Equivalent Permeability Research Articles

Use of Symbolic Regression for Developing Petrophysical Interpretation Models

Complex lithology petrophysical interpretation with multiphysics logging tools has been and continues to be a major challenge in formation evaluation. Many currently used data-driven approaches, such as a neural network (NN), deliver predicted results in numerical quantities rather than analytical equations. It is more challenging if multiphysics logging measurements are collectively used to estimate a petrophysical parameter. To overcome these problems, a physics-guided, artificial intelligence (AI) machine-learning (ML) method for petrophysical interpretation model development is described. The workflow consists of the following five constituents: (1) statistical tools such as correlation heatmaps are employed to select the best candidate input variables for the target petrophysical equations; (2) a genetic programming-based symbolic regression approach is used to fuse multiphysics measurements data for training the petrophysical prediction equations; (3) an optional ensemble modeling procedure is applied for maximally utilizing all available training data by integrating multiple instances of prediction equations objectively, which is especially useful for a small training data set; (4) a means of obtaining conditional branching in prediction equations is enabled in symbolic regression to handle certain formation heterogeneity; and (5) a model discrimination framework is introduced to finalize the model selection based on mathematical complexity, physics complexity, and model performance. The efficacy of the five-constituents petrophysical interpretation development process is demonstrated on a data set collected from six wells with the goal of obtaining formation resistivity factor (F) and permeability (k) equations for heterogeneous carbonate reservoirs. We show quantitatively how individual constituents of the workflow improve the model performance with two error metrics. A comparison of NN-method-predicted permeability values vs. SR-based-workflow-predicted permeability equation is included to showcase many advantages of the latter. Beyond the transparency of an analytical form of the prediction equations, the SR method intrinsically has a more relaxed requirement on the training data size, is less prone to overfitting, yet can deliver superior model performance rival to the NN approach.

A fractal model for estimating the permeability of tortuous fracture networks with correlated fracture length and aperture

Many fractures are present in the crust and dominate fluid flow and mass transport. This study proposes a fractal model of permeability for fractured rock masses that includes fractal properties of both fracture networks and fracture surface tortuosity. Using this model, a mathematical expression is derived based on the traditional parallel-plate cubic law and fractal theory. This expression functions as the equivalent permeability of the tortuous fracture network in terms of the maximum fracture length lmax, the fractal dimension of the length distribution Df, porosity ϕ, fracture orientation θ, and the proportionality coefficient between fracture length and aperture β. The fractal scaling law of the fracture length distribution and fractal permeability model is verified by comparison with published studies and fluid dynamic computation, respectively. The results indicate that the deviation of permeability values predicted by the models that do or do not consider the fracture surface tortuosity are as large as three orders of magnitude, which emphasizes that the role of tortuosity should be considered to avoid the overestimation of permeability due to the smooth fracture surface assumption. Further analyses show that the permeability increases with increasing fractal dimension Df, proportionality coefficient β, maximum fracture length lmax, and effective porosity ϕ but decreases with increasing tortuosity dimension Dtf and orientation θ. The fractal dimension of the fracture length distribution Df has the most significant influence on the permeability of the fracture network, followed by Dtf, β, lmax, θ, and ϕ, sequentially.

A study on equivalent permeability models for tree-shaped fractures with different geometric parameters

A study on equivalent permeability models for tree-shaped fractures with different geometric parameters

Equivalent permeability model of dual-porosity and bi-dispersed porous media based on the intermingled fractal units

Dual-porosity and bi-dispersed porous media (DBPM) widely exist in geotechnical engineering, material engineering, soil science, and groundwater exploitation. Therefore, it is significant to quantify the relationship between permeability and matrix–fracture structure parameters for mastering fluid's seepage and transport characteristics. Hence, this paper derives an analytical solution of equivalent permeability for DBPM based on the intermingled fractal units (IFU). The developed model considers the capillary pressure of fractures and capillaries and the tortuosity of fractures and capillaries. Specifically, the number of porous matrix fractal units in IFU is quantified, and then, the dimensionless permeability is calculated, defined as the ratio of the permeability of np matrix fractal units to a single fracture fractal unit. The results reveal that equivalent permeability is mainly contributed by fracture permeability. Next, the second dimensionless permeability is defined to compare further and quantify the permeable ability of fracture and porous matrix. The results highlight that the permeability difference between a single fracture fractal unit and a single porous matrix fractal unit is approximately 7–11 orders of magnitude. Overall, through this paper, the preferential flow mechanism of DBPM can be better described and understood by introducing the above two dimensionless permeabilities and analyzing the influence of structural parameters on them.

Importance of shear dilation to two-phase flow in naturally fractured geological media: A numerical study using zero-thickness interface elements

In this paper, a hydro-mechanical coupling model is proposed to investigate the geomechanical effects on two-phase flow in fractured rocks. The discrete fracture model (DFM) and modified Barton-Bandis's model are used to model two-phase fluid flow and mechanical behavior of fractures, respectively. The coupled problem is solved by a mixed finite volume/finite element method, together with the fixed-stress split algorithm. Our model is able to capture the complex coupled behavior induced by the interaction between pore pressure and in-situ stress loadings. Then the proposed model is applied to investigate the two-phase flow behaviors in the fracture networks with different fracture configurations under various in-situ stress conditions. The results illustrate significant influences of shear dilation on the hydraulic properties of fractured rock. With the increase of differential stress, fracture aperture gradually increases due to shear dilation, which accordingly enhances the equivalent permeability of the fractured medium. The degree of shear dilation is highly dependent on the fracture orientation and length distribution. Consequently, channelized flow is formed through the dilated fractures, which results in early water breakthrough and reduces the water sweep efficiency. This study highlights the necessity of considering hydro-mechanical coupling effect for accurate prediction of saturation distributions in fractured rocks.

The Surface Settlement Law of Precipitation in Pile-Beam-Arch Station Adjacent to Pile Foundation

The pile foundations of adjacent buildings can have an impact on the precipitation process in subway stations. In this paper, surface settlement monitoring is carried out by precipitation of horizontal wells in subway stations, and the law of surface settlement change with and without neighboring pile foundations is studied. The influence law of surface settlement in subway stations under different pile depth, different pile spacing and different distance between piles and station boundary is obtained through numerical simulation. Three models of groundwater infiltration in the adjacent pile foundation were established, the principle of pile-soil permeability coefficient equivalence was proposed, and the function relationship between the equivalent soil permeability discount factor and pile parameters was derived. The research results show that the surface settlement is smaller when there are piles in the measured precipitation process than when there are no piles. The greater the pile depth, the smaller the pile spacing and the smaller the pile-station boundary, the stronger the barrier effect and the smaller the final settlement of the ground surface. The accuracy of the formula is also analyzed to show that changing the pile parameters also changes the permeability coefficient of the piles and the soil between the piles. The station surface settlement law is related to the pile parameters, and the study can provide an important theoretical reference for the construction of precipitation with adjacent pile foundations.

Modeling Permeability in Different Carbonate Rock Types

In carbonate reservoirs, permeability prediction is often difficult due to the influence of various geological variables that control fluid flow. Many attempts have been made to estimate permeability from porosity by using theoretical and empirical equations. The suggested permeability models have been questionable in carbonates due to inherent heterogeneity and complex pore systems. The main objective of this paper is to provide a workflow to improve the use of existing models (e.g., Kozeny, Lucia, and Winland) to predict permeability in carbonate reservoirs. More than 1,000 core plugs were studied from seven different carbonate reservoirs across the Middle East: mainly Cretaceous reservoirs. The plugs were carefully selected to represent a wide range of properties within the cored intervals. The data set available included laboratory-measured helium porosity, gas permeability, thin-section photomicrographs, and high-pressure mercury injection. Rock textures were analyzed in the thin-section photomicrographs and were classified based on their content as grainy, muddy, and mixed. Special attention was given to the diagenesis effects, mainly compaction, cementation, and dissolution. The texture information was plotted in the porosity-permeability domain and was found to produce three distinct porosity-permeability relationships. Each texture gave a unique porosity-permeability trend, where the extent of the trend was controlled by diagenesis. Rock types were defined on each trend by detailed texture analysis and capillary pressure. Three different permeability equations (Kozeny, Winland, and Lucia) were evaluated to study their effectiveness in complex carbonate reservoirs. Both Kozeny and Lucia models honored the geology of the samples and showed similar trends to the porosity-permeability relationships, whereas the Winland model gave different slopes to the experimental data. The prediction of the permeability was improved by using different model parameters per RRT within each texture. This work presents a systematic approach to construct correlations between porosity and permeability in complex carbonate reservoirs. Model parameters (i.e., FZI, RFN, and r35) were suggested within different geological rock types to estimate permeability. Based on the workflow presented in the paper, the predicted permeability was improved to less than a factor of 2 compared to the measured values. Moreover, the same workflow was applied using the data from seven different reservoirs, and the same rock typing scheme was applicable to all the reservoirs. Such work is not abundant in the literature and would serve to improve permeability prediction in heterogeneous carbonate reservoirs, which is one of the main uncertainties in modeling carbonates.

A semi-analytical approach to nonlinear multi-scale seepage problem in fractured shale gas reservoirs with uncertain permeability distributions

To investigate the influence of the inhomogeneity and uncertainty of porosity and permeability on gas production, a mathematical model for shale gas nonlinear seepage problem with random distributions of inhomogeneous equivalent porosity and permeability is established. Based on the statistical characteristic of the artificial fracture density distribution and incompletely controllable hydraulic fracturing technology, the random distribution models of continuous inhomogeneous equivalent porosity and permeability are proposed to simulate the uncertainty of the porosity and permeability distribution in SRV. Coupled with multi-scale flow and adsorption effects, a mathematical model of shale gas nonlinear seepage problem is presented to address the inhomogeneity and uncertainty of equivalent porosity and permeability. By Boltzmann transformation and local homogenization approximation, a semi-analytical method and the corresponding explicit iterative scheme are developed.Simulation results match well with field data, which verifies the validity of the presented mathematical model and approach. The morphology of the SRV has significant influence on gas production, followed by the uncertainly of the permeability distribution and the hydraulic fracture half-length. When the horizontal width reaches its maximum in the front part of the macro-fracture zone, shale gas production reaches its peak and increases by 60%. The uncertainty of the permeability distribution is resulted from the deviation of the hydraulic fracture. As the deviation angle increases from 0° to 10°, shale gas production drops by 16%. The greater the gas production, the bigger the drop rate. When the hydraulic fracture half-length is 50 m, gas production reaches its peak, however it drops by 13% if the hydraulic fracture half-length increases to 70 m.

Modeling Water Injectivity Tests under Multiple Rate Schedule: An Approximate Solution

Summary An injectivity test consists of continuously injecting a phase (water or gas) into an oil-saturated reservoir during a period. According to the analysis of the wellbore pressure behavior, this procedure estimates reservoir parameters, such as permeability and skin factor, and the volume of recoverable oil. In this context, this study proposes an approximate analytical solution for the pressure behavior during a water injectivity test on a multilayer reservoir considering multiple injection flow rates. The accuracy of the proposed solution was evaluated through comparison with a commercial finite-difference-based flow simulator in different scenarios. The results indicate a considerable agreement between the data provided by the numerical simulator and the proposed model. In addition, we successfully estimated the equivalent reservoir permeability using the proposed model with satisfactory results.

Numerical simulating of pre-grouting in multi-jointed rock mass in deep coalmine roadway excavated via TBM

Tunnel Boring Machines (TBMs) have been used in many underground coal mines in China in recent years, most of which are open-type. The open-type TBM is unfavorable to advance in complex geological conditions, especially in heavily jointed rock mass. When the TBM advance in multi-jointed soft rock mass, the surrounding rock frequently damages and deforms around the tunneling face uncontrollably, which brings risk for the workers and reduces the advance rate. Pre-grouting the rock mass ahead before excavating is a common measure used to handle the above-mentioned condition. However, it is difficult to evaluate how the slurry diffuses and how it enhances the rock mass. Therefore, it is short of guidance to determine the parameters of the pre-grouting. In the present research, a 3D Ubiquitous-Multiple-Joint (UMJ) model was first proposed to simulate the mechanical behavior of rock mass affected by multiple sets of joints around the excavating face of TBM. Then, a seepage model with an equivalent anisotropic permeability tensor was proposed to simulate the diffusion of slurry in the jointed rock mass for pre-grouting. Following the in-situ action, the numerical model was first excavated using the UMJ model and then grouted according to the equivalent permeability tensor. After that, the grouted rock mass was enhanced by improving the mechanical parameters of joints. The effect of pre-grouting reinforcement could thus be simulated. Using the method, this paper studied the effect of grouting parameters and proposed some suggestions for TBM tunneling in multi-jointed rock mass.

Design and analysis methods of a novel inductor with metamaterial core

The core of the inductor used in a high-frequency switching power converter usually operates on a local magnetization curve under DC bias, and is easily saturated. In this paper, a novel inductor with a metamaterial core is proposed, in which the magnetic core is replaced by coil units. The coil units can achieve an equivalent permeability comparable to that of the commonly used magnetic powder core material at the working frequency. The design and analysis methods for an inductor with a metamaterial core are described in detail. The proposed inductor is insensitive to the DC component, but can store and transmit the energy of the pulse component. The inductor with a metamaterial core has low loss and is lightweight, which helps increase the power density.

3D numerical modelling and simulation of the impact of fault zones on fluid flow in sandstones of the Rio do Peixe Basin, NE Brazil

Deformation bands are usually responsible for a reduction in permeability perpendicular to the structure planes of up to three orders of magnitude, while the fault core represent a reduction of up to seven orders of magnitude in cross-fault permeability, imposing large anisotropies to fluid flow. As deformation bands occur distributed along the damage zone, they impact not only the across-fault flow but also the along-fault flow. The fault core is usually represented using fault-transmissibility multipliers (TMs) along the fault planes, using well-established workflows. However, there is a lack of methods to represent fault damage zones in any direction and grid-cell sizes. In this context, we proposed new methods to: (1) estimate the deformation intensity in damage zones; (2) calculate their most representative value within the cell domain; and (3) calculate the equivalent permeability of a cell containing oblique deformation bands. The workflow is applied to a 3D numerical model for the Santa Helena High in the Rio do Peixe Basin, NE Brazil. We performed streamline simulations in four models to evaluate the impact of fault damage zones and the fault core in fluid flow. Our models show that the fault core and damage zone negatively affected the performance of the reservoir. Supplementary material : A description of the method developed to estimate the deformation intensity in damage zones and of the method developed to calculate the equivalent permeability are available at https://doi.org/10.6084/m9.figshare.c.6251469

Permeability model establishment and experimental verification of a tree-like branching structure with conventional scale

Permeability is usually only used in the study of porous media with micro-scale characteristics. It is a challenge to accurately calculate permeability for T-shaped branching structures with conventional sizes and non-uniform flow channels. To solve this problem, we derived a theoretical model for the equivalent permeability of the T-shaped tree-like branching structure (TLBS) and the entire microchannel plate (EMP) based on the generalized Darcy's law and the principle of series and parallel solution of equivalent permeability. In order to verify the correctness of the proposed theoretical model, six T-shaped TLBSs with different scale ratios were randomly designed. Experimental values of the permeability of all structures were obtained indirectly by measuring the pressure at both ends of the entrance and exit. The experiments showed that the calculated results of the theoretical model are in good agreement with the experimental results. The average relative error between theoretical calculations results and experimental results is only 3% minimum and 12% maximum in the study range. This study demonstrates that the proposed theoretical model of permeability is reasonable for TLBSs with conventional scales. This provides a new perspective for analyzing the fluid transport performance of fractal-like networks with conventional scales.

A multiscale neural network model for the prediction on the equivalent permeability of discrete fracture network

An equivalent permeability approach can upscale the discrete fracture network (DFN) model to an equivalent DFN model and significantly reduce the gas flow simulations in a large-scale fractured gas reservoir. Current equivalent permeability prediction models are only applicable to the reservoir with a simple fracture network. However, an equivalent permeability prediction model has not been available for a reservoir with a multiscale discrete fracture network. This study proposes a multiscale convolutional neural network model (called MsNet) and introduces three mainstream structures with high performance convolutional neural network (CNN) (ResNet-18, VGG-16 and GoogLeNet) to efficiently predict the equivalent permeability of a complex multiscale fracture network. These CNN models use both the images and features of DFN as their input and the equivalent permeability as their output. This MsNet model is validated with the simulation results simulated by Lattice Boltzmann method and compared with the three mainstream CNN structures and an existing permeability prediction model (CNN-4). It is found that this MsNet model innovatively considers the multiscale characteristics of DFN by a multiscale convolution feature fusion and combines the residual connection for further performance enhancement. Both DFN dataset and MsNet model structure affect the model prediction ability. A deeper network structure of MsNet model can enhance its prediction ability, but significantly increases training time. The MsNet-8-4 (a MsNet structure with 8 multiscale connection modules and 4 sub-networks in each module) has the least convergence time and the lowest mean absolute error on the test set. It performs obviously better than other four models on the DFN dataset with higher fracture density. The MsNet model can well accelerate the simulation on the gas flow in a complex discrete fracture network.

Characterization of large strain consolidation parameters of soft materials with time-variant compressibility at low effective stress

The one-dimensional (1D) large strain consolidation (LSC) of saturated soft materials that are deposited at very low-density usually exhibit time-variant compressibility (void ratio vs vertical effective stress (e-σv′)) relation. The 1D column-like model test serves as an effective approach for characterizing this consolidation characteristic if all the physical parameters (including the settlement rate, pore pressure and density) are measured. Unfortunately, the density measurement is not always realistic due to its high cost (e.g., with X-rays) and the time-effect must be roughly neglected by using an average compressibility relation. This can further lead to erroneous estimations of the materials’ permeability (k) relation (permeability vs void ratio (k-e)) in the LSC analysis. This paper presents two modifications on two conventional equations for compressibility and permeability, respectively. The first one describes the compressibility curve’s movement in the lne-lnσv′ plane, and the other quantifies the ratio between the permeability calibrated by neglecting time-effect and its true value. These modifications originate from deep comparative analyses of several physical parameters between the column test and numerical prediction. Meanwhile, a simple hand-calculation procedure is proposed to estimate the new constants.

Analysis and Research on No-Load Air Gap Magnetic Field and System Multiobjective Optimization of Interior PM Motor

In this article, considering the influence of iron core saturation and complex boundary conditions, a novel no-load magnetic field analytical method and system multiobjective optimization model for interior permanent magnet synchronous motor (IPMSM) are proposed. First, an analytical method based on magnetic potential permeance method is proposed. The complex cogging effect and equivalent air gap permeability function are fully considered in this method. Furthermore, on the basis of Taguchi optimization design based on Fuzzy theory, an improved optimization model considering multivariable interaction is proposed. Among them, in order to comprehensively consider the performance of electromagnetic, vibration, and temperature rise, five optimization objectives including torque, pulsation, cogging torque, loss, and flux density are optimized. The sensitivity of design variables and the interaction between variables are not ignored. Finally, a 6-pole 36-slot IPMSM is manufactured for prototype experiments. A large number of prototype experiments, finite element, and computational fluid dynamics simulation analysis including no-load magnetic field, back electromotive force, temperature rise, electromagnetic torque, and so on are carried out. The accuracy and rationality of the proposed no-load magnetic field analysis method and global optimization model are well-verified.

A two‐phase flow extended finite element technology modeling CO2 in fractured porous media

AbstractCarbon dioxide (CO2) geological sequestration technology is an attractive means of reducing greenhouse gas emissions. This work aims to seek the transport mechanism of CO2 in fractured porous media. A two‐phase flow extended finite element model is developed in this paper. The two‐phase flow in the fracture and porous matrix are all considered in the model development. We present numerical simulations under 2D linear flow conditions with specific and sensitivity analysis about fracture properties, capillary pressure, relative permeability, reservoir heterogeneity, and properties. It is found that fracture will dominate the CO2 transport when the intersection angle of fracture orientation and injection direction is less than 90°, and CO2 flows into deeper regions along the fracture. It is significant to find a suitable capillary pressure and relative permeability model which matches the microscopic pore structure when simulating CO2 geological sequestration in fractured media. Using the Brooks–Corey equation for capillary pressure and relative permeability, the CO2 transport efficiency would increase by around 16% compared to the Van Gneuchten equation. Besides, the CO2 phase and properties evolution during CO2 geological sequestration has a great influence on CO2 transport efficiency., CO2 transport efficiency is reduced by around 18% due to an increase in CO2 density and viscosity when considering CO2 properties evolution. © 2022 Society of Chemical Industry and John Wiley & Sons, Ltd.

A new calculation approach of heterogeneous fractal dimensions in complex hydraulic fractures and its application

Fractal dimension is often used to characterize the distribution of complexity fractures. However, the existing fractal dimension calculation method can only obtain one fractal dimension for a specific fracture network, which cannot accurately describe the complexity hydraulic fractures with strong heterogeneity. This work considers the actual distribution of complex hydraulic fractures in stimulated reservoir area, calculate fractal dimensions of complex hydraulic fractures at different regions. A new quantitative model is established to describes the distribution of heterogeneous fractal dimensions (QM-FD), and obtains the relationship between fractal dimensions and fracture locations. Combined with the heterogeneous fractal dimensions for complex hydraulic fractures, the porosity/permeability/compressibility of complex fractures is determined, and a more general distribution of flow capacity parameters of complex fractures is obtained. A conceptual reservoir model with complex hydraulic fractures is given in the work, and the fractal dimension distribution is described with the proposed approach, then the permeability is calculated with the fractal dimensions. The permeability calculated by heterogeneous fractal dimensions and the actual permeability is compared, and results show that the fitting rate between the calculated permeability and the real value is higher than 93%. The proposed approach is used in four typical reservoirs with complex hydraulic fractures, the fractal dimensions and equivalent permeability is calculated and compared with that from conventional fractal model. The conventional fractal model is poor in dealing with the description of complex hydraulic fractures with non-monotonic changes, and the fitting rate is below then 60%. While the fitting rate of the equivalent permeability calculated by the heterogeneous fractal dimensions in this work and the actual permeability is higher than 92%. The calculation approach of heterogeneous fractal dimensions proposed in the work can be applied to parameter characterization of complex fractures, which provides effective help for flow simulation and productivity prediction of unconventional reservoirs.

Elevating the water/salt selectivity of polybenzimidazole to the empirical upper bound of desalting polymers by marrying N-substitution with chlorination

Polybenzimidazole (PBI) was modified by marrying N-substitution with chlorination in order to improve its desalination properties. Based on the solution-diffusion theory, the effect of N-substitution and then chlorination on the water/salt transport properties of PBI was systematically analyzed. Although the introduction of hydrophobic p-toluenesulfonyl groups decreases water sorption, significantly increases water diffusion by weakening the hydrogen bonding between polymer chains. As a result, the water permeability of substituted PBI (SPBI) was 20 times higher than that of PBI. In terms of salt transport, the increase in salt diffusion of SPBI caused by the weaker inter-chain interactions was offset by a substantial decrease in salt sorption due to the combined effect of electrostatic repulsion and increased accessibility of hydrophilic groups to bind water, leading to equivalent salt permeability of PBI to SPBI. Controlled chlorination of SPBI under acidic conditions can further improve polymer water diffusion and thus water/salt diffusivity selectivity. The permeation of both SPBI and chlorinated SPBI is diffusion dependent. Therefore, combining N-substitution and chlorination is an effective strategy to simultaneously improve PBI permeability and water/salt selectivity for desalination applications.

Blast damage zone influence on groundwater fluxes through backfilled open pits

The blast damage zone (BDZ) surrounding backfilled pits could significantly influence groundwater flow, increase flowrate through backfilled wastes, and potentially increase the dispersion of contaminants to the environment. A conceptual model of the fracture network of the BDZ was first proposed, which consisted of circular fracture oriented along the pit walls, and with a decreasing density with the distance to it. Realistic model parameters were also estimated based on the literature; BDZ equivalent permeability was analytically derived and used in 3D numerical models of conic open pits considering different backfilling scenarios. A total of 16 500 BDZ models were generated to assess the general influence of the BDZ on the flow and the influence of each parameter (fracture trace and attenuation length and fracture radius, aperture, and anisotropy). Results showed that the BDZ generally contributes to reduce (up to three orders of magnitude) the groundwater flow through backfilled tailings, but contributes to increase (by up to 250%) the flow in backfilled waste rocks, especially for wide and fractured BDZ in small pits. This study therefore suggests that the BDZ could be a self-sufficient containment structure to limit contaminant transports from backfilled tailings to the environment, but increases the risk considering waste rock.