Total Stresses Research Articles

Integrated analysis of cyclic steam stimulation in a heavy oil reservoir: Microseismicity, brittleness, and landing height

We compare microseismic observations against pumping information, landing heights, and various well logs. The data were acquired during cyclic-steam injection between September 2002 and December 2005. Ninety-five percent of the microseismicity occurred during injection and in the overburden; 70% of the events happened during the first cycle. Microseismicity in the overburden is likely caused by a greater brittleness than in the reservoir and a cluster of microseismic events in regions with a smaller landing height, thereby facilitating dry cracking due to the volumetric expansion of the reservoir. Yet, other areas with equally shallow landing heights displayed little to no microseismicity, pointing to an inhomogeneous steam front. Furthermore, recorded microseismicity is subject to the Kaiser effect in that event rates are low in subsequent cycles until the current injection pressure exceeds the previous maximum, explaining why 70% of the events occurred during the first cycle and possibly why microseismicity during production accounted for only 5%. Microseismicity in brittle formations can be caused by pore-pressure variations (wet cracking) and/or changes in the total stresses (dry cracking). Identification of pore-pressure variations in the overburden is important because it may indicate containment challenges. Analysis of the growth rate of the microseismic cloud combined with the shallow landing height indicated dry cracking to be more likely than wet cracking but analysis of additional data is required to strengthen this conclusion.

Incorporating Unresolved Stresses in Blood Damage Modeling: Energy Dissipation More Accurate Than Reynolds Stress Formulation.

Reynolds Averaged Navier Stokes (RANS) models are often used as the basis for modeling blood damage in turbulent flows. To predict blood damage by turbulence stresses that are not resolved in RANS, a stress formulation that represents the corresponding scales is required. Here, we compare two commonly employed stress formulations: a scalar stress representation that uses Reynolds stresses as a surrogate for unresolved fluid stresses, and an effective stress formulation based on energy dissipation. We conducted unsteady RANS simulations of the CentriMag blood pump with three different closure models and a Large Eddy Simulation (LES) for reference. We implemented both stress representations in all models and compared the resulting total stress distributions in Eulerian and Lagrangian frameworks. The Reynolds-stress-based approach overestimated the contribution of unresolved stresses in RANS, with differences between closure models of up to several orders of magnitude. With the dissipation-based approach, the total stresses predicted with RANS deviated by about 50% from the LES reference, which was more accurate than only considering resolved stresses. The Reynolds-stress-based formulation proved unreliable for estimating scalar stresses in our RANS simulations, while the dissipation-based approach provided an accuracy improvement over simply neglecting unresolved stresses. Our results suggest that dissipation-based inclusion of unresolved stresses should be the preferred choice for blood damage modeling in RANS.

Higher-order shear deformation theory for accurate prediction of vibration behavior of thick piezoelectric disks and design of efficient surface electrodes

A modified higher-order shear deformation plate theory (MHSDT) is proposed to predict the resonance frequencies and related mode shapes of piezoelectric disks of various thicknesses. The theory explicitly accounts for the effects of the electrical potential on the displacement of the disk. The equations of motion for the disk are derived using Hamilton's principle in conjunction with a variational approach. It is shown analytically that the distribution of the electrical field on the surface of the disk is quantitatively similar to the distribution of the sum of the total stresses within the disk. Thus, the deformation of the disk can be approximated directly from the stress distribution on the disk, thereby greatly simplifying the solution process. The validity of the proposed MHSDT is confirmed by comparing the solutions for the natural frequencies, displacement fields, and electric field distributions of the disk for various radius-to-thickness ratios and boundary conditions with those obtained using Reddy’s third-order plate theory and COMSOL finite element simulations, respectively. The results show that the solutions obtained from the MHSDT method are consistently closer to the exact solutions than those determined using Reddy’s third-order method. Finally, the linear relationship between the electric field and the flexural displacement of the disk is leveraged to design surface electrodes capable of exciting specific flexural mode shapes of the disk under a free boundary condition. The efficiency of the designed electrodes is experimentally demonstrated using an amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI).

Influence of non-linear slip dependent zeta potential on the bi-layered electrohydrodynamic flow in an electrically actuated microsystem

In this paper, the electro-osmotic flow (EOF) of two immiscible fluids through an electrically actuated micro-slit with a non-linear slip-dependent zeta potential is considered. The main objective of this work is to enhance the EOF velocity of non-polar fluid with an intensive flow variation in micro-confinements. We present the comparison between the Poisson–Nernst–Planck (P–N–P) and the Poisson–Boltzmann model and illustrate the impact of different parameters, such as Debye–Hückel parameter, slip coefficient, and interfacial potential drop, on the ionic concentration and the induced potential in detail. In this study, the interface between two immiscible fluids is considered to be planar, and a potential drop is observed close to the interface due to the development of back-to-back diffuse layers. At the interface between two layers, the continuity of the velocity and total stresses (Maxwell stress and shear stress) are taken into account to investigate the flow field in the system of immiscible electrolytes. The P–N–P model, linked with the ion transport equation and the Poisson equation, is employed to describe the motion of electrolyte solutions. The non-linear governing equations are numerically computed using a pressure-correction-based finite volume technique based on a staggered grid algorithm. Closed-form analytical solutions are derived for both steady and unsteady EOF field of two immiscible fluids for the low wall zeta potential and non-overlapping electrical double layer. The analytical solutions are well validated with our numerical results under suitable assumptions. It is also observed that the slip-dependent zeta potential successfully provides a considerable enhancement in EOF velocity over a broad range of parameters such as zeta potential, Debye–Hückel parameter, as well as boundary slip parameter. The results for both layers are highly affected by unequal wall potential, and the ion concentrations are highly actuated by the external electric field.

An active hybrid Reynolds-averaged Navier–Stokes/large eddy simulation approach for gray area mitigation

During the transition from the RANS (Reynolds-Averaged Navier–Stokes) mode to the LES (Large Eddy Simulation) mode, i.e., in the so-called gray area, continuous hybrid RANS/LES approaches suffer from the well-known problem of excessively slow generation of resolved structures. Indeed, when the mesh is refined in the direction of the flow, the model is designed to reduce the modeled energy, but there is no mechanism to transfer the equivalent amount of energy into the resolved motion. Hence, the total turbulent energy and turbulent stresses are underestimated, which strongly affects the prediction of the mean flow. This also constitutes a violation of the conservation of mechanical energy, which can only be corrected by an active approach, i.e., an approach that allows the injection of resolved energy. The aim of this work is to develop such an active approach based on the introduction of a fluctuating volume force into the resolved momentum equation, similar to the anisotropic linear forcing (ALF) method proposed previously. The major difference with ALF is that the new method does not require target statistics obtained by a RANS computation but is based on a simple analysis of the rate of energy transfer related to variations in resolution, enabling the forcing to be extended to continuous hybrid RANS/LES. The application of the new method to the cases of a channel with or without periodic constriction shows a drastic improvement over the case without forcing. Although the method is applied herein to a particular hybrid RANS/LES approach (HTLES, hybrid temporal LES), it can easily be extended to any other approach, as long as a parameter identifies variations in resolution, and thus offers vast application prospects.

Comparison of different internal fixation implants in the treatment of talar neck fractures: A finite element analysis.

This study aimed to analyze the biomechanics of cannulated screws (CS) with or without a lateral locking plate (LLP) in talar neck fractures through a finite element analysis. The computed tomography image of the talus from a healthy volunteer (adult male) was used to reconstruct a three-dimensional talar model. The method for fixing talar neck fractures with CS and an LLP was planned using computer-aided design software. Afterward, the three-dimensional models of comminuted talar neck fractures were used to simulate fixation with anteroposterior parallel dual CS, single CS+LLP, and dual CS+LLP. Finally, finite element analysis was carried out to compare the outcomes of dual CS+LLP to those of single CS+LLP and to those of using dual CS alone. The displacement and von Mises stress values of the three groups with different internal fixation were analyzed. For a simple talar neck fracture, the lowest amount of displacement was obtained with CS+LLP (0.407 mm), while dual CS (0.459 mm) showed the highest amount of total displacement; the lowest amount of peak stresses was obtained with CS+LLP (5.38 MPa), while dual CS (8.749 MPa) showed the highest amount of total peak stresses. For a comminuted talar neck fracture, the lowest amount of displacement was obtained with CS+LLP (0.398 mm), while dual CS (0.408 mm) showed the highest amount of total displacement; the lowest amount of peak stresses was obtained with CS+LLP (129.9 MPa), while dual CS (205.9 MPa) showed the highest amount of peak stresses. Compared to the other two groups, the dual CS+LLP group had better biomechanics properties in the displacement and stress peak of the talus and implant. Thus, the use of dual CS+LLP fixation is recommended for the surgical treatment of comminuted talar neck fractures.

Topology optimization of fluid‐structure interaction problems with total stress equilibrium

AbstractThis work extends force coupling in the topology optimization of fluid‐structure interaction problems from hydrostatic to total stresses through the inclusion of viscous stress components. The superconvergent patch recovery technique is implemented to remove the discontinuities in velocity derivatives over the finite elements boundaries. The sensitivity analysis is derived analytically for the superconvergent patch recovery approach and further verified through the use of the complex‐step derivative approximation method. Numerical examples demonstrate a differentiation in the optimized designs using pressure versus total stress coupling depending on the flow characteristics of the design problem.

Design Improvement for a Maritime Training Polygon using 3D Terrestrial Laser Scanning Technology

In the current geo-strategic context, the North Atlantic Treaty Organization (NATO) and the International Maritime Organization (IMO) impose increasingly high standards for seafarers' preparation in the ship's vitality training centers (elaborate constructions, designed and equipped specifically for training personnel in scenarios involving flooding and fires that may occur on board ships). The buildings housing these training areas experience considerable mechanical and thermal stress, which, over time, affects their structural integrity. Therefore, repairs and modernization work become necessary. This paper provides a case study on the application of technologies for terrestrial laser scanning of buildings, with a focus on the vitality training polygon. The originality of the approach comes from achieving "as-built" documentation for the vitality polygon using 3D laser scanning technology. This includes both three-dimensional modeling based on digital information from the scanning process, as well as structural analysis using finite element techniques for the buildings where crew training takes place. The study also analyzes the distribution of total deformations and stresses in the walls of exercise compartments for flooding and water fight scenarios. Moreover, temperature distribution in the walls and interior atmosphere of these compartments is examined for fire scenarios and firefighting.

Mechanical Behavior of Fiber-Reinforced Soils under Undrained Triaxial Loading Conditions

The design of fiber-reinforced soil structures, such as embankments and pavements, can be carried out using the results of unconsolidated, undrained triaxial compression tests conducted on specimens at their “as-compacted” water content and analyzed in terms of total stresses. The effects of soil and fiber type on the mechanical behavior of fiber-reinforced soils have not been methodically or adequately examined in the past under these conditions, and the effects of fiber length and content on the shear strength parameters of fiber-reinforced soils need further experimental documentation. Accordingly, five soils ranging from “excellent” to “poor” materials for use in earthwork structures were tested in the present study, in combination with five types of polypropylene fibers having lengths ranging from 9 to 50 mm. Unconsolidated undrained triaxial compression tests were conducted on specimens at their “as-compacted” water content, with fiber contents ranging from 0.5 to 2% by weight of dry soil. Fiber reinforcement reduces the stiffness and increases the deformability of the soil. The fiber-reinforced soils exhibit a more ductile behavior in comparison with the unreinforced soils. A Mohr–Coulomb type linear failure criterion satisfactorily describes the shear strength behavior of fiber-reinforced soils in total stress terms. The cohesion values of the fiber-reinforced soils range between 61 kPa and 301 kPa and increase up to seven times in comparison with the cohesion values of the unreinforced soils. The variations of the angle of internal friction of soils due to fiber reinforcement are generally limited to ±25%. The cohesion improvement due to fiber reinforcement is increased with increasing fiber content and fiber length up to 30 mm and is inversely proportional to the fine-grained fraction and the cohesion of the unreinforced soil.

Temperature Evolution, Material Flow, and Resulting Mechanical Properties as a Function of Tool Geometry during Friction Stir Welding of AA6082

The friction stir welding process was simulated for joining AA6082 aluminum alloy with the use of the computational fluid dynamics method. Two different tool geometries were used—a tapered cylindrical pin (simple pin) and a hexagonal pin with grooves (complex pin). The analysis of the simulations performed was discussed in terms of temperature evolution during the process, total heat input, residual stresses and material flow. Simulations revealed that a 5% higher temperature, equal to maximum 406 °C, was provided when using the complex pin than with the simple pin. Higher temperature and higher shear stresses during the welding with the complex pin caused the introduction of higher residual stresses in the weld. Experimental results on the produced welds allowed observation of the microstructure of the joints, hardness tests in cross sections and tensile strength tests. Due to the higher temperature during the process with the complex pin and the more efficient recrystallization process, grain refinement in the SZ was more pronounced. The average grain size in the stir zone for the weld produced with the complex pin was equal to 11 ± 1 µm, and in the case of the simple pin 14 ± 1 µm. The presented hardness profiles revealed that the weld produced with a complex pin had higher hardness in the stir zone, equal to 89.5 ± 1.3 HV, which is consistent with the Hall-Petch relationship. The obtained UTS values corresponded to the joint efficiency of 72.5 ± 4.9% and 55.8 ± 8.6% for the weld produced with the complex pin and the simple pin.

Optimization of Deformation Behaviors during Continuous Forming Extrusion of C18150 Copper Alloy through Response Surface Methodology

Continuous extrusion (CE) is a method of creating endless profiles of high-quality products of dimensional accurateness, high productivity, and excellent material properties. The main objective of this study is to investigate the influence of CE input process parameters on optimal overall extrusion load requirement and effective stress induced. The input parameters considered were extrusion driving wheel speed, feed metal temperature, tool temperature, and factor of friction proceeding. Numerical simulations of a copper alloy (C18150) were carried out using DEFORM-3D to investigate the impact of the input variables on total load and effective stresses. A mathematical model based on response surface methodology (RSM) was developed for optimized results. The optimized parameters in terms of wheel extrusion velocities, feedstock temperatures, tool temperatures, and friction factors expressed. The ANOVA test was performed to assess the suitability and appropriateness of the model. Using RSM, the optimal load value of 408.167 kN and effective stress of 1241.0 MPa were achieved within the composite preference of 1.0. A load of 408.167 kN had been obtained if the velocity of the wheel, temperatures of feedstock, tool temperatures, and factors of friction are 4 rpm, 500°C, 400°C, and 0.85, respectively. The minimum effective stress of 1241.0 MPa is induced in the feedstock due to the CE process if the velocity of the wheel, temperature of the feedstock, die temperature, and frictional factor were 4 rpm, 500°C, 400°C, and 0.95, respectively.

Analysis of interfacial stresses on narrow beams strengthened by FRP straps with a side anchorage system

AbstractInterfacial stress concentrations at fiber‐reinforced polymer (FRP) ends can easily cause premature debonding failure of FRP‐strengthened beams. To solve this problem in narrow beams, a hybrid strengthening method was proposed in our previous study, but the theoretical stress analysis was lacking due to the complexity at the anchorage zone. In this study, the interfacial stresses of the hybrid strengthened beam were initially analyzed by decomposing it into a purely bonded beam under three different effects, including the load on the beam, the tensile force at the end of the FRP strap released from anchor plates and a pair of eccentric forces induced by the side anchorage system towards the midspan. By summing the special solutions of the three cases, the expressions for the total interfacial stresses of a hybrid strengthened beam were determined. Moreover, a formula for the shear stiffness of the side anchorage system was proposed, in which the side plate and the bolt were compared to a deep cantilever beam and an elastic foundation beam, respectively. Finally, a finite element model was developed to verify the applicability of the proposed theoretical model. The theoretical and simulated results were in satisfactory agreement for a hybrid strengthened beam.

Dynamic response of a shallow lined circular tunnel by incident P1 and SV waves in an unsaturated half-plane

The dynamic response of a shallow lined circular tunnel in the unsaturated half-plane during earthquakes is investigated in this paper using the complex function method. The wave fields of the unsaturated medium with unknown coefficients are solved by Fredlund’s theory and Helmholtz’s decomposition based on the momentum and mass balance equations. General expressions for the displacements and total stresses of the solid skeleton, pore pressures, and seepage are derived using the complex function method and conformal mapping technique. The half-plane surface and the tunnel in the physical domain are mapped onto two bonded ring regions in the image plane using two conformal mapping functions. The boundary value problem yields a series of algebraic equations by the boundary conditions and the continuity conditions along the liner-medium interface. The unknown coefficients in the infinite series of algebraic equations can be solved numerically by truncating the series number. A parametric study is performed to investigate the dynamic responses of the medium and the liner by the incident P1 waves. Numerical results reveal that the embedded depth of the tunnel, the saturation degree of the medium, the liner-medium rigidity ratio, the frequency and angle of the incident P1 waves significantly affect the dynamic responses of the medium and the liner. The shielding effect on the tunnel to the incident P1 waves is obvious. The response of the liner-medium system to the incident waves is independent of the saturation degree of the unsaturated medium when its value is high. The effect of the liner-medium rigidity ratio is great on the dynamic stress concentrations of the liner-medium system and weak on the displacements along the half surface. The frequency of the incident excitation has a great influence on the displacements of the half surface.

Single-phase dual-input split-source inverter for photovoltaic systems

This paper proposes dual-input configuration of split-source inverter (abbreviated as DSSI) to transfer the power of two photovoltaic (PV) modules simultaneously or individually. The proposed DSSI keeps the continuity of the input current and the voltage boosting capability offered by the conventional split-source inverter (SSI). Thus, independent as well as efficient tracking of the maximum power points (MPPs) is realized. In addition, low capacitance is utilized for power decoupling in single-phase circuits. Compared to using one SSI per PV source to feed a common load, the proposed DSSI uses lower number of switches at reduced total volt-ampere stresses. Therefore, decreased cost, size, and control circuitry are realized. Simple unipolar modulation technique is presented. Accordingly, the modes of operation and the Volt-ampere stresses of the switches are explained. A closed loop control system is explained to fulfill operation at MPP along with regulation of DC-link voltage and load current. Comparative study with other dual-input topologies is presented. Experimental results as well as real-time simulation results for dual- and single-input scenarios are presented. Also, the dynamic performance under step change in the radiation is investigated. Finally, the capability of operation under different power factors is verified.

Complex function method for the scattering of harmonic plane waves by an arbitrary-shaped cavity in an unsaturated medium

General formulations of the complex function method for the scattering of harmonic waves by an arbitrary-shaped cavity in the unsaturated medium are constructed. In light of the momentum and the mass balance equations, the wave fields of the unsaturated medium with unknown coefficients are solved by Fredlund’s theory and Helmholtz’s decomposition. Complex-valued expressions for the displacements and the total stresses of the solid skeleton, the pore pressures, and the seepages of the unsaturated medium are obtained by the conformal transformation technique to map the arbitrary cavity onto a unit circle in the complex plane. The boundary value problem results in a series of infinite algebraic equations with unknown coefficients by considering the stress, displacement, and permeable conditions along the cavity. A parametric study for the scattering of the incident P1 waves around a rectangular cavity in an unsaturated medium is performed to investigate the dynamic stress concentrations, displacements, and pore pressures. Numerical results show that the saturation degree of the unsaturated medium has a considerable influence on the dynamic response of the cavity by the incident P1 waves. With the increase of the saturation degree, the dynamic stress concentration around the rectangular cavity decreases, and the equivalent pore pressure increases. For obliquely incident P1 waves, the shielding effect of the rectangular cavity is obvious.

Variant of strain gradient elasticity with simplified formulation of traction boundary value problems

AbstractIn this paper, we show that within the class of isotropic Mindlin‐Toupin gradient theories there exist a particular variant of the theory, which allows a completely simplified form of traction boundary value problems. This theory can be obtained assuming that the high‐grade part of the strain energy density depends only on the vector‐type quantities, that are the gradient of dilatation and the curl of small rotation. Such incomplete gradient theory becomes positive semi‐definite and at the same time it obeys the strong ellipticity conditions of the general Mindlin‐Toupin first strain gradient elasticity. Based on the variational approach it is shown that the equilibrium equations of the developed theory and its definition for the surface traction can be given only in terms of the total stresses (like in classical elasticity). Such formulation can be useful for derivation of the closed form solutions for the problems with traction‐type boundary conditions. Examples of the solutions for the problem of cylindrical bending and for the inplane crack tip fields are presented. It is shown that considered theory allows to obtain a regularized solution for the crack problems and at the same time it does not predict a non‐physical infinite increase of the material's rigidity under bending.

INFLUENCE OF OPERATIONAL LOADS ON BALLASTING PARAMETERS OF UNDERWATER CROSSINGS OF TRUNK OIL AND GAS PIPELINES

The trunk oil and gas pipelines are under operational loads affecting the strength and reliability of the structure. The most significant effect of loads is manifested at pipeline crossings across rivers. The stability of underwater pipeline design position and thus its reliability is achieved by ballasting with the required intensity. The current regulatory document SP 36.13330.2012 provides for the required intensity determination, taking into account the water buoyancy force and pipeline crossing along the longitudinal profile of the transition. At the same time, it should be noted that ensuring pipeline bending along the longitudinal profile of the crossing with the use of ballasting of the required intensity also depends on the internal forces acting in the pipeline wall during operation. However, this document does not take into account operational loads when determining the required ballasting intensity. In addition, ballasting with the ensuring pipeline bending along the longitudinal profile of the crossing leads to the occurrence of bending stresses, in some cases quite high, which affect the level of total stresses, strength and reliability of the structure. In many cases, water occupies only part of the concave section of the pipeline at crossings. Taking into account the above conditions and features, studies of the ballasting of underwater crossings of trunk oil and gas pipelines were carried out.
 The aim of the work is to obtain analytical dependences of the ballasting intensity and bending stress in an underwater pipeline at different altitude water levels at the transition from operational loads and geometric characteristics of the crossing.
 The research method is a theoretical study of the stress-strain state of an underwater pipeline under the action of operational loads and weights.
 The dependences of ballasting intensity with weights of the underwater pipeline on the operational loads and the geometric characteristics of the transition at different elevations of the water level at the transition are obtained, and the highest bending stresses in the pipeline wall at the underwater crossing are determined. Recommendations to reduce the total longitudinal stress level in an underwater pipeline are given.

Influence of the filling process on the behaviour of geotextile tubes

Geotextile tubes can be used to dewater materials such as sludge, sediments or residues aiming at reducing their moisture contents to acceptable levels. The tube filling process can be carried out using one or several filling stages, and the number of stages can influence the tube behaviour in terms of dewatering rate, final shape and geotextile strains, for instance. In this research, laboratory tests were carried out on nonwoven geotextile tubes for the dewatering of a fine-grained material using different numbers of filling stages. The behaviour of the tube was monitored by instrumentation to assess tube geometry, pore pressures, total stresses at the tube base, geotextile strains and retention capacity. Evaluations of the accuracy of some available methods for the prediction of tube behaviour were also made. The results obtained showed that the increase in the number of filling stages resulted in larger final tube height, volume, geotextile strains as well as larger diameters of the soil particles that piped through the geotextile. Predictions of tube behaviour by available methods showed varying degrees of accuracy depending on the tube parameter considered.

Numerical and theoretical analysis of normal stress distribution in flanges of I-girders with trapezoidal corrugated webs under uniform bending

I-girders with corrugated webs (IGCWs) are a new type of structure that has been increasingly used in engineering. The normal stress distribution in flanges of IGCWs is quite complicated compared with that of conventional I-girders under in-plane loads. This study numerically and theoretically investigates the normal stress distribution rules and mechanisms in flanges of I-girders with trapezoidal corrugated webs (IGTCWs) under uniform bending in the elastic stage. First, finite element (FE) modelling was developed using ABAQUS software and verified by the available experimental results. Based on the verified FE models, a detailed analysis of the normal stress distribution in flanges of IGTCWs was performed. The results show that the normal stresses are nearly uniformly distributed over the flange in cross-sections with inclined web folds. Normal stresses in cross-sections with parallel web folds are approximately linearly distributed along the flange width, with stresses away from the corrugated web side being much larger than those near it. Its mechanism is explored using a secondary transverse moment theory, which is improved and completed by considering the proposed effective cross-section coefficient and equivalent width coefficient for the action of shear flow. The reliability of this theory was verified through comparison with the normal stress difference and distribution results of the supplementary FE models. Finally, the formulae for calculating the normal stresses in flanges of different cross-sections of IGTCWs under uniform bending are presented. These formulae suggest that secondary normal stresses generated by secondary transverse moments in cross-sections with parallel web folds account for 7% to 13% of the total normal stresses and cannot be ignored in the stress analysis.

Mechanism of casing deformation relieved under densified-multi-cluster fracturing based on an analytical model

The densified-multi-cluster fracturing, a technology being used to enhance shale gas production, also shows a special advantage on casing deformation relieved in WeiRong oil field, southwest China, compared to the conventional multi-cluster fracturing. In view of this, an analytical model was established to illustrate the mechanism of casing deformation relieved under densified-multi-cluster fracturing from the viewpoint of casing stress decreased. Two steps of critical calculations were included in the analytical model, namely, calculation of fracture-induced stresses, calculation of casing total stresses. Based on the fracture mechanics, the fracture-induced stresses under the stress-shadow were respectively calculated in reservoir plane and vertical profile. Then, the fracture-induced stresses were transformed into the casing additional loads, under which the casing additional stresses were calculated. Adding the casing additional stresses to initial stresses, the casing total stresses were obtained under densified-multi-cluster fracturing. Furthermore, the analytical model was validated by comparison of analytical results to numerical results on fracture-induced stresses and casing stresses, with the maximum deviation less than around 7.4 %. By the analytical model, it can be found that the reservoir re-distributed stresses (i.e., casing loads) were decreased and furthermore the casing stresses were decreased with increasing of fracture cluster amounts, which validates the phenomenon of casing deformation relieved under densified-multi-cluster fracturing. Finally, the mechanism of casing deformation relieved was comprehensively revealed.