Conventional Equation Research Articles

An anisotropic equation of state for high-pressure, high-temperature applications

SUMMARY This paper presents a strategy for extending scalar (P–V–T) equations of state to self-consistently model anisotropic materials over a wide range of pressures and temperatures under nearly hydrostatic conditions. The method involves defining a conventional scalar equation of state (V(P, T) or P(V, T)) and a fourth-rank tensor state variable $\boldsymbol {\Psi }(V,T)$ whose derivatives can be used to determine the anisotropic properties of materials of arbitrary symmetry. This paper proposes two functional forms for $\boldsymbol {\Psi }(V,T)$ and provides expressions describing the relationship between $\boldsymbol {\Psi }$ and physical properties including the deformation gradient tensor, the lattice parameters, the isothermal elastic compliance tensor and thermal expansivity tensor. The isothermal and isentropic stiffness tensors, the Grüneisen tensor and anisotropic seismic velocities can be derived from these properties. To illustrate the use of the formulations, anisotropic models are parametrized using numerical simulations of cubic periclase and experimental data on orthorhombic San Carlos olivine.

Work-Energy Principle Based Characteristic Mode Theory With Solution Domain Compression for Metal-Material Composite Scattering Systems

This article focuses on further improving the characteristic mode theory (CMT) for metal-material composite scattering objects. The improved CMT is alternatively established in a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">work-energy principle</i> (WEP) framework. Under the WEP framework, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">driving power operator</i> (DPO) is introduced as the generating operator for characteristic modes (CMs). Then, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">diagonalizing DPO method</i> is proposed to construct the CMs. In addition, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">solution domain compression</i> (SDC) scheme is developed to suppress spurious modes. Compared with the conventional integral equation (IE) framework, the novel WEP framework more clearly reveals the core physical feature of CMs, and then easily derives the famous Parseval’s identity. Compared with the conventional impedance matrix operator (IMO), the novel DPO is more advantageous in the aspect of distinguishing independent variables from dependent variables, and has a smaller computational burden, and has a more acceptable numerical performance in the aspect of calculating CMs. Compared with the conventional dependent variable elimination (DVE) scheme, the novel SDC scheme is equally effective in the aspect of suppressing spurious mode, and reconfirms the same conclusion that the spurious modes result from ignoring the dependence relationships among the currents contained in DPO.

A New Approach for Storage Capacity and Allowable Injection Time Calculation in Carbon Capture and Storage (CCS) Reservoir

In reducing the CO2 emissions in the atmosphere, some energy industry start to implement the Carbon Capture Storage (CCS). However, there still less restriction and considerations in calculating the storage capacity and allowable injection time of CCS reservoir which might lead into leakage problem. In overcoming this problem, there is a study conducted to make a new calculation with higher accuracy. In this study, the reservoir model is using CMG software with base model of East Natuna Field. Then, this model is used to make a sensitivity analysis for 10 CCS fields with a support of CMG-CMOST and Eureqa software. The result is an equation with an R-Square of 96.6% for storage capacity and 95.4% for allowable injection time. The considered parameter in the new equation are permeability, porosity, fracture pressure, thickness, area, reservoir temperature, and reservoir pressure. To validate the new equation, a case study is made for 3 CCS fields. The result shows that the new equation shows up to 21% smaller storage capacity and 7 years reduction in allowable injection time. The result is reasonable due to additional restrictions. Therefore, it is believed that the generated equation is valid and more beneficial compared to the conventional equation.

Wavefield simulation of fractured porous media and propagation characteristics analysis

ABSTRACTMost fractured reservoirs are two‐phase media, that is, mixture of solid matrix and void. It contains rock skeleton and fractures or pores filled with oil, gas and water. These fractures are the channels of oil and gas storage and migration. In two‐phase media, the interaction between the fluid and solid phases will further complicate the seismic wave propagation. Natural fractures are typically irregular in shape, thereby causing difficulties in the exploration of fractured reservoirs. Therefore, the key to the prediction of fractures is to study the equation of motion of seismic waves and energy distribution of seismic waves at the fracture interface. To derive the propagation law for complex irregular shape fractures in two‐phase media, we combined the stiffness matrix of the media with linear slip theory and derived a numerical simulation scheme. The simulation scheme considers the fractures in the two‐phase media to be in any direction. In addition, seismic wave energy distribution at the fracture interface was obtained. The linear slip boundary condition was introduced into the conventional Zoeppritz equation, and a modified Zoeppritz equation was proposed for two‐phase fractured media. The reflection and transmission due to the fracture interface were considered in the new equation, thereby making the equation more flexible. Using the new numerical simulation scheme, we analysed the elastic waves produced by the linear slip fracture interface in two‐phase media and provided the long‐term stability results of the new scheme. Moreover, we provided the relationship between the reflection and transmission coefficients of the linear slip fracture interface and the incident angle and compliance in two‐phase media using the new Zoeppritz equation. The results show that the reflection wave of two‐phase fractured media can be divided into wave impedance and fracture parts to accurately describe the properties of underground rocks.

On the Kaczmarz methods based on relaxed greedy selection for solving matrix equation [formula omitted

For solving a consistent system of linear equations, the Kaczmarz method is a popular representative among iterative algorithms due to its simplicity and efficiency. Based on the Petrov–Galerkin conditions and the relaxed greedy selection strategy, we introduce two kinds of Kaczmarz-type methods to solve matrix equation AXB=C, including randomized and deterministic versions and establish the corresponding convergence theories. We prove that our algorithms all have exponential convergence rates. Then we apply our algorithms to real-world application, such as the tensor product surface fitting in computer-aided geometric design (CAGD), it is shown that our algorithms are more effective than some often used conventional matrix equation solvers.

Improved Integral Equation Method for Rapid 3-D Forward Modeling of Magnetotelluric

Computational cost tremendously restricts the wide application of conventional integral equation (IE) method in large-scale magnetotelluric (MT) modeling. A couple of obstacles limit the developments of traditional MT modeling based on the IE method. They are: O (N2) space complexity of memory requirements for storing coefficients of dense matrix; singularity of Dyadic Green’s function; low efficiency of using digital filtering, such as Hankel transform, to calculate the Bessel function integral within the dyadic Green’s function, as well as inefficiency of accumulative calculation of 3-D discrete convolution. To solve these problems, we use an analytical formula instead of the Hankel transform to compute the integral of the Bessel function and replace a block cell by a spherical cell with the same volume to integrate through the singularity. Because the coefficient matrices are symmetric and antisymmetric three-level block-Toeplitz (BT) and Toeplitz + Hankel matrices, only non-redundant entities of the matrix are computed and stored. Afterwards, 3-D fast Fourier transform (FFT) is used to expedite matrix–vector multiplication at each successive iteration when using the contraction iterative method to solve the system of equations, which decreases memory and time consumption sharply compared with the traditional IE method.

Modeling phase behavior of nano-confined fluids in shale reservoirs with a modified Soave-Redlich-Kwong equation of state

• A m-SRK EOS for modeling the phase behavior of nano-confined fluids is proposed. • A more accurate empirical formula for critical temperature shift is developed. • Phase transformation of nano-confined fluid is revealed by nano-confined index. • There is a critical pore radius around 2 nm that phase behavior deviates sharply. The conventional cubic equation of state (EOS), as a robust and efficient method for phase behavior prediction of bulk fluids, is no longer applicable for nano-confined fluids because of the fluid phase behavior deviation induced by the confinement effect in shale reservoirs. In this work, a modified Soave-Redlich-Kwong (m-SRK) EOS is proposed for modeling the fluid phase behavior in nanopores. Three parameters, including the fluid distribution coefficient, relative effective molecular volume coefficient, and nano-confined index, are defined to revise the molar volume term in the SRK EOS. Then, the empirical formulas of adsorption layer thickness and critical temperature shift are established by correlating the data points collected from the literature to solve the m-SRK EOS. Finally, two analytical formulas are derived for calculating critical property shifts. Computational accuracy of critical temperature shift in this work is remarkably improved for the dimensionless pore radius more than 0.2 compared with analytic models proposed by other scholars. Besides, the nano-confined index can comprehensively reflect the phase transformation law of nano-confined fluids. Modifying the molar volume term in EOSs is crucial for the phase behavior prediction when the pore radius is less than 100 nm, especially for the pores less than 10 nm. There is a critical pore radius around 2 nm that the molar volume of nano-confined fluids changes sharply. When the pore radius equal 1 nm, the descend range of the bubble point pressure increases sharply as the temperature increases compared with that of bulk fluids.

A Spurious-Free Characteristic Mode Formulation Based on Surface Integral Equation for Patch Antenna Structures

Conventional surface integral equation (SIE)-based characteristic mode formulation for the patch antenna structure with a finite substrate is susceptible to the spurious (nonphysical) modes due to the dielectric part. To avoid the contamination of spurious modes, we propose a novel generalized eigenvalue formulation based on the electric field integral equation coupled Poggio-Miller-Chang-Harrington-Wu-Tsai (EFIE-PMCHWT) equation. In this formulation, the real and imaginary parts of the exterior integral operators are chosen to construct the finalized weighting matrices, to connect radiated power of the characteristic current. Compared with other SIE-based methods, this equation doesn't need additional post-processing since it can effectively avoid spurious modes. Numerical results compared with the volume-surface integral equation (VSIE) are investigated to validate the accuracy.

Dynamic algorithmic conversion of compressed sward height to dry matter yield by a rising plate meter

The strategic allocation of pasture grazing area to dairy cows is essential for optimal management and increased outputs. Rising plate meters are frequently used to estimate pasture herbage mass, i.e. dry matter yield per hectare, by employing simple regression equations that relate compressed sward height to herbage mass. However, to improve the accuracy and precision of these equations, so that inherent variation of grasslands is captured, there is a need to incorporate differences in grass types and seasonal growth. Using a total of 308 grass plots, the variation of growth for both perennial ryegrass and hybrid ryegrass was recorded over the seven-month growing season, i.e. March–September. From these data three dynamic equations were derived. The models showed reduced levels of error in comparison to most other conventional equations. As such, the derived models represent a considerable advance for predictive assessment of herbage mass and will support more efficient grassland utilisation by farmers. Although all equations were found to be highly accurate and precise, only a single equation was considered the most effective (R2 = 0.7; RMSE = 248.05), allowing herbage mass to be predicted reliably from compressed sward height data in relation to ryegrass type and calendar month. Although further research will be required, the results presented allow farm operators to calculate herbage mass, as well as support the development of decision support tools to improve on-farm grassland management, particularly at the local paddock rather than national level.

Numerical analysis of fractional viscoelastic fluid problem solved by finite difference scheme

The finite difference schemes for equations depicting the fluid flow, heat and mass transfer of viscoelastic fluid in porous media based on fractional constitutive model are analyzed, when Soret and Dufour effects are taken into account. Different from the conventional fractional partial differential equations, the governing momentum equation of this system not only contains convection terms, but also a term: Dtβ0(∂2u/∂y2), which are great challenges when we do the stability analysis. In addition, the temperature equation and concentration equation are coupled with each other, which increases the difficulty of numerical analysis. By using the energy method, the stability and convergence of the finite difference schemes employed for the governing equations are obtained. The accuracy of the scheme for momentum equation is O(τ+hx2+hy2), while the one for temperature and concentration equations is O(τ2+hx2+hy). Finally, numerical examples are presented to verify methods and demonstrate the analysis effectiveness.

Fast-running model for high-volume hydraulic fracturing

We develop a rapid simulator of hydraulic fracture growth in complex geologic layered conditions and extensively drilled multistage fractured wells in a reservoir. The simulator allows running independent simulations for thousands hydraulic fractures within a minute on modern personal computers. The model is meshless and has a simplified rectangular fracture shape; however, coupling all conventional hydraulic fracture mechanics equations makes the model accurate enough. We demonstrate it via comparison with exact analytical solutions in limiting toughness-, viscosity-, and leakoff-dominated regimes, as well as via comparison with other commercial simulators on real field cases. Field validation of the model by temperature log measurements is also provided.To demonstrate the practical implications of the model for reservoir engineering, we show several examples of numerical simulations for a huge number of hydraulic fractures in a reservoir, which interact with each other due to the closeness of stimulated wells and perforation clusters. Rapid simulations allow obtaining the field-scale simulations of fracture growth in the range of a minute and efficiently building distributions of asymmetric fracture growth both in length and in height. We show that the stress shadow effect causes undesired outbreaks to adjacent gas- or water-saturated geological layers if the fracture placement in a reservoir is dense enough. Even small spatial misalignment of stages in the offset wells may result in changes of geometry of transverse fractures growing from neighbored wells. Vertical true vertical depths mismatch in offset wells; their deviations in the dip and azimuth angles are shown to cause the asymmetry of fracture growth as well, which hardly can be predicted in field development workflows without rigorous fracture simulations. Examples of longitudinal fracturing in horizontal wells demonstrate the strongest distortion of conventionally assumed lateral symmetry and height growth similarity of fractures created in one horizontal well. Our simulations also show that these results are affected by the viscosity of pumped fracturing fluids. The presented model of field-scale interactive fracture growth in a reservoir enables finding the best fracture placement characteristics and pumping schedule for the needs of field development with hundreds to thousands of fractures in a short time frame.

A novel modified Modica–Mortola equation with a phase-dependent interfacial function

In this study, we propose a novel modified Modica–Mortola (MM) equation with a phase-dependent interfacial function. The classical MM functional has a multiple-well potential and the derived classical MM equation allows multiple local minima. The MM equation has a good property compared to its counterpart vector-valued Allen–Cahn (AC) system because the MM equation consists of a single equation. In the MM equation, one phase-field can represent multiple states. However, with a constant interfacial parameter, the interfacial transition layer is getting wider as the jump across two adjacent phases is higher. To overcome this unwanted phenomenon, we propose a phase-dependent interfacial function which has smaller value if gradient of phase-field is larger. We present several computational experiments to demonstrate the superior performance of the proposed modified MM equation over the conventional MM equation.

Peristaltic Transport of Carreau Nanofluid in Presence of Triple Diffusion in an Asymmetric Channel by Multi-Step Differential Transformation Method

The present work investigates the influence of triple diffusion on Carreau nanoliquid in peristaltic flow through an asymmetric channel. By using appropriate non-dimensional parameters, governing equations are transformed to conventional non-linear partial differential equations. The Ms-DTM is used to find solutions to developing equations. Because of the buoyancy force that prevails inside the boundary layer, velocity is impacted by the buoyancy ratio. The current investigation found that as the varied values of the modified Dufour parameter were increased, the temperature profile increased. The thermal conductivity increases as thermal diffusivity increases. It has also been discovered that the existence of triple-diffusing components with low diffusivity might alter the type of convection in the system. Graphs depict the influence of several parameters on velocity, salt1 and salt2 concentrations, solute concentration, and temperature.

Stability Factors F c , F s , and F γ for Twin Tunnels in Three Dimensions

This paper aims to study the three-dimensional (3D) tunnel heading pressure of twin circular tunnels in a drained cohesive–frictional soil using a stability factor approach and finite-element limit analysis (FELA). The primary concept adopted will be the use of a conventional equation that is similar to the traditional bearing capacity factors (e.g., Nc, Ns, and Nγ) in Terzaghi’s bearing capacity equation. For various spacing ratios (S/D) between the tunnels, the stability of the tunnels is expressed in terms of nondimensional tunnel stability factors (e.g., Fc, Fs, and Fγ), which are functions of the depth ratio (C/D) and soil internal friction angle (ϕ). For conciseness, the results will be presented in charts and tables with the tunnel stability factors that used the rigorous FELA upper (UB) and lower bound (LB) solutions.

Adsorption of copper, and zinc onto novel Ca-alginate-biochar composite prepared by biochars produced from pyrolysis of groundnut husk

Alginate-based composites have been studied for adsorption technology as adsorbents due to their biocompatible, non-toxic, and cost-effective properties. In this work, groundnut husk biochar (GHB), calcium alginate (CA), and groundnut husk biochar/calcium alginate novel composites (%10) (CA-GHB1) and (% 20) (CA-GHB2) are synthesized and characterized using BET, SEM, EDX, FTIR, TGA. Adsorption performance is compared among GHB, CA, CA-GHB1, and CA-GHB2 composites to remove Cu(II), Zn (II) from aqueous solutions. Factors affecting adsorption, as well as kinetics, equilibrium, and thermal properties of adsorption, were studied using conventional equations. Adsorption isotherm models were used for two and three-parameter isotherm models to understand the interaction between the adsorbent and the adsorbate. 24.3, 44.6, 45.6, and 40.73 mg g−1 for removal of Cu(II) on GHB, CA, CA-GHB1, and CA-GHB2 and 32.16, 25.07, 36.09, and 40.55 mg g−1 for removal of Zn(II) on GHB, CA, CA-GHB1, and CA-GHB2 found maximum adsorption capacity (Qm) calculated from Langmuir isotherm. According to D-R isotherm data, the adsorption process is classified as physical adsorption. Thermodynamically, the adsorption process is non-spontaneous and endothermic.

Evaluation and Comparison of SEM, ESEM, and BSEM in Estimating Structural Models with Potentially Unknown Cross-loadings

ABSTRACT Cross-loadings are common in multidimensional instruments; however, they cannot be appropriately addressed in conventional structural equation modeling (SEM) owing to the assumption of zero cross-loadings in standard confirmatory factor analysis (CFA). Although it has been proposed that exploratory structural equation modeling (ESEM) and Bayesian structural equation modeling (BSEM) can address this issue more flexibly, their performance in structural parameter estimation has not been adequately compared. This study uses simulated data to evaluate and compare SEM, ESEM, and BSEM in estimating structural models under different manipulation conditions (i.e., sample size, target loading, cross-loading, and path coefficient). The results demonstrated that the performances of these approaches were similar in the case of zero cross-loadings. SEM performed worse as cross-loadings increased, and the performance of BSEM significantly depended on the accuracy of the priors for cross-loadings. ESEM was inferior to BSEM with correctly specified prior means for cross-loadings in most evaluation measures and exhibits unstable performance in conditions with small target loadings. Recommended strategies for selecting an appropriate modeling approach are discussed based on our findings.

Cartesian Operator Factorization Method for Hydrogen

We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations.

Predicting High Temperature Flow Stress of Nickel Alloy A230 Based on an Artificial Neural Network

The high-temperature deformation behavior of metals and alloys undergoes complex mechanisms depending on the deformation conditions. The microstructure and mechanical properties after deformation are important factors that determine the strength and durability of the final product. Therefore, many studies to predict the microstructure and mechanical properties have been conducted. In this regard, numerous mathematical approaches for predicting microstructure and flow stress have been proposed over the past half century. Accordingly, many advances have been made in the field of material science. Nevertheless, there are limitations in the mathematical modeling method as there is a complex relationship between the deformation conditions and the mechanical properties. Therefore, in this study, flow stress prediction was performed by applying conventional constitutive equation and artificial intelligence technology, which is known to be effective in modeling complex relationships. As a result, it was confirmed that the flow stresses modeled by the artificial neural network showed a higher accuracy than the flow stresses modeled by the conventional Arrhenius hyperbolic sine equation.

Deep learning models for improved reliability of tree aboveground biomass prediction in the tropical evergreen broadleaf forests

Aboveground biomass (AGB) and carbon uptake of a forest are key ecological indicators for various technical and scientific applications and sustainable forest management. Deep Learning (DL) methods have been considered as alternative to regression techniques to increase the reliability of tree AGB prediction. The objectives were to develop DL models to predict AGB in the tropical evergreen broadleaf forests and compare DL models with traditional regression equations for their reliability in AGB prediction. A total of 968 individual trees were destructively sampled from fourteen 1-ha and twenty-six 0.2-ha plots distributed across five ecoregions of Viet Nam to get a dataset of tree predictors of diameter at breast height (DBH), tree height (H), wood density (WD) and the response variable of AGB along with forest stand factors of basal area (BA) and density (N); ecological and environmental variables such as ecoregion, slope, altitude, soil type, averaged annual temperature (T), averaged annual rainfall (P) and averaged dry season length. The DL models were developed using different combinations of variables selected by factor analysis for mixed data and compared with traditional regression equations by using cross-validation. Trees AGB in tropical rainforest predicted by DL models had significantly higher reliability than the conventional regression equations when both had the same input variables. Of the 16 developed DL models with 1 to 9 predictors, the model with 9 predictors (DBH, H, Ecoregion, Altitude, P, T, Soil type, N and WD) was the best DL model which reduced root mean square percent error (RMSPE) and mean absolute percent error (MAPE) by up to 7.7% and 6.1%, respectively, compared to traditional allometric equations. The DL models created in this study should be applied for measured tree data following factors of the forest stand, ecology, and environment in sampled plots to predict the tree AGB and total AGB, carbon on a large scale with variation in the value of these factors. Thus, we recommend that the DL models apply for the Measurement, Reporting, and Verification (MRV) system of the Reducing Emissions from Deforestation and forest Degradation (REDD+) program at a large regional level, national or territorial level scale.

Effect of radiation physics on inherent statistics of glow curves from small samples or low doses

A theory is developed to predict statistical noise in the trap populations of small samples or single grains subjected to high-energy ionizing irradiation. Using a model of the radiation process and a one-trap one-center model of a thermoluminescent (TL) material, the statistical behavior of the number of occupied traps during irradiation is predicted. The model focuses on the inherent physics of the process. Experimental sources of error are not considered. The interaction of radiation with the TL material is modeled in a simple way using the Bethe equation. The trap and center populations in the TL material are modeled both with the conventional phenomenological equations and also the more general Master Equation approach. The theory predicts, as the irradiation process proceeds, the mean, standard deviation, dispersion, skewness, and kurtosis of the probability distribution of occupied traps in the TL material. For the same applied dose, the standard deviation and dispersion of the trap population depend strongly on the type of radiation as well as the shape and orientation of the material. High-energy radiation sources, such as alpha, beta, or gamma rays, are found to produce standard deviations and dispersion much larger than low-energy sources, such as UV radiation. The results are summarized in tables which enable, for useful limiting cases, easy calculation of not just standard deviation but also skewness and kurtosis for various radiation sources and geometries.