Prediction Of Phase Behavior Research Articles

Characterization of multi-fractured horizontal shale wells using drill cuttings: 1. Fluid-in-place estimation

The most common method to produce shale gas and liquid-rich (gas condensate and oil) shale reservoirs is through long horizontal wells completed in multiple hydraulic fracturing stages. Usually the only reservoir samples gathered from these wells are drill cuttings obtained at multiple intervals along the well. While drill cuttings collection is inexpensive, and can allow for high density sampling in heterogeneous reservoirs, quantitative analysis of cuttings for reservoir properties remains a challenge.An important property to assess from drill cuttings is fluid storage capacity. In shales, fluid storage may occur via multiple mechanisms within the complex pore structure, including adsorption in microporosity (pore width < 2 nm), multi-mechanism storage (free gas and adsorption) in mesopores (pore width between 2 and 50 nm), and free gas storage in macroporosity (pore width greater than 50 nm). Rigorous models that account for these complex storage mechanisms over the wide range of pore sizes typically observed in shales are therefore required. A further complication, particularly important for liquid-rich shales, is the alteration of fluid properties, which can affect both fluid storage and producibility, due to pore confinement. Rigorous models are also required to account for these effects so that resource in-place and recovery can be properly predicted.In this work, the use of high-precision, low-pressure adsorption data, combined with the simplified local density model (SLD) is explored for the purpose of 1) estimating high-pressure hydrocarbon storage and 2) calculating fluid property alteration using artificial drill cuttings. Use of low-pressure adsorption equipment, commonly used for estimating internal surface area and pore size distributions, is necessary to allow quantitative analysis of small sample sizes typical of cuttings. Artificial cuttings, created by crushing larger core samples, are used to enable comparison with more conventional methods applicable to large sample sizes.It is demonstrated herein for the first time that the SLD model, after calibration to low-pressure, non-hydrocarbon gas adsorption data (CO2 and N2), may be used to predict high-pressure hydrocarbon gas adsorption. Further, it is demonstrated that the SLD model may be used to predict phase behavior changes of liquid-rich gases in confined pores. The end result is that fluid storage properties from small masses of cuttings may be determined rapidly and accurately enabling the quantification of fluid-in-place along the length of a horizontal well.

Predicting phase behavior of metallic mercury in liquid and compressed gaseous hydrocarbons

This study presents the first predictive approach for estimating phase behavior of metallic mercury in hydrocarbons and compressed gases. This approach is based on the recently developed CP-PC-SAFT EoS attached by a universal value of k12=0.3. The CP method of solving SAFT parameters is not applicable for metallic mercury. Therefore it has been treated as a monoatomic substance, while σ and ε/k have been adjusted to the experimental vapor pressure and density data. It has been demonstrated that the proposed approach accurately predicts the solubility enhancement of metallic mercury in the compressed gases up to ∼400bar. The solubility data in liquid n-alkanes, aromatic and oxygenated hydrocarbons in wide range of temperatures have been truthfully estimated as well. The available binary diffusion coefficients of mercury in gases can also be accurately predicted by the Chapman–Enskog model when applied with the model’s molecular parameters. This study also shows that the analogous implementation of the original version of PC-SAFT results in a slightly lesser over-all accuracy, which could probably be attributed to the not entirely systematic practice of evaluating its molecular parameters for various hydrocarbons. A doubtless advantage of both considered versions of PC-SAFT over Peng–Robinson EoS is demonstrated as well.

Prediction of carbon dioxide solubility in ionic liquids using MLP and radial basis function (RBF) neural networks

Elimination of carbon dioxide from gas mixtures is a common commercial step in natural gas refineries. Nowadays, room-temperature ionic liquids, which are a relatively novel type of compounds have gained attention in recent years and have potential to be considered as a substitution for conventional volatile organic solvents in reaction and separation processes. No flammability, high thermal stability, a wide liquid range, and electric conductivity are some properties of ILs, which make them interesting more and more.Information about the solubility and the rate of solubility is a crucial factor for consideration of ILs for potential industrial processes. Because of some difficulties associated with experimental measurements and expenses spent on ILs, developing predictive methods for prognostication of the phase behavior of such types of systems are more favorable. Thermodynamic models are relatively complex and require complicated mathematical operations.Due to such difficulties there is a need to develop general models capable to predict phase behavior of systems such as CO2 with various kinds of ILs. In this study, four different methods based on artificial intelligence are proposed to predict CO2 solubility in different ionic liquids. The results showed that the predicted values are in great agreement with the experimental data and the maximum absolute error deviation for the best predictor is no more than 3.5%. A comparison between developed models and previously published ones reveals the superiority of the proposed models in this study.

Self-Assembly of Asymmetrically Interacting ABC Star Triblock Copolymer Melts.

The phase behavior of asymmetrically interacting ABC star triblock copolymer melts is investigated by the self-consistent field theory (SCFT). Motivated by the experimental systems, in this study, we focus on the systems in which the Flory-Huggins interaction parameters satisfy χAC > χ BC ≈ χAB. Using various initialization strategies, a large number of periodic structures have been obtained in our calculations. A fourth-order pseudospectral algorithm combined with Anderson mixing method is used to compute the free energy of candidate structures carefully. The stability has been analyzed in detail by splitting the free energy into internal and entropic parts. A complete and complex triangular phase diagram is presented for a model with χAC > χBC = χAB in which 15 ordered phases, including two- and three-dimensional structures, have been predicted to be stable from the SCFT calculations. Generally speaking, with the asymmetrical interactions, the hierarchical structures tend to be formed near the B-rich corner of the triangular phase diagram. This work broadens the previous theoretical results from equal interaction systems to unequal interaction systems. The predicted phase behavior is in good agreement with experimental observations and previous theoretical results.

Solubility of gypsum in aqueous NaCl + K2SO4 solution using calcium ion selective electrode-investigation of ionic interactions

Gypsum is a sparingly soluble salt, often precipitates in some industrial equipment and imposes instrumental damage. Prediction of phase behavior and scaling tendency of gypsum requires recognition of all ionic interactions in an electrolyte aqueous solution. In this respect, Ion Selective Electrode (ISE) is progressively applied in industry and laboratory as a fast and online monitoring tool to measure concentration of individual ions. This study is an attempt to advance the current application of ISEs for reliable measurement of concentration of ions with more precision. The solubility of gypsum in the aqueous solution of K2SO4 and NaCl is measured at the different temperatures. We carried out a two-part experimental approach that at the first step we focus on effect of the ionic interactions on activity of Ca2+ in multi-electrolyte solutions. Following determination of the ionic interaction parameters, the calcium ISE is used to measure the concentration of Ca2+ in a saturated solution of gypsum so that the influence of the other ions on the gypsum formation is investigated. The Khoshkbarchi and Vera's (KV) and Lin and Lee (LL) models are applied to calculate the activity coefficient of Ca2+ and discussed in this work.

PC-SAFT predictions of VLE and LLE of systems related to biodiesel production

Biodiesel has risen as a potential energy carrier and alternative biofuel. Understanding the phase equilibria of systems formed by alkyl esters of fatty acids and alcohols is of great interest to biodiesel industry. Predictive thermodynamic models are essential for process simulators used for plant design and optimization. The Perturbed Chain form of the Statistical Associating Fluid Theory (PC-SAFT) is used in this work to predict the thermophysical properties, vapor–liquid equilibrium (VLE) and liquid–liquid equilibrium (LLE) of binary and ternary systems related to biodiesel processing. The PC-SAFT was able to accurately predict the VLE and LLE of mixtures involving different types of biodiesel. The phase behavior prediction for mixtures investigated in this work seems to be not greatly influenced by the dipole–dipole. Furthermore, the phase behavior of binary and ternary mixtures of biodiesel was successfully predicted using a transferable cross–association parameters approach.

Sequence heuristics to encode phase behaviour in intrinsically disordered protein polymers.

Proteins and synthetic polymers that undergo aqueous phase transitions mediate self-assembly in nature and in man-made material systems. Yet little is known about how the phase behaviour of a protein is encoded in its amino acid sequence. Here, by synthesizing intrinsically disordered, repeat proteins to test motifs that we hypothesized would encode phase behaviour, we show that the proteins can be designed to exhibit tunable lower or upper critical solution temperature (LCST and UCST, respectively) transitions in physiological solutions. We also show that mutation of key residues at the repeat level abolishes phase behaviour or encodes an orthogonal transition. Furthermore, we provide heuristics to identify, at the proteome level, proteins that might exhibit phase behaviour and to design novel protein polymers consisting of biologically active peptide repeats that exhibit LCST or UCST transitions. These findings set the foundation for the prediction and encoding of phase behaviour at the sequence level.

Systematic Misprediction of n-Alkane + Aromatic and Naphthenic Hydrocarbon Phase Behavior Using Common Equations of State

Phase equilibrium calculations based on 150 n-alkane + aromatic and n-alkane + naphthenic hydrocarbon binary mixtures were performed. These calculations were compared with experimental measurements whenever possible, and additional measurements were made as part of this work. The widely used Peng–Robinson (PR) and Soave–Redlich–Kwong (SRK) equations of state are shown to predict nonphysical liquid–liquid phase behavior for long-chain n-alkane + aromatic and long-chain n-alkane + naphthenic hydrocarbon binary mixtures with standard pure-component parameters (Tc, Pc, ω). Incorrect phase behavior prediction is shown to be insensitive to the selection of correlations for estimating pure-component properties for n-alkanes that are not available from experimental data. For cubic equations of state, correct phase behaviors are obtained only when negative values of the binary interaction parameters (kij) are used. For PC-SAFT, a noncubic equation of state (with standard parameter values defining molecules and wit...

A Priori Prediction of Dissociation Phenomena and Phase Behaviors of Ionic Liquids

Many unique properties of ionic liquids (ILs) are closely related to the extent of dissociation (α) in solution. However, most of the existing models for ILs assume either full dissociation or full nondissociation regardless of mixture compositions. In this work, the dissociation of ILs (CA = C+ + A–) is described as a chemical reaction. Together with the predictive COSMO-SAC model, the dissociation constant can be determined based on the value of α in the pure state. Our results show that the predicted composition dependence of α is in good agreement with experiment over the entire concentration range. We further examine the prediction of a variety of thermodynamic properties and phase behaviors of IL solutions, covering high (infinite dilution activity coefficient of solvent), medium (vapor–liquid and liquid–liquid), and low (osmotic coefficient and mean ionic activity coefficient) IL concentrations (a total of 9857 data points). Our results show that the composition dependence of IL dissociation has a ...

An evolved cubic equation of state with a new attractive term

A large variety of modifications have been presented for the temperature dependent function (α) existing in the attractive term of cubic equations of state (CEOS). Most of α-functions attempted to modify the vapor pressure prediction of polar components while other modifications have focused on both polar and non-polar compounds and other relations have considered an expansion of polynomials in the acentric factor (ω) and reduced temperature (Tr) to predict vapor pressure more accurately. In most cases such as Soave and Peng–Robinson equations of state, the suggested α-functions do not show a limiting behavior when temperature increases infinitely. In addition, the incompetency of many α-functions in describing the supercritical behavior of fluids have been modified by defining specific parameters for some different components or switching α-functions. Such approaches can create difficulties for the calculation of properties in which, there are equations of state derivatives. In this work, a new α-function has been put forward to improve the attractive term of Peng–Robinson EOS. The proposed model is capable of predicting the saturation vapor pressure of light gases, polar, nonpolar and heavy hydrocarbon compounds. The normal boiling point (Tb) is used as an additional parameter along with acentric factor (ω) and reduced temperature (Tr) to enhance the ability of the proposed α-function in the prediction of fluid phase behavior especially for petroleum fractions and heavy cuts existing in reservoir fluids samples. Moreover, two discriminating indexes (λ1, λ2) are introduced, to handle all pure substances and classify them under distinct groups for the allocation of special coefficients to each group. Experimental vapor pressure of 31 different pure components, 11 binary mixtures and 12 reservoir oil samples were collected, in order to evaluate the performance of the proposed model compared to the original Peng–Robinson. A comparison between the proposed model and other powerful equations of state such as modified PR, PSRK and PC-SAFT model was also performed, and the results of which indicate that the new proposed model has better outcomes over the mentioned methods, especially for complex mixtures and reservoir fluids. Furthermore, the proposed α-function shows a reasonable limiting behavior as the temperature approaches infinity and is capable of describing the fluid phase behavior at supercritical region.

Interactions in protein mixtures. Part II: A virial approach to predict phase behavior

A virial theory was used to relate molecular interactions (in terms of second virial coefficients, B′) and molecular size ratios to liquid–liquid phase separation. Application of the virial theory to binary hard sphere mixtures (additive and non-additive) confirmed the applicability of this simple approach towards predicting phase behavior based on two-particle interactions.Experimentally, second cross virial coefficients were obtained for dextran/gelatin, whey protein isolate (WPI)/gelatin mixtures and whey protein aggregate (WPA)/gelatin mixtures using membrane osmometry at varying ionic strength. From this, solvent conditions where interactions between proteins are dominated by electrostatics and solvent conditions where interactions are dominated by hard body interactions could be determined.Using experimentally obtained second virial coefficients, the liquid–liquid phase separation for gelatin/dextran mixtures was successfully predicted. Second cross virial coefficients for gelatin/whey protein isolate and for gelatin/whey protein aggregate could be related to the absence of phase separation in these mixtures. This could be related to a similar size of the proteins and their non-additive behavior at conditions where they mainly interact via hard body interactions.

Comparison of Cubic-Plus-Association and Perturbed-Chain Statistical Associating Fluid Theory Methods for Modeling Asphaltene Phase Behavior and Pressure–Volume–Temperature Properties

The tendency of asphaltene, the heaviest and most polarizable component of crude oil, to deposit and block wellbores and pipelines can potentially lead to production loss and significant cost of remediation. Asphaltenes constitute a particular interest in oil production because of the lack of understanding of their molecular structure and the mechanisms by which they precipitate and deposit. Because prevention is far less expensive than removal, a better understanding of asphaltene precipitation and deposition phenomena is of great importance for the oil industry. Precipitation, which is a necessary but not sufficient condition for deposition, requires accurate modeling of asphaltene phase behavior with respect to variations in the temperature, pressure, and composition. Over the past decade, cubic-plus-association (CPA) and perturbed-chain statistical associating fluid theory (PC-SAFT) equations of state (EOSs) have been proposed for modeling complex systems, such as asphaltenic crude oils. In this work, a comparison between CPA and PC-SAFT EOSs is presented to illustrate their potential and limitations on the prediction of asphaltene phase behavior and pressure–volume–temperature (PVT) properties of crude oils over a range of pressures and temperatures. With an optimized characterization, both EOSs are able to give acceptable predictions of the phase behavior and asphaltene precipitation tendency. However, PC-SAFT is superior in the prediction of derivative thermodynamic properties, especially at high pressures.

Prediction of liquid–liquid equilibrium behavior for aliphatic + aromatic + ionic liquid using two different neural network-based models

In this study, the liquid–liquid phase behavior of aromatic compound+aliphatic compound+ionic liquid (IL) ternary systems was estimated by using two artificial neural networks (ANN) developed based on back propagation (BP) and hybrid group method of data handling (GMDH). Molar ratio of aliphatic compound, aromatic compound, and IL as well as temperature, molecular weight ratio of aliphatic compound to IL, and molecular weight ratio of aromatic compound to IL were chosen as the inputs to the networks. Additionally, the mole fraction of components in final alkane-rich phase and IL-rich phase was considered as desired outputs. The best topology of the BP-ANN model was found as (6-8-4). Besides, corresponding quadratic expression for each neuron of the hybrid GMDH model was determined. The obtained results by the hybrid GMDH model provided a near to accurate prediction of phase behavior with average relative deviation (ARD%) of 7.27%. However, the BP-ANN model presented a slightly more accurate prediction with ARD% of 4.49%. Furthermore, the comparison of the results with those obtained by the famous thermodynamic models, proved the preciseness of the proposed networks.

Vapor-liquid equilibria of CH4, CO2 and their binary system CH4 + CO2: A comparison between the molecular simulation and equation of state

An accurate knowledge about phase behaviors of CH4, CO2 and their binary mixture is crucial in fields of natural gas liquefaction and refrigeration applications. In this work, two all-atom force fields of TraPPE-EH and EMP2 were used for the components CH4 and CO2, respectively. Then the vapor-liquid equilibria (VLE) of CH4, CO2 and their binary system were calculated via the NVT- and NpT Gibbs Ensemble Monte Carlo Simulations. Meanwhile the traditional method using Equation of State (EoS) to correlate the VLE properties was also investigated. The EoSs considered in this work were three classic cubic RK, SRK, PR and another advanced molecular-based PC-SAFT equations. For pure components, both molecular simulations and the PC-SAFT EoS could obtain satisfactory predictions for all the saturated properties. However, the saturated liquid densities calculated by the cubic EoSs were not so good. It was also observed that the TraPPE-EH force field had a good representation for CH4 molecule, while the EMP2 force field was not enough accurate to represent CO2 molecules. For the mixture CH4 + CO2, SRK and PR showed the best predictions for the saturated pressure-component property, while good results were also obtained via molecular simulations and PC-SAFT EoS. It was suggested that special combining rules or binary interaction parameters were important to obtain enough accurate prediction of the mixed phase behavior. Compared with the cubic EoS, the PC-SAFT and molecular simulation method showed better adaptabilities for both the pure and mixture systems. Besides, the accurate molecular parameters used in the PC-SAFT and molecular simulations could bring about direct and deep understanding about the molecular characteristics.

Direct perturbation of the Peng–Robinson attraction and covolume parameters for reservoir fluid characterization

Numerical simulation of enhanced oil recovery processes, such as gas and steam floods, requires that the reservoir fluid of interest is reliably modeled by use of an equation of state (EOS). Reservoir fluid characterization using an EOS can be challenging because of the uncertainties associated with non-identifiable heavy fractions, which are characterized as a few pseudo components.This paper presents a new method for reservoir fluid characterization that directly perturbs the attraction and covolume parameters of pseudo components from the n-alkanes’ values (direct perturbation from n-alkanes, or direct PnA). The direct PnA method systematically adjusts the attraction and covolume parameters of all pseudo components within a given reservoir fluid through a few adjustable parameters. As in our prior research, the ψ(=a/b2) parameter is used to ensure the proper interrelationship between the attraction (a) and covolume (b) parameters. A new approach to adjustment of the ψ parameters is developed with a linear relationship of ψ with molecular weight. The Peng–Robinson (PR) EOS is used throughout the research.The direct PnA method is applied to 84 different reservoir fluids, such as gas condensates, volatile oils, and heavy oils. Characterization of each fluid uses the saturation pressure and liquid density data at different pressures at the reservoir temperature. Other phase behavior data available are used to test the predictive capability of the fluid models developed using the PR EOS with the direct PnA method. Results show that the reservoir fluids tested are reliably characterized by the systematic control over phase behavior predictions in the direct PnA method. Unlike prior characterization methods, the direct PnA method does not require estimation of apparent critical parameters of non-identifiable heavy fractions.

Hydrate and Phase Behavior Modeling in CO2-Rich Pipelines

Carbon dioxide transport will be a key part of any Carbon Capture and Storage (CCS) system. Generally for CCS, flow assurance modeling of these pipelines has been restricted to pure CO2 systems; however CO2 streams coming from capture processes are not pure and contain various components. These fluids have a certain level of moisture, so dehydration is needed before delivering CO2 rich fluid to the export pipeline, in order to prevent potential hydrate formation, two-phase flow and corrosion in the export line. In this contribution, the hydrate phase equilibria of the binary CO2 + (N2 or CH4 or O2 or Ar or CO) systems in the presence of a free water phase were determined as well as direct hydrate dissociation conditions of liquid CO2 in absence of a free phase. A rigorous and generalized model is presented to predict the phase behavior, hydrate dissociation pressures and the dehydration requirements of CO2 rich gases. A statistical thermodynamic approach, with the Cubic-Plus-Association equation of state, is employed to model the phase equilibria. The hydrate-forming conditions are modeled by the solid solution theory of van der Waals and Platteeuw. Predictions of the developed model are first validated using simple systems and then for more complicated synthetic multicomponent systems. Showing that accurate phase behavior prediction of CO2 rich stream is critical for accurate predictions of the hydrate phase behavior of these fluids.

A modified continuous flow apparatus for gas solubility measurements at high pressure and temperature with camera system

A continuous flow apparatus with a camera system was modified for high-pressure gas–liquid equilibrium measurements. The continuous flow method is suitable for phase equilibrium measurements of heat-sensitive materials, such as heavy oils and bio-based feedstocks, which are refined in hydroprocesses at high temperatures and pressures. Benefits of the modified apparatus are a short residence time of a sample in the heated zone and no sampling is required. In contrast, the static method is usually impractical with the heat-sensitive materials due to the long residence time of sample in the equilibrium cell. In this study, the continuous flow apparatus and the measurement method was adopted and validated. The validation was conducted by measuring hydrogen solubility in toluene and n-hexadecane at high temperatures and pressures (T=461–575K, P=5–10MPa) and comparing the results with the literature values. Relative average pressure deviation was found to be 3.8% for measurements of this work when the PC-SAFT equation of state was optimized against literature values while the relative average pressure deviations for the literature data sets varied from 0.9% to 4.7%. Thus, the quality of measurements was shown to be excellent. In addition, hydrogen solubility was also measured in n-octadecane in order to obtain new data for this system. All measured systems were modeled with the PC-SAFT and the original and modified Peng–Robinson equations of state in order to simplify the validation of the measurement method and to compare the performance of the selected models. PC-SAFT was found to predict phase behavior of the systems precisely whereas the predictions of the Peng–Robinson modifications were less precise.

Thermodynamic model for biomass processing in pressure intensified technologies

Pressure intensified technologies have a great potential in the context of biomass refining. A thermodynamic model able to predict phase behavior of typical mixtures found in biomass processing technologies, containing for instance hydrocarbons, organo-oxygenated compounds and water, is required for the development of a biorefinery process simulator. Moreover, the design of particular fuel/biofuel blends also requires the support of a thermodynamic model to predict the properties of the final products. These types of mixtures are highly non-ideal due to the presence of association and solvation effects. It has already been proved that the Group Contribution with Association Equation of State (GCA-EoS) is able to predict the complex phase behavior of mixtures containing natural products and biofuels. In the last few years, several contributions agree that 2,5-dimethylfuran has a great potential as a sugar-derived fuel additive. In this work, as a case study, we extend the GCA-EoS to represent the phase equilibria of furan derivatives with hydrocarbons and alcohols. In addition, we show that the GCA-EoS is able to predict, based on the performed parameterization, high pressure data of 2,5-hydroxymethylfurfural solubility in CO2 and ethanol as co-solvent.

Prediction of Thermodynamic Properties and Phase Behavior of Fluids and Mixtures with the SAFT-γ Mie Group-Contribution Equation of State

Group contribution (GC) approaches are based on the premise that the properties of a molecule or a mixture can be determined from the appropriate contributions of the functional chemical groups present in the system of interest. Although this is clearly an approximation, GC methods can provide accurate estimates of the properties of many systems and are often used as predictive tools when experimental data are scarce or not available. Our focus is on the SAFT-γ Mie approach [Papaioannou, V.; Lafitte, T.; Avendaño, C.; Adjiman, C. S.; Jackson, G.; Müller, E. A.; Galindo, A. Group contribution methodology based on the statistical associating fluid theory for heteronuclear molecules formed from Mie segments. J. Chem. Phys. 2014, 140, 054107–29] which incorporates a detailed heteronuclear molecular model specifically designed for use as a GC thermodynamic platform. It is based on a formulation of the recent statistical associating fluid theory for Mie potentials of variable range, where a formal statistical–mechanical perturbation theory is used to maintain a firm link between the molecular model and the macroscopic thermodynamic properties. Here we summarize the current status of the SAFT-γ Mie approach, presenting a compilation of the parameters for all functional groups developed to date and a number of new groups. Examples of the capability of the GC method in describing experimental data accurately are provided, both as a correlative and as a predictive tool for the phase behavior and the thermodynamic properties of a broad range of complex fluids.

Polyelectrolyte Blends and Nontrivial Behavior in Effective Flory-Huggins Parameters.

The Flory χ-parameter is a ubiquitous description of the extent of immiscibility that is apparent between two or more polymeric species. While the formalism is a powerful one in most systems of technological interest, experimentally obtaining this parameter requires the assumption of an underlying theoretical model. For charged systems, mapping to analogous uncharged systems is often assumed by introducing an "effective χ", χeff. Random phase approximation (RPA) analysis based on the Hamiltonian used for recent self-consistent field theory-liquid-state theories (SCFT-LS) demonstrates that χeff incorporates molecular-level details such as charge ordering. Even for simple polyelectrolyte blends where the bare χ is kept constant, the observed χeff will drastically change as a function of composition; prediction of heterogeneous polyelectrolyte material phase behavior using χeff is thus highly nontrivial since an understanding of local charge structure is required.