Aperture Profile Research Articles

The Influence of the Aperture Stop Fractal Shape of an Optical System on the Illuminance Distribution

In the article the options for the application of aperture shapes with fractal properties in the design of optical systems are considered. Calculations of mathematical models of point spread functions of a diffraction-limited optical system are performed. The diffraction patterns of the light distribution in these systems are presented, and the point spread functions are considered for various shapes of the aperture stop. Analytical expressions are obtained for the light distribution depending on the pupil shape, which can be used to control the process of image formation. The pupil shape, which has the shape of an equilateral triangle, is chosen as the basic one, and the shape of the pupil as a "Koch snowflake" curve is also considered. Using the Fraunhofer integral, the dependences of the distribution of the spectral density of the complex amplitude on the aperture located on an opaque screen are derived in the Fraunhofer approximation and under the condition of illumination by a plane monochromatic wave. Using the relationship with the complex amplitude, the sought-for intensity distribution in the plane of the diffraction pattern is obtained. Taking into account the simplifications adopted in this article, the solution of the Fraunhofer integral is found, by setting the integration limits, depending on: the selected aperture profile, the coordinate system chosen for it, and the position of nodal points in this system

A New High Aperture Efficiency Transmitarray Antenna Based on Huygens Metasurfaces

In this article, a new transmitarray (TA) antenna is designed and fabricated based on Huygens metasurfaces (HMSs). An analytical method has been used to achieve the distribution of surface parameters equivalent to HMS. In this method, the required equations for computing the surface parameters are derived using ray tracing, generalized sheet transition conditions, and two conditions of local power consumption and local impedance equalization. In order to implement these surface parameters, wire and loop unit cells are utilized, which are exposed to the radiation transversely. The array of these unit cells offers high aperture efficiency broadband TA antennas in low aperture profile. The fabricated TA antenna with a low aperture thickness of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.2\lambda $ </tex-math></inline-formula> exhibits the bandwidth, sidelobe level (SLL), and cross-polarization level of 14%, −19 dB, and −19 dB, respectively. In addition, the high aperture efficiency of 65% is reached with this TA, which shows great improvement in comparison to previous works.

Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

Electrochemically exfoliated functionalized graphene flakes: Facile synthesis, 3rd order optical nonlinearity and optical limiting response

High quality graphene production is prerequisite for good performance in nonlinear optics application such as fast optical communications, all-optical switching, and optical limiting. Graphene journey begins with the method of synthesizing graphene which need to be simple, fast, and environmentally friendly. Hence, we introduce the method by exfoliating graphite by electrochemical route to produce good quality functionalized graphene for various nonlinear optics application. In this work, functionalized graphene flakes are synthesis by using two different electrodes; furnaced graphite rod (Gr-FG) and non-furnaced graphite electrode (Gr-NFG). Visual inspection on the synthesized Gr-FG and Gr-NFG show dark murky color solutions give the impression of high yield functionalized graphene flakes. Further observation under transmission electron microscopy (TEM), selected area electron diffraction (SAED), and UV–vis and Raman analysis confirm the good quality functionalized graphene structure. The nonlinear optical behavior of the functionalized graphene was accessed via Z-scan technique with 637 nm laser source operating in continuous mode with simultaneous monitoring of the close and open aperture signal. Close aperture profile of Gr-FG and Gr-NFG display nonlinear refraction, whereas open aperture profile shows reverse saturable absorption (RSA). Equation fitting reveals higher n2 magnitude for Gr-FG compared to Gr-NFG, but the later possess higher magnitude of β. Further analysis on the 3rd order of optical nonlinearity by z-scan technique reveal the admirable value at the range of 10−6 esu. Optical limiting performance conducted via transmittance-based measurement shows superior limiting of Gr-NFG compared to Gr-FG.

A compact and high sensitive photometer based on axial multi-reflection inside metal capillary: Application to detecting nitrite of nanomolar concentration

As for photometry analysis, increasing the optical-path (OP) via multi-reflection is an effective method for improving the detection sensitivity. In this paper, a photometer based on axial multi-reflection between two end-faces of capillary was proposed and investigated. The probe light can be back and forth reflected inside the capillary, by employing apertured silver-mirrors at both ends of a metal-capillary. With designed aperture profile (i.e., a central disc-aperture and a peripheral ring-aperture) of the mirror, a 4.2-fold enhancement on OP was realized (i.e., OP is 4.2 times the capillary length). Moreover, the photometer was applied to detecting nitrite with a detection limit of 1 nM, which exhibits 8.6~15 fold improvement compared with that of previous photometers with similar capillary length and the same chromogenic reagent without sample enrichment. The photometer features compact, low cost and easy fabrication.

Antenna Aperture Synthesis Using Mode-Converting Metasurfaces

Mode-converting metasurfaces are passive, lossless devices that can be designed to transform a set of incident modes to a desired set of transmitted modes. In this paper, mode-converting metasurfaces are utilized to synthesize arbitrary, azimuthally-invariant TM apertures. The methods presented in this work can be used to design antennas that can meet specific near-field and far-field criteria unlike most metasurfaces which solely manipulate the far field. The proposed antennas consist of a coaxially-excited, radial cavity topped by the mode-converting metasurface. The main role of the mode-converting metasurface is to establish the desired aperture by converting the modal distribution of the excitation to that of desired aperture. Its secondary role is to impedance match the coaxial feed to the radial cavity. Using modal network theory, an optimization-based design procedure is developed to synthesize the proposed metasurface antennas. The admittance profiles of the electric sheets that comprise the metasurface are optimized to establish the desired aperture profile. To illustrate the design procedure, a radial Gaussian beam antenna is synthesized at 10 GHz and its performance is verified using a full wave electromagnetic solver. The proposed antenna has a height and weight advantage over Gaussian beam horn antennas.

Investigation of the aperture-dependent flow characteristics of a supercritical carbon dioxide-induced fracture under high-temperature and high-pressure conditions: A numerical study

A numerical model was developed to simulate heat extraction and flow behaviour of fractures under deep geological conditions by incorporating a real fracture morphology induced by supercritical carbon dioxide (ScCO2) injection. 3-D scanning technology was used to generate the geometry of the fracture in the model in order to provide a realistic reflection of the tortuous and rough fractures induced by ScCO2 fracturing. The fracture exhibits a tortuosity of 1.44 and a JRC (Joint Roughness Coefficient) of 14.6. The flow characteristics along the ScCO2-induced fracture was evaluated under different in-situ stresses and at high-temperature of 100 °C, by integrating coupled thermal-hydraulic (TH) processes into the model. In addition, the influence of shearing on aperture alteration and flow enhancement was evaluated using the model. The model accurately simulates the permeability characteristics at 100 °C and under varying confining pressures (20–60 MPa) and injection pressures (1–8 MPa) with more than 87% accuracy compared with the corresponding experimental results. The results indicate a non-linear pressure gradient over the highly varying aperture profile along the fracture. Furthermore, the orientation of flow channels and the streamline network heavily depends upon the aperture profile. The fluid injected at 20 °C into a narrow, tortuous fracture is heated to 100 °C while flowing along a distance 63% of the fracture length towards the outlet, whereas the heat transfer efficiency is decreased by 26% with the increase of injection pressure from 1 MPa to 8 MPa. The increase in confining pressure up to 60 MPa results in reduction of fracture permeability by 90% due to stress-induced fracture closure. Shearing of the fracture surface by 5% results in increments in mean aperture, outlet flowrate and average fracture permeability by 44%, 45% and 113%, respectively.

A radial hydraulic fracture with pressure-dependent leak-off

This paper investigates the problem of a radial (or a penny-shaped) hydraulic fracture propagating in a permeable reservoir. In particular, we consider the fluid exchange between the crack and ambient porous media as a pressure-dependent process. In most of the existing models, the fluid exchange process is represented as one-dimensional pressure-independent leak-off described by Carter’s law. We modify this mechanism by including the dependence of the fluid-exchange rate on the fluid pressure inside the fracture. The proposed approach allows the liquid not only to flow out of the fracture but also to leak-in along the region adjacent to the fracture front. A complete model for hydraulic fracturing involves several physical processes that determine the crack propagation, namely, brittle rock failure, elastic equilibrium of rock, viscous fluid flow inside the fracture channel, and fluid exchange. In order to resolve these phenomena near the fracture front in the numerical scheme, we utilise a special asymptotic multi-scale model for the near-tip region, which is used as a propagation condition for the finite fracture. The main aspect of the analysis is a comparison of fracture characteristics such as aperture profile, radius and others calculated via the proposed model with the results of the radial fracture model that assumes standard pressure-independent fluid exchange mechanism governed by Carter’s law. Based on the comparison, we determine parameter ranges, for which the effect of the pressure-dependent fluid exchange mechanism is essential, and, on the other hand, we outline the zones for which Carter’s leak-off model provides accurate results. Finally, by using an approximate solution for the radial fracture with Carter’s leak-off, we obtain estimates of the effect in the entire parametric space, which allows one to evaluate the influence of the pressure-dependent leak-off for any problem parameters.

Numerical Analysis of Complex Fracture Propagation Under Temporary Plugging Conditions in a Naturally Fractured Reservoir

Summary Temporary plugging during fracturing operations has become an efficient method to create a complex fracture network in tight reservoirs with natural fractures (NFs). Accurate prediction of the network-propagation process plays a critical role in plugging- and fracturing-parameter optimization. In this paper, the interaction between one single hydraulic fracture (HF) within a temporary plugging segment and multiple NFs was simulated using a complex fracture-development model. A new opening criterion for an NF penetrated by a nonorthogonal HF already was implemented to identify the dominant propagation direction of a HF under plugging conditions. Fracture displacements and the induced-stress field were determined by the 3D displacement-discontinuity method, and the Gauss-Jordan and Levenberg-Marquardt methods were combined to handle the coupling between rock mechanics and fluid flow numerically. Numerical results demonstrate that the opening and development of an NF are mainly dominated by its approaching angle and relative location. For a certain NF crossed by an HF within the plugging segment, the HF tends to propagate along the NF branch inclined in the main HF direction. The farther the interaction position is away from the HF tip, the easier the NF with an approaching angle less than 30° or larger than 150° can be open, and the outcome will be opposite if the approaching angle is larger than 45° or less than 135°. Higher approaching angle and plugging strength is necessary for expanding the position scope of NF that can be opened around HF. Under the effect of plugging, fluid pressure in the HF plummets at the beginning of the NF opening and keeps decreasing until the NF extends for a certain distance or encounters secondary NFs. Fluid-pressure drop occurs mainly in the unturned NF, together with the width of unturned NF being significantly lower than that of the turned NF and HF. Sensitivity analysis shows that the main factors, such as geometry, aperture profile, and fluid-pressure distribution, affecting the network progress under the temporary plugging conditions are the horizontal differential stress, NF position, approaching angle, plugging time, and plugging-segment length. The simulation results provide critical insight into complex fracture-propagation progress under temporary plugging conditions, which should serve as guidelines for well choosing and plugging optimization in temporary plugging fracturing.

An integrated analytical and experimental study of contact acoustic nonlinearity at rough interfaces of fatigue cracks

This paper presents an analytical model that describes the relationship between interface stiffness and separation of rough interfaces such as fatigue cracks. The contact acoustic nonlinearity at the interface is simulated by a quasi-static model based on Hertzian contact theory. The model is validated using the results of dynamic acoustoelastic testing (DAET) with a Rayleigh wave probe on fatigue cracks in two aluminum alloy samples. One novel aspect of this work is that all the required geometrical parameters for the model are acquired directly from the aperture profile of real cracks extracted from their scanning electron microscopy (SEM) images. Also, the separation of the crack faces during dynamic perturbation is independently measured using a 3D laser Doppler vibrometer (LDV). In addition, using DAET allows for an unprecedented direct comparison between experimental and analytical results. This is because unlike conventional nonlinear ultrasonic tests that are based on measuring the amplitude of higher harmonics, DAET outputs directly the strain-dependency of transmission and time delay of ultrasonic waves propagating across the interface. The model is found in good qualitative agreement with the experimental results, although the predictions tend to underestimate the variation of transmission coefficient and time delay. We conduct a sensitivity analysis to investigate the influence of different assumptions and simplifications on model predictions.

FEM analysis of birefringence, dispersion and nonlinearity of graphene coated photonic crystal fiber

Abstract A novel D-shaped photonic crystal fiber (D-PCF) is proposed to concurrently achieve ultra-high nonlinearity, birefringence, and numerical aperture profile. Also, the model consists of a single core design with a series of air holes that are arranged in a rectangular manner. However, the entire analysis is finished through by COMSOL Multiphysics software with 235430 mesh elements using the finite element method (FEM) for 0.1 μm–1.5 μm wavelength. In fact, the evanescent fields of the structure help to the simulation process and that exhibit the performance profile with an ultra-high birefringence of 0.17, the outstanding high numerical aperture of 0.80 and ultra-high nonlinearity of 7.1 × 1024 W−1Km−1. From the designed model, an outstanding negative dispersion of −6533 ps/(nm.km) has been derived which is highly beneficial for optical communication. Nonetheless, graphene is used as a core material and silica is used in the core-cladding region to compensate for unwanted radiation. Simplicity in geometry makes it more acceptable in the fabrication process. To sum up, the overall performance of the PCF makes it more appropriate in optical propagation of non-telecom and telecom applications.

Compact Vertically Polarized Omnidirectional Ultra-Wideband Antenna and its Band-Notched Filtering Application

This paper first proposes a 3D printed, compact, low-profile, ultra-wideband (UWB) antenna with vertically polarized omnidirectional radiation characteristics. The antenna is composed of one triangular feeding structure, one triangular shorted structure, a common capacitive top hat, and two parasitic metallic pins. Compared with traditional UWB antennas with symmetrical configuration, the proposed UWB antenna is a three-dimensional version of inverted “F”with asymmetrical structure. The proposed antenna can realize compact aperture and low profile (0.19λ L × 0.19λ L × 0.06λ L , where λ L is the free-space wavelength of the lowest operating frequency). Two parasitic pins are utilized to enhance the omnidirectionality of the antenna in the azimuth plane. The impedance bandwidth is greater than 10:1 (2.96-30 GHz) for VSWR <; 3, and the bandwidth with horizontal radiation pattern ripple better than ±5 dB is more than 4.9:1 (2.96-14.5 GHz). Second, this UWB design is modified to realize dual bands with a deep band-notch by adding a split gap inside the antenna. The proposed band-notched antenna is designed with the bandwidth of 2.4-2.5 and 5-10 GHz for VSWR <; 2 with good horizontal radiation pattern omnidirectionality, which can be used for 2.4/5 GHz WLAN applications. The prototypes of the proposed UWB and band-notched antennas are fabricated by metallic 3D printing, and the simulated and measured results meet well.

Role of proppant distribution on the evolution of hydraulic fracture conductivity

The residual opening of fluid-driven fractures is conditioned by proppant distribution and has a significant impact on fracture conductivity - a key parameter to determine fluid production rate and well performance. A 2D model follows the evolution of the residual aperture profile and conductivity of fractures partially/fully filled with proppant packs. The model accommodates the mechanical response of proppant packs in response to closure of arbitrarily rough fractures and the evolution of proppant embedment. The numerical model is validated against existing models and an analytic solution. Proppant may accumulate in a bank at the fracture base during slick water fracturing, and as hydraulic pressure is released, an arched zone forms at the top of the proppant bank as a result of partial closure of the overlaying unpropped fracture. The width and height of the arched zone decreases as the fluid pressure declines, and is further reduced where low concentrations of proppant fill the fracture or where the formation is highly compressible. This high-conductivity arch represents a preferential flow channel and significantly influences the distribution of fluid transport and overall fracture transmissivity. However, elevated compacting stresses and evolving proppant embedment at the top of the settled proppant bed reduce the aperture and diminish the effectiveness of this highly-conductive zone, with time. Two-dimensional analyses are performed on the fractures created by channel fracturing, showing that the open channels formed between proppant pillars dramatically improve fracture transmissivity if they are maintained throughout the lifetime of the fracture. However, for a fixed proppant pillar height, a large proppant pillar spacing results in the premature closure of the flow channels, while a small spacing narrows the existing channels. Such a model provides a rational means to design optimal distribution of the proppant pillars using deformation moduli of the host to control pillar deformation and flexural spans of the fracture wall.

A direct effect of perception on action when grasping a cup

Affordances represent features of an object that trigger specific actions. Here we tested whether the presence and orientation of a handle on a cup could bias grasping movements towards it in conditions where subjects were explicitly told to ignore the handle. We quantified the grip aperture profile of twelve healthy participants instructed to grasp a cup from its body while it either had no handle, a handle pointing towards, or away from the grasping hand (3 ‘move’ conditions, with large grip aperture). To ensure the smaller grip aperture afforded by the handle was implicitly processed, we interspersed trials in which participants had to grasp the cup from its handle or a handle not attached to a cup with a small grip aperture. We found that grip aperture was smaller in the presence of a handle in the ‘move’ conditions, independently of its orientation. Our finding, of an effect of the handle during the execution of a grasp action, extends previous evidence of such an influence measured during motor preparation using simple reaction times. It suggests that the specific action elicited by an object’s attribute can affect movement performance in a sustained manner throughout movement execution.

Coupled Viscous Fluid Flow and Joint Deformation Analysis for Grout Injection in a Rock Joint

Fluid flow modeling is a major area of interest within the field of rock mechanics. The main objective of this study is to gain insight into the performance of grout injection inside jointed rock masses by numerical modeling of grout flow through a single rock joint. Grout flow has been widely simulated using non-Newtonian Bingham fluid characterized by two main parameters of dynamic viscosity and shear yield strength both of which are time dependent. The increasing value of these properties with injection time will apparently affect the parameters representing the grouting performance including grout penetration length and volumetric injection rate. In addition, through hydromechanical coupling a mutual influence between the injection pressure from the one side and the joint opening/closing behavior and the aperture profile variation on the other side is anticipated. This is capable of producing a considerable impact on grout spread within the rock joints. In this study based on the Bingham fluid model, a series of numerical analysis has been conducted using UDEC to simulate the flow of viscous grout in a single rock joint with smooth parallel surfaces. In these analyses, the time-dependent evolution of the grout fluid properties and the hydromechanical coupling have been considered to investigate their impact on grouting performance. In order to verify the validity of these simulations, the results of analyses including the grout penetration length and the injection flow rate were compared with a well-known analytical solution which is available for the simple case of constant grout properties and non-coupled hydraulic analysis. The comparison demonstrated that the grout penetration length can be overestimated when the time-dependent hardening of grout material is not considered. Moreover, due to the HM coupling, it was shown that the joint opening induced by injection pressure may have a considerable increasing impression on the values of penetration length and injected grout volume.

Modelling of mineral precipitation in fractures with variable aperture

Fractures act as a highly permeable conduit for flow in naturally fractured reservoir. Geo-chemical reactions inside fractures may lead to partial or complete filling of pore spaces inside fractures over time. This reduction in fracture aperture directly affects its capability to transport fluid. The current study presents a 1-D mathematical model to simulate the geochemical filling of natural fractures. The fracture walls have been represented by simple mathematical functions to reflect variable aperture of natural fractures. Mass transfer through convection/diffusion and mineral precipitation due to precipitation/dissolution reaction were solved as a simplified mathematical representation of the actual processes. Precipitation reaction is coupled with mass transport by the fluid to ensure mass conservation of reacting components. For simplification, calcite precipitation in fracture has been modelled. The effect of pressure drop, diffusion constant, type of fracture aperture profile on evolution of fracture aperture was simulated in this study. Comparison is made between increasing, decreasing and constant fracture profiles to understand their effect on fracture evolution. This study aims to present a methodology to model variable fractures and study its effect on deposition inside the fracture. The model shows that the simple parallel model over predicts the precipitation that occurs in a fracture.

Hand-use and tool-use in grasping control

The goal of this study was to elucidate the underlying mechanisms of hand and tool grasping control. We assumed that there is a single principle-governing grasping control irrespective of its effectors and that the degree of prior experience of the effector determines the smoothness of aperture control. Eight participants performed a reach-to-grasp task with four different effectors: index finger and thumb, middle finger and thumb, chopsticks, and a scissor-like tool. Although we employed different effectors with large mechanical variations and different degrees of prior use, maximum grip aperture was scaled as a function of object size and appeared at almost the same timing in all four types of grasping movements. Moreover, reaching time did not substantially differ among grasping conditions. However, plateau duration of the aperture profile differed by effector. Plateau duration was the longest in the unfamiliar scissor-like tool grasping condition. There was no difference between the unfamiliar hand-use grasp with the thumb and the middle finger and the familiar tool-grasp with chopsticks. The familiar hand-use grasp with the thumb and the index finger had the shortest plateau duration. These results supported the idea that there is an effector-independent continuity between hand-use and tool-use in motor control as a function of prior degree of use, rather than the conventionally assumed dichotomy between them.

Hydrodynamic coupling in microbially mediated fracture mineralization: Formation of self-organized groundwater flow channels

[1] Evidence of fossilized microorganisms embedded within mineral veins and mineral-filled fractures has been observed in a wide range of geological environments. Microorganisms can act as sites for mineral nucleation and also contribute to mineral precipitation by inducing local geochemical changes. In this study, we explore fundamental controls on microbially induced mineralization in rock fractures. Specifically, we systematically investigate the influence of hydrodynamics (velocity, flow rate, and aperture) on microbially mediated calcite precipitation. Our experimental results demonstrate that a feedback mechanism exists between the gradual reduction in fracture aperture due to precipitation, and its effect on the local fluid velocity. This feedback results in mineral-fill distributions that focus flow into a small number of self-organizing channels that remain open, ultimately controlling the final aperture profile that governs flow within the fracture. This hydrodynamic coupling can explain field observations of discrete groundwater flow channeling within fracture-fill mineral geometries where strong evidence of microbial activity is reported.

Enhancing the Strength of an Optical Trap by Truncation

Optical traps (tweezers) are beginning to be used with increasing efficacy in diverse studies in the biological and biomedical sciences. We report here results of a systematic study aimed at enhancing the efficiency with which dielectric (transparent) materials can be optically trapped. Specifically, we investigate how truncation of the incident laser beam affects the strength of an optical trap in the presence of a circular aperture. Apertures of various sizes have been used by us to alter the beam radius, thereby changing the effective numerical aperture and intensity profile. We observe significant enhancement of the radial and axial trap stiffness when an aperture is used to truncate the beam compared to when no aperture was used, keeping incident laser power constant. Enhancement in trap stiffness persists even when the beam intensity profile is modulated. The possibility of applying truncation to multiple traps is explored; to this end a wire mesh is utilized to produce multiple trapping that also alters the effective numerical aperture. The use of a mesh leads to reduction in trap stiffness compared to the case when no wire mesh is used. Our findings lead to a simple-to-implement and inexpensive method of significantly enhancing optical trapping efficiency under a wide range of circumstances.

Effects of overpressure variations on fracture apertures and fluid transport

For an isolated rock fracture in a homogeneous, isotropic rock subject to constant internal fluid overpressure or constant external driving stress, the ideal opening-displacement (aperture) profile or variation is that of a flat ellipse. For many mineral veins, dykes, tension fractures, normal faults, and other rock fractures, however, the opening-displacements show irregular aperture (thickness) variations very different from that of a flat ellipse. Here we present field data on typical fracture-aperture variations, as well as new numerical and analytical models to explain these data. We present the overpressure variation by Fourier cosine series, a very flexible method that can be used to model abrupt overpressure and driving-stress variations in vertical and lateral sections for fractures of various sizes and types. We calculate the opening-displacements of typical hydrofractures, and discuss the results with reference to mineral veins and dykes. We also present numerical models showing that when a fracture dissects layer, or parts of a single layer (such as a lava flow), with different stiffnesses (Young's moduli), the opening-displacement may show irregular variation even when the overpressure is constant. From the cubic law, the volumetric flow rates in the large-aperture segments of the mineral veins and dykes discussed in the paper may have been as much as 3–5-times the flow rates in the small-aperture segments. We propose that differences in volumetric flow rates are related to irregular opening-displacement variations in feeder-dykes is one principal reason for the development of crater cones, a universal feature of volcanic fissures.