Main Groups Research Articles

From Building Blocks to Catalysts: The Underinvestigated Potential of Boronic Acid Esters.

Lewis acids are crucial in chemistry, with applications in pharmaceuticals, agrochemicals, and materials science. In main-group chemistry, they offer alternatives to transition metals, prompting our study of halogenated boronic acid esters (BAEs). Although BAEs are well-known, their catalytic potential has been overlooked. Our investigation found their Lewis acidity superior to that of boron trifluoride and comparable to that of tris(pentafluorophenyl)borane. Additionally, their catalysis of the Sakurai allylation of aldehydes has been documented, paving the way for future advancements.

Ab initio investigation of the geometrical behavior in solution and electronic structure of the anion complexes [bis(1,3-dithiole-2-thione-4,5-dithiolate)M], for M = Bi(III), Sb(III), and Zn(II).

1,3-Dithiole-2-thione-4,5-dithiolate (dmit) ligands are known for their conductive and optical properties. Dmit compounds have been assessed for use in sensor devices, information storage, spintronics, and optical material applications. Associations with various metallic centers endow dmit complexes with magnetic, optical, conductive, and antioxidant properties. Optical doping can facilitate the fabrication of magnetic conductor materials from ground-state nonmagnetic cations. While most studied complexes involve transition-metal centers due to their diverse chemistry, compounds with representative elements are less explored in the literature. This study investigated the structural and electronic properties of bisdmit complexes with representative Bi(III), Sb(III), and Zn(II) cations. AIMD calculations revealed two new geometries for Bi(III) and Zn(II) complexes, diverging from the isolated geometry typically used in quantum chemical calculations. The coordination of acetonitrile molecules to the cationic centers of the complexes resulted in unstable structures, while the dimerization of the complexes was stable. SA-CASSCF/NEVPT2 calculations were applied to the structures of the isolated complexes and stable dimers, confirming the multireference character of the electronic structure of the three systems and the multiconfigurational character of the Bi(III) complex. The electronic spectra simulated by the STEOM-DLPNO-CCSD calculations accurately reproduced the experimental UV‒Vis spectra indicating the participation of the isolated Bi(III) dmit complex and its dimeric form in solution. AIMD calculations of the dmit salts were conducted using the GFN2-xTB method with 60 explicit acetonitrile molecules as the solvent at 300K for a total simulation time of 50.0ps, with printing intervals of 0.5fs. The final geometries were optimized employing the PBEh-3c compound method, incorporating implicit conductor-like polarizable continuum model (CPCM) solvation for acetonitrile. Local energy decomposition (LED) analysis at the DLPNO-CCSD(T)/Def2-TZVP level of theory was utilized to investigate the stability of the complex geometries identified by AIMD. The electronic structures of the complexes were assessed using the SA-CASSCF/NEVPT2/Def2-TZVP method to confirm the multiconfigurational and multireference nature of their electronic structures. Electronic spectra were analyzed using the STEOM-DLPNO-CCSD/Def2-TZVP method, with CPCM used to simulate an acetonitrile medium.

Crack Initiation in Compacted Graphite Iron with Random Microstructure: Effect of Volume Fraction and Distribution of Particles.

Thanks to the distinctive morphology of graphite particles in its microstructure, compacted graphite iron (CGI) exhibits excellent thermal conductivity together with high strength and durability. CGI is extensively used in many applications, e.g., engine cylinder heads and brakes. The structural integrity of such metal-matrix materials is controlled by the generation and growth of microcracks. Although the effects of the volume fraction and morphology of graphite inclusions on the tensile response of CGI were investigated in recent years, their influence on crack initiation is still unknown. Experimental studies of crack initiation require a considerable amount of time and resources due to the highly complicated geometries of graphite inclusions scattered throughout the metallic matrix. Therefore, developing a 2D computational framework for CGI with a random microstructure capable of predicting the crack initiation and path is desirable. In this work, an integrated numerical model is developed for the analysis of the effects of volume fraction and nodularity on the mechanical properties of CGI as well as its damage and failure behaviours. Finite-element models of random microstructure are generated using an in-house Python script. The determination of spacings between a graphite inclusion and its four adjacent particles is performed with a plugin, written in Java and implemented in ImageJ. To analyse the orientation effect of inclusions, a statistical analysis is implemented for representative elements in this research. Further, Johnson-Cook damage criteria are used to predict crack initiation in the developed models. The numerical simulations are validated with conventional tensile-test data. The created models can support the understanding of the fracture behaviour of CGI under mechanical load, and the proposed approach can be utilised to design metal-matrix composites with optimised mechanical properties and performance.

Comparative Analysis of Water-Induced Response in 3D-Printed SCF/ABS Composites under Controlled Diffusion

Additive manufacturing (AM) or 3D-printing of fiber-reinforced composites (FRCs) has garnered significant interests for its versatility in creating intricate parts and rapid prototyping due to cost-effectiveness. Although short fiber-reinforced thermoplastic composites are challenging to manufacture, their mechanical properties can be easily tailored by adjusting fiber type, orientation, and volume fraction. However, void formation during printing is a key issue, impacting mechanical properties and facilitating water ingression, affecting long-term durability. This work studies water diffusion characteristics and the associated hydro-aging of 3D-printed short carbon fiber (SCF)/acrylonitrile butadiene styrene (ABS) composites with controlled water diffusion. Effects of material type (ABS and SCF/ABS), 3D-printing path (horizontal and vertical filament orientation), and diffusion surface (uni-directional and bi-directional diffusion) on water diffusion coefficient and maximum water absorption level are characterized to ensure the long-term durability of 3D-printed ABS and SCF/ABS composites. Baseline representative volume element-based finite element (RVE-FE) diffusion models were developed based on micro-computed tomography (micro-CT) image analysis to understand water diffusion characteristics. This work proves that the SCF/ABS composite is more resistive to hydro-aging than neat ABS due to the SCFs’ hydrophobic nature. SCF/ABS composites, while providing distinct advantages over pure ABS in terms of mechanical properties, could also be more effective against water environments.

Exploring the Use of "Honorary Transition Metals" To Push the Boundaries of Planar Hypercoordinate Alkaline-Earth Metals.

The quest for planar hypercoordinate atoms (phA) beyond six has predominantly focused on transition metals, with dodecacoordination being the highest reported thus far. Extending this bonding scenario to main-group elements, which typically lack d orbitals despite their larger atomic radius, has posed significant challenges. Intrigued by the potentiality of covalent bonding formation using the d orbitals of the heavier alkaline-earth metals (Ae = Ca, Sr, Ba), the so-called "honorary transition metals", we aim to push the boundaries of planar hypercoordination. By including rings formed by 12-15 atoms of boron-carbon and Ae centers, we propose a design scheme of 180 candidates with a phA. Further systematic screening, structural examination, and stability assessments identified 10 potential clusters with a planar hypercoordinate alkaline-earth metal (phAe) as the lowest-energy form. These unconventional structures embody planar dodeca-, trideca-, tetradeca-, and pentadecacoordinate atoms. Chemical bonding analyses reveal the important role of Ae d orbitals in facilitating covalent interactions between the central Ae atom and the surrounding boron-carbon rings, thereby establishing a new record for coordination numbers in the two-dimensional realm.

Geomechanical characterization and mineralogical correlation of compositionally diverse world-class Kazakhstani source rocks: Insights from nanoindentation testing

Extensive nanoindentation testing over a range of deflection depths of up to 4 μm yielded a large dataset, providing a viable framework for the statistical assessment of the mechanical properties, specifically elastic modulus (E) and hardness (H), of compositionally diverse organic-rich mudstone samples. The data from indentations as shallow as 300–400 nm were clustered using the k-means algorithm to identify three mechanical categories in the samples: a soft pseudophase (e.g., organic matter, gypsum, and clay minerals), a stiff pseudophase (e.g., quartz and feldspar), and a transitional composite-like pseudophase bridging the soft and hard minerals. The initially diverse values of E and H for the mechanical pseudophases were observed to converge to a constant value at indentations beyond 2–2.5 μm (varying between different samples), implying the existence of a minimal probing depth for assessing the bulk E and H of heterogeneous mudstone samples. The obtained bulk E and H values (8–21 GPa and 0.3–0.9 GPa, respectively) demonstrated a strong correlation with the mineralogical composition of the indented samples. Despite containing a notable proportion of mechanically stiff components (>45 vol%), the bulk mechanical parameters determined in this study were significantly lower than those reported for major shale formations such as the Barnett and Longmaxi Shale. This discrepancy is primarily due to the presence of organic matter with low thermal maturity (Ro < 0.6%), which constitutes <36 vol% of the samples, and a significant gypsum content, accounting for <15 vol%.The employed approach not only demonstrated the importance of choosing the proper indentation depths for investigating the mechanical properties of highly heterogeneous mudstone rocks and their constituent minerals, but it also illustrated the capability of examining various volumes of investigation using nanoindentation, approaching macroscopic values, and identifying a representative element volume (REV). The findings also provided crucial insights into the fracability and overall producibility of the investigated formations, thereby enhancing our understanding of their extraction potential.

Examining How High School Athletic Directors Leverage Communication With Key Stakeholder Groups to Inform Performance Appraisals of Head Coaches

To ensure that head coaches are effective in leading athletic programs, interscholastic athletic directors engage in a performance appraisal process that reviews coaching efforts. Given the demands of the athletic director role, these leaders are reliant on stakeholders to provide insight that informs the coaching evaluation. Therefore, using the tenets of stakeholder theory, the purpose of this study was to analyze the role that stakeholder feedback plays as athletic directors develop the full picture of coaching performance during an evaluation. Participants (N = 25) featured high school athletic directors represented across school classification (i.e., 1A, 2A, 3A, and 4A) and school type (i.e., public and private). Through semistructured interviews and a subsequent thematic analysis, saturation was achieved at this sample size. Two main themes (i.e., main stakeholder groups and leveraging stakeholder feedback) emerged and demonstrate how key stakeholders should be considered as important sources of information guiding interscholastic athletic directors when leading coaching evaluations.

Finite element modelling of pipe piles driven in low-to-medium density chalk under monotonic axial loading

Percussive driving in low-to-medium density chalk creates a thin annulus of fully de-structured ‘putty’ chalk around pile shafts and an outer annular zone where fracturing is more intense than in the natural chalk. This damage and the related generation, and subsequent equalisation, of excess pore pressures impacts the piles’ time-dependent axial loading behaviour, as do other ageing processes. This paper presents finite element analyses of open steel tubular piles driven for the recent ALPACA and ALPACA Plus research projects under monotonic axial loading to failure after extended ageing periods. A nonlinear elastic stiffness model with a nonlocal deviatoric strain-based Mohr-Coulomb failure criterion was employed, with different sets of properties to represent the de-structured, fractured chalk and intact chalk. It is shown that the piles’ axial responses are mainly controlled by the puttified chalk annuli. A simplified but efficient means is adopted to impose pre-loading chalk effective stress conditions that explicitly capture the effects of installation damage and subsequent ageing. The potential for strain-softening in the brittle chalk is examined and the effective stress paths developed in representative chalk elements throughout the loading are considered in conjunction with the mobilisation of accumulated deviatoric strains. The simulations indicate 264% higher shaft capacity in compression than in tension, which is mainly attributed to the internal chalk plug.

Atomically Dispersed p-Block Aluminum-Based Catalysts for Oxygen Reduction Reaction.

The main group metals are commonly perceived as catalytically inert in the context of oxygen reduction reactions (ORR) due to the delocalized valence orbitals. Regulating the local environment and structure of metal center coordinated by nitrogen ligands (M-Nx) is a promising approach to accelerate catalytic dynamics. Herein, we, for the first time, report the atomically dispersed Al catalysts coordinated with N and C atoms for 4-electron ORR. The axial coordinated pyrrolyl N group (No) is constructed in the Al-N4-No moiety to regulate the p-band structure of Al center, effectively steering the local environment and structure of the square planar Al-N4 sites, which typically exhibit too strong interaction with ORR intermediates. The dynamic covalency competition of axial Al-No and Al-O bonding could endow the Al center with moderate hybridization between Al 3p orbital and O 2p orbital, alleviating the binding energy of ORR intermediates. The as-prepared Al-N4-No electrocatalyst exhibits excellent ORR activity, selectivity, and durability, along with the rapid kinetics as demonstrated by in situ Raman spectroscopy. This work offers a fundamental comprehension of the fine regulation on p-band and guides the rational design of main-group metal-based single atom catalysts.

N2 cleavage by silylene and formation of H2Si(μ-N)2SiH2

Fixation and functionalisation of N2 by main-group elements has remained scarce. Herein, we report a fixation and cleavage of the N ≡ N triple bond achieved in a dinitrogen (N2) matrix by the reaction of hydrogen and laser-ablated silicon atoms. The four-membered heterocycle H2Si(μ-N)2SiH2, the H2SiNN(H2) and HNSiNH complexes are characterized by infrared spectroscopy in conjunction with quantum-chemical calculations. The synergistic interaction of the two SiH2 moieties with N2 results in the formation of final product H2Si(μ-N)2SiH2, and theoretical calculations reveal the donation of electron density of Si to π* antibonding orbitals and the removal of electron density from the π bonding orbitals of N2, leading to cleave the non-polar and strong NN triple bond.

In-silico-assisted derivatization of triarylboranes for the catalytic reductive functionalization of aniline-derived amino acids and peptides with H2

Cheminformatics-based machine learning (ML) has been employed to determine optimal reaction conditions, including catalyst structures, in the field of synthetic chemistry. However, such ML-focused strategies have remained largely unexplored in the context of catalytic molecular transformations using Lewis-acidic main-group elements, probably due to the absence of a candidate library and effective guidelines (parameters) for the prediction of the activity of main-group elements. Here, the construction of a triarylborane library and its application to an ML-assisted approach for the catalytic reductive alkylation of aniline-derived amino acids and C-terminal-protected peptides with aldehydes and H2 is reported. A combined theoretical and experimental approach identified the optimal borane, i.e., B(2,3,5,6-Cl4-C6H)(2,6-F2-3,5-(CF3)2-C6H)2, which exhibits remarkable functional-group compatibility toward aniline derivatives in the presence of 4-methyltetrahydropyran. The present catalytic system generates H2O as the sole byproduct.

Convective heat transfer improvement for a single row of continuous air jets by the acoustic-actuator integration

An experimental study is performed to identify the roles of acoustic-actuator integration on improving convective heat transfer from a row of continuous air jets onto a flat target surface, by using a representative element that involves three continuous-jet pipes with a fixed jet-to-jet pitch (X/d = 5) and two acoustic actuators positioned at the intermediate centers of adjacent continuous jets. The tests are performed under Re = 3000–7000 (continuous jet), f = 150Hz–250Hz (acoustic actuator) and H/d = 2–10 (jet-to-target distance). In addition, several numerical simulations regarding the flow fields are also made to illustrate the mutual interaction mechanism between neighboring synthetic jet and continuous jet. The results clearly demonstrate that three featured local-Nu distribution patterns would happen, dependent on the reference velocity ratio (UR) and the jet-to-target distance. Within the limit of this study, the most possibilities in accordance with different featured local-Nu distribution patterns are put forward. As far as spatially-averaged Nusselt number (Nus-av) and heat transfer uniformity indicator (HTU) for a specified zone, the roles of acoustic-actuator integration on heat transfer performance are thoroughly evaluated. Its positive role on heat transfer enhancement is identified to behave significantly in the situations of UR > 1 and H/d ≥ 6, wherein Nus-av could be raised by 20 % at least in terms of the baseline continuous-jet configurations. However, when UR is far less than 1.0, a little negative influence on Nus-av would happen at larger jet-to-target distances. Meanwhile it is confirmed that the acoustic-actuator integration could always improve the heat transfer uniformity, especially at larger jet-to-target distances wherein HTU could be mostly reduced beyond 30 % with respect to the baseline continuous-jet situations.

Recent Advances in Nanobiotechnology for the Treatment of Non-Hodgkin's Lymphoma.

Lymphoma is the eighth most common type of cancer worldwide. Currently, lymphoma is mainly classified into two main groups: Hodgkin's lymphoma (HL) and non-Hodgkin's lymphoma (NHL), with NHL accounting for 80% to 90% of the cases. NHL is primarily divided into B, T, and natural killer (NK) cell lymphoma. Nanotechnology is developing rapidly and has made significant contributions to the field of medicine. This review summarizes the advancements of nanobiotechnology in recent years and its applications in the treatment of NHL, especially in diffuse large B cell lymphoma (DLBCL), primary central nervous system lymphoma (PCNSL), and follicular lymphoma (FL). The technologies discussed include clinical imaging, targeted drug delivery, photodynamic therapy (PDT), and thermodynamic therapy (TDT) for lymphoma. This review aims to provide a better understanding of the use of nanotechnology in the treatment of non-Hodgkin's lymphoma.

Bacterial microbiota shifts in vacuum-packed beef during storage at different temperatures: Impacts on blown pack spoilage

We aimed to evaluate the bacterial growth and diversity in vacuum-packed beef bags stored at different temperatures and to monitor blown-pack spoilage. We used culture-based methods and high-throughput sequencing to study the development of the main bacterial groups naturally present in beef stored at 4 and 15 °C for 28 days. The growth of sulfite-reducing clostridium (SRC) was impaired in beef bags stored at 4 °C; significant differences among SRC counts were observed in beef bags stored at 4 and 15 °C on days 14, 21, and 28 (P = 0.001). Blown pack was observed in most beef bags stored at 15 °C, from day 14 to day 28, but not in beef bags stored at 4 °C. A storage temperature of 4 °C was able to maintain a stable bacterial microbiota (most prevalent: Photobacterium, Hafnia-Obesumbacterium, and Lactococcus). Remarkable changes in microbial abundance occurred at 15 °C from day 14 to day 28, with a predominance of strict anaerobes (Bacteroides) and the presence of Clostridium spp. The relative frequencies of strict anaerobes and Clostridium were statistically higher in the beef bags stored at 15 °C (P < 0.001 and P = 0.004, respectively). The temperature influenced the microbial counts and relative abundance of spoilage bacteria, leading to blown pack spoilage.

Salt extrusion, a new function of lenticels in red mangrove leaves

Halophytes are characterized by their ability to survive in environments with high salinity, and many of these species eliminate excess salts through secretory structures, such as salt glands or bladders. Rhizophora mangle, however, is a halophyte that does not have specialized structures for eliminating salts, but it does have lenticels located on the lower surface of the leaves. These lenticels are not only involved in oxygenating plant tissues, but they can also eliminate toxic substances. Thus, we hypothesized that the lenticels could act as structures involved in the elimination of excess salts. In this study, we used light microscopy to observe the occurrence of lenticels on the stipules and both young and adult leaves of R. mangle. Using scanning electron microscopy (SEM) and energy dispersive X-ray analysis (EDX), we also analyzed the composition of chemical elements found in the extrusion content of lenticels. The origin of lenticels, both stipular and foliar, is stomatal, and, at the end of development, the stomatal complex is degraded, generating the opening pore. Sea salt was observed on the surface of leaf lenticels, as evidenced by EDX-SEM. The most representative elements on the surface of the lenticels were Cl, Ca, and Na, respectively. In this way, we suggest that the lenticels in R. mangle act as pores through which minerals, such as Na, Cl, Ca, K, Mg, P, S, and Si, can be eliminated by the leaves, an important strategy to maintain the osmotic balance of a typical mangrove species growing in a tropical region.

Synthesis and Reactivity of Bis-tris(pyrazolyl)borate Lanthanide/Aluminum Heterobimetallic Trihydride Complexes.

Molecular heterobimetallic hydride complexes of lanthanide (Ln) and main-group (MG) metals exhibit chemical properties unique from their monometallic counterparts and are highly reactive species, making their synthesis and isolation challenging. Herein, molecular Ln/Al heterobimetallic trihydrides [Ln(Tp)2(μ-H)2Al(H)(N″)] [2-Ln; Ln = Y, Sm, Dy, Yb; Tp = hydrotris(1-pyrazolyl)borate; N″ = N(SiMe3)2] have been synthesized by facile insertion of aminoalane [Me3N·AlH3] into the Ln-N amide bonds of [Ln(Tp)2(N″)] (1-Ln). Thus, this is a simple synthetic strategy to access a range of Ln/Al hydrides. Reactivity studies demonstrate that 2-Ln is a heterobimetallic hydride, with evidence for the cooperative nature of 2-Ln shown by the catalytic amine-borane dehydrocoupling under ambient conditions in contrast to its monomeric counterparts.

An Ice Protection System Based on Phased Piezoelectric Transducers

This study focuses on a system constituted of two piezoelectric transducers installed on a slat representative element, with ice protection purposes. The waves generated by these actuators can cause, in fact, shear actions between the slat panel and the ice accretion, with the final effect of breaking and detaching it. A property of the system is, however, the possibility of regulating the phase between the excitation signals of the two transducers. This capability can be exploited to produce local advantageous wave interference with a consequent amplification of the shear actions. Benefits can be obtained in terms of: (1) reduction of needed power; (2) recovery of signal intensity losses due to distance, geometric, and mechanic discontinuities; (3) recovery of non-optimal functionality due to off-design conditions. The work starts with an overview of the impact of the ice on the aeronautic and other sectors. Then, attention is paid to the systems currently used to protect aircraft, with a specific focus on ultrasounds generated by piezoelectric transducers. The concept proposed in this work is then presented, illustrating the main components and the working modality. On this basis and considering the specific nature of the physical phenomenon, the modeling approach was defined and implemented. At first, the impact of some critical parameters, such as the temperature and the thickness of the ice, was investigated. Then, the impact of the phase delay parameter was considered, estimating the increase of magnitude potentially reachable by means of optimal tuning. Finally, a preliminary experimental campaign was organized and a comparison with the numerical predictions was performed.

Calcium-Ligand Cooperation Promoted Activation of N2O, Amine, and H2 as well as Catalytic Hydrogenation of Imines, Quinoline, and Alkenes.

Bond activation and catalysis using s-block metals are of great significance. Herein, a series of calcium pincer complexes with deprotonated side arms have been prepared using pyridine-based PNP and PNN ligands. The complexes were characterized by NMR and X-ray crystal diffraction. Utilizing the obtained calcium complexes, unprecedented N2O activation by metal-ligand cooperation (MLC) involving dearomatization-aromatization of the pyridine ligand was achieved, generating aromatized calcium diazotate complexes as products. Additionally, the dearomatized calcium complexes were able to activate the N-H bond as well as reversibly activate H2, offering an opportunity for the catalytic hydrogenation of various unsaturated molecules. DFT calculations were applied to analyze the electronic structures of the synthesized complexes and explore possible reaction mechanisms. This study is an important complement to the area of MLC and main-group metal chemistry.

A computational method for calculating the electrical and thermal conductivity of random composites

In the present work, for the first time, the electrical–thermal conductivity of a composite random material is correlated with the number of conductive paths, the mean path length, and the mean effective path width measured in a representative surface element (RSE) by Monte Carlo sampling. This work is organized in two parts. The theoretical calculation takes place in the first part for the correlation of conductive paths by adopting three approaches (incremental with respect to accuracy and complexity) based on: (i) one-component straight paths; (ii) one-component non-straight paths; (iii) multi-component non-straight paths. In the second part, a novel numerical methodology for the calculation of the conductance of the RSE through the conductive paths, mean path length and the mean effective path is developed for the general model of multi-component non-straight line paths based on Ohm’s law. In addition, a methodology that reduces the calculated conductivity from the microscale to the macroscale is proposed, that takes into account the probability of percolation for finite sizes and the average value of the conductivity from the samples in which percolation occurs. After the necessary consistency steps and verification of the method and its implementation on random binary material systems, it is further applied to solve synthetic continuous percolation problems at the microscale and macroscale. The results and the accuracy of the calculations are in close agreement with existing models. The electrical Direct Current (DC) and thermal conductivity follow the same scaling relation and the proposed methodology can be applied at the same stage without additional cost. In a random binary medium the simulations showed that a scaling law of conductivity is followed by an exponent value close to 1.3. In addition, the number of conductive paths, the average length of the paths, and the effective width follow scaling laws with exponents 1.44±0.01 , −0.25±0.01 and −0.39±0.01, respectively.

Analytic representative element rate decline models for naturally fractured reservoir depletion

Representative single anisotropic matrix block 2D Green’s function models for depletion through fully-penetrating, vertical fractures through different numbers of fracture faces are constructed that analytically capture both fracture and block depletion with fracture-matrix mass transfer. The 1D Green’s function for a fracture system is likewise solved in terms of the time evolution of average fracture pressure. While transient average pressure values are not inherently measurable, they are transformed into cumulative production or instantaneous flowrate values, thus producing new rate decline model functional forms. Primary variables in assembling the interacting systems model are the volume ratio, Vf /Vm, permeability ratio, kf /kx, and geometry, (a/b)(ky/kx), with the last term accounting for both block shape and permeability anisotropy. We construct interacting systems models in terms of various ratios of Vf /Vm, and kf /kx for three fracture architecture prototypes: representative matrix blocks depleted by 4, 2, or 1 contacting fractures. The single matrix block models can be migrated to ones for heterogeneous systems using superposition and matrix block distributions, as demonstrated with a binary distribution of block sizes with variable fractions. Analytic solutions for rate decline problems can be used to understand the production signatures of naturally fractured reservoirs and interpretation of fracture volume fraction, permeability ratio, average matrix block size, and measures of heterogeneity.