Gravity Gradient Research Articles

Noise assessment of cold atom interferometry gradiometry observations

Over the past few decades, the development of interferometric techniques utilizing cold atoms has significantly advanced, particularly in the realm of gravimetry. This paper explores the characterization of noise within atomic gradiometry systems, which is essential for enhancing measurement accuracy as new methodologies and tools are introduced. Specifically, we examine the noise components using advanced techniques such as least squares component estimation and Allan variance analysis. Applying these sophisticated methods to two distinct datasets, our results robustly indicate that white noise is the predominant and most resilient type of noise in gravity gradiometry measurements based on atomic interferometry, with variances of σP12=2.97×10−8 for dataset 1 and σP22=5.66×10−7 for dataset 2. The predominance of white noise simplifies the system dynamics and enhances the reliability of the estimation methods. Furthermore, our calculations yield an uncertainty in the gravity gradient resolution of σ Γ1 = 0.13 E (where 1 E = 10−9 s−2) and σ Γ2 = 3.55 E over time, underscoring the high precision achievable with current interferometric techniques.

Crustal silica content of East China: A seismological perspective and its significance

East China has experienced multiple periods of tectonic movements, which have contributed to the composition and rheological properties of its present crust. Estimating the composition of the crust is crucial for understanding the tectonic processes. Based on the teleseismic receiver functions with data from the National Seismic Network of China, we applied the H-κ-c method to obtain the crustal bulk VP/VS ratio and to constrain the SiO2 content in the crust. We estimated the SiO2 content to range from 50.89 wt% to 73.51 wt%, with an average value of 65.87 wt%, indicating a predominant felsic composition of the East China's crust. Our study suggests that the North-South Gravity Lineament (NSGL) is an approximate delimitator of the felsic and mafic crust in East China, hinting at a widespread deficiency of mafic lower crust in the east of the NSGL. The mafic crust is extensively distributed in the Taihang orogenic region (TSR) and WuLingshan gravity gradient belts (WLG), particularly in the Datong volcanic area, which manifests the mantle materials intraplating. The scatteredly distributed mafic crust at the east of the NSGL is mainly concentrated in the southeast coast of China and in the intersection region of the Tanlu Fault (TLF) and Sulu region. In Sulu region, the TLF may primarily provide a channel for the thermal intrusion from the underlying mantle lithosphere, which has increased the mafic content. The Pacific/Philippine Sea plate subduction has triggered a significant amount of crust-mantle material exchange below southeast China that resulted in a high degree of mafic crustal composition.

Разработка и валидация ПИТС-индекса для оценки тяжести синдрома последствий интенсивной терапии: описательное проспективное несравнительное когортное исследование

INTRODUCTION: The post intensive care syndrome (PICS) is a clinical manifestation of “learned non-use” phenomena, resulting from the application of metabolic rest strategies such as bed rest, sedation, and ventilation. This leads to prolonged, multifocal dysfunction in vital functions. The lack of standardized diagnostic methods for this syndrome limits its study. For the diagnosis of PICS, we combined 15 signs of vital function disturbances that have the greatest impact on quality of life, with a scoring value of 0.5–1. The sum of these symptoms has been designated as the PICS-index. OBJECTIVE: To develop a scale for calculating the PICS-index with stratification of its values into three ranges of severity: mild, moderate and severe. MATERIALS AND METHODS: A descriptive, prospective, and unmatched cohort study was conducted on 169 patients who had a high dependency on medical care and were admitted to a rehabilitation center between 2022 and 2023. RESULTS: Using the PICS-index, showed that 100 % of patients who spent more than 72 hours in the ICU during the early recovery period (up to 6 months) developed PICS, with a predominance of moderate cases (35–52 %). The proportion of severe PICS forms reached its maximum (37 %) between 3 and 6 months. In the late recovery phase (more than 6 months), mild PICS forms were predominant (45 %). The severity of PICS was influenced by the duration of ICU stay and the timing of mechanical ventilation, but not the age of patients. CONCLUSIONS: The PICS-index is a clinically effective tool for the multidisciplinary diagnosis and assessment of the severity and progression of PICS. Using the PICS-index, we have devised a method for evaluating the predominance of specific signs. The most common signs are disturbance of the gravitational gradient, polyneuromyopathy, dysphagia. This confirms the essence of PICS as a clinical manifestation of the "learned non-use" phenomenon.

Impacts of Digital Elevation Model Elevation Error on Terrain Gravity Field Calculations: A Case Study in the Wudalianchi Airborne Gravity Gradiometer Test Site, China

Accurate Digital Elevation Models (DEMs) are essential for precise terrain gravity field calculations, which are critical in gravity field modeling, airborne gravimeter and gradiometer calibration, and geophysical inversion. This study evaluates the accuracy of various satellite DEMs by comparing them with a LiDAR DEM at the Wudalianchi test site, a location requiring ultra-accurate terrain gravity fields. Major DEM error sources, particularly those related to vegetation, were identified and corrected using a least squares method that integrates canopy height, vegetation cover, NDVI, and airborne LiDAR DEM data. The impact of DEM vegetation errors on terrain gravity anomalies and gravity gradients was quantified using a partitioned adaptive gravity forward-modeling method at different measurement heights. The results indicate that the TanDEM-X DEM and AW3D30 DEM exhibit the highest vertical accuracy among the satellite DEMs evaluated in the Wudalianchi area. Vegetation significantly affects DEM accuracy, with vegetation-related errors causing an impact of approximately 0.17 mGal (RMS) on surface gravity anomalies. This effect is more pronounced in densely vegetated and volcanic regions. At 100 m above the surface and at an altitude of 1 km, vegetation height affects gravity anomalies by approximately 0.12 mGal and 0.07 mGal, respectively. Additionally, vegetation height impacts the vertical gravity gradient at 100 m above the surface by approximately 4.20 E (RMS), with errors up to 48.84 E over vegetation covered areas. The findings underscore the critical importance of using DEMs with vegetation errors removed for high-precision terrain gravity and gravity gradient modeling, particularly in applications such as airborne gravimeter and gradiometer calibration.

Investigation of Churn and Annular Flow Transition Boundaries in Vertical Upward Gas-Water Two-Phase Flow

ABSTRACT Flow patterns in two-phase flow in vertical upward pipes are crucial for predicting phase distribution, flow transitions, and stability. Existing studies on churn to annular flow transitions primarily focus on flow rates, physical parameters, pressure, and temperature, often overlooking pipe diameter. This study addresses this gap using a multiphase pipe flow experimental setup to examine transitions under varying pipe diameters, gas flow rates, and liquid flow rates. Results show that with a constant liquid flow rate, increasing pipe diameter requires a higher gas flow rate to form annular flow, leading to greater pressure gradients and smoother fluctuations. Conversely, with a constant gas flow rate, larger pipe diameters decrease the liquid flow rate needed for annular flow, resulting in smaller pressure gradients and smoother fluctuations. Smaller pipe diameters make annular flow formation easier, with smaller pressure gradients but larger fluctuations. A novel churn-annular flow transition boundary model was developed, considering factors like gas shear force, surface tension, gravity, viscous force, and pressure gradient. Evaluated with 804 experimental data sets, the new model’s prediction accuracy exceeds 98.19%, outperforming traditional models with 53.92%-86.27% accuracy, enhancing applicability across varying diameters.

EDT demonstration for keeping low altitude orbit using carbon nanotube tether

This paper describes an electrodynamic tether system (EDT). Conductive tethers are used to overcome atmospheric resistance in low orbit and extend orbit life using EDT propulsion. First, based on the conceptual design, the feasibility of the EDT system has been evaluated through simulation. Especially, the induced electromotive force of the conductive tether has been clarified, the amount of current required to keep the orbital altitude, and the requirements for each device. Then, feasibilities of the key mission devices have been evaluated experimentally. Those are the conductive tether, the electron emission device, and the tether extension control mechanism. The conductive tether is planned to be made of carbon nanotubes, which are attracting attention as a new material, and its properties are evaluated. The electron-emitting device has been evaluated as a device that can be mounted on a micro/nano satellite. Tether extension mechanism was developed in our past project, then it should be evaluated with the purpose of adapting to conductive tether systems. Based on the evaluation, the orbital demonstration mission plan using a micro/nano satellite has been introduced. The mission is as follows. The conductive tether is extended and stabilized by the gravity gradient. By emitting electrons from an electron emitter mounted on a high-altitude satellite and collecting electrons on bare tether at low-altitude end, a current is generated from the high-altitude side to the low-altitude side. According to Fleming's law, the Lorentz force is generated in the direction of the orbiting accelerated, so it can be propelled in the upward direction of the orbit.

Revealing the dynamics of stagnant rings of third-grade fluid film with heat transfer in the presence of surface tension

In many engineering applications, including coating and lubrication operations, analyzing the temperature behavior of thin film flows on a vertically upward-moving tube is crucial to improving predictive models. This paper examines a steady third-grade fluid film flow with a surface tension gradient on a vertical tube. The mechanisms responsible for the fluid motion are upward tube motion, gravity, and surface tension gradient. This analysis focuses on heat transfer and stagnant ring dynamics. The formulated highly nonlinear ordinary differential equations are solved using the Adomian decomposition method. The conditions for stagnant rings and uniform film thickness are attained and discussed. The inverse capillary number C, Stokes number St, Deborah number De, and Brinkman number Br emerged as flow control parameters. The temperature of the fluid film rises with an increase in the C, St, De, and Br, whereas it decreases with an increase in thermal diffusion rate. The radius of stagnant rings tends to shrink by the increase in C, St, and De. When the value of De is high, third-grade fluid behaves like solids; only free drainage happens with smaller radius stagnant rings and high temperatures. A comparison between Newtonian and third-grade fluids regarding surface tension, velocity, temperature, stationary rings, and fluid film thickness is also provided.

The Distribution of Surface Heat Flow on the Tibetan Plateau Revealed by Data‐Driven Methods

AbstractSurface heat flow (SHF) serves as a vital parameter for assessing the heat transfer from deep Earth to the surface, which can provide crucial insights into internal geodynamic processes. As the “roof of the world,” the Tibetan Plateau and its tectonic evolution are highly important in terms of global climate change and geodynamic study. However, a comprehensive understanding of the SHF distribution across most regions of the Tibetan Plateau is limited due to sparse measurement data. To surmount this limitation, a spatially intelligent approach has been developed: The geographically neural network weighted regression with enhanced interpretability (EI‐GNNWR). This method integrates spatial heterogeneity and nonlinear interactions between geophysical and geological factors to predict the SHF distribution across the Tibetan Plateau. In this study, the EI‐GNNWR model is used to accurately predict SHF across the entire region. After evaluating the effectiveness and interpretability of the EI‐GNNWR model, our results demonstrate that medium to high SHF values are predominantly concentrated in the southern, northeastern, and southeastern sectors of the Tibetan Plateau. These observations suggest that the formation of zones with high SHF values may be strongly influenced by the Moho depth, ridges, topography, and average curvature of satellite gravity gradients. Especially, higher SHF values may indicate more profound geodynamic activities such as collisional orogeny, shear deformation zones, or lithospheric extension. These findings offer novel insights into the spatial patterns of SHF and deepen our understanding of the geothermal formation mechanisms driven by underlying tectonic activities.

Modeling Gravity Gradients from Surface Gravity Anomaly Data

Gravity gradients are useful to characterize near mass anomalies since they are much more sensitive to short wavelength anomalies than gravitational accelerations. Estimating gravity gradients from surface gravity data is based on numerical implementations of solutions to geodetic boundary value problem for determination of disturbing potential. One of methods to solve this problem is least-squares collocation which is basically based on data and a defined covariance function. This study deals with estimating gravity gradient tensor components from along track surface gravity anomaly data. The Least-Squares Collocation solution is based on a stationary local covariance function defined for the disturbing potential which allows upward continuation of the observations to a desired altitude. The modeling method is evaluated in using Earth Gravitational Model 2008(EGM2008) and real airborne gravity gradiometry data collected over Southern Texas, Oklahoma region. The results show that modeled gravity gradients estimated in both on the ground and at a certain altitude have basically good agreement with EGM08 gradients. Modeled gradients including horizontal components in the east-west direction exhibit some discrepancies in comparison to the airborne gradiometry data, which may be attributed to some measurement errors in the gradient data.

Stress heterogeneity in the eastern Tibetan Plateau and implications for the present-day plateau expansion

The eastward expansion of the Tibetan Plateau has resulted in different earthquake types in the eastern Tibetan Plateau, but the mechanism remains unclear. Here, we construct a three-dimensional visco-elastoplastic finite element model considering the topography to investigate the influence of fault geometry and rheological heterogeneity on stress fields. In our best-fitting model, the minimum principal stress is nearly vertical around the southern Huya fault zone, which is adjacent to the Longmen Shan fault zone, due to the significant mid-lower crust lateral rheological heterogeneity, and the thrust stress regime accounts for the reverse fault and thrust-dominated earthquakes. In this scenario, the eastward horizontal motion of the mid-lower crust is obstructed and facilitates thrust faulting, suggesting the limited eastward expansion of the Tibetan Plateau. In contrast, the northern Huya fault zone, one of the terminal branches of the East Kunlun fault, accommodates the continuous eastward extrusion of the East Kunlun fault, where the stress regime under a more homogenized crust favors the strike-slip faulting process, along with the dominant strike-slip earthquakes. Moreover, the best-fitting of stress regime explains the thrust-dominated 2008 Ms. 8.0 Wenchuan and 2013 Ms. 7.0 Lushan earthquakes on the Longmen Shan fault zone. Combining geophysical and geodetic observations and model analyses, we propose that the hybrid deformation mode in the eastern Tibetan Plateau is accommodated by upper crustal shear and thrusting deformation and mid-lower crustal thickening driven by the gravitational potential energy gradient. Our results elucidate the mechanism for differences in strong historical earthquakes and, more importantly, isolate the effect of fault geometry from those of heterogeneous viscosity on crustal deformation and stress heterogeneity in the eastern Tibetan Plateau.

Modeling and control of perturbation torques and mass distribution impact on a tethered system for a 12U CubeSat in sun-synchronous orbit

Tethered CubeSat systems are characterized by momentum exchange and gravity gradient stabilization, and encounter stability challenges, especially with shorter tethers in highly inclined and eccentric orbits. This study investigates the libration dynamics of a 12U CubeSat in its stowed configuration, which separates into two satellites connected via tether, throughout its deployment, station-keeping, and retrieval phases. Two deployed configurations, symmetrical 6U-6U, and asymmetrical 8U-4U connected by a non-conductive tether with a maximum length of 100m are analyzed. The study accounts for perturbations including Earth’s oblateness, atmospheric drag, solar radiation pressure, and lunisolar gravitation, modeling the tether as a rigid, extendable rod and the satellites as lumped masses. The translational and rotational dynamics are decoupled, assuming the system’s center of mass follows an unperturbed Keplerian Sun-synchronous orbit at altitudes between 400 and 600km. A tension control strategy based on Lyapunov’s direct method and supplemented by external actuation torques is explored. Results show high orbit eccentricity significantly affects the maximum in-plane libration angle. The 6U-6U system experiences smaller disturbance torques but greater tether tension variations than the 8U-4U system. Both configurations exhibit larger in-plane oscillations compared to near-zero out-of-plane oscillations, nonetheless, eclipse passages exacerbate the out-of-plane libration of the 8U-4U system. Relative stability is maintained during deployment, however, retrieval is chaotic, with notable oscillations in libration angles and tether tension. The tension control strategy effectively dampens oscillations during retrieval but loses effectiveness as tether tension approaches zero at retrieval’s conclusion. External actuation torques, ranging from 1-2mNm for deployment to 15mNm for retrieval, complement tension control.

High-Efficiency Forward Modeling of Gravitational Fields in Spherical Harmonic Domain with Application to Lunar Topography Correction

Gravity forward modeling as a basic tool has been widely used for topography correction and 3D density inversion. The source region is usually discretized into tesseroids (i.e., spherical prisms) to consider the influence of the curvature of planets in global or large-scale problems. Traditional gravity forward modeling methods in spherical coordinates, including the Taylor expansion and Gaussian–Legendre quadrature, are all based on spatial domains, which mostly have low computational efficiency. This study proposes a high-efficiency forward modeling method of gravitational fields in the spherical harmonic domain, in which the gravity anomalies and gradient tensors can be expressed as spherical harmonic synthesis forms of spherical harmonic coefficients of 3D density distribution. A homogeneous spherical shell model is used to test its effectiveness compared with traditional spatial domain methods. It demonstrates that the computational efficiency of the proposed spherical harmonic domain method is improved by four orders of magnitude with a similar level of computational accuracy compared with the optimized 3D GLQ method. The test also shows that the computational time of the proposed method is not affected by the observation height. Finally, the proposed forward method is applied to the topography correction of the Moon. The results show that the gravity response of the topography obtained with our method is close to that of the optimized 3D GLQ method and is also consistent with previous results.

Improved frequency shift compensation technique and pulse sequence for multi-loop atom interference experiments

With the rapid development of atom interference technology, multi-loop atom interferometers are widely used in the high-precision measurement of various physical constants and testing of various gravity-related effects. However, in ground-based multi-loop atom interference experiments, the systematic error contribution by classical effects is an important factor that affects the experimental measurement accuracy and gravitational effect detection. Based on this, we used the atomic wave-function evolution-phase accumulation method to provide a high-order interference phase of multi-loop atom interferometers in an inhomogeneous gravitational field containing Earth’s rotation. Furthermore, we propose a new scheme that combines optimised frequency-shift compensation technology with an improved pulse sequence to eliminate systematic errors due to the gravity gradient, Earth’s rotation, and their coupling effect with the pulse duration, as well as the coupling effect of laser detuning with the pulse duration. This study lays a theoretical foundation for experiments on multi-loop atom interferometers with higher precision.

Perspective on Quantum Sensors from Basic Research to Commercial Applications

Quantum sensors represent a new generation of sensors with improved precision, accuracy, stability, and robustness to environmental effects compared to their classical predecessors. After decades of laboratory development, several types of quantum sensors are now commercially available or are part-way through the commercialization process. This paper provides a brief description of the operation of a selection of quantum sensors that employ the principles of atom–light interactions and discusses progress toward packaging those sensors into products. This paper covers quantum inertial and gravitational sensors, including gyroscopes, accelerometers, gravimeters, and gravity gradiometers that employ atom interferometry, nuclear magnetic resonance gyroscopes, atomic and spin-defect magnetometers, and Rydberg electric field sensors.

The Benefits of Future Quantum Accelerometers for Satellite Gravimetry

AbstractWe investigate the benefits of future quantum accelerometers based on cold atom interferometry (CAI) on current and upcoming satellite gravity mission concepts. These mission concepts include satellite‐to‐satellite tracking (SST) in a single‐pair (GRACE‐like) and double‐pair constellation as well as satellite gravity gradiometry (SGG, single satellite, GOCE‐like). Regarding instruments, four scenarios are considered: current‐generation electrostatic (GRACE‐, GOCE‐like), next‐generation electrostatic, conservative hybrid/CAI and optimistic hybrid/CAI. For SST, it is shown that temporal aliasing poses currently the dominating error source in simulated global gravity field solutions independent of the investigated instrument and constellation. To still quantify the advantages of CAI instruments on the gravity functional itself, additional simulations are performed where the impact of temporal aliasing is synthetically reduced. When neglecting temporal aliasing, future accelerometers in conjunction with future ranging instruments can substantially improve the retrieval performance of the Earth's gravity field (depending on instrument and constellation). These simulation results are further investigated regarding possible benefit for hydrological use cases where these improvements can also be observed (when omitting temporal aliasing). For SGG, it is demonstrated that, with realistic instrument assumptions, one is still mostly insensitive to time‐variable gravity and not competitive with the SST principle. However, due to the improved instrument sensitivity of quantum gradiometers compared to the GOCE mission, static gravity field solutions can be improved significantly.

Vertical gravity gradient in volcano monitoring – In situ measured or theoretical? (Campi Flegrei study)

We analyse the vertical gravity gradient (VGG) properties at calderas using the Campi Flegrei (CF) site in Italy. In situ observed VGG values can depart significantly from the theoretical (normal) value of −308.6 μGal/m, particularly in areas of rugged relief. It is assumed that in sufficiently flat areas, the effect of geology, i.e., of the subsurface density heterogeneities, on VGG could prevail over the effect of terrain (topography), which can subsequently be neglected. With respect to the CF caldera, which is often considered as ‘reasonably flat area’, according to our findings the effect of topography on VGG is usually underestimated, while the effect of deeper geology is overestimated. We model the effect of the near topography on VGG at CF and subsequently verify the results of modelling by in situ observations to support our predictions. The results show that, in terms of VGG, the topographic relief plays a more significant role than the assumed geological sources even at ‘flat’ calderas such as CF. For a better understanding, in addition to CF, we analyse the effect of deeper geological sources on VGG also in the territory of Slovakia using a detailed gravimetric database of Slovakia. As a result, we question the use of in situ observed VGG values when processing and interpreting observed time-lapse gravity changes in volcanic areas accompanied by surface deformation.

Observation and Local Prediction of the Vertical Gravity Gradient: Review Paper

Observation and Local Prediction of the Vertical Gravity Gradient: Review Paper

Vibration-attitude integrated control of large membrane diffraction space telescope

The development of telescopes with large apertures is driven by the increasing demand for higher imaging resolution and coverage. Large membrane diffraction space telescopes are particularly attractive due to their lightweight nature, ease of folding and unfolding, and high-precision imaging capabilities. However, their substantial size and flexibility make them susceptible to long-duration, low-frequency vibrations during attitude maneuvers, which degrade attitude and imaging accuracy and risk instrument damage. Consequently, an urgent need exists for a high-precision integrated control scheme. In this paper, the integrated control of vibration and attitude in a large space telescope is studied using cable actuators and control moment gyroscopes (CMGs). Since cables can only withstand limited tension, the unilateral and constrained characteristics of the control input are taken into account during the control design process. Firstly, a rigid-flexible coupling dynamic model of the space telescope is established using the velocity variation principle with hybrid coordinates. Additionally, a two-body astrodynamic model is developed to assess the impact of gravity gradient torque. Next, combining the computational torque method and H∞ control theory, an integrated control scheme is proposed, which achieves vibration and attitude control during maneuvers. Then, this paper optimizes the cable actuator positions via the discrete particle swarm optimization (PSO) algorithm and plans various attitude maneuver paths. Finally, numerical simulations are adopted to demonstrate the effectiveness of the control scheme. The results show that this integrated control scheme swiftly suppresses structural vibrations during attitude maneuvers, enabling high-precision attitude control of the space telescope.

Gravity and magnetism of southern Africa in relation to Craton structures and belts

The general tectonic structural architecture of southern Africa is an ensemble comprised of Kalahari Craton, a vast range of mosaics of the best-preserved geological Belts, and exposed crustal blocks. In the region, demarcation of geological boundaries, tracing of magnetic and gravitational bodies, and detrending lineaments are essential to understanding the structural limits. The study presents the prevalent gravitational and magnetic investigation of major geological Belts, enhancing edges of Cratons, and evaluation of surface invariants to improve the geological understanding and evaluate the correlation of structures with the general structural tectonic framework. The utilized filter methods in the gravity maps are the Directional derivatives along x, y, and z-directions, the Modulus tensor, the Total horizontal derivative, and the Tilt angle techniques.The phase-based filters used in the magnetic section include the Total horizontal derivative, Analytical signal, Tilt angle method, Tilt of total horizontal derivative, Theta method, Horizontal tilt angle method, Enhanced tilt filter, and Enhanced total derivative of the tilt angle. The Directional derivatives of the gravity field and the Modulus of the gravity gradient tensor demarcate the boundaries of geological structures. The Total horizontal derivative method gives an immediate and easy-to-read image of the linear structures and fault systems. The signal cluster on the Analytical signal and Tilt angle maps delineate the boundaries of causative geological sources. The Theta map is comparable with the enhanced total derivative of the tilt angle. The Tilt of the total horizontal derivative, Theta map, and Enhanced tilt filter accentuate the traces of magnetic bodies, including the Cratons. The study finds regional lineaments surrounding the Cratons and concentrated along the geological Belts. The approach of using joint elegant filters is effective and increases the certainty of the interpretation.

Study on the influence of rock vertical heterogeneity on the vertical extension law of hydraulic fractures

The vertical expansion of fractures during hydraulic fracturing in low-permeability bottom-water oil and gas reservoirs is a crucial factor that significantly influences stimulation effectiveness. Fractures propagate concurrently in three dimensions; however, as operation time for fracturing and fracture length increase, there is a gradual reduction in fracture height. In the absence of a barrier layer or with inadequate strength and thickness, fractures may extend vertically or oscillate through it leading to an “unyielding” fractured state that impedes successful operations. This study introduces a mathematical model delineating the vertical expansion of hydraulic fractures (HFs) while computing stress intensity factors at upper and lower tips of each individual fracture. We use numerical methods to examine how vertical heterogeneity within reservoir rocks impacts laws governing vertical fracture propagation while conducting sensitivity analysis for making informed decisions on design parameters for hydraulic fracturing operations. The research results indicate that an escalation in stress disparity and increased toughness results in diminished height for HFs. Hydraulic fracturing augments fracture lengths along their respective axes. Nevertheless, if gap stress disparity surpasses net fluid pressure, a reduction in fracture length is observed. With an increasing ratio between local stress gradient versus fluid gravity gradient, HFs persistently advance vertically across cap and bed formations.