Effective Flow Properties Research Articles

Scattering by source-type flows in disordered media

Scattering through a natural porous formation (by far the most ubiquitous example of disordered medium) represents a formidable tool to identify effective flow and transport properties. In particular, we are interested here in the scattering of a passive scalar as determined by a steady velocity field, which is generated by a line of singularity. The velocity undergoes erratic spatial variations, and concurrently, the evolution of the scattering is conveniently described within a stochastic framework that regards the conductivity of the hosting medium as a stationary, Gaussian, random field. Unlike the similar problem in uniform (in the mean) flow fields, the problem at stake results much more complex. Central to the present study is the fluctuation of the driving field that is computed in closed (analytical) form as a large time limit of the same quantity in the unsteady-state flow regime. The structure of the second-order moment Xrr, quantifying the scattering along the radial direction, is explained by the rapid change of the distance along which the velocities of two fluid particles become uncorrelated. Moreover, two approximate, analytical expressions are shown to be quite accurate in reproducing full simulations of Xrr. Finally, the same problem is encountered in other fields, belonging both to classical and to quantum physics. As such, our results lend themselves to being used within a context much wider than that exploited in the present study.

Digital rock analysis to estimate stress-sensitive rock permeabilities

Reservoir deformations due to stress changes may have a substantial impact on the effective flow properties of reservoir rocks, which may potentially impact productivity and injectivity performance. Absolute permeability has been a primary factor in measuring hydraulic conductivity of porous materials. In order to investigate the impact of the granular structure and the extent of intergranular cementation on stress-dependent absolute permeability, we present a workflow that utilizes limited knowledge about the rock sample in the form of micro-Computed Tomography images to simulate stress-induced deformations. This analysis is then followed by Lattice Boltzmann Methods to simulate fluid flow in the deformed medium. While previous works using digital rocks ignored the impact of intergranular cements, we should consider the impact of these cements on the stress sensitivity of absolute permeability of the rock. It is notable that changes in the rock properties like the nature of stress would be a continuous change, so measurements at few stress points may not be enough to describe the overall response in any arbitrary loadings. Despite empirical formulae that are limited to few measurements, we proposed a method based on the equation of states to estimate rock permeability through three independent strain invariants. Digital rock physics, finite element analysis, and Lattice Boltzmann Methods were integrated to obtain a quick estimation of the rock permeability using micro-CT images without the need for repetitive laboratory experiments. Our results show that absolute permeability decreases significantly with increasing first and third strain invariants while it slightly increases with the second strain invariant. The stress sensitivity of absolute permeability was found to be negatively correlated with Young’s modulus of the intergranular cement. We observed that incorporating more intergranular cement will make the absolute permeability more sensitive to stress changes.

Aquifer Characterization and Uncertainty in Multi-Frequency Oscillatory Flow Tests: Approach and Insights.

Characterizing aquifer properties and their associated uncertainty remains a fundamental challenge in hydrogeology. Recent studies demonstrate the use of oscillatory flow interference testing to characterize effective aquifer flow properties. These characterization efforts relate the relative amplitude and phase of an observation signal with a single frequency component to aquifer diffusivity and transmissivity. Here, we present a generalized workflow that relates extracted Fourier coefficients for observation signals with single and multiple frequency components to aquifer flow properties and their associated uncertainty. Through synthetic analytical modeling we show that multi‐frequency oscillatory flow interference testing adds information that improves inversion performance and decreases parameter uncertainty. We show increased observation signal length, sampling frequency, and pressure sensor accuracy all produce decreased parameter uncertainty. This work represents the first attempt we are aware of to quantify effective aquifer parameters and their associated uncertainty using multi‐frequency oscillatory flow interference testing.

Effective flow and transport properties of heterogeneous unsaturated soils

The heterogeneity of field scale soils poses a challenge to predictive large scale flow and transport modeling. The theory of effective macroscale parameters holds good and is applicable in dealing with such problems. But the validity of the analytic stochastic solutions obtained for randomly heterogeneous soils is debatable, as the test cases under which they are validated are of limited scope due to linearization and perturbation approximations. In this study, samples of heterogeneous soils are generated using sets of spatially correlated random field parameters that are either geometrically isotropic, or else, geometrically anisotropic with either horizontal or vertical stratification (perfect or imperfect). Several combinations of ratios of correlation length and capillary dispersion lengths are considered. Numerical simulations of unsaturated flow are performed on each randomly heterogeneous soil sample. The principal components K^ii(Ψ) of the macroscale effective unsaturated conductivity are then obtained as a function of the mean suction Ψ of the sample. They are compared to stochastic spectral perturbation theory, and to a probabilistic semi-empirical Power Average Model (PAM). They are also compared with arithmetic, geometric and harmonic mean conductivity-suction curves. The numerically upscaled principal conductivity curves match quite well the PAM, better than the classical means (Arithmetic, Geometric, Harmonic), and also somewhat better than the curves obtained from stochastic spectral perturbation theory. It is observed that the upscaled principal components Kii(ψ), obtained numerically and with the PAM along directions “i” orthogonal/parallel to perfect stratification coincide with the harmonic/arithmetic mean curves at low suctions (i.e., near saturation), but deviate from it and come closer to the geometric mean at higher suctions. The PAM appears suitable for generation of approximate upscaled conductivity curves, e.g., for obtaining the mesh-scale or block-scale conductivity curves in large scale simulation codes. Transient solute transport simulations are then performed on the detailed random velocity fields obtained from the steady state simulations of unsaturated flow in the randomly heterogeneous soil samples. Snapshots of solute concentration C(x,z,t) are taken at different times. The temporal evolution of spatial moments of concentration is analyzed in order to characterize the macroscale advection and dispersion of the unsaturated concentration plume, and in particular, its macro-dispersion coefficient (D) and dispersivity length scale (A). For the synthetic soil samples considered in this study, the macro-dispersive spreading of the solute is stronger for flow parallel to vertical stratification, compared to flow perpendicular to horizontal stratification, and also, compared to flow in statistically isotropic non-stratified soil.

Micromechanical modeling of the effect of phase distribution topology on the plastic behavior of dual-phase steels

In this paper, a topology optimization based numerical method is proposed to investigate the micromechanical plastic behavior of dual-phase (DP) steels. Representative volume elements (RVEs) are constructed using the topology optimization based artificial microstructures. Micromechanical behavior under various loading conditions are predicted for the RVEs to investigate the plasticity, strain localization and strengthening mechanism affected by the microstructural characteristics of DP steels. Plastic strain patterns including shear band are found during the deformation. Due to the twisting movement of martensite grains, the direction of the strain localization bands in the shear loading case is 0° or 90° to the loading direction, while it is 45° in the tensile and compressive cases. Moreover, the effective flow behavior of the material under shear loading is lower than those found in tensile and compressive cases. The influence of various microstructural features, such as, martensite fraction, distribution of each phase, on the effective flow properties and the local strain partitioning has also been identified. Both of the effective flow properties and strain localization exhibit the tendency to be strengthened with the increase of martensite phase fraction. Furthermore, the RVE with more uniform martensite distribution leads to the decrease of effective flow properties and strain localization. Longer martensite-ferrite interface results from the clustering of martensite, which increases the strain localization effect during the plastic deformation.

Structure-Dependent Analysis of Nanoporous Metals: Clues from Mechanical, Conduction, and Flow Properties

Nanoporous metals have emerged as promising electrocatalysts for electrochemical systems. The rational design needs us to understand the structure–property relations of such bicontinuous porous structures in detail. In this study, we investigate the effect of microstructures on effective macro-properties of three types of nanoporous architectures, namely dealloying structure, Voronoi structure, and diamond structure. These structures have similar values of nodal connectivity and relative density. Finite volume simulations are carried out to predict the effective mechanical, conduction and flow properties. Quantitative structural analysis reveals that the dealloying and Voronoi structures are distinct in terms of ligament curliness, whereas the diamond structure is a long-range order topology. The results suggest that the specific surface area, Young’s modulus, yield strength, conductivity, and permeability are highly sensitive to the curliness factor, while being relatively much less sensitive to the dist...

Conductivity and Transmissivity of a Single Fracture

The effective flow and conduction properties of fractures with Gaussian spatial correlations are investigated by solving the microscale governing equations in three-dimensional samples, along the lines initiated by Mourzenko et al. (J Phys II(5):465–482, 1995), Volik et al. (Trasnp Porous Media 27:305–325, 1997) but in greater details, over a wider range of the parameters, and with greatly improved accuracy. The effective transport coefficients are related to intrinsic geometrical characteristics, quantified by the mean aperture, the surface roughness RMS amplitude, its correlation length, and the intercorrelation coefficient of the roughness on the two surfaces. Extensive results are presented and analyzed. An empirical relationship between the transmissivity and conductivity is formulated, the validity of the Reynolds approximation is assessed, and heuristic expressions are proposed for the direct estimation of the transport coefficients as functions of the fracture geometrical characteristics.

Effective Flow Properties for Cells Containing Fractures of Arbitrary Geometry

Summary Hydrocarbon production from unconventional resources such as shale usually entails stimulation by hydraulic fracturing, which results in nonplanar (curved) fractures. However, most reservoir models assume that the induced fractures are planar for the sake of simplicity. Considering the growing interest of the petroleum industry in better understanding production from these resources, we develop a fracture-cell model to capture the effects of fracture nonplanarity on transport properties. To build a realistic reservoir model for a fractured formation, we must account for three types of interactions: matrix-matrix (M-M), matrix-fracture (M-F), and fracture-fracture (F-F). The transport properties of the M-M interaction are based on laboratory measurements. In this study, we analytically determine the transport properties of the two other types of interactions (M-F and F-F). For this purpose, we account for the aperture size and spatial location of the fracture. As a result, we provide effective porosity and effective anisotropic permeabilities for a reservoir cell that contains a fracture inside it. The reservoir cell with transport properties that are modified is a fracture cell. We implement the fracture-cell model in a reservoir simulator and perform analyses for a single fracture and for multiple intersecting fractures; these fractures are nonplanar. The analyses include both single- and multiphase flow models and show that the hydrocarbon pressure inside the reservoir is strongly dependent on the fracture geometry when the matrix permeability is smaller than 1 µd. Thus, it is crucial to model the fracture geometry more accurately in unconventional reservoirs with ultralow permeabilities such as shale. One can easily implement the developed fracture-cell model in reservoir simulators, and there is no local refinement around the fracture. The main advantage of the proposed model is its simplicity, conjoined with its ability to capture the nonplanarity of the fracture. The developed model has major applications for understanding production from formations that are heavily fractured.

Effective flow properties heterolithic, cross-bedded tidal sandstones: Part 1. Surface-based modeling

Tidal heterolithic sandstones are commonly characterized by millimeter- to centimeter-scale intercalations of mudstone and sandstone. Consequently, their effective flow properties are poorly predicted by (1) data that do not sample a representative volume or (2) models that fail to capture the complex three-dimensional architecture of sandstone and mudstone layers. We present a modeling approach in which surfaces are used to represent all geologic heterogeneities that control the spatial distribution of reservoir rock properties (surface-based modeling). The workflow uses template surfaces to represent heterogeneities classified by geometry instead of length scale. The topology of the template surfaces is described mathematically by a small number of geometric input parameters, and models are constructed stochastically. The methodology has been applied to generate generic, three-dimensional minimodels (9 m3 [∼318 ft3] volume) of cross-bedded heterolithic sandstones representing trough and tabular cross bedding with differing proportions of sandstone and mudstone, using conditioning data from two outcrop analogs from a tide-dominated deltaic deposit. The minimodels capture the cross-stratified architectures observed in outcrop and are suitable for flow simulation, allowing computation of effective permeability values for use in larger-scale models. We show that mudstone drapes in cross-bedded heterolithic sandstones significantly reduce effective permeability and also impart permeability anisotropy in the horizontal as well as vertical flow directions. The workflow can be used with subsurface data, supplemented by outcrop analog observations, to generate effective permeability values to be derived for use in larger-scale reservoir models. The methodology could be applied to the characterization and modeling of heterogeneities in other types of sandstone reservoirs.

Numerical Results for the Effective Flow and Thermal Properties of Idealized Graphite Foam

To simulate the heat transfer performance of devices incorporating high-conductivity porous materials, it is necessary to determine the relevant effective properties to close the volume-averaged momentum and energy equations. In this work, we determine these effective properties by conducting direct simulations in an idealized spherical void phase geometry and use the results to establish closure relations to be employed in a volume-averaged framework. To close the volume-averaged momentum equation, we determine the permeability as defined by Darcy’s law as well as a non-Darcy term, which characterizes the departure from Darcy’s law at higher Reynolds numbers. Results indicate that the non-Darcy term is nonlinearly related to Reynolds number, not only confirming previous evidence regarding such behavior in the weak inertia flow regime, but demonstrating that this is generally true at higher Reynolds numbers as well. The volume-averaged energy equation in the fluid phase is closed by the thermal dispersion conductivity tensor, the convecting velocity, and the interfacial Nusselt number. Overall, it has been found that many existing correlations for the effective thermal properties of graphite foams are oversimplified. In particular, it has been found that the dispersion conductivity is not well characterized using the Péclet number alone, rather the Reynolds and Prandtl numbers must be considered as separate influences. Additionally, the convecting velocity modification, which is not typically considered, was found to be significant, while the interfacial Nusselt number was found to exhibit a nonzero asymptote at low Péclet numbers. Finally, simulations using the closed volume-averaged equations reveal significant differences in heat transfer when employing the present dispersion model in comparison to a simpler dispersion model commonly used for metallic foams, particularly at high Péclet numbers and for thicker foam blocks.

Design criteria for a printed tissue engineering construct: A mathematical homogenization approach

Cartilage tissue repair procedures currently under development aim to create a construct in which patient-derived cells are seeded and expanded ex vivo before implantation back into the body. The key challenge is producing physiologically realistic constructs that mimic real tissue structure and function. One option with vast potential is to print strands of material in a 3D structure called a scaffold that imitates the real tissue structure; the strands are composed of gel seeded with cells and so provide a template for cartilaginous tissue growth. The scaffold is placed in the construct and pumped with nutrient-rich culture medium to supply nutrients to the cells and remove waste products, thus promoting tissue growth. In this paper we use asymptotic homogenization to determine the effective flow and transport properties of such a printed scaffold system. These properties are used to predict the distribution of nutrient/waste products through the construct, and to specify design criteria for the scaffold that will optimize the growth of functional tissue.

The effect of carbonate cementation on permeability heterogeneity in fluvial aquifers: An outcrop analog study

Spatial variations in permeability have been widely recognized as a dominant control on solute transport and petroleum migration. Outcrop studies are one means of improving our understanding of the nature of permeability heterogeneity and the underlying geologic controls. To date, these studies have either neglected the importance of cementation or investigated only the modern cemented permeability without explicitly considering how cementation has affected permeability. Two outcrops of channel sand deposits in the Plio-Pleistocene Sierra Ladrones Formation, New Mexico are investigated to quantify (1) the effect of cementation on permeability at a given location, and (2) the patterns of cementation and their impact on permeability heterogeneity. We analyzed approximately 400 spatially located samples (∼ 10 cm 3 each) from the two outcrops. Cementation is typically weak to moderate, filling less than 60% of the original pore space. Permeability and cement content are bimodally distributed and negatively correlated. Permeability reduction is investigated by measuring modern permeability, minus-cement permeability, minus-cement porosity, and volumetric cement content. Permeability is reduced locally by up to three orders of magnitude. By combining models of permeability for uncemented sediments and cemented sediments, we derive a model of cemented permeability ( k c) that depends primarily on three factors: uncemented permeability ( k u), original porosity, and cement fraction. Our results suggest that accounting for porosity reduction only may result in an overestimation of k c. Conversely, accounting for porosity reduction, increased specific surface area, and increased tortuosity appears to result in an underestimation of k c. Accounting for porosity reduction and increased specific surface area provide the best estimate of k c. Cementation is highly variable spatially and is poorly correlated with lithofacies. Variogram analysis of cementation and permeability indicates that the spatial distribution of cementation is the dominant control on the permeability correlation structure. The spatially variable cementation also results in an order-of-magnitude increase in the variance of log-permeability. Applying stochastic models of effective flow and transport properties to the results of this study suggest that cementation has a relatively small effect on effective permeability but can potentially increase dispersivity by two orders of magnitude.

Mechanical Property Evaluation Using the Small Punch Test

Abstract The small punch (SP) test technique is an attractive method for evaluating mechanical behavior in materials where only small volumes of materials are available. An SP-test machine was developed where miniature disc specimens 5 and 3 mm in diameter and with thickness ranging between 50 to 400 µm can be investigated. For two materials, a 1Cr-0.5Mo low alloy steel and an 18Cr-9Ni austenitic stainless steel, the stress-strain flow curve was assessed and compared with that in tensile tests. The yield strength was determined from the SP-test by analyzing the initial elastic deformation with help of classical plate bending theory. The effective flow properties in the biaxial stress state were evaluated through analytical modelling of the stretching deformation. Good agreement between the uniaxial and biaxial flow properties was found for the 1Cr-0.5Mo steel and acceptable agreement for the 18Cr-9Ni steel.

Fluid flow and conduction in two‐dimensional fractures with rough, self‐affine surfaces: A comparative study

We study fluid flow and conduction in a two‐dimensional model of a fracture with rough, self‐affine internal surfaces. The model consists of two parallel flat plates on which two rough, self‐affine surfaces, characterized by a roughness exponent H, are superimposed. The methods that we use for computing the effective flow and transport properties of the fracture include the lattice Boltzmann method for computing the flow properties, a random walk method for determining the effective conductivity of the (fluid‐saturated) fracture, and the Reynolds approximation. We also develop an asymptotic expression for the effective conductivity. The aperture of the fracture, as well as the roughness of its rough surface, are systematically varied in order to assess their effect on the effective permeability and conductivity of the fracture, and also test the accuracy and consistency of the methods. For large mean apertures, and all values of the roughness exponent H, all the methods yield essentially the same results. However, as the mean aperture decreases, the differences between the predictions of the methods increase significantly. We find that the Reynolds approximation provides relatively accurate estimates of the (hydraulic or electrical) apertures only if the fracture is at least moderately wide and that, similar to real three‐dimensional fractures, the electrical aperture is always smaller than the hydraulic fracture.

Upscaled flow and transport properties for heterogeneous unsaturated media

To represent a heterogeneous unsaturated medium by its homogeneous equivalent, stochastic theory‐based analytical formulas and numerical Monte Carlo simulations are used to obtain upscaled (effective) flow and transport properties. The Monte Carlo experiments simulate steady state flow and transient transport of nonreactive solute in two‐dimensional heterogeneous media. Constitutive relations for unsaturated media at the mesh‐size scale are based on the van Genuchten‐Mualem relationships. A unit‐mean‐gradient approach is used to derive upscaled properties for flow parallel and perpendicular to bedding. Upscaled moisture retention and unsaturated conductivities (K) and longitudinal dispersivities are obtained by simulating steady gravity drainage conditions for a series of applied infiltration rates, characteristic of relatively dry conditions in coarse‐textured sediments. Macrodispersivities are calculated on the basis of spatial moments of the ensemble‐mean plume. Results show that the calculated effective unsaturated K based on the analytical formulas compare well with those based on Monte Carlo simulations. The macroscopic anisotropy (ratio of K parallel to bedding to K perpendicular to bedding) increases with increasing tension, although the increase is rather mild. The longitudinal macrodispersivity for the equivalent homogeneous medium also increases with increasing tension. A comparison of numerical results with stochastic solutions suggests that the computed dispersivities are of the same order of magnitude at low tensions. At higher tensions the dispersivity estimates deviate significantly. Nonetheless, both numerical and analytical results show that the longitudinal dispersivities for flow parallel to bedding are higher than those for flow perpendicular to bedding. In addition, our results suggest that the Fickian regime is reached much earlier for cases with flow perpendicular to bedding than for those with flow parallel to bedding.

Percolation and flow in geological formations: upscaling from microscopic to megascopic scales

Modelling transport processes in a disordered system with broadly distributed heterogeneities, such as a field-scale porous medium, is often hampered by the fact that a very large set of equations (of the order of several millions) has to be solved repeatedly if detailed information about the system's morphology is available. Solving such a large set is often not practical. We discuss a novel and very efficient method for scale up of such systems such that no important information about their characteristics is lost at any length scale, and at the same time the number of transport equations to be solved is reduced drastically to a manageable level. The method, which is applicable to any type of heterogeneous medium, including one with percolation disorder, uses a wavelet transformation to coarsen the original fine-scale description of the system, such that finer resolution is maintained in regions of high transport properties, whereas coarser property description is applied to the rest of the system. The computational cost of the method is several orders of magnitude less than those of the most efficient methods currently available. The performance of the method is demonstrated by its application to calculation of the effective flow properties of several models of field-scale porous media. The method is equally applicable to other disordered media with broadly distributed heterogeneities, such as amorphous semiconductors.

Theory for the semi-analytical calculation of oil recovery and effective relative permeabilities using streamtubes

A semi-analytical method has been developed for calculating oil recovery in two and three dimensions, and for calculating effective relative permeabilities for coarse grids. The calculations are based on the assumption that the effects of a changing mobility field can be accounted for by using fixed streamtube geometries with flowrates updated to account for the changing mobility distribution. The single-phase pressure distribution from a numerical solution of Laplace's equation is used to calculate the pressure distribution for a two-phase flow based on a mapping of the solution of the Buckley-Leverett equation onto the streamtubes derived from the single-phase solution. The displacement calculations for oil recovery are based on theory previously developed by Dykstra and Parsons, extended to include the effects of spatially varying permeability and continuously changing mobilities, as occurs in solutions of the Buckley-Leverett equation for typical values of the mobility ratio. This idea has also been extended to the calculation of effective relative permeabilities for coarse-grid simulation and finally establishes the proper rules for averaging the results of fine-grid numerical simulations of two-phase flow for the definition of effective two-phase flow properties on coarse grids. These calculations have been generalized to three-dimensional flows by the simple device of conceptually inserting a gridded plane across the flow and defining each streamtube at that location as those streamlines which pass through any one of the grid cells. When combined with time-of-flight calculations from the gridded plane to both the producer and injector, the distribution of pore volume along each streamtube can be calculated. This information, combined with a tabulation of the single-phase, steady-state pressure distribution along each streamtube, provides all of the information needed for the semi-analytical calculation of oil recovery and effective flow properties in three-dimensional flows. © 1997 Elsevier Science Ltd. All rights reserved

Scale-up of two-phase flow in heterogeneous porous media

The objective of this work is to study the scale-up of multiphase flow properties obtained from core-scale experiments to the grid-scale required for typical field simulations of oil recovery. Typical naturally-occurring porous media represented by field simulation grid-blocks are internally heterogeneous. Heterogeneity is modeled here as a correlated random field parameterized in terms of its variance and two-point variogram. Variogram models of both finite (spherical) and infinite (fractal) correlation length are included as special cases. Local core-scale porosity, permeability, capillary pressure function, relative permeability functions, and initial water saturation are assumed to be correlated. Capillary pressure and relative permeability hysteresis are modeled at the core-scale. Water injection is simulated and effective flow properties (not pseudo-functions) and flow equations are calculated. Conditions under which the traditional relative permeability formulation is valid are identified. For strongly water-wet media and small variance in permeability, flow can be described by an effective relative permeability. In water-wet grid-blocks of large permeability variance and finite correlation length, effective relative permeabilities can describe the average front movements, but a stochastic term is needed to describe the detailed fluctuations. In fractal media, effective relative permeabilities cannot describe even the average front movements.

Computer simulation of concentrated fluid-particle suspension flows in axisymmetric geometries

To investigate the particle migration effects and fluid-particle interaction occurring in the flow of highly concentrated fluid-particle suspensions, a numerical method has been developed for effective computer simulation in arbitrary axisymmetric geometries. In the mathematical flow model the suspension is treated as a generalized Newtonian fluid where the effective flow properties of the suspension (density and viscosity) are determined by the local volume fraction of the particles. The description of the particle motion is governed by a modified transport equation with diffusion coefficients accounting for the effects of shear-induced particle migrations. The strongly coupled system of flow and transport equations is solved by applying the Galerkin finite element method and a velocity-pressure projection scheme. The numerical results in tube flow demonstrate strong particle migration towards the center of the tube and an increasing blunting of the velocity profiles which is in good agreement with an available analytical solution. In the case of flow through a stenosed tube model, particle concentration is lowest at the site of maximum constriction whereas a strong accumulation of particles can be seen in the recirculation zone downstream of the stenosis.

Upscaling permeability: Error analysis for renormalization

In this paper we briefly discuss the background to the problems of finding effective flow properties when moving from a detailed representation of reservoir geology to a coarse gridded model required for reservoir performance simulation. The basic requirements for the upscaled properties are also discussed. We then consider one technique, renormalization, that in recent years has shown promise as an accurate, yet fast, method. The mathematical background of the renormalization approach is examined. A rigorous formalism is developed that allows an explicit calculation of the error terms to be made. In a very simple case use of the correction terms is shown to produce a dramatic improvement in accuracy of the method.