Cylindrical Matrix Research Articles

Two-dimensional counter-current capillary imbibition of a wetting phase into a partially submerged porous cylindrical matrix block

The purpose of this study is to address the two-dimensional counter-current capillary dominant imbibition of a wetting phase into a water-wet porous cylindrical matrix block partially submerged in the wetting phase. A two-dimensional unsteady-state diffusion equation is used to model the process. The governing equation is solved using a combination of the Laplace and the finite Fourier sine transforms to find and analyze the solutions for the normalized water saturation and the volume of the imbibed wetting phase. The results reveal that the volume of the imbibed wetting phase and the capillary diffusion shape factor for a partially submerged matrix block are significantly lower compared to those of a fully submerged matrix block, highlighting the overestimation of imbibed volume using available models based on full immersion in the wetting phase. It has been observed that the volume of the imbibed wetting phase increases over time until reaching a state of equilibrium. In the case of a partially submerged matrix block, the shape factor is inversely proportional to the square root of time (σ ∼ 1/t) during the early time and decreases sharply as the imbibed wetting phase reaches an equilibrium. In the case of a fully submerged matrix block, the shape factor is inversely proportional to the square root of time (σ ∼ 1/t) during the early time and later reaches a pseudo-steady-state value. The proposed model, along with the findings obtained, advances our understanding of capillary imbibition in porous media.

Numerical study of TRISO particles with random size and location distribution in cuboid and cylindrical matrix: A validation for two-regime heat conduction model

During reactor operation, the precise size values or specific size distribution of tri-structural isotropic (TRISO) fuel particles randomly dispersed within the fully ceramic microencapsulated (FCM) fuel elements are often unknown. To investigate the impact of the random distribution of positions and sizes of TRISO particles within a cuboid or cylindrical matrix on the temperature distribution of fuel elements and to establish a practical heat conduction model, TRISO fuel particles with randomly distributed positions, diameters following Gaussian or uniform distributions are generated using relevant algorithms. A cuboid and a cylindrical TRISO fuel element are modeled in ANSYS, and a three-dimensional temperature field distribution of the fuel element is obtained through numerical simulation. The probability density of the maximum temperature of the fuel element is calculated and fitted. Finally, the applicability of the two-regime heat conduction model over a broader range is validated based on the simulated data. The conclusions of this study provide reference information for predicting the temperature distribution of FCM fuel elements containing TRISO particles with random size and location distribution.

Color and surface roughness alterations of bulk-fill resin composites submitted to simulated toothbrushing with whitening dentifrices

Aim: To evaluate the surface roughness and color stability of bulk-fill resin composites after simulated toothbrushing with whitening dentifrices. The radioactive/relative dentin abrasion (RDA) and radioactive/relative enamel abrasion (REA) of dentifrices were also assessed. Methods: Specimens (n=10) of Tetric N Ceram Bulk Fill (TNCB), Filtek One Bulk Fill (FOB) resin composites, and Z100(Control) were prepared using a cylindrical Teflon matrix. Surface roughness (Ra, μm) was assessed by a roughness meter and the color evaluations (ΔEab , ΔE00 , WID ) were performed using a digital spectrophotometer based on the CIELAB system. Three measurements were performed per sample, before and after simulated toothbrushing with 3D Oral-B White Perfection (3DW) and Black is White (BW) dentifrices. The abrasivity (REA and RDA values) of the used dentifrices was also determined by the Hefferren abrasivity test. Results: The Ra values increased significantly in all resin composites after 3DW and BW toothbrushing. The acceptable threshold color varied among resin composites, and TNCB and Z100 presented the highest ΔEab and ΔE00 for BW dentifrice. The 3DW dentifrice was significantly more abrasive than BW dentifrice on enamel and dentin. Conclusions: simulated toothbrushing with tested whitening dentifrices increased the surface roughness at acceptable levels. The Tetric N Ceram Bulk-fill and Z100 composite showed the highest color alteration in BW. 3D White Perfection dentifrice was more abrasive on dentin and enamel than Black is White.

3D-printed composites with a programmable response to tension and torsion: A design guide

3D-printed composite structures employed in applications such as soft robotics, actuators, and artificial muscles typically experience different loading and deformation modes throughout their life cycle. Tailoring the mechanical properties and responses to those modes in an independent manner can significantly enhance performance. In this work, we propose a helical fiber embedded cylindrical matrix as a platform to independently tune the behavior under two loading modes — uniaxial extension without twist and torsion at a constant length. Using finite elements (FE), we study the influence of the helical geometry, the volume fraction, and the moduli of the matrix and the fiber on the stiffness and the constitutive behavior. Our findings reveal that the responses to tension and torsion can be programmed by an appropriate choice of geometrical and material parameters. For example, we show that one can achieve composites with the same stiffness and behavior under uniaxial extension but a wide range of responses to torsion. As a design guide, the wide range of properties and potential strengths that can be obtained in these composites are mapped. To demonstrate the validity of this design, composites embedded with a helical fiber were 3D-printed and tested under tension and torsion. The experimentally observed trends agree with the FE simulations. The design and the insights from this work can be used to enhance and optimize the performance of structures in various applications. For example, this composite design can be used in lattice structures to tune the local response.

Влияние продолжительности механической активации на структуру и механические свойства порошкового композита W – Cu

Mixtures of tungsten-copper powders (88.2 wt. % W + 11.8 wt. % Cu) with different granulometric composition after mechanical activation at different process times were studied. Before mechanical activation, W particles have a size of about 4 μm, and large agglomerates of tungsten particles are ~20 μm. The structures and elemental composition of W-Cu powder samples before and after mechanical activation (1, 3, 5, and 10 min) were studied by scanning electron microscopy and X-ray microanalysis. The strength (compression) tests of pressed samples in a cylindrical matrix 12 mm in diameter were carried out on an Instron 3382 testing machine. Powder compaction curves were obtained, the dependences of the pressing stress on density were plotted, and the dependence of the sample density on the mechanical activation time was determined. As a result of strength tests, the dependences of stress on strain were obtained. The highest density of the mixture is achieved when the duration of treatment in the mechanical activator is 5 min. A further increase in the processing time of the mixture leads to the grinding of the powder and the destruction of the agglomerates. The average particle size of the powder decreases, which leads to a deterioration in the compaction of the samples. At the same time, with an increase in the treatment time to 10 min, the distribution of copper particles over the sample improves simultaneously with a decrease in the size of its particles to 100 – 400 nm, which, apparently, affects the increase in strength.

Superplastic forming of EK61 nickel-based superalloy with ultrafine-grained structure

The formation of an ultrafine-grained structure in the heat-resistant nickel-based EK61 superalloy makes it possible to realize the effect of low-temperature superplasticity. This is important for the development of resource-saving technologies for forming hollow products of complex configuration. The study of mechanical properties of the nickel-based EK61 superalloy with ultrafine-grained structure under conditions of uniaxial tension and compression at temperatures of 650 – 850°C and strain rates of 10−3 – 10−4 s−1 was carried out. The process of superplastic forming (SPF) under conditions of low-temperature superplasticity of ultrafine-grained sheet blanks from the EK61 superalloy using cylindrical matrix has been studied. It was found that after SPF, a homogeneous equiaxed ultrafine-grained duplex structure is preserved in the samples.

Experimental simulation of volumetric compacts formation from spherical waxy elements

The growth in metal intensity of industrial production and the volume of consumption of finished metal products determine the relevance of development and research of energy efficient technological processes aimed at reducing costs by reducing the number of operations while maintaining product performance. In mechanical engineering, the problem of obtaining blanks with increased dimensional and geometric accuracy and complex configuration is solved by using a common method of investment casting. Expansion of the use of such technological approach to produce blanks in mechanical engineering is hindered by a number of physical phenomena associated with the thermal expansion of investment and ceramic materials, which leads to an increase in the product final cost. A significant number of defect-forming factors can be eliminated by applying an innovative solution consisting in the formation of porous removable models by compacting mixtures based on waxy materials. This solves the problem of material shrinkage and increases the crack resistance of ceramic molds, which significantly reduces the share of machining in the overall volume of technological operations. Technical tests of the new method have revealed the reason why the machining of castings cannot be completely eliminated at present. The problem mainly lies in elastic response of compacted material of the model mixture, which, in some cases, affects the increase in the compacts size. This paper considers the effect of initial packing of spherical-shaped elements simulating one- and two-component model mixtures on the stress-strain state of a powder body subjected to unilateral compaction in a rigid cylindrical matrix to technologically justified density values. The results of the experiment are presented in the form of stress-strain relations. Preferable conditions of compact formation with minimal values of elastic response of the compacted material are considered.

Mechanical Properties of SDR™ and Biodentine™ as Dentin Replacement Materials: An In Vitro Study

The aim of this study is to evaluate the shear bond strength of nanohybrid composite resins (NCR) and microhybrid composite resins (MCR) placed over three different dentin replacement materials: SDR-Smart Dentin Replacement™, Biodentine™, and resin-modified glass ionomer cement (RMGIC). Thirty acrylic blocks (50 mm × 20 mm × 15 mm), each with a central hole, were prepared, which were randomly distributed into three equal groups, each corresponding to one of the three dentin replacement materials-SDR, Biodentine, and RMGIC. The central holes were then filled with these materials. After setting and application of the respective adhesive system, the specimens were further divided into two subgroups each of NCR or MCR. The respective composite material was then applied to the dentin replacement materials using a cylindrical plastic matrix. Shear bond strength was tested on a universal testing machine (Instron 3366), at a crosshead speed of 1.0 mm/minute. SDR attained consistently higher shear bond strength (means: 21.18, 22.19 Mpa) values than RMGIC and Biodentine, with both types of composite resins (MCR and NCR), which were statistically significant (p <0.001). When considering the means of the shear bond strength measurements obtained by the two types of the composite resin, no significant difference (p <0.05) was found between them with all three types of dentin replacement materials. There is no significant difference in the bond strengths achieved between MCR and NCR to the different dentine replacement materials. Hence, either type of composite resin may be expected to achieve similar bond strengths to the underlying substrate. SDR™ is a suitable dentine replacement material for placing below a composite resin veneer as it can achieve immediate higher bond strengths. SDR can be used as an effective bulk fill material in deep dentinal caries which can be capped with composite resins.

Experimental investigation on the diffusion property of different form coal: Implication for the selection of CO2 storage reservoir

Gas diffusion is an important migration behavior for coal, and the diffusion ability serves as an important parameter to evaluate whether CH4 extraction in a coal reservoir is effective and whether the reservoir is suitable for CO2 storage. Granular coal, whose morphology differs greatly from that of coal mass in actual gas extraction, remains the research object of most current studies on the diffusion characteristics of coal. Consequently, problems occur in the application of the experimental results to engineering practice. In this study, first, different shape units and gas diffusion paths were proposed for coal matrix. Subsequently, an improved Uni-pore spherical diffusion model considering the gas loss at the beginning of the experiment was established based on the Uni-pore spherical diffusion model of granular coal. Then, a gas diffusion model suitable for coal mass was established based on Fick’s Second Law of Diffusion and the cylindrical matrix shape. In addition, a contrastive analysis was conducted on the desorption laws of coal with different forms under different adsorption equilibrium pressures. It is found that granular coal corresponds to a great gas desorption amount than coal mass. The gas diffusion coefficients of coal under different conditions were calculated based on the gas desorption curves and the corresponding diffusion models. To better grasp the diffusion coefficient law of coal with different forms, coal mass and 0.2–0.25 mm granular coal at the same matrix scale were contrasted. The results suggest that they are in the same order of magnitude and differ insignificantly. Finally, the primary reason why the desorption amounts of coal mass and 0.2–0.25 mm granular coal differ notably was revealed: The shape factor of spherical matrix of granular coal is larger than that of coal mass matrix. Furthermore, considering that different matrix shape factors caused by the structures of granular coal and coal mass lead to notably different gas adsorption capacities, this study held that tectonic coal seams may be more suitable for CO2 storage than intact coal seams under the same condition.

Dimensional stability of stored extended-pour irreversible hydrocolloids materials

The purpose of this study was to evaluate the linear dimensional stability of four extended-pour irreversible hydrocolloids (EPIHs). Material and Methods: Five samples per material (Cavex ColorChange, Cavex Orthotrace, Jeltrate Plus, and Orthoprint) were prepared following the manufacturers’ instructions. The samples were prepared using a cylindrical matrix coupled with a nylon-polyamide ring. Two parallel, 25-mm equidistant lines were made on its surface following ANSI/American Dental Association (ADA) Specification 18 for plaster reproducibility and compatibility and Specification 19 for linear dimensional change. The samples were stored in an environment with a relative humidity of 70% (± 3) and temperature of 28°C (± 2). Photo images were obtained using a digital camera to record images for 120 hours, with a standardized distance of 80cm between the lens and the specimen. Adobe Photoshop CS3 software was used for the measurement of the recorded images. The measurements refer to the equivalent distance between the two parallel lines printed on the samples. Data were analyzed using one-way analysis of variance (ANOVA) and Tukey’s test for multiple comparisons between the means of the groups. Results: There was no statistically significant difference (p &gt; 0.05) when EPIHs were compared at the same time of evaluation. Orthoprint, Cavex Orthotrace, and Cavex Colorchange presented with least dimensional stability up to 24 hours (p &gt; 0.05) of storage, followed by Jeltrate Plus (48 hours). Conclusions: Storage of EPIHs for more than 24 hours for Cavex ColorChange and 48 hours for others EPIHs studied produces significant dimensional changes in the impressions stored at a humidity of 70% (± 3) and temperature of 28°C (± 2). Extended storage times produce large dimensional changes.

Long-term bond strength of glass fiber post to composite resin does not depend on surface treatment with silane coupling agent or universal adhesive

This study evaluated the effect of glass fiber post (GFP) surface treatment with or without a silane coupling agent, associated with adhesive systems with or without silane in their composition, on the long-term bond strength to resin composite. Sixty GFPs (Whitepost DC no. 2, FGM) were cleaned using 70% ethanol, and randomly divided into two groups, either with or without pretreatment with silane (Ceramic Primer, 3 M ESPE), for 60 s. Each group was further subdivided into three groups, according to the adhesive system applied over the GFP (n = 10): UAS (Scotchbond Universal, 3 M ESPE), E&R – etch-and-rinse adhesive system (Adper Single Bond Plus, 3 M ESPE), and a control group with no adhesive system. The GFPs were centralized in a cylindrical matrix, and a nanoparticulate resin composite (Filtek Supreme Ultra, 3 M ESPE) was inserted and photoactivated. After 48 h, the samples were sectioned in slices, two of which were immediately (48 h) submitted to the push-out bond strength test, and another two, after 6 months of storage in water. The failure mode was evaluated with a stereomicroscope under 40× magnification, and classified using scores. Three-way ANOVA (α = 0.05) showed that the triple interaction of pretreatment with silane, use of adhesive and time period were significant (p = 0.019). After 48 h, the application of silane resulted in a significant increase in bond strength values, regardless of the adhesive application (p = 0.013), but this effect was no longer observed after 6 months (p = 0.677). Application of UAS did not influence bond strength values, regardless of silane application or storage time (p > 0.05). As for E&R, significantly higher bond strength was observed after 6 months, compared with 48 h (p = 0.040). When post silanization was performed with no adhesive system, there was no difference in bond strength between 48 h and 6 months storage time (p > 0.05). When there was neither silanization, nor an adhesive system, a significantly higher bond strength was observed after 6 months, compared with 48 h (p < 0.05). Most of the failure types were classified as mixed in both of the water storage time periods. The long-term bond strength of the GFP to composite resin did not depend on post surface treatment with silane, or application of either a UAS or a E&R system.

Omnidirectional cylindrical graphene-based Bragg fiber in terahertz

The optical properties of a graphene-based cylindrical photonic crystal are theoretically investigated based on the generalized cylindrical transfer matrix method to include the out-of-plane propagation of the cylindrical waves. It is found that Maxwell's equations have solutions for both E- and H-polarized waves only at the azimuthal mode number m = 1 . We study the reflectance spectrum of the structure to obtain its photonic bandgaps. We show that the reflection spectrum, in addition to the usual Bragg gaps, contains an omnidirectional bandgap at low terahertz frequencies for both E and H-polarized cylindrical waves. The robustness of this bandgap under random variations of geometrical parameters is investigated. This few-period structure is an ideal candidate for guiding waves at the low terahertz region because of its weak absorption and transmission losses. In contrast to the usual Bragg gap, the results show that in both polarizations, the width of graphene-produces bandgap reaches its maximum value when the refractive indices of the two cylindrical shells have their lowest possible values. The width of this unusual omnidirectional bandgap increases by decreasing the thicknesses of the cylindrical shells and increasing the chemical potential of the graphene monolayers.

Deformable scintillation dosimeter: II. Real-time simultaneous measurements of dose and tracking of deformation vector fields

Anatomical motion and deformation pose challenges to the understanding of the delivered dose distribution during radiotherapy treatments. Hence, deformable image registration (DIR) algorithms are increasingly used to map contours and dose distributions from one image set to another. However, the lack of validation tools slows their clinical adoption, despite their commercial availability. This work presents a novel water-equivalent deformable dosimeter that simultaneously measures the dose distribution and tracks deformation vector fields (DVF). The dosimeter in made of an array of 19 scintillating fiber detectors embedded in a cylindrical elastomer matrix. It is imaged by two pairs of stereoscopic cameras tracking the position and angulation of the scintillators, while measuring the dose. The resulting system provides a precision of 0.3 mm on DVF measurements. The dosimeter was irradiated with 5 × 3, 4 × 3 and 3 × 3 cm2 6 MV photon beams in both fixed and deformed conditions. The measured DVF was compared to the one computed with a DIR algorithm (Plastimatch). The deviations between the computed and measured DVFs was below 1.5 mm. As for dose measurements, the dosimeter acquired the dose distribution in fixed and deformed conditions within 1% of the treatment planning system calculation and complementary dose validation using the Hyperscint dosimetry system. Using the demonstrated qualities of scintillating detectors, we developed a real-time, water-equivalent deformable dosimeter. Given it’s sensor tracking position precision and dose measurements accuracy, the developed detector is a promising tools for the validation of DIR algorithms as well as dose distribution measurements under fixed and deformed conditions.

Theoretical study of diffusional release of a dispersed solute from a hollow cylindrical polymeric matrix

An exact solution for the release kinetic of a solute from inside a hollow cylindrical polymeric matrix into an infinite medium has been developed, when the initial concentration of the solute (A) is greater than the solubility limit (Cs). A combination of analytical and numerical methods was used to calculate the solute concentration profile and the release rate. The model was developed for two different strategies: 1) the release medium is flowing through the hollow cylinder in which boundary layer may be neglected, 2) the release medium inside the hollow cylinder is stagnant and the boundary layer should be considered. The results indicated that the release profiles were close to the constant release rate after a typical burst release. Also, the release profile from a solid cylindrical matrix and a hollow cylinder was compared. The results indicated that the hollow cylindrical matrix is a promising carrier when zero-order release is desirable. The present model demonstrates the potential of the hollow cylindrical matrix as a suitable geometry for sustained drug delivery systems.

Анализ формирования геометрии изделий при вытяжке квадратных заготовок на профильной матрице

There is considered square billet stretching in a cylindrical matrix. To ensure minimization of an inequality effect of billet ends in the matrix there was used a leading-in profile area. The investigation of matrix profile impact upon geometry of products manufactured is carried out. The efficient geometrical ratios of a tool allowing the achievement of a lesser inequality in the geometry of parts end area are revealed.

Degree of conversion and microhardness of resin cements photoactivated through glass ceramic.

BackgroundTo assess whether glass-ceramic shade, thickness and translucency affect degree of conversion (DC) and Knoop microhardness (KHN) of resin cements photoactivated using light-emitting diode (LED) or quartz-tungsten-halogen (QTH) units. Material and MethodsGlass-ceramic blocks were cut (2, 3 and 4mm) and sintered. For DC FT Raman spectroscopy (n=3), film specimens of cements (RelyX ARC, U200, Veneer, C&B) were obtained. For KHN test (n=3), cements were inserted in cylindrical matrix and covered by polyester strip. Specimens were photoactivated (30s) using LED or QTH according to each group: direct photoactivation (DP), interposing ceramic specimens or no photoactivation (NP). Data were analysed by ANOVA and Tukey’s test, Kruskal-Wallis and Dunn’s tests (p<0.05). ResultsCeramic features had significant effect on DC of RelyX ARC, U200 and Veneer (p<0.0017). Light source had no effect (p=0.9512). C&B and Veneer had higher DC, followed by dual cements. NP dual cements showed the lowest DC. For KHN, ceramic shade (p=0.1717) and light source (p=0.1421) were not significant, but ceramic translucency, thickness and resin cement were significant (p=0.0001). KHN was higher for U200 followed by ARC, and lowest for Veneer. ConclusionsDC was affected by ceramic shade, translucency and thickness. KHN was dependent on ceramic translucency and thickness. Higher DC and KHN were achieved for dual-cured cements photoactivated through 2mm-thick low translucent or 3mm-thick high translucent glass-ceramic. Key words:Cementation, composite resin cements, dental curing lights, glass ceramics.

Influence of different power densities during polymerization on the color of composite resins

This study evaluated the influence of different levels of power density on resin polymerization in the staining of composite resins. Thirty-six specimens were made with composite resin (Z350 3M ESPE) with cylindrical Teflon matrix (8 mm X 1 mm). The light curing was performed with different degrees of power density for each group (n=12): 300mW/cm², 700mW/cm² e 1000mW/cm². Afterwards, the specimens had their color checked, with the help of a spectrophotometer and immersed in coffee solution during 7 days. After, a new measuring was done (CIEL*a*b* color system). Results showed, for Δa, only G1000 group obtained a significantly lower value, and the same result was found in the evaluation of Δb. For ΔE and ΔL, no significant differences were found between groups. It is concluded that, low and medium power density entails more reddish and bluish composite resin, while the specimens submitted to high power density and then immersed in coffee did not present color change in a short term.

Superplastic forming of hemispheres out of AZ61 magnesium alloy

The paper proposes a method for calculating the rheological parameters included in the standard power ratio of superplasticity based on the results of test forming of hemispheres using a cylindrical matrix. The method takes into account the value of the entry radius of the matrix. The analytical model of the process is developed based on the main assumptions of the theory of thin shells, the finite element model is executed using the ANSYS10 ED software package. As compared to the traditional approach, the proposed method allows improving the modeling accuracy with an error less than 1%, not only for the forming time, but also for the thickness at the pole of the dome.

The Effect of Ultrasonic Cleaning Upon Mechanical Properties of Metal Matrix Composites

The aim of this study is to produce composite materials by recycling metallic chips, which are found in industry as a large amount of waste. In addition, it is aimed to investigate the effect of ultrasonic cleaning process as the consolidation behavior and mechanical properties of bulk material directly depend on the cleaning of waste metallic chips. In the present investigation, spheroidal graphite cast iron (GGG-40) was employed as reinforcement material in tin bronze (CuSn10) matrix system. GGG-40 and CuSn10 chips were cleaned by ultrasonic agitation in water for 20 and 40 min. Consolidation of the cleaned metallic chips was achieved with a hot press by applying 820 MPa pressure under 450 °C, and the cylindrical and prismatic metal matrix composite materials with different reinforcement ratios were successfully produced. Energy-dispersive X-ray and scanning electron microscopy analyses were carried out to determine the amount of the oxide removed from the surfaces of chips. The mechanical properties of the samples were determined by hardness, porosity, compression and three-point bending tests. According to the results of the analyses, it was found that CuSn10 surfaces were cleaned from 20%, 50% and GGG-40 surfaces from 35%, 39% oxides during 20- and 40-min cleaning time, respectively. In addition, the results of the mechanical tests revealed that increased ultrasonic cleaning time improves the consolidation quality of metallic chips and it provides successful covering of GGG-40 chips by the CuSn10 chips as a result of a better structural integrity. New machinery parts with high mechanical properties can be produced as a result of recycling of the metallic chips which are available as waste in industry by appropriate cleaning process and this situation makes this study more innovative, economical and environmentally friendly research.

The performance of complex-structure fractured reservoirs considering natural and induced matrix block size, shape, and distribution

This paper introduces an analytical approach for the performance of the fractured reservoir considering the impact of the matrix block size, shape, and distribution. The objective is better understanding the roles of the variable interporosity flow systems, fracture flowing systems, and reservoir matrix heterogeneity in the responses of the wellbore pressure drop, flow rate, cumulative production, and productivity index. This understanding could help to eliminate the uncertainties and increase the accuracy in characterizing the conventional and unconventional reservoirs where very complex structures of a significant disorder in the matrix block size, shape and distribution could be existed in the porous media.The study uses the well documented multilinear flow model to describe the pressure distribution in the naturally fractured reservoirs depleted by multiple hydraulic fractures considering different matrix block sizes, shapes, and distributions. The analytical models of the flow rate, cumulative production, and the productivity index are also developed in this study from the multilinear flow model. The reservoirs of interest are assumed to have stimulated porous media extending between hydraulic fracture and unstimulated volume extending beyond the fracture tips toward the reservoir boundary. The unstimulated reservoir volume is assumed consisting of uniformly distributed single matrix block size and shape because there is no impact for the hydraulic fracturing process. While the stimulated reservoir volume is considered consisting of a variable matrix block size, shape, and distribution because of the impact of hydraulic fracture propagation. The petrophysical properties of the two porous media, as well as the storativity and interporosity flow coefficient, are assumed functions of the matrix characteristics in the two volumes. The matrix block dimensions (radius or length) are correlated exponentially with the distance from the hydraulic fracture face toward the no-flow boundary between these fractures. Three fracture-matrix fluid flux functions are developed for the reservoirs where the matrix blocks could have a spherical, slab, and cylindrical configuration.The study has reached several conclusions. A positive impact on the reservoir performance is seen when the matrix block size becomes small or when the small size matrix blocks control the distribution of the blocks in the porous media. Reservoir performance is enhanced also when there are several matrix block sizes compared to a single size or very few sizes. It is observed that the reservoirs with spherical blocks may give better performance than the reservoir with slab or cylindrical matrix blocks. In terms of matrix block size and distribution, heterogeneous reservoirs may have a higher productivity index than homogenous reservoirs. The variable matrix block size and distribution in the stimulated reservoir volume is more beneficial than the unstimulated volume. The novel point presented in this paper is developing several analytical models for the wellbore pressure drop, flow rate, cumulative production, and productivity index of hydraulically fractured reservoirs with a consideration given to the matrix block size, shape, and distribution.