Tree-like Network Research Articles

A NOVEL KOZENY–CARMAN CONSTANT MODEL FOR POROUS MEDIA EMBEDDED WITH TREE-LIKE BRANCHING NETWORKS WITH ROUGHENED SURFACES

Although the hydraulic features of the tree-like branching network have been widely investigated, the seepage characteristics of the networks have not been studied sufficiently. In this study, the seepage characteristics of porous media embedded with a tree-like branching network with the effects of roughness are studied based on fractal theory. Then, the Kozeny–Carman (KC) constant of the composite network is derived. The KC constant of porous media embedded with a tree-like branching network with roughened surfaces is in good agreement with the experimental data in the literature. The effects of structural parameters on seepage characteristics are also discussed. Notably, the results show that the KC constant of the composite network increases with an increasing volume porosity, and decreases with an increase in the relative roughness. Besides, the model established in this paper contains no empirical constants to ensure that each parameter has its physical significance. Thus, the proposed model can facilitate a better understanding of the seepage characteristics of fluid transport through a tree-like branching network embedded in porous media.

Permeability Model of Unsaturated Fractured Porous Media Based on Tree-like Network and Finite Element Method

In order to characterize the multi-scale structures of fractures and understand the multiphase fluid flow mechanisms in unconventional reservoirs, a physical conceptual model is developed for the gas-water flow through fractured porous media based on tree-like networks. The finite element method is carried out to explore the local flow field properties based on the level set method. The results show that the successive branching ratio and branching angle of the network take significant effect on the absolute permeability and relative permeability of fractured porous media. The proposed model for fractured porous media may provide theoretical basis for the development of unconventional oil and gas resources, carbon dioxide geological sequestration, ground water mining etc.

A FRACTAL ELECTRICAL CONDUCTIVITY MODEL FOR WATER-SATURATED TREE-LIKE BRANCHING NETWORK

Electrical conductivity is an important physical property of porous media, and has great significance to rock physics and reservoir engineering. In this work, a conductivity model including pore water conductivity and surface conductivity is derived for water-saturated tree-like branching network. In addition, combined with Archie’s law, a general analytical formula for the formation factor is presented. Through the numerical simulation of the analytical formula above, we discuss the impact of some structural parameters ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] in tree-like branching network on the resistance, conductivity and formation factor. The results show that the total resistance [Formula: see text] is proportional to [Formula: see text], [Formula: see text], and inversely proportional to [Formula: see text], [Formula: see text]. The relation between conductivity and porosity in this model is contrasted with previous models and experimental data, and the results show considerable consistency at lower porosity. It is worth noting that when [Formula: see text], the conductivity and porosity curve of this model overlap exactly with those plotted by the parallel model. The fractal conductance model proposed in this work reveals the operation of the current in the tree-like branching network more comprehensively.

FRACTAL ANALYSIS FOR THERMAL CONDUCTIVITY OF DUAL POROUS MEDIA EMBEDDED WITH ASYMMETRIC TREE-LIKE BIFURCATION NETWORKS

Heat transport in tree-like bifurcation networks has been widely studied in various fields. In this work, we investigate heat conduction in the dual porous media embedded with asymmetric tree-like bifurcation networks. In addition, considering the effects of nonuniform tube shape, we assume that the bifurcated tube shows sinusoidal fluctuations. Based on the fractal distribution of pore size and bifurcation structure, we established a dimensionless effective thermal conductivity (ETC) model of the dual porous media. The dimensionless ETC ([Formula: see text] obtained is related to the porosity ([Formula: see text], the fluid–solid thermal conductivity ratio ([Formula: see text], the pore area fractal dimension [Formula: see text] and the structural parameters of the bifurcation network (bifurcation level [Formula: see text], length ratio [Formula: see text], radius ratio [Formula: see text], fluctuation amplitude factor [Formula: see text], bifurcation angle [Formula: see text]. To verify the validity of this model, a comparison of the present dimensionless ETC model with available experimental data was carried out and the results were in good agreement. We have discussed the effects of each parameter on the dimensionless thermal conductivity in detail and constructed parametric planes to evaluate the structural parameters more directly. The model has positive implications for revealing the heat transport mechanism in asymmetric tree-like bifurcation dual porous media.

PREDICTING THE ELECTRICAL CONDUCTIVITY OF DUAL-POROSITY MEDIA WITH FRACTAL THEORY

The microspatial structure of porous media affects the electrical properties of reservoir rocks significantly. In this work, a dual-porosity model is established to investigate the electrical properties of porous media, in which tree-like networks and capillary channels represent fractures and pores. By using fractal theory, we established an analytical equation for the conductivity of water-saturated dual-porosity media. The analytical equation, devoid of any empirical constants, expresses the electrical properties of the porous media as a function of some structural parameters ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. We also examine the impact of various matrix structural parameters on conductivity. It is found that increasing the length of mother channel ([Formula: see text], length ratio ([Formula: see text], the number of branching layers ([Formula: see text], and tortuosity fractal dimension ([Formula: see text] leads to a decrease in conductivity, whereas increasing the diameter of mother channel ([Formula: see text], diameter ratio ([Formula: see text], the cross-sectional porosity ([Formula: see text], [Formula: see text], and the channel bifurcation number ([Formula: see text] enhances conductivity. Furthermore, we validated this analytical model by comparing it with the experimental data available, and the results demonstrate good agreement. This research has proposed an advanced conductivity model that enables us to better understand the underlying physical mechanisms of the electrical properties in porous media.

A FRACTAL MODEL FOR ELECTRICAL CONDUCTIVITY OF POROUS MEDIA EMBEDDED WITH A DAMAGED TREE-LIKE NETWORK

Predicting the electrical conductivity of porous media is important for oil reservoirs, rock physics, and fuel cells. In this work, a cuboid embedded with a damaged tree-like network is employed to denote a portion of porous media. Analytical expressions for the electrical conductivity are then derived with the fractal theory. Various structural parameters have been examined in detail for the influence of electrical conductivity. It is found that the increased number of damaged channels means more difficult ion migration and lower electrical conductivity of porous media. In addition, a decreasing length ratio or an increasing diameter ratio will increase electrical conductivity. Moreover, both the channel distribution fractal dimension and tortuosity fractal dimension result in a decrease in electrical conductivity. A comparison of the results predicted by other models yields a good agreement, validating our proposed model. These results may further interpret the transport properties of porous media.

A NOVEL FRACTAL MODEL FOR SPONTANEOUS IMBIBITION IN DAMAGED TREE-LIKE BRANCHING NETWORKS

Spontaneous imbibition has attracted considerable attention due to its extensive existence in nature. In this study, we theoretically explored the spontaneous imbibition dynamics in a damaged V-shaped tree-like branching network by comparing with a parallel net with fixed constraints. Additionally, the imbibition capacity is characterized by two dimensionless quantities: imbibition potential and dimensionless imbibition time. The fractal theory is then used to generate the analytical expressions of these two dimensionless quantities. After that, the influence of structural parameters on the imbibition process is systematically investigated. It is found that a larger number of damaged channels will correspond to the lower imbibition potential and dimensionless imbibition time. Notably, the branching number N has an evident enhancement effect on the imbibition potential. A parameter plane is introduced to visualize parameter combinations, enabling the direct evaluation of the imbibition process in a specific network system. The physical mechanisms revealed by the proposed model provide effective guidance for imbibition process analysis in the damaged tree-like networks.

Thermal Conductivity Model of Porous Media Embedded with a Damaged Tree-like Branching Network Considering the Influence of Roughness

In the study of heat transfer in tree-like branching network, neither the heat convection caused by fluid flow in the tree-like branching network nor the asymmetric structure of the tree-like branching network can be ignored. In this work, we assume the porous media is embedded with a tree-like branching network that are characterized by damaged pipes. We investigated the effects of surface roughness on heat conduction and heat convection in the porous media embedded with the damaged tree-like branching network based on the fractal features of tree-like branching networks and the basic theory of thermodynamics. The proposed model for thermal conductivity can be expressed as a function of micro-structural parameters of the composite, such as the relative roughness, the ratio of thermal conductivity of the wall to that of the fluid in the micro-channel, the diameter ratio, the length ratio, the branching level, the number of damaged channels, the total number of branching levels, and the main tube porosity of the porous media. The effects of the micro-structural parameters of the model on its effective thermal conductivity have been analyzed in detail. It is believed that the joint expression of heat conduction and heat convection could enrich and develop the physical study of heat transport in porous media.

A novel probabilistic model to explain drainage network evolution

River channels self-organize to form tree-like networks that follow certain statistical laws, such as Horton's laws and Hack's law, which is perhaps the reason why simple statistical models are often used to explain river network evolution. However, statistical models generally do not attempt to explain the differences between network structures. Here we address this issue by incorporating certain physically meaningful variables in a statistical modeling framework. The model allows the network to grow in a headward direction via probabilistic decisions based on two free parameters. Simulations performed over a planar matrix result in tree-like networks exhibiting power-law scaling relationships as displayed by real river networks. The two parameters are shown to be capable of explaining the variation of key network characteristics such as compactness coefficient and Hack's exponent. The uniqueness of the model thus lies in its ability to generate networks with different shapes and characteristics.

Linear response theory for renewable fluctuations in power grids with transmission losses.

We study the spreading of renewable power fluctuations through grids with Ohmic losses on the lines. By formulating a network-adapted linear response theory, we find that vulnerability patterns are linked to the left Laplacian eigenvectors of the overdamped eigenmodes. We show that for tree-like networks, fluctuations are amplified in the opposite direction of the power flow. This novel mechanism explains vulnerability patterns that were observed in previous numerical simulations of renewable microgrids. While exact mathematical derivations are only possible for tree-like networks with a homogeneous response, we show that the mechanisms discovered also explain vulnerability patterns in realistic heterogeneous meshed grids by studying the IEEE RTS-1996 test system.

Global piecewise classical solutions to quasilinear hyperbolic systems on a tree-like network

In this paper, we discuss the existence, uniqueness and asymptotic stability of global piecewise [Formula: see text] solution to the mixed initial-boundary value problem for 1-D quasilinear hyperbolic systems on a tree-like network. Under the assumption of boundary dissipation, when the given boundary and interface functions possess suitably small [Formula: see text] norm, we obtain the existence and uniqueness of global piecewise [Formula: see text] solution. Moreover, when they further possess a polynomial or exponential decaying property with respect to [Formula: see text], then the corresponding global piecewise [Formula: see text] solution possesses the same or similar decaying property. These results will be used to show the asymptotic stability of the exact boundary controllability of nodal profile on a tree-like network.

Efficient eigenvalue counts for tree-like networks

AbstractEstimating the number of eigenvalues $\mu_{[a,b]}$ of a network’s adjacency matrix in a given interval $[a,b]$ is essential in several fields. The straightforward approach consists of calculating all the eigenvalues in $O(n^3)$ (where $n$ is the number of nodes in the network) and then counting the ones that belong to the interval $[a,b]$. Another approach is to use Sylvester’s law of inertia, which also requires $O(n^3)$. Although both methods provide the exact number of eigenvalues in $[a,b]$, their application for large networks is computationally infeasible. Sometimes, an approximation of $\mu_{[a,b]}$ is enough. In this case, Chebyshev’s method approximates $\mu_{[a,b]}$ in $O(|E|)$ (where $|E|$ is the number of edges). This study presents two alternatives to compute $\mu_{[a,b]}$ for locally tree-like networks: edge- and degree-based algorithms. The former presented a better accuracy than Chebyshev’s method. It runs in $O(d|E|)$, where $d$ is the number of iterations. The latter presented slightly lower accuracy but ran linearly ($O(n)$).

A NOVEL FRACTAL MODEL FOR GAS DIFFUSION COEFFICIENT IN DRY POROUS MEDIA EMBEDDED WITH A DAMAGED TREE-LIKE BRANCHING NETWORK

A novel gas diffusivity model for dry porous media with a damaged tree-like branching network is proposed by using the fractal theory in this study. We systematically investigated the effects of the number of damaged channels and the other structural parameters on the dimensionless gas diffusivity (DGD) and concentration drop. As the number of damaged channels increases, the DGD presents a decreasing trend, while the ratio of concentration drop shows a rising tendency. Meanwhile, the DGD is negatively correlated to the length exponent, the total number of branching levels, and the branching angle, respectively. On the other hand, the DGD is positively correlated with the diameter exponent. Besides, the ratio of concentration drop is negatively correlated with the length exponent and the total number of branching levels. However, it is positively associated with the diameter exponent and branching levels. In addition, during the calculation of the value of concentration drop, the total concentration drop can be disassembled into two equal-ratio sequences. And the scale factors in sequences are constants that are independent of the number of damaged channels. The reliability of the model predictions was verified by a comparison with the experimental data available in the literature. The physical mechanism of gas diffusion in the damaged network may be well explained by the proposed model.

On Improving Resource Allocations by Sharing

Given an initial resource allocation, where some agents may envy others or where a different distribution of resources might lead to higher social welfare, our goal is to improve the allocation without reassigning resources. We consider a sharing concept allowing resources being shared with social network neighbors of the resource owners. To this end, we introduce a formal model that allows a central authority to compute an optimal sharing between neighbors based on an initial allocation. Advocating this point of view, we focus on the most basic scenario where a resource may be shared by two neighbors in a social network and each agent can participate in a bounded number of sharings. We present algorithms for optimizing utilitarian and egalitarian social welfare of allocations and for reducing the number of envious agents. In particular, we examine the computational complexity with respect to several natural parameters. Furthermore, we study cases with restricted social network structures and, among others, devise polynomial-time algorithms in path- and tree-like (hierarchical) social networks.

ANALYSIS OF THERMAL CONDUCTIVITY OF DAMAGED TREE-LIKE BIFURCATION NETWORK WITH FRACTAL ROUGHENED SURFACES

Tree bifurcation network has always been a hot issue concerning scientists all over the world. However, the research on the heat conduction properties of damaged tree-like bifurcation networks is insufficient, and many issues remain unsolved. Therefore, this paper systematically studies the heat conduction phenomenon of a new damaged tree-like bifurcation network with fractal roughened surfaces (FRS), and derives the equivalent thermal conductivity of damaged network. According to the study, the dimensionless thermal conductivity (DTC) shows a downward trend with an increase in roughness. In addition, it can be seen that an increase in the number of damaged channels leads to a downward trend in the optimal thermal conductivity, but a slow upward trend in the critical diameter. Another interesting phenomenon is that the factors leading to the change in the optimal thermal conductivity do not include the total number of bifurcation levels, bifurcation number or the fractal dimension of length distribution. The model established in this paper does not contain any empirical constants to ensure that each parameter has its physical significance, revealing the heat transfer mechanism of tree-like bifurcation network more profoundly.

Dependability of water disposal systems of major cities

The Aim of the paper is to develop an analytical procedure for calculating the dependability indicator of a specific tree-like transporting network, i.e., the water disposal system of a major city.Methods. It is suggested using the relative volume of sewage water released from a sewage system into the environment as the result of the failure of the former’s components as the measure of dependability. This paper presents a simple and convenient method for calculating such volume. The method is based on the representation of a tree-like sewage network as a combination of Y fragments referred to as structural elements. Every such element is formally replaced with its pseudo equivalent with a failure rate calculated assuming that the volumes of untreated waste water released by actual and fictitious elements are equal. A consistent application of this approach reduces the problem of estimation of the volume of released waste water to elementary subproblems that are solved using probability methods.Results. The problem of identifying the operational dependability indicator comes down to a recurrent stepwise procedure for equalization of Y fragments of the network, at each step of which the calculation data from the preceding step are used as the input data. Each such step, starting from the entry network, leads to another (virtual) network, for which the procedure repeats. The equalization process ends when the original network is represented by only one pseudo element, for which the dependability indicator is defined in an elementary way.Conclusion. The authors developed a convenient method for calculating the operational dependability indicator of a tree-like sewer network of a major city. Some practical problems are referred to that can be solved using the proposed method. Future lines of research in the domain were identified.

Liver ductal organoids reconstruct intrahepatic biliary trees in decellularized liver grafts

Three-dimensional scaffolds decellularized from native organs are a promising technique to establish engineered liver grafts and overcome the current shortage of donor organs. However, limited sources of bile duct cells and inappropriate cell distribution in bioengineered liver grafts have hindered their practical application. Organoid technology is anticipated to be an excellent tool for the advancement of regenerative medicine. In the present study, we reconstructed intrahepatic bile ducts in a rat decellularized liver graft by recellularization with liver ductal organoids. Using an ex vivo perfusion culture system, we demonstrated the biliary characteristics of repopulated mouse liver organoids, which maintained bile duct markers and reconstructed biliary tree-like networks with luminal structures. We also established a method for the co-recellularization with engineered bile ducts and primary hepatocytes, revealing the appropriate cell distribution to mimic the native liver. We then utilized this model in human organoids to demonstrate the reconstructed bile ducts. Our results show that liver ductal organoids are a potential cell source for bile ducts from bioengineered liver grafts using three-dimensional scaffolds.

Cooper pairs localization in tree-like networks of superconducting islands

We study inhomogeneous Cooper pairs distribution and localization effects in tree-like networks of superconducting islands coupled via Josephson weak links. Using a generalized Feynman’s approach, reminiscent of the Bose–Hubbard model, we demonstrate that the Cooper pairs fraction which localizes on a specific network’s island is limited by the network topology and, if present, by the repulsive interaction. These findings contribute to clarify the interplay between confinement effects induced by the network’s topology and interaction and shed some light on recent experiments dealing with networks of Josephson junctions.

Parcel delivery using the hyperconnected service network

The small parcel delivery industry is experiencing rapid growth, primarily driven by the growth of the e-commerce sector. We introduce a logistic model for delivering parcels to service points that are used as drop-off, pickup, and intermediate storage locations. A parcel may be carried from its origin to its destination in several legs via several possible intermediate service points. Such a system constitutes a hyperconnected service network (HCSN). The HCSN is a generalization of the current practice of using a hierarchical, tree-like service network (TLSN), where a parcel can switch vehicles only in a large sorting facility (hub) and a service point is served by a single route. The HCSN topology presents an opportunity to improve the delivery process by reducing the total distance that parcels are carried while still exploiting the possibility of shipment consolidation. In addition, such a system may save resources that are associated (and tied for an extended period) with the construction and operation of a large sorting facility. In this paper, we develop tools to design and operate a successful HCSN for parcel delivery – a parcel routing mechanism and a math heuristic for routing and scheduling of the vehicles that transfer the parcels in the network. The effectiveness of the proposed tools and the advantages of the HCSN are demonstrated by a simulation study.

Spectral properties of a class of treelike networks generated by motifs with single root node

In this paper, we introduce a class of treelike networks generated by motifs with single root node. We study the spectral properties of treelike networks by studying the behaviors of the corresponding adjacency matrix and Laplacian matrix. First, we get the adjacency matrix and Laplacian matrix of this treelike network. Then, we analyze symmetry of eigenvalues, multiplicity of eigenvalue [Formula: see text] and prove that its largest eigenvalue is strictly monotonically increasing and convergent. Finally, we introduce a new method to prove that the largest eigenvalue of the Laplacian matrix is strictly monotonically increasing and convergent. Also, we obtain that its smallest nonzero eigenvalue is strictly monotonically decreasing and convergent.