Fluid Bulk Modulus Research Articles

The Role of the Working Fluid and Non-Ideal Thermodynamic Effects on Performance of Gas Lubricated Bearings

Abstract Small-scale turbomachinery operating at high rotational speed is a key technology for increasing the power density of energy and propulsion systems. A notable example is the turbine of an organic Rankine cycle turbogenerator for thermal recuperation from prime engines and industrial processes. Such systems typically operate with organic compounds characterized by complex molecular structures, to allow the design of efficient fluid machinery and flexibility in matching the heat source and sink temperature profiles. Gas bearings are considered advantageous compared to traditional oil-lubricated rolling element bearings for supporting the turbine rotor, enabling greater machine compactness and reduced complexity, and to avoid contamination of the working fluid. In certain operating conditions, however, the lubricant of the bearing is in thermodynamic states near the saturated vapor line or in the vicinity of the fluid critical point whereby non-ideal effects are relevant and may affect bearing performance. This work investigates the physics of thin film flows in gas bearings operating with fluids made by complex molecules. The influence of non-ideal thermodynamic effects on gas bearing performance is discussed by analysis of the fluid bulk modulus. Reduced values of the non-dimensional bulk modulus near the critical point or saturated vapor line decrease bearing performance. The main parameter characterizing the influence of molecular complexity on bearing performance is shown to be the acentric factor. For complex fluids with large acentric factors, the impact of non-ideal thermodynamic effects on non-dimensional bearing load capacity and rotor-dynamic characteristics is less pronounced.

Theoretical and Experimental Study on the Performance of Hermetic Diaphragm Squeeze Film Dampers for Gas-Lubricated Bearings

Low damping characteristics have always been a key sticking points in the development of gas bearings. The application of squeeze film dampers can significantly improve the damping performance of gas lubricated bearings. This paper proposed a novel hermetic diaphragm squeeze film damper (HDSFD) for oil-free turbomachinery supported by gas lubricated bearings. Several types of HDSFDs with symmetrical structure were proposed for good damping performance. By considering the compressibility of the damper fluid, based on hydraulic fluid mechanics theory, a dynamic model of HDSFDs under medium is proposed, which successfully reflects the frequency dependence of force coefficients. Based on the dynamic model, the effects of damper fluid viscosity, bulk modulus of damper fluid, thickness of damper fluid film and plunger thickness on the dynamic stiffness and damping of HDSFDs were analyzed. An experimental test rig was assembled and series of experimental studies on HDSFDs were conducted. The damper fluid transverse flow is added to the existing HDSFD concept, which aims to make the dynamic force coefficients independent of frequency. Although the force coefficient is still frequency dependent, the damping coefficient at high frequency excitation with damper fluid supply twice as that without damper fluid supply. The results serve as a benchmark for the calibration of analytical tools under development.

Gas prediction in tight sandstones based on the rock-physics-derived seismic amplitude variation versus offset method

Estimating gas enrichments is a key objective in exploring sweet spots within tight sandstone gas reservoirs. However, the low sensitivity of elastic parameters to gas saturations in such formations makes it a significant challenge to reliably estimate gas enrichments using seismic methods. Through rock physical modeling and reservoir parameter analyses conducted in this study, a more suitable indicator for estimating gas enrichment, termed the gas content indicator, has been proposed. This indicator is formulated based on effective fluid bulk modulus and shear modulus and demonstrates a clear positive correlation with gas content in tight sandstones. Moreover, a new seismic amplitude variation versus offset (AVO) equation is derived to directly extract reservoir properties, such as the gas content indicator and porosity, from prestack seismic data. The accuracy of this proposed AVO equation is validated through comparison with the exact solutions provided by the Zoeppritz equation. To ensure reliable estimations of reservoir properties from partial angle-stacked seismic data, the proposed AVO equation is reformulated within the elastic impedance inversion framework. The estimated gas content indicator and porosity exhibit favorable agreement with logging data, suggesting that the obtained results are suitable for reliable predictions of tight sandstones with high gas enrichments. Furthermore, the proposed methods have the potential to stimulate the advancement of other suitable inversion techniques for directly estimating reservoir properties from seismic data across various petroleum resources.

Effect of undissolved gas on fluid bulk modulus

This document describes the effect of undissolved gas on the bulk modulus of the fluid. The concept of liquid compressibility and dependence on composition, temperature and pressure is considered. Various methods of measuring the modulus of volumetric elasticity of the liquid were considered. It was also proposed to improve the mathematical model for calculating the module of volumetric elasticity of a liquid. The results of the study were used to plot the effect of the gas phase content modulus of volumetric elasticity of working fluid at different pressures. It was also concluded that the obtained values describe the need to equip hydraulic drives with devices for degassing working liquids.

Fluid Identification Method of Nuclear Magnetic Resonance and Array Acoustic Logging for Complex Oil and Water Layers in Tight Sandstone Reservoir

In order to improve the logging interpretation accuracy for complex oil and water layers developed in tight sandstone reservoirs, this study takes the Chang 8 member of the Yanchang Formation in the Huanxian area as the research object, and two new fluid identification methods were constructed based on nuclear magnetic resonance (NMR) logging and array acoustic logging. Firstly, the reservoir characteristics of physical properties and conductivity were studied in the research area, and the limitations of conventional logging methods in identifying complex oil and water layers were clarified. Then, the sensitive parameters for identifying different pore fluids were established by analyzing the relationship between NMR logging and array acoustic logging with different pore fluids. On this basis, the fluid identification plate, composed of movable fluid apparent diffusion coefficient and effective porosity difference (Da-Δφe) by NMR logging data of D9TWE3 observation mode, and the other fluid identification plate, composed of apparent bulk modulus of pore fluid and elastic parameter sensitive factor (Kf-Fac), were constructed, respectively. Finally, these two fluid identification methods were used for reservoir interpretation of actual logging data. This study shows that the two new fluid identification methods constructed by NMR logging and array acoustic logging can effectively eliminate the interference of rock skeleton on logging interpretation, which make them more effective in identifying complex oil and water layers than the conventional logging method. Additionally, the two methods have their own advantages and disadvantages when used separately for interpreting complex oil and water layers, and the comprehensive interpretation of the two methods provides a technical development direction for further improving the accuracy of logging the interpretation of complex oil and water layers.

Numerical validation of Gassmann’s equations

Derived several decades ago, Gassmann’s equations continue to be widely used in applied geophysics. Gassmann’s equations allow us to calculate the elastic moduli of a fully saturated rock from dry rock moduli knowing the porosity, fluid bulk modulus, and bulk modulus of the solid grains. These equations are treated as exact in the scientific community, but there is a lack of comprehensive numerical validation. Furthermore, recently several publications have appeared in the literature postulating a logical error in the derivation of Gassmann’s equations. Therefore, I develop a numerical validation of Gassmann’s equations. For that, I use a 3D finite-element approach to resolve the conservation of linear momentum that is coupled with the stress-strain relations for the solid phase and the quasistatic linearized compressible Navier-Stokes momentum equation for the fluid phase. Finally, a convergence study validating the correctness of Gassmann’s equations for a particular yet arbitrarily chosen “generic” pore geometry is presented. The arbitrary model geometry is simple as compared with real rocks; however, it is sufficiently complex with elements resembling wider pore bodies and narrower pore throats to, in general, validate Gassmann’s equations. MATLAB routines to reproduce the presented results are provided.

Water Influence on the Determination of the Rock Matrix Bulk Modulus in Reservoir Engineering and Rock-Fluid Coupling Projects

This research was conducted to determine how the incorporation of different poroelastic equations would affect the measured rock matrix bulk modulus in the laboratory. To do this, three experimental methods were used to measure the matrix bulk modulus, Ks, of seven sandstone specimens taken from the Świętokrzyskie mine in Poland. Those experimental methods were based on the different governing equations in poroelasticty theory. The matrix bulk modulus has a substantial impact on the rock strength against external stresses. Moreover, the rock bulk modulus depends directly on two components: the pore fluid bulk modulus and matrix bulk modulus. The second one is more important as it is much higher than the first one. In this study, the accuracy of those three methods in the measurement of the matrix bulk modulus was evaluated. For this purpose, an acoustic wave propagation apparatus was used to perform the required tests. For each method, an empirical correlation was extracted between the matrix bulk modulus and the applied hydrostatic stress. In all the experiments, an exponential correlation was observed between the matrix bulk modulus and the hydrostatic stress applied on the rock. Furthermore, it was found that the incorporation of the dry bulk modulus in the calculations led to an underestimation of the matrix bulk modulus. In addition, as the hydrostatic stress was raised, the matrix bulk modulus also increased. The applied methodology can be deployed to determine the matrix bulk modulus in coupled rock-fluid problems such as reservoir depletion, hydraulic fracturing, oil recovery enhancement, underground gas storage and land subsidence.

Elastic wave velocity of marine sediments with free gas: Insights from CT-acoustic observation and theoretical analysis

The existence of free gas in pores will significantly affect the elastic wave velocity and attenuation coefficient of aqueous sediments. However, because the gas content and distribution are difficult to be accurately obtained through experiments, it is difficult to accurately characterize the relationship between elastic wave velocity and gas content, which may lead to the evaluation error of gas saturation in sediments. In this study, we applied the combined detection technology of X-CT and ultrasonic detection to simultaneously obtain the volume proportion of each phase and acoustic velocity of the selected marine sediments, respectively. On the basis of experiments, the applicability of effective medium theory (EMT), Biot-Gassmann theory by Lee (BGTL) and simplified three-phase equation (STPE) is verified. Meanwhile, the bulk modulus of pore fluid (Kfl) is recalculated by integrating Wood's and Domenico's equations by introducing calibration constant J to adapt to different reservoir conditions. The results show that the elastic wave velocities predicted by BGTL and STPE are more accurate compared with other gas estimation models. When J = −0.8, the acoustic velocity of sedimentary medium predicted by BGTL is more consistent with the experimental data. In addition, we apply the model to the elastic wave velocity data from ODP 204 Hole 1245 and obtain the free gas saturation. The results provide a theoretical and practical application reference for accurately estimating the free gas saturation in marine sediments.

Technology Focus: Seismic (February 2023)

The energy transition and disruption in the energy supply/demand chains fueled by the combined effect of the COVID-19 pandemic and the ongoing Russia/Ukraine war have given rise to more-innovative geoscience approaches geared toward carbon-footprint reduction with efficient and sufficient energy supply despite the increasing difficulty in finding new or remaining hydrocarbon accumulations. This year’s titles have a good mix of new frontier exploration, expanding heartlands, and brownfield-development themes. Improved seismic survey techniques, seismic image uplift, enhanced seismic attribute analysis, reservoir characterization, fracture diagnostics, mitigation of induced seismicity, and carbon capture and storage topics also are featured. As in previous years, more achievements in machine learning and automation of seismic processing and attribute analysis on both new or existing data to redefine geological plays, improve understanding of risks and uncertainties, enhance well placement and hydrocarbon recovery, reduce cost, and increase production are worthy of note. The selected top three papers for synopsis and those recommended for reading provide a taste of the themes mentioned above. I hope you find them as informative and transformational as I did. Recommended additional reading at OnePetro: www.onepetro.org. SPE 208174 Accurate Pore Fluid Indicator Prediction Using Seismic Fluid Bulk Modulus Inversion by Makky Sandra Jaya, Petronas, et al. SPE 211627 Seismic Motion Inversion Based on Geological Conditioning and Its Application in Thin Reservoir Prediction, Middle East by Chen Xin, China National Petroleum Corporation, et al. IPTC 22631 First Time-Lapse Walkaway Virtual Seismic Profile Monitoring of CO2 Water-Alternating-Gas Enhanced Oil Recovery Pilot: Challenges and Learnings From Onshore Carbonate Field UAE by Moza Mohamed, ADNOC, et al. SPE 207253 4D Seismic Detectability in a Brazilian Presalt Carbonate Field by Cyril Agut, TotalEnergies, et al. SPE 207719 Benefits of Ultradense 3D Spatial Sampling for Seismic Processing and Interpretation by Mohamed Mahgoub, ADNOC, et al.

A signal-processing approach to assess viscous-damper absorbing boundary conditions for dry and saturated soils in time domain dynamic problems

For numerical modelling of dynamic problems, absorbing boundary conditions (ABC) are necessary to attenuate reflected waves (’box effect’). Different approaches have been developed for this numerical issue, being the viscous damper (VD-ABC) widely employed. The VD-ABC is not frequency dependent but a total attenuation is never achieved, and hence a tentative enlargement of the computational domain is adopted. Relevant issues on VD-ABC are integrative assessment by a signal processing approach. One and two-phases (dry and saturated soil, respectively) time-domain responses have been considered besides spectral properties and energy content. P-wave velocity has been modified to consider the fluid bulk modulus. ABC affect differently to each variable, being the vertical displacement the less sensible. Higher frequencies present great discrepancies although their energy contribution to the overall signal is not relevant. The results with VD-ABC yield to a feasible reduction of the size of the computational domain, maintaining the required accuracy.

Numerical investigation of poroelastic effects during hydraulic fracturing using XFEM combined with cohesive zone model

Hydraulic fracturing is an effective technology used to stimulate hydrocarbon production from unconventional reservoirs. Numerical simulation of the hydraulic fracturing process plays a key role in the design of hydraulic fracturing. However, most of existing models are overly simplified by neglecting the poroelasticity of rock matrix, which may have a substantial effect on the growth of hydraulic fractures, especially in oil and/or water saturated formations. In this study, we developed a fully coupled finite element model for hydraulic fracture propagation in permeable formations by combining XFEM technique with cohesive zone model, and used it to investigate the effects of poroelasticity on the geometry evolution of single fracture which initiated from an injection point. Fluid flow within the fractures, Darcy flow within the rock matrix, hydraulic fractures propagation, and fluid leak-off into the formation are simultaneously taking into account in the model. Then, the model is validated by comparing the results with available analytical solutions. To understand and quantify the poroelastic effects on the propagation of hydraulic fracture, several cases with different matrix permeability, leak-off coefficients, and bulk modulus of pore fluid are performed. The simulation results show that the total volume of leakage is controlled by the combined action of matrix permeability and leak-off coefficient. The fracture aperture and length decrease with the increase of matrix permeability or leak-off coefficient, while as the bulk moduli of pore fluid increases, the fracture aperture and length tend to increase.

Direct hydrocarbon identification in shale oil reservoirs using fluid dispersion attribute based on an extended frequency-dependent seismic inversion scheme

The identification of hydrocarbons using seismic methods is critical in the prediction of shale oil reservoirs. However, delineating shales of high oil saturation is challenging owing to the similarity in the elastic properties of oil- and water-bearing shales. The complexity of the organic matter properties associated with kerogen and hydrocarbon further complicates the characterization of shale oil reservoirs using seismic methods. Nevertheless, the inelastic shale properties associated with oil saturation can enable the utilization of velocity dispersion for hydrocarbon identification in shales. In this study, a seismic inversion scheme based on the fluid dispersion attribute was proposed for the estimation of hydrocarbon enrichment. In the proposed approach, the conventional frequency-dependent inversion scheme was extended by incorporating the PP-wave reflection coefficient presented in terms of the effective fluid bulk modulus. A rock physics model for shale oil reservoirs was constructed to describe the relationship between hydrocarbon saturation and shale inelasticity. According to the modeling results, the hydrocarbon sensitivity of the frequency-dependent effective fluid bulk modulus is superior to the traditional compressional wave velocity dispersion of shales. Quantitative analysis of the inversion results based on synthetics also reveals that the proposed approach identifies the oil saturation and related hydrocarbon enrichment better than the above-mentioned conventional approach. Meanwhile, in real data applications, actual drilling results validate the superiority of the proposed fluid dispersion attribute as a useful hydrocarbon indicator in shale oil reservoirs.

Direct inversion for the equivalent pore aspect ratio based on the theory of ellipsoid modelling

AbstractPore aspect ratio, together with porosity, is a structural parameter that represents the geometric property of rock reservoirs. We have adopted the theory of ellipsoid modelling in material mechanics to derive the dry rock modulus. Based on this derivation, the Gassmann equation, which is a constitutional equation for a fully saturated rock model, can be linearized in terms of the pore structure parameters and the elastic parameters. We have established a relationship between the seismic reflection coefficient and the pore structure parameters, where the equivalent pore aspect ratio and porosity are two key parameters. Based on this relationship, the equivalent pore aspect ratio can be inverted directly from seismic reflection data instead of being converted from the intermediate parameters of conventional seismic inversion. Therefore, seismic inversion is a simultaneous inversion in which seven parameters are inverted: the elastic moduli (matrix bulk modulus and shear modulus, fluid bulk modulus), the densities (matrix and fluid densities) and the pore structure parameters (the equivalent pore aspect ratio and porosity). We applied this direct inversion scheme to a carbonate reservoir to predict the fracture development zone with low porosity and low aspect ratio.

Analysis of liquid spring damper for vertical landing reusable launch vehicle with network-based methodology

This paper presents the network-based modeling, validation and analysis of the nonlinear liquid spring damper model under vertical landing conditions of reusable launch vehicle. The impedance function of damper model is derived first. Then, its mechanical and hydraulic networks are newly established based on the hydro-mechanical analogy and network-based analysis. By comparing the networks between the corresponding symmetric and asymmetric structures, the meaning of each branch in the network is elucidated. After that, the validity of the network-based model for the liquid spring damper is confirmed by comparison against the experimentally verified nonlinear model in both frequency and time domain. The force and energy absorption characteristics of the damper model are further decomposed, and, specifically, the influence of the orifice area and orifice length on the attenuation performance is studied. The results show that the network-based model provides predictions consistent with those generated by the nonlinear model. The main discrepancy is attributed to the inaccuracy caused by the equivalent fluid bulk modulus. The network-based analysis indicates that the orifice area mainly influences the damping force in the network, which further affects the loads and efficiency of the damper. The orifice length mainly influences the inertia force in the network, which should be limited to a small value. The proposed novel interpretation of the damper models and responses under impact conditions constitutes a framework suitable for systematic design of typically highly nonlinear landing systems in reusable launch vehicles.

Seismic characterization of in situ stress in orthorhombic shale reservoirs using anisotropic extended elastic impedance inversion

Seismic characterization of anisotropic in situ stress is of great importance in improving the success of planning drilling and hydraulic fracturing treatment. Shales generally exhibit orthorhombic elastic symmetry due to the presence of vertical fractures and horizontal fine layering. We propose a novel anisotropic extended elastic impedance (AEEI) inversion approach for in situ stress estimates in shale gas reservoirs with weakly orthorhombic symmetry. Considering an orthorhombic model formed by a system of aligned vertical fractures embedded in a vertical transverse isotropic (VTI) background rock, we first derive the in situ stress expression by combining poroelastic Hooke’s law and linear slip theory. Next, we deduce a linearized PP-wave reflection coefficient as a function of fluid bulk modulus, vertical effective stress-sensitive parameter, dry-rock P- and S-wave moduli, density, dry normal and tangential weaknesses of the VTI background, and dry normal and tangential weaknesses of vertical fractures. To estimate in situ stress from the observed seismic data, we derive the AEEI and Fourier coefficients (FCs) expression and establish a three-step AEEI inversion workflow involving (1) Bayesian seismic inversion for intercept impedance, gradient impedance, and curvature impedance estimates; (2) estimating model parameters using the FCs of AEEI; and (3) estimating in situ stress using the inverted model parameters. The synthetic examples demonstrate that in situ stress can be reliably estimated even with moderate noise. Test on a real data set implies that the proposed method can generate reasonable results of in situ stress that are helpful for optimizing the horizontal drilling and hydraulic fracture stimulations of shale gas reservoirs.

The Frequency Dependence of Poisson’s Ratio in Fluid-Saturated Rocks

We investigated the frequency dependence of Poisson’s ratio ν in partially/fully fluid-saturated rocks. Based on one dominant fluid flow mechanism at each condition, we theoretically summarized that (1) when a rock is partially saturated or transits from drained state to undrained state at full saturation, ν increases monotonously with frequency, and the associated attenuation 1 / Q ν is positive with one peak. (2) When the rock transits from undrained state to unrelaxed state at full saturation, there are three cases: 1) ν increases monotonously with frequency and has positive 1 / Q ν with one peak, 2) ν keeps constant with frequency and has no attenuation, 3) and ν decreases monotonously with frequency and has negative 1 / Q ν with one peak. In this condition, the dependence is influenced by the concentrations of stiff and soft pores, the aspect ratio of soft pores, and the pore fluid bulk modulus. (3) When it comes to the transition from drained state to unrelaxed state at full saturation, ν can exhibit two shapes with frequency: 1) step shape with two positive attenuation peaks and 2) bell shape with one positive attenuation peak and one negative attenuation peak. Then, we conducted a numerical example to indicate the effect of influence factors (the concentrations of stiff and soft pores, the aspect ratio of soft pores, and the pore fluid bulk modulus) on Poisson’s ratio from undrained state to unrelaxed state, and validated the theoretical analysis by the published experimental data. In addition, based on 1 / Q ν , we reanalyzed and validated the relationship between different attenuation modes (i.e., bulk attenuation 1 / Q K , P-wave attenuation 1 / Q P , extensional attenuation 1 / Q E , and S-wave attenuation 1 / Q S ): (1) when 1 / Q ν is positive, the relationship between them is 1 / Q K > 1 / Q P > 1 / Q E > 1 / Q S ; when 1 / Q ν is 0, the relationship between them is 1 / Q K = 1 / Q P = 1 / Q E = 1 / Q S ; and when 1 / Q ν is negative, the relationship between them is 1 / Q K < 1 / Q P < 1 / Q E < 1 / Q S . The relationship between different attenuation modes does not depend on saturation state (partial or full saturation) or ν but on 1 / Q ν . This research provides the frequency dependence of Poisson’s ratio in partially/fully saturated rocks, which helps better understand Poisson’s ratio at different frequencies and saturation states and can be used to improve the accuracy of geophysical data interpretation, such as lithology identification, hydrocarbon characterization in conventional reservoir, and brittleness evaluation of shale/tight sandstones in unconventional reservoir.

Theoretical modelling of seismic dispersion, attenuation and frequency-dependent anisotropy in a fluid-saturated porous rock with intersecting fractures

SUMMARYDetection of intersecting open fractures is an important task in many earth science domains. To quantify the seismic responses in the fluid-saturated porous rock with intersecting open fractures, we develop a theoretical model based on Biot's equations of dynamic poroelasticity. The seismic dispersion, attenuation and frequency-dependent anisotropy due to joint effects of fracture–background wave-induced fluid flow (FB-WIFF), and fracture–fracture wave-induced fluid flow (FF-WIFF), as well as elastic scattering are investigated. The numerical results on a fluid-saturated porous and fractured sandstone show that the characteristic frequency of FF-WIFF is controlled by fracture connectivity and fluid viscosity. Variations of fracture connectivity and fluid viscosity may result in the coupling of FF-WIFF with FB-WIFF or elastic scattering. When the fracture connectivity tends to zero, the FF-WIFF vanishes and FB-WIFF becomes most significant. Besides fracture connectivity and fluid viscosity, the fracture geometry and fluid bulk modulus also affect the magnitudes of these three mechanisms and their interplay. Due to effects of these three mechanisms, the P-wave anisotropy varies greatly with frequencies. Furthermore, the fracture intersection angle also influences the P-wave anisotropy significantly. Our model agrees well with previous models in the frequency limits and for the special case with parallel fractures. Since our model incorporates the effects of FF-WIFF, it has a great potential to be applied in the detection for effective fracture networks for fluid flow.

Société de Biomécanique Young Investigator Award 2021: Numerical investigation of the time-dependent stress–strain mechanical behaviour of skeletal muscle tissue in the context of pressure ulcer prevention

BackgroundPressure-induced tissue strain is one major pathway for Pressure Ulcer development and, especially, Deep Tissue Injury. Biomechanical investigation of the time-dependent stress–strain mechanical behaviour of skeletal muscle tissue is therefore essential. In the literature, a viscoelastic formulation is generally assumed for the experimental characterization of skeletal muscles, with the limitation that the underlying physical mechanisms that give rise to the time dependent stress–strain behaviour are not known. The objective of this study is to explore the capability of poroelasticity to reproduce the apparent viscoelastic behaviour of passive muscle tissue under confined compression. MethodsExperimental stress-relaxation response of 31 cylindrical porcine samples tested under fast and slow confined compression by Vaidya and collaborators were used. An axisymmetric Finite Element model was developed in ABAQUS and, for each sample a one-to-one inverse analysis was performed to calibrate the specimen-specific constitutive parameters, namely, the drained Young's modulus, the void ratio, hydraulic permeability, the Poisson's ratio, the solid grain's and fluid's bulk moduli. FindingsThe peak stress and consolidation were recovered for most of the samples (N=25) by the poroelastic model (normalised root-mean-square error ≤0.03 for fast and slow confined compression conditions). InterpretationThe strength of the proposed model is its fewer number of variables (N=6 for the proposed poroelastic model versus N=18 for the viscohyperelastic model proposed by Vaidya and collaborators). The incorporation of poroelasticity to clinical models of Pessure Ulcer formation could lead to more precise and mechanistic explorations of soft tissue injury risk factors.

A mathematical model of cavitation behaviour in a single-ended magnetorheological damper: experimental validation

This paper proposes a mathematical model of the cavitation behavior to occur in a single-ended magnetorheological (MR) damper (MRD), and the effectiveness of the model is validated through the comparison with experimental results. Several causes of the cavitation behavior of MRD are discussed with different conditions of the initial pressure of the gas chamber and the piston stroke speed. The model to capture the cavitation behavior is then formulated considering differential equations for gas volume, internal pressure, ideal gas law, and bulk modulus of MR fluid. To calculate the flow rate, which is difficult to solve from the differential equations, the model is approximated as a nondimensional equation the parameters of the yield stress and pole length. Subsequently, the field-dependent damping force of MRD is computed using the gaseous cavitation model and nondimensional equation. To validate the proposed cavitation model, a single-ended MRD is designed, manufactured, and tested. It is observed that the damping force characteristics under cavitation are revealed to be much different from those under regular operation without cavitation. More specifically, it is hard to calculate the dissipation energy and hysteretic damping due to highly nonlinear characteristics with respect to the stroke and velocity. However, the proposed model can fairly capture the cavitation behavior showing an excellent agreement between simulation and experiment. In this work, to confirm the internal influence of MRD by cavitation, which is difficult to confirm experimentally, the changes of the pressure distribution and the gas-to-liquid volume ratio are analyzed through the simulation of the nondimensional equation. In addition, the bubbles representing the cavitation behavior are visually observed from the lower chamber of MRD.

Anisotropic Poroelasticity and AVAZ Inversion for in Situ Stress Estimate in Fractured Shale-Gas Reservoirs

Knowledge of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress is of great significance for hydraulic fracture stimulating of unconventional reservoirs. Quantitative estimation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress, especially in the far field, is a major challenge. This research mainly focuses on developing a novel Bayesian AVAZ inversion approach to estimate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress of fractured shale-gas reservoirs with horizontal transversely isotropic (HTI) symmetry. Using the generalized Hooke’s law and Schoenberg’s linear slip model, we first deduce the anisotropic horizontal stress equation in an HTI medium formed by a single set of vertically aligned fractures embedded in an isotropic background rock. Based on the critical porosity model, we then obtain the saturated stiffness matrix with vertical effective stress-sensitive parameters and dry fracture weaknesses using the relationship between vertical effective stress and porosity. Combining the scattering function and the perturbed stiffness matrix, we deduce a linearized PP-wave reflection coefficient as a function of fluid bulk modulus, vertical effective stress-sensitive parameter, dry-rock P- and S-wave moduli, density, and fracture density. Finally, we propose a novel Bayesian amplitude variation with azimuth (AVAZ) inversion constrained by the Cauchy regularization and low-frequency regularization to estimate these model parameters, which are further used to calculate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress. The analysis of synthetic data demonstrates that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress can be reasonably estimated even with moderate noise. A field data set test reveals that the inversion results agree well with the well log interpretation, and the proposed approach can generate meaningful results that are useful for seismic identification of potential fracturing barriers during fracturing stimulation of shale-gas reservoirs.