGD operations. A commercial reservoir simulator is used to calculate changes of pore pressure and temperature during steam injection. These results are used as a part of input data for the geomechanical model that considers poro-elasto-plastic stress-strain relations of the formations. The shear failure of the rocks is determined by the Drucker-Prager criterion while the tensile failure is judged by the tensile strength of the rocks, which are used to assess the integrity of the caprock. Our simulation results indicate that the temperature change can be extended deep into the overlying formations while the steam chamber is developed in the reservoir interval. Because the caprock is expected to have low permeability, these temperature changes lead to notable pore pressure changes in the caprock interval, which plays an important role in the stability of the caprock in the geomechancial analysis. The simulation results also suggest the importance of considering free surface, underburden, and sideburdens as well as assigning appropriate boundary conditions in the model. Using the model developed in this work, the Joslyn field case is investigated showing the existence of failure region in the caprock layer during the steam circulation phase. These findings may explain the mechanism of the caprock failure and the resultant steam release at the surface experienced in the field. It should be noted that the analysis results indicate, not only possible shear failure events but also a possibility of tensile failure developed in the caprock interval above the steam chamber. It is also found that the geological complexity including the existence of a mudstone layer between the reservoir and the caprock affects the likelihood of the steam release event. The caprock integrity analysis method presented in this work can help engineers evaluate risks of the containment breach during a planning phase of SAGD project. Also, using the simulation model developed in this work as a forward model, the integrity of the caprock and the development of steam chamber during SAGD operation can be monitored by surface displacement measurements by In-SAR or tiltmeters. These study results can enable effective and safe operation for future SAGD production.

Copyright 2018, Society of Petroleum Engineers. The legacy of conventional fields has resulted in many low permeability reservoirs deemed sub-commercial without an appropriate stimulation strategy. With low permeabilities and potentially heterogeneous reservoir characteristics, an optimal development approach would highly depend on their specific reservoir properties that may well require stimulation methods other than hydraulic fracturing. In this paper, we present a fully integrated characterization and modeling workflow applied to the Kita-Akita oil field in northern Japan, demonstrating the screening process for multiple completion and stimulation methods in a highly heterogeneous, low permeability sandstone reservoir. To select a best completion and stimulation candidate from multiple methods, we constructed an evaluation matrix including the maturity of technologies, applicability to our reservoir, productivity, and economics. Multi-branch type completions such as radial drilling and fishbone drilling, as well as hydraulic fracturing were simulated and subsequently compared based on their productivities. Especially for the radial drilling and the fishbone drilling, a 3D FEM model was built for their complex laterals, and the inflow performances were evaluated with homogenous reservoir properties, respectively. Besides, due to the highly heterogeneous nature of the reservoir, we built a full-physics subsurface model based on a pilot-hole data acquisition and legacy 2D seismic lines. The 3D model served as a canvas to assess reservoir flow and geomechanical behavior, calibrated with production history from past producing wells in the 1950's to 1970's. Based on these models, the best infill drilling location was selected and multiple well completion and stimulation practices were evaluated. Through the screening methodology, the multi-stage hydraulic fracturing was identified as the best suited from an instantaneous productivity perspective. Yet, even though hydraulic fracturing would enhance the accessibility into multiple distinctively isolated sandstones occurring in the deepwater slope channel setting, the treatment costs exceeded the economic threshold significantly in our case. Inflow performance evaluation based on the 3D FEM modeling illustrates multi-branch type completions such as radial drilling and fishbone drilling were identified with a good stimulation skin factor. As a result of 3D simulation study, multi-branch completion was revealed as a technical and economically viable stimulation option in the heterogeneously distributed sandstone reservoirs. The advent of recent completion and stimulation techniques now renders low permeability reservoirs with relatively large development potential. Even with the development challenges quite different from conventional reservoirs, the approach shown in this paper provides a helpful reference for the study and decision-making process when the legacy field needs an optimal stimulation strategy.

Copyright 2018, Society of Petroleum Engineers. Many applications in the petroleum industry require both an understanding of the porous flow of reservoir fluids and an understanding of reservoir stresses and displacements. Historically reservoir simulation has accounted for geomechanical effects by simple use of a rock compressibility. This assumption results in pore volume to change only with pore pressure. On the other hand, according to the poroelasticity theory, pore volume should change not only with pore pressure but also with confining stresses or volumetric strains induced by rock deformation. This difference in the governing equations poses a great challenge when coupling reservoir flow and geomechanics models. In this study, we develop a mathematical expression that relates the pore volume compressibility used in the porous flow equation to poroelasticity parameters defined in the geomechanics model. Secondly, in order to implement consistent pore volume changes between the reservoir flow and geomechanics models, we derive a pore volume correction term for the porous flow equation, which accounts for volumetric strain changes and rock matrix deformation. As demonstrated in the paper, the correction term can be easily implemented in sink/source terms (or "fictitious well" term), which are readily available for most commercial reservoir flow models. With this simple implementation, virtually any existing commercial reservoir simulation models can account for geomechanical effects via modular coupling techniques. In this work, we compare three different techniques for coupling reservoir flow and geomechanics. One technique uses an explicit algorithm to couple reservoir flow and displacements in which flow calculations are performed every time step followed by displacement calculations (i.e., One-way coupling method). A second technique uses an iteratively coupled algorithm in which flow calculations and displacement calculations are performed sequentially for the nonlinear iterations during each time step (i.e., Iterative partitioned coupling method). The third technique uses a fully coupled approach in which the program's linear solver must solve simultaneously for fluid-flow variables and displacement variables (i.e., Monolithic coupling method). Using Mandel's problem, example simulations are presented to highlight accuracy and computational efficiencies in these coupling techniques. To the best of the author's knowledge, this is the first paper to present a coupling technique to consider rigorous geomechanical effects in the porous flow equations. The coupling method proposed in this study can be applicable for virtually any existing reservoir and geomechanics simulation models. The proposed coupling techniques are easily extended to multiphase flow and poroelastoplastic problems. All problems in this paper are described in detail, so the results presented here may be used for comparison with other geomechanical / porous-flow simulators.

Copyright 2018, Society of Petroleum Engineers. Many casing failure incidents have been reported in oil and gas fields around the world. These casing failure events can occur not only within reservoirs but also in surrounding formations. Engineers must evaluate risks of casing failure when drilling and completing wells especially in highly compacting reservoirs. However, one of the challenges encountered during the evaluation of casing failure risks is that field-scale stress changes and displacements as a result of drilling wells and producing hydrocarbon from reservoirs must be properly taken into account for casing stability analysis. The objective of this study is to develop an efficient integration method for large-scale reservoir compaction and small-scale casing stability analyses for the evaluation of casing deformation and failure. The numerical model developed in this work is based on 3D elasto-plastic finite element method (FEM). Reservoir compaction and subsidence are analyzed using a large-scale FEM model considering details of geological settings while casing stability is analyzed separately by a small-scale FEM model. The two FEM models are integrated by interpolating displacements calculated by the large-scale model and assigning resultant displacements for boundaries of the small-scale casing stability analysis model. The validation of the proposed integration method is also presented in the paper. Our study results indicate that the integration method presented in this paper significantly improves computational efficiencies on an order of 5 times faster than the conventional simulation method that requires a large number of finite elements for reservoir, surrounding formations, cement, and casing. Also it is demonstrated that the integrated model can be applied to inclined wells completed in highly heterogeneous formations at sufficient accuracy. The field case study also indicates that the risk of casing deformation highly depends on its inclination and the position relative to the compacting formation. The small and large scale coupling method developed in this work helps engineers evaluate casing deformation and failure in various locations in reservoir and surrounding formations in an efficient manner and also develop safe and efficient drilling and completion programs to reduce risk of casing mechanical problems.

© Copyright 2018, Society of Petroleum Engineers. A comprehensive study on wormholing has been conducted to improve the understanding of matrix acidizing in carbonate reservoirs. This work is a continuation of previous work (Furui et al. 2012a, 2012b). An analysis of additional experimental results as well as field measurements is provided to reinforce and extend the wormhole penetration model and productivity benefits provided by Furui et al. (2012b). A series of small block tests and one large block test under geomechanical stresses has been conducted to characterize wormholing in outcrop chalk samples. In addition, field data including acid pumping data as well as post-stimulation pressure falloff data has been collected and analyzed to evaluate stimulation effectiveness. Pressure buildup data from stimulated wells has also been analyzed to evaluate the sustainability of the acid induced skin benefits. Production logging data has been used to investigate whether created wormhole networks have remained stable or have collapsed under production stresses. To statistically analyze the data more comprehensively, the new data was also compared to the data from other field data available in the literature. The following conclusions are drawn from an analysis of the laboratory data and field data: 1) a skin value of −4 is achievable in carbonate reservoirs by matrix acidizing, 2) the negative acid skin is relatively stable under production stresses, 3) the wormhole penetration model is proven to successfully simulate matrix acidizing processes in both laboratory scale and field scale work, 4) the small and large block laboratory tests re-confirmed wormholing efficiency which was discussed as a scale effect in previous studies, 5) an understanding of the possible range of wormhole penetration has allowed us to improve field acid treatments and reduce the risk of connecting to water. This comprehensive study includes acid linear core flooding tests, small block tests, large block tests and field measurements to thoroughly analyze acid wormholing in carbonate rock. The database can be very useful information for understanding, benchmarking and optimizing future completion/stimulation design.

Copyright 2017, Society of Petroleum Engineers. The objectives of this paper are: 1) to evaluate clean-up flow effects on High Rate Water Pack (HRWP) and Cased Hole Frac Pack (CHFP) completions and 2) to develop predictive models to quantify clean-up effects on well inflow performance in these types of sand control completions. Analysis methods include: data acquisition and well test design for reservoir property estimation, multirate test and non-Darcy skin factor analysis, and perforation tunnel permeability factor (Kpt/Kr) analysis to evaluate gravel pack flow performance. Eight zones from five gas wells have been investigated using this procedure. All pressure data were obtained by intelligent well systems' (IWS) permanent downhole pressure gauges located at reservoir depth. The study results show a two-fold improvement in the average perforation tunnel permeability after clean-up flow. Based upon the study findings, two empirical correlations that can predict postclean-up well inflow performance are developed. For planning purposes, Kpt/Kr improvement factors can be estimated based on formation permeability while for wells tested using short-term clean-up flows Kpt/Kr improvement factors and stabilized well flow potential can be calculated from Water/Gas Ratio (WGR) data collected during the clean-up flow period. These predictive models are expected to help engineers assess future well performance for high-rate water pack and frac-pack completions in high permeability reservoirs. A thorough review and comparative well performance analysis of a series of HRWP and CHFP wells has been conducted using high quality downhole pressure data. The perforation tunnel permeability (Kpt/Kr) database and correlations presented in this work can be used for predicting both initial and long-term well flow performance of gravel and frac-packed wells, which can be very useful information for optimizing future completion designs and field development plans.

© Copyright 2015, Society of Petroleum Engineers. It is common practice to complete long carbonate intervals with multi-stage stimulation treatments especially in horizontal wells. Each zone tends to be mechanically isolated using cement or openhole packers and then acid stimulated. Zonal isolation effectiveness is judged on the basis of job pressure response or post-job production logging. The recent introduction of intelligent well systems (IWS) with zone specific pressure and temperature gauges allows more effective review of zonal isolation during stimulation. This paper reviews zonal isolation results from a series of high-rate acid jobs conducted in wells equipped with zone-specific pressure and temperature gauges. 21 acid stimulation jobs from 13 different wells are reviewed to investigate the effectiveness of zonal isolation during and after treatment. Well examples presented in this paper cover several different completion types: cemented and un-cemented; intelligent well system, plug-and-perf completion and ball-activated sliding sleeve completion. The analysis revealed several different pressure and/or flow communication patterns. Field examples and analysis results presented in this work will help engineers design and optimize cemented and un-cemented wells requiring multi-stage stimulation in carbonate fields.

This paper describes the design, testing, installation, and performance of the first fully completed well by use of an intelligent inner completion inside an uncemented liner with openhole packers for zonal isolation. The well-design concept evolved from technical challenges associated with completing long cased-and-cemented laterals in the mature Ekofisk waterflood. The term fully completed implies full reservoir access along the pay length for production and high-rate matrix acid stimulation by use of limited entry for fluid diversion within well segments. The paper covers the development and qualification of custom openhole 75=8-in.-liner components that can handle high differential pressures and severe temperature fluctuations of 200°F; the marriage of this complex liner with a five-zone intelligent-completion system; and results from 1 year of system-integration testing. The paper also discusses the strategic placement of both mechanical openhole and inner-string packers based on caliper and drilling logs; challenges met and overcome during installation; and comprehensive downhole-gauge data that confirms the performance of each component before, during, and after the stimulation. The Ekofisk field waterflood began in 1987 and continues to date, exceeding expectations for improved oil recovery while mitigating reservoir compaction. As the waterflood matures, new wells are more often found partially water-swept. Limited infrastructure for lifting and handling the high water production has led to increased emphasis on isolating these water-swept intervals. Cased, cemented, and perforated completions have traditionally been used for this service. Effective placement of cement is challenging in horizontals 4,000-8,000 ft in length, where rotation of the liner is not possible and high effective-circulating densities limit rates during cementing. Wide variations in reservoir pore pressures, often in excess of 2,000-psi difference along the lateral, are typical of the Ekofisk chalk and compound the difficulties of cementing. As a result, a new method for zonal isolation has been developed to ensure the success of future infill-drilling campaigns. The design and careful planning that went into the fully completed openhole uncemented-liner strategy resulted in a successful field trial and has proved this solution to be an effective alternative to cemented reservoir liners in long horizontals where zonal isolation is critical. Use of the intelligent-well system (IWS) allowed offline acid stimulation without rig, coiled-tubing, or wireline intervention. What would have traditionally been a significant water producer, with three water-swept zones totaling nearly 2,000 ft across a 4,000-ft reservoir section, has turned out to be one of the best oil producers in the field, with nearly zero water cut. Production results show high productivity with highly negative acidized-completion skins. With large investments in intelligent completions to provide zone-specific inflow control and water shutoff, isolation outside the liner becomes much more important. Over recent years, the Ekofisk wells have illustrated the difficulty of achieving effective cement along lengthy reservoir targets. The openhole fully completed solution combining an accessorized uncemented liner with an inner intelligent-completion string will allow operators to push the limits in terms of lateral length while maintaining full control over producing and nonproducing zones. © 2014 Society of Petroleum Engineers.

The paper investigates the capability of a simplified Mohr-Coulomb (MC) analytical model to predict satisfactorily fully anisotropic loading experimental results on hole failure, as well as numerical predictions from several parametric analyses of field cases based on a 3D non-linear FEM borehole and perforation failure analysis model (Sand3D), which employs the critical plastic strain failure criterion coupled with a Drucker-Prager failure surface. The comparisons with the experimental data and the field simulations demonstrate that the MC model is insufficient for satisfactory predictions under anisotropic loadings. A Drucker-Prager type model is advanced and shows significantly improved analytical predictions under any loading condition. Two field case comparison examples show that the developed analytical sand production initiation model accounts for the effect of intermediate stress and gives good matches to the formation failure envelopes predicted by the numerical model. Copyright 2013 ARMA, American Rock Mechanics Association.

This paper describes the design, testing, installation, and performance of the first 'fully-completed' well using an intelligent inner completion inside an un-cemented liner with openhole packers for zonal isolation. The well design concept evolved from technical challenges associated with completing long cased and cemented laterals in the mature Ekofisk waterflood. The term 'fully-completed' implies full reservoir access along the pay length for production and high rate matrix acid stimulation using limited entry for fluid diversion within well segments. The paper covers the development and qualification of custom openhole 7 5/8 in. liner components that can handle high differential pressures and extreme temperature fluctuations, the marriage of this complex liner with a five zone intelligent completion system, and results from a year of system integration testing. The paper also discusses the strategic placement of both mechanical openhole and inner string packers based on caliper and drilling logs; challenges met and overcome during installation; and a remarkable collection of down-hole gauge data that confirms the performance of each component before, during, and after the stimulation. The Ekofisk field waterflood began in 1987 and continues to date, exceeding expectations for improved oil recovery while mitigating reservoir compaction. As the waterflood matures, new wells are more often found partially water-swept. Limited infrastructure for lifting and handling the high water production has led to increased emphasis on isolating these water-swept intervals. Cased, cemented and perforated completions have traditionally been used for this service. It has become increasingly difficult to execute a successful cement job in longer horizontals with 4,000 ft to 8,000 ft laterals where rotation of the liner is impossible and high effective circulating densities (ECDs) limit rates during cementing. Wide variations in reservoir pore pressures, often in excess of 2,000 psi difference along the lateral, are typical of the Ekofisk chalk and exacerbate the difficulties of cementing. As a result, a new method for zonal isolation has been developed to ensure the success of future infill drilling campaigns. The design and careful planning that went into the fully-completed openhole un-cemented liner strategy resulted in a successful field trial and has proven this solution to be an effective alternative to cemented reservoir liners in long horizontals where zonal isolation is critical. Use of the intelligent well system (IWS) allowed offline acid stimulation without rig, coiledtubing, or wireline intervention. What would have traditionally been a significant water producer, with three water-swept zones totaling nearly 2,000 ft across a 4,000 ft reservoir section, has turned out to be one of the best oil producers in the field with nearly zero water cut. Production results show high productivity with highly negative acidized completion skins. With large investments in intelligent completions to provide zone-specific inflow control and water shut-off, isolation outside the liner becomes much more important. Over recent years, the Ekofisk wells have illustrated the difficulty of achieving effective cement along lengthy reservoir targets. The openhole fully-completed solution combining an accessorized uncemented liner with an inner intelligent completion string will allow operators to push the limits in terms of lateral length while maintaining full control over producing and non-producing zones. Copyright 2013, Society of Petroleum Engineers.

Many casing- and screen-damage incidents have been reported in deepwater oil and gas fields in the Gulf of Mexico and other locations around the world. We reviewed historical casing/well failure events in a highly compacting sandstone field and performed a comprehensive geomechanics analysis of various casing- damage mechanisms (tension, axial compression, shear, and bending) related to large reservoir depletion. Among five wells that experienced mechanical well-integrity issues, two of them showed casing restrictions in the caprock at intervals approximately 1,000- to 1,600-ft true vertical depth (TVD) above the top of the depleting (main) reservoir. A multi-finger caliper log obtained from one of the wells indicates that the overburden casing failure occurred at a highly geopressured, thin sand layer approximately 1,100-ft TVD above the top of the compacting reservoir. The remaining casingfailure events occurred near (less than 200-ft TVD) or within the compacting reservoir interval. A 3D nonlinear finite-element-method (FEM) model has been developed for simulating stress changes in the overburden and the reservoir intervals and evaluating the effect of lithological anomalies on casing stability. The simulation results indicate that large tensile and shear strains could develop within a thin, weakstrength layer in the overburden and at the interface between caprock and the depleting reservoir interval. Casing damage by bending/ shear could also occur at these thin-layered sands saturated with overpressured gas. In the reservoir interval, shear stresses acting on the screens can be relatively high because of the difference of the movements between the internal base pipe and the external shroud and gravel. Screen failure may also occur at the welded points. If casing failure occurs in the unperforated sand layer just above the compacting reservoir, it induces localized high-velocity flow on the upper part of the screen, causing potential screen erosion. Casing failure caused by fault slip near the reservoir occurs only if a fault has sealing capability while maintaining a large pressure differential across the fault plane. The numerical-analysis results presented in this work help engineers understand possible casing- and screen- deformation and -failure mechanisms experienced in highly compacting sandstone fields. On the basis of the study findings, we also present some completion- design guidelines to avoid or mitigate compaction-induced casing damage in both the overburden and reservoir intervals. copyright © Society of Petroleum Engineers.

Matrix-acidizing models have traditionally underpredicted acidstimulation benefits because of underprediction of wormhole penetration and the corresponding magnitude of completion-skin factors in vertical wells. For long horizontal wells drilled in carbonate reservoirs, productivity enhancement is a function of acid placement and effective wormhole penetration. However, prediction of wormhole penetration requires more effective analysis than that provided by current industry models. This paper presents results of matrix-acid modeling work for horizontal wells and describes a practical engineering tool for analyzing the progress of matrix-acid stimulation in carbonate reservoirs. The wormhole-growth model is based on the Buijse and Glasbergen empirical correlation. Combining with the mechanistic model of the wormhole propagation based on acid transport and fluid loss from a single wormhole, a modified Buijse-Glasbergen wormholegrowth model is developed that relates the wormhole growth rate to the in-situ injection velocity at the tip of the dominant wormhole. The wormhole constitutive model developed in this study also accounts for core-size dependencies seen in laboratory acid-flood experiments. A semianalytical flow correlation is derived for estimating interstitial velocities at the tip of the dominant wormholes based on a number of 3D FEM simulation analyses, accounting for more realistic flow regimes (radial and spherical flow) typically observed in field application. The scaleup procedure developed in this study extends the wormhole geometry and penetration from laboratory flow tests on small cores to field-sized treatments. The scaleup procedure developed in this work can be applied to cemented and uncemented horizontal wells, including barefoot and perforation-cluster completions typically employed in carbonate reservoirs. Application of this modeling shows that acid wormholing through carbonate formations can provide significant stimulation, resulting in post-stimulation skins as low as -3.5 to -4.0 vs. previously predicted values in the -1.0 to -2.0 range. Copyright © 2012 Society of Petroleum Engineers.

Successful acid stimulation of long-horizontal-well intervals in carbonate reservoirs requires effective acid distribution along the entire reservoir length. Such treatments also require large volumes of acid and seawater/brine injection at sufficiently high injection rates to drive the acid wormholes deep into the reservoir. Under these flowing conditions, significantly large tubing friction loss is anticipated unless optimal friction reducer performance in the tubing is maintained throughout the pumping operation. Because prediction of wormhole penetration and corresponding skin factor depends on analysis of downhole-injection pressures at the reservoir face, it is crucial to properly account for these hydrostatic and friction changes prior to evaluation of wormhole length and skin factor. In this study, an integrated flow model has been developed to predict the wellbore-pressure profile and wormhole distribution by tracking the movement of the acid in the wellbore and the formation. The wellbore-flow model is based on steady-state, 1D, pressure-based nodal method. The segmented wellbore in the reservoir interval is then coupled with analytical transient reservoir-flow models. The wormhole propagation in the formation is calculated based on the modified Buijse-Glasbergen correlation and upscaling model developed in our earlier work. The resultant wormholing skin factor is calculated by simulating and updating the changing well injectivity along the entire injection interval at every timestep. The model developed in this work is applicable for both fully completed wells (i.e., radial flow) and selectively completed perforation-cluster wells (i.e., spherical flow) typically employed in carbonate reservoirs. Analysis of injection rates and pressures during acid treatment provides engineers with a way to determine the varying injectivity and tubing friction as stimulation proceeds. The model presented here can be used as a forward model for analyzing real-time treatment rate and pressure histories and can also be used to review past treatments to improve future treatment designs. Using actual field-stimulation data, we also discuss key elements to successful stimulation planning and the diagnosis of matrix-acid treatments to achieve effective wormhole coverage for horizontal completions in carbonate formations. Copyright © 2012 Society of Petroleum Engineers.

Many casing and screen damage incidents have been reported in deep-water oil and gas fields in the Gulf of Mexico and other locations around the world. We reviewed historical casing/well failure events in a highly compacting sandstone field and performed a comprehensive geomechanics analysis of various casing damage mechanisms (tension, axial compression, shear, and bending) related to large reservoir depletion. Among five wells that experienced mechanical well integrity issues, two of them showed casing restrictions in the cap rock at intervals around 1,000 to 1,600 ft-TVD above the top of the depleting (main) reservoir. A multi-finger caliper log obtained from one of the wells indicates that the overburden casing failure occurred at a highly geo-pressured, thin sand layer approximately 1,100 ft-TVD above the top of the compacting reservoir. The remaining casing failure events occurred near (less than 200 ft-TVD) or within the compacting reservoir interval. A 3D non-linear finite element model has been developed for simulating stress changes in the overburden and the reservoir intervals and evaluating the effect of lithological anomalies on casing stability. The simulation results indicate that large tensile and shear strains could develop within a thin, weak-strength layer in the overburden and at the interface between cap rock and depleting reservoir interval. Casing damage by bending/shear could also occur at these thin-layered sands saturated with overpressured gas. In the reservoir interval, shear stresses acting on the screens can be relatively high due to the difference of the movements between the internal base pipe and the external shroud and gravel. Screen failure may also occur at the welded points. If casing failure occurs in the unperforated sand layer just above the compacting reservoir, it induces localized high velocity flow on the upper part of the screen causing potential screen erosion. Casing failure due to fault slip near the reservoir occurs only if a fault has sealing capability while maintaining a large pressure differential across the fault plane. The numerical analysis results presented in this work help engineers understand possible casing and screen deformation and failure mechanisms experienced in highly compacting sandstone fields. Based on the study findings, we also present completion design guidelines to avoid or mitigate compaction-induced casing damage in both the overburden and reservoir intervals. Copyright 2011, Society of Petroleum Engineers.

Numerous casing and production-liner deformation/failure problems have been reported in high-porosity chalk formations in both the over- • burden and the reservoir sections, causing costly operation problems that prevent workovers and recompletions. This paper presents the results of studies performed to investigate stability of an openhole, cemented liner and uncemented-liner completions in a highly compacting chalk formation. The effects of critical cavity dimensions caused by various acid-stimulation techniques were also investigated. On the basis of the review of historical caliper-survey data, we ascertain that axial-compression collapse is a major liner-deformation mechanism in the reservoir zones. Axial-compression collapse has been found in both low-angle wells (also in buildup sections of horizontal wells) and horizontal laterals. The casing deformation in low-angle sections is a result of reservoir compaction (i.e., change in the vertical formation strain), while the deformation in horizontal sections is primarily induced by increased axial loading because of cavity deformation. The current completion practice using cluster perforations and high-volume acid treatments causes vertically enlarged cavities, resulting in poor radial constraint. A series of laboratory triaxial tests was performed on selected reservoir chalk samples to measure the stress/strain and failure behavior of the chalk formation considering a wide range of porosity and water saturation and different levels of confining pressures. Using the chalk-failure criteria and constitutive relations developed from the analysis of laboratory triaxial-compression-test data, a 3D nonlinear poroelastic/plastic finite-element-method (FEM) model was developed for the openhole stability analysis. The simulation results show that the abnormally high ductility of chalks after pore collapse around a borehole could actually enhance borehole stability, with a magnitude beyond expectation. In this study, analytical and numerical models are also developed for evaluating cavity-induced axial- compression collapse of production liners. Model results indicate that the risk of the cavity-induced axial- compression collapse substantially increases for short perforated intervals stimulated with large acid treatments. However, increasing the perforation-interval lengths along the entire liner axis results in more-uniform acid distribution and will greatly reduce the chance of axial-compression collapse caused by localized cavity deformation. On the basis of these analysis results, key completion design criteria and stimulation strategies were developed for wells completed in highly compacting chalk reservoirs to reduce risk of casing and liner mechanical problems. Copyright © 2010 Society of Petroleum Engineers.

The manufacturer's specifications of sand screen usually provide information such as base pipe collapse pressure, maximum tensile load, and bending strength. However, the problem is that a very heavy base pipe is thus required to satisfy such specifications if the pipe collapse pressure indicated by manufacturer is used for field application. The reason is that the screen collapse tests are conducted under hydrostatic loading for screens wrapped with rubber jacket by screen manufacturers. Observation of these tests shows that screen collapse occurs immediately after the base pipe starts yielding. However, this is not the case if screens are installed in the wellbore under geotectonic load. A series of laboratory tests are conducted using either straight pipes without screen or pipes with screen installed in a drilled borehole of some large rock sample. Since the base pipe failure is the dominant factor for screen failure, half of the experiments shown in this paper used straight pipe without including screen or shroud. These tests show that tubular strings may initiate yielding earlier than the specified yielding data provided by screen manufacturers. However, after the initial yielding, the base pipe deforms gradually and it never collapses even after exceeding the ordinary base pipe yielding (0.2-0.4% deformational strain) by more than 10 times.

Based on these laboratory tests, a numerical model is constructed for screen design along with gravel pack or stand-alone screen in openhole completions. The numerical model first simulates hydrostatic screen collapse tests conducted by manufacturers to confirm the design specifications as measured. Then, it is extended to simulate screen behavior after it is installed downhole. It assumes that a screen is placed in the borehole, and then gravel-pack covers the screen, followed by reservoir depletion and drawdown in production mode.

This paper sheds light on the sand screen collapse resistance under three typical loading types: hydrostatic, geotechtonic load without void around sand screen and geotectonic load with void for openhole standalone screen applications. Distinctly different failures criteria are proposed for these three types of loadings. Empirical data under such high stress levels are rarely found in the literatures.

Using a combination of analytical calculations and 3D finite-element simulation, we have developed a comprehensive skin-factor model for perforated horizontal wells. In this paper, we present the mathematical model development and validation by comparison with finite-element Simulation results. With the new perforation skin model, we then show how to optimize horizontal well perforating to maximize well productivity.
A cased, perforated well may have lower productivity (as characterized by a positive skin factor) relative to the equivalent openhole completion because of two factors: the convergence of the flow to the perforations, and the blockage of the flow by the wellbore itself. Because of the orientation of a horizontal well relative to the anisotropic permeability field, perforation skin models for vertical wells that consider these effects, notably the Karakas and Tariq model (1991), are not directly applicable to perforated horizontal completions. Using appropriate variable transformations, we derived a skin-factor model for a horizontal perforated completion that is analogous to the Karakas and Tariq (1991) vertical-well model. The empirical parameters in the model were determined from an extensive 3D finite-element simulation study.
The results of the new model show that the azimuth of a perforation (the angle between the perforation tunnel and the maximum permeability direction, usually thought to be in the horizontal direction) affects the performance of perforated completions in anisotropic reservoirs. When perforations are normal to the maximum-permeability direction, perforations will enhance horizontal-well flow compared with an openhole completion (a negative skin factor). But if perforations are in the same direction as the maximum permeability, significant positive skin will result. The new skin-factor model provides a clear guide to the shot density, perforation orientation, and level of perforation damage that is tolerable to create high-productivity perforated completions in horizontal wells.

Accurate and reliable downhole data acquisition has been made possible by advanced permanent monitoring systems such as downhole pressure and temperature gauges and fiber optic sensors. These downhole measurement instruments are increasingly incorporated as part of the intelligent completion in complex (highly slanted, horizontal, and multilateral) wells where they provide bottomhole temperature, pressure and sometimes volumetric flow rate along the wellbore. To fully realize the value of these
intelligent completions, there is a need for a systematic data analysis process to improve our understanding of reservoir and production conditions using the acquired data and to make decisions for well performance optimization.

We have successfully developed a model to predict well flowing pressure and temperature (i.e. the forward model), and applied inversion method to detect water and gas entry into wellbore using the synthetic data generated by the forward model (i.e. the inversion model) in the previous study. It is concluded that temperature profiles could provide sufficient information to identify fluid entries, especially in gas wells. However, both the mathematical complexity and advanced well structure lead to challenges in model validation and application. In this study, we applied the wellbore-reservoir flow coupled thermal simulation model to high-rate gas wells with field data. The main objectives are to evaluate applicability of the model to field problems, to study the sensitivity of parameters such as permeability and reservoir pressure on accuracy of interpretation, and to generate practical guidelines on how to initialize the inversion process. The model is applied to highly-slanted gas wells with water produced from a bottom aquifer. The interpretation result was compared against production logging data. The sensitivity of interpretation error to input reservoir properties are examined and the results showed that temperature and pressure anomalies caused by water production and flow rate changes can be detected theoretically and also practically. Judgments should be used based on the understanding of temperature and pressure behavior when initializing the forward model and this can increase efficiency
of model application. The study results and guidelines developed in this study will help us to design permanent monitoring systems and set realistic expectation for predictive capability of intelligent well systems.

A well completion is a critical interface between the productive formation and the wellbore. An effective completion must maintain the mechanical integrity of the borehole without creating any significant restrictions on the flow capacity of the well. In this paper, a process is outlined to design optimal completions for horizontal wells by applying comprehensive skin-factor models that include damage and turbulence effects for all common types of completions.
Slotted or perforated liner, cased, perforated, or gravel-pack completions have been used in horizontal wells for borehole stability and sand-control purposes. However, these completions may have lower productivity (as characterized by a positive skin) relative to an equivalent openhole completion, because the convergent flow to perforations or slots increases fluid velocity in the near-well vicinity. In addition, any reduced permeability zones (formation damage caused by drilling, completion, or other processes) magnify the convergent flow effects and therefore may result in substantially increased skin factors. Compound effects of formation damage around the well completion, a crushed zone because of perforating, plugging of slots, and turbulent flow, as well as interactions among these effects, are included in the model.
This paper illustrates how to use skin factor models to screen the available completion types for cased/perforated and slotted liner completions. This screening approach considers reservoir permeability, permeability anisotropy, fluid properties, formation damage effects, and rock mechanical characteristics as the key parameters. The types of completion that yield the most productive well performance for this matrix of properties are presented.
A more detailed completion design is then illustrated by showing how the skin-factor models were used to redesign the slot configuration of liner completions for viscous oil reservoirs on the North Slope of Alaska. Application of the slotted or perforated liner models to the readily available liners showed that the completion skin factor can vary by as much as 40%, depending on the detailed characteristics of the slots or perforations in the liner (slot or perforation size, density, and distribution). The example showed that optimizing the performance of the completion can increase well productivity at little or no cost and with no loss in liner mechanical strength.

In several places around the world, notably the North Sea and the Middle East, carbonate reservoirs are being accessed with very long horizontal wells (2000 to 20,000 feet of reservoir section.) These wells are often acid stimulated to remove drilling fluid filter cakes and to overcome formation damage effects, or to create acid fractures or deep matrix stimulation to enhance productivity. Good acid coverage with a relatively small acid volume is required to economically obtain the desired broad reservoir access.

We have developed a model to predict the placement of injected acid in a long horizontal well, and to predict the subsequent effect of the acid in creating wormholes, overcoming damage effects, and stimulating productivity. The model tracks the interface between the acid and the completion fluid in the wellbore, models transient flow in the reservoir during acid injection, considers frictional effects in the tubulars, and predicts the depth of penetration of acid as a function of the acid volume and injection rate at all locations along the completion.

We have used this model to simulate treatments that are typical of those performed in the North Sea and in the Middle East. We present a hypothetical example of acid placement in a long horizontal section and an example of using the model to history match actual treatment data from a North Sea chalk well.

Horizontal wells or laterals are completed as openhole, slotted-liner, cased and perforated, or gravel-pack completions. We have developed a comprehensive skin-factor model to predict the performance of any of these completion types and have calibrated this model with extensive finite-element simulations of flow for a horizontal-well completion. This model can be used to predict the performance of virtually any horizontal-well completion.
The new completion skin-factor model accounts for the effects of formation damage, convergent flow to perforations and slots, and flow through slots, with interaction among these effects. To account for formation damage, we extended our previous rigorous model of a damaged horizontal well to include the presence of perforations within, or extending through, the damage zone. The formation damage model is also integrated with the models of slotted-liner performance to model these completions. The model of slotted- or perforated-liner performance is made on the basis of the relationship between pressure drop and flow rate for turbulent flow in these geometries. The slotted-liner model accounts for partial plugging of the slots by grains of formation minerals or precipitates (scale). Turbulence effects are a major part of the apparent skin factor for these completion types.
The model shows the recommended conditions to obtain high-productivity (i.e., low skin factor) completions in horizontal wells. In particular, the interactions among damage effects and skin effects caused by perforations or slots are shown to greatly affect horizontal-well completion performance. The models developed can be applied to design optimal completions for horizontal wells or laterals.

Accelerated production, increased ultimate recovery, and reduced interventions are goals of any operating company. It is now possible to attain all of these goals simultaneously by retrofitting intelligent well technology to an existing, conventional completion. This task is much more complex when the new technology must be installed in conjunction with artificial lift, such as an electric submersible pump (ESP). However, new technology and procedures have overcome this complexity and offer a viable option for optimizing production in existing completions that use artificial lift.

Commingled or selective production from two or more zones is an ideal method of accelerating production from a single well. Traditional designs require interventions into the well to select the intervals to be produced. In most ESP wells, commingling is achieved by using "Y" blocks which allow production tubing to be run from the tubing hanger to the zones of interest. The ESP must then be downsized to accommodate this side-string, which significantly reduces available pump horsepower. This paper focuses on single-ESP wells producing from multiple pay zones. Various application patterns for use of intelligent well technology beneath ESPs are presented, especially focusing on immediate and future benefits. Theoretical examples are presented to illustrate how intelligent completions can enhance the ESP performance, add flexibility, and extend the range of application for a given pump.

Intelligent well technology, referring to the implementation of fundamental process control downhole, has advanced rapidly over the last decade in the upstream oil and gas industry. Application of this technology has continued to expand since recent installation results have demonstrated the value of these completions with a high success rate that operators find attractive.

In this paper, we present an economic evaluation procedure to quantify the benefit of Intelligent Well Systems based on net present value (NPV) calculations. Reducing well count and eliminating or reducing interventions by using Intelligent Well Systems has the potential to add many millions of dollars to well NPV, since drilling rig and workover costs may be very expensive, especially in offshore deepwater environments. Delayed revenue associated with production shut-in also must be considered and reduced by discount rate at the same time that intervention and well operating costs increase by inflation rate.

The use of Intelligent Well Systems not only reduces or eliminates some capital and operating costs, but also allows operators to accelerate oil production, increase the ultimate recovery and reduce water handling cost, which result in significant benefits in NPV of a project. Thus, Intelligent Well Systems are becoming economically attractive for modest-cost onshore operations as well as for the high-cost offshore environments for which they were initially deemed applicable.

A well completion is a critical interface between the productive formation and the wellbore. An effective completion must maintain mechanical integrity of the borehole without creating any significant restrictions in the flow capacity of the well. The full-length paper outlines a process to design optimal completions for horizontal wells by applying comprehensive skin-factor models that include damage and turbulence effects for all common types of completions.

Gravel packing is widely used in well completions to prevent sand production. An efficient gravel pack completion retains formation sand without creating significant additional pressure drop through the completion itself. To predict gravel-packed well performance, a completion model that calculates pressure drop for a gravel pack completion is necessary. In this paper, a theoretical turbulence skin factor model for cased-hole gravel-packed wells is presented that can be used to determine the effects of the gravel pack on well performance. The overall pressure drop through the gravel pack completion is divided into three parts; the pressure drop through the gravel between the casing and the screen, which is usually small and can be neglected; the pressure drop occurring through gravel in the perforation tunnels penetrating through the casing and cement and out into the formation; and the pressure drop outside the casing caused by flow converging to the perforations. This study is focused on the latter two pressure drops. Based on extensive 3D finite element simulation studies, the flow field in and around a gravel pack completion is approximated by a series of linear, radial, and hemispherical flow geometries, and the Forchheimer equation is then integrated along a simplified flowpath to obtain the pressure drop both inside and outside of the casing.

The FEM simulation results show that the flow geometry of a cased, perforated, and gravel packed well greatly depends on the ratio of the formation permeability to the permeability of the gravel in the perforation tunnels. As the perforation tunnel permeability approaches the formation permeability, the flow geometry near the perforation through the casing is no longer linear but spherical as if there is no perforation in the formation. As a result, the conventional model of flow in the gravel packed perforations may give inaccurate results under these conditions. The new skin factor model accounts for the transition from linear to spherical flow as a function of the ratio of the formation to perforation tunnel permeabilities. This model also includes permeability damage in a perforation tunnel that significantly increases rate dependent (turbulent) skin effects. We used the model to determine the gravel pack conditions needed to insure that gravel pack skin factor is low and to minimize the effects of turbulence in the gravel pack in high rate wells. Guidelines for gravel permeability are presented based on these results.

In this paper, we present a new analytical model for formation damage skin factor and the resulting reservoir inflow, including the effect of reservoir anisotropy and damage heterogeneity. The shape of the damaged region perpendicular to the well is based on the pressure equation for an anisotropic medium and, thus, is circular near the well and elliptical far from the well. The new model can be used for various distributions of damage along the well, depending on the time of exposure during drilling and completion. The inflow equation for a damaged, parallel-piped-shape reservoir illustrates the importance of the ratio of the reservoir thickness to the drainage length perpendicular to the well on the influence of formation damage for horizontal well productivity. Our model gives a simple, analytical expression for determining this effect.