Treatment of Individual Wells and Grids in Reservoir Modeling

1968 ◽  
Vol 8 (04) ◽  
pp. 341-346 ◽  
Author(s):  
H.K. Van Poollen ◽  
E.A. Breitenbach ◽  
D.H. Thurnau
Keyword(s):  
Three Dimensional ◽  
Computer Time ◽  
Small Time ◽  
Economic Consequences ◽  
Productivity Index ◽  
Reservoir Modeling ◽  
Two Dimensional ◽  
Flow Equations ◽  
General Utility ◽  
Partially Penetrating Well

Abstract Reservoir modeling, mathematical modeling, or simulation of a petroleum or natural gas reservoir enables the engineer to examine and evaluate the physical a-nd economic consequences of various physical a-nd economic consequences of various alternative production policies. Approximations are inherent in all workable, economical simulators. This paper describes three workable, useful approximations. (1)a method to compare observed field pressures with those calculated by a numerical simulator, (2) a method to reduce three-dimensional problems to two space dimensions with pseudo-third-dimensional features, and (3) a method to calculate the productivity index (PI) and the water-oil ratio (WOR) in a partially penetrating well partially penetrating well These methods, although admittedly approximations, are workable and have been found to be very useful. Their general utility will, however, depend upon the extent to which any underlying assumptions used in their formulation apply to a particular problem. particular problem Introduction The objectives, applications and mathematical background of reservoir modeling have been described in other works. Ideally networks should be as shown in Fig. 1. Here, the grids are smaller near the wellbore than farther away. However, the number of grid points becomes large, even in a two-dimensional grid. Also, the small block sizes force one to use very small time steps, which can increase the computer time to the point of rendering the study economically unfeasible. Fig. 1 shows an example where the wells are located on a regular pattern. If that pattern becomes irregular enough, all cells pattern becomes irregular enough, all cells eventually will have to be small. In order to proceed with a study, modelers are forced to use linger grid sizes, as shown in Fig. 2. We realize that, by using large grid sizes, the fundamental flow equations are not truly represented. The network approaches a set of interconnected material balances with flow terms as a function of pressures and saturations. This paper describes the present method of handling wellbores in models with grid sizes many times the wellbore diameters. A method to compare pressures observed in the field with those calculated in the model is presented. A method also is given to reduce three- dimensional problems to two-dimensional grids. SPEJ P. 341

Design Methods of Volute Casings for Turbocharger Turbine Applications

1996 ◽  
Vol 210 (2) ◽  
pp. 149-156 ◽  
Author(s):  
H Chen
Keyword(s):  
Experimental Data ◽  
Potential Flow ◽  
Design Method ◽  
Three Dimensional ◽  
Design Methods ◽  
Two Dimensional ◽  
Aerodynamic Design ◽  
3D Design ◽  
Flow Equations ◽  
Good Agreement

This paper discusses aerodynamic design methods of volute casings used in turbocharger turbines. A quasi-three-dimensional (Q-3D) design method is proposed in which a group of extended two-dimensional potential flow equations and the streamline equation are numerically solved to obtain the geometry of spiral volutes. A tongue loss model, based on the turbulence wake theory, is also presented, and good agreement with experimental data is shown.

A Three-Dimensional Version of the Free Surface Capturing Discretization for Staggered Grid Finite Difference Schemes: Implementation into StagYY

2021 ◽  
Author(s):  
Paul Tackley
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Three Dimensional ◽  
Small Time ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Finite Volume Scheme ◽  
Two Dimensional ◽  
Simulation Code ◽  
Free Surface Capturing

<p>In order to treat a free surface in models of lithosphere and mantle dynamics that use a fixed Eulerian grid it is typical to use "sticky air", a layer of low-viscosity, low-density material above the solid surface (e.g. Crameri et al., 2012). This can, however, cause numerical problems, including poor solver convergence due to the huge viscosity jump and small time-steps due to high velocities in the air. Additionally, it is not completely realistic because the assumed viscosity of the air layer is typically similar to that of rock in the asthenosphere so the surface is not stress free.  </p><p>In order to overcome these problems, Duretz et al. (2016) introduced and tested a method for treating the free surface that instead detects and applies special conditions at the free surface. This avoids the huge viscosity jump and having to solve for velocities in the air. They applied it to a two-dimensional staggered grid finite difference / finite volume scheme, a discretization that is in common use for modelling mantle and lithosphere dynamics. Here I document the application of this approach to a three-dimensional spherical staggered grid solver in the mantle simulation code StagYY. Some adjustments had to be made to the two-dimensional scheme documented in Duretz et al. (2016) in order to avoid problems due to undefined velocities for certain boundary topographies. The approach was applied not only to the Stokes solver but also to the temperature solver, including the implementation of a mixed radiative/conductive boundary condition applicable to surface magma oceans/lakes.</p><p><strong>References</strong></p><p>Crameri, F., H. Schmeling, G. J. Golabek, T. Duretz, R. Orendt, S. J. H. Buiter, D. A. May, B. J. P. Kaus, T. V. Gerya, and P. J. Tackley (2012), A comparison of numerical surface topography calculations in geodynamic modelling: an evaluation of the ‘sticky air’ method, Geophysical Journal International,189(1), 38-54, doi:10.1111/j.1365-246X.2012.05388.x.</p><p>Duretz, T., D. A. May, and P. Yamato (2016), A free surface capturing discretization for the staggered grid finite difference scheme, Geophysical Journal International, 204(3), 1518-1530, doi:10.1093/gji/ggv526.</p>

Three-Dimensional Lifting-Surface Theory for an Annular Blade Row

1979 ◽  
Author(s):  
G. F. Homicz ◽  
J. A. Lordi
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Computer Time ◽  
Two Dimensional ◽  
Blade Loading ◽  
Surface Theory ◽  
Lifting Surface ◽  
Strip Theory ◽  
Blade Row ◽  
Flow Through

A lifting-surface analysis is presented for the steady, three-dimensional, compressible flow through an annular blade row. A kernel-function procedure is used to solve the linearized integral equation which relates the unknown blade loading to a specified camber line. The unknown loading is expanded in a finite series of prescribed loading functions which allows the required integrations to be performed analytically, leading to a great savings in computer time. Numerical results are reported for a range of solidities and hub-to-tip ratios; comparisons are made with both two-dimensional strip theory and other three-dimensional results.

Three-Dimensional Non-Reflecting Boundary Condition for Linearized Flow Solvers

2010 ◽  
Author(s):  
Paul J. Petrie-Repar
Keyword(s):  
Boundary Condition ◽  
Unsteady Flow ◽  
Steady Flow ◽  
Three Dimensional ◽  
Far Field ◽  
Two Dimensional ◽  
Flow Equations ◽  
Flow Simulations ◽  
Unsteady Aerodynamic ◽  
2D And 3D

A three-dimensional (3D) non-reflecting boundary condition for linearized flow solvers is presented. The unsteady aerodynamic modes at the inlet and outlet (far-field) are numerically determined by solving an eigen problem for the semi-discretized flow equations on a two-dimensional mesh. Unlike previous methods the shape of the far-field can be general and the non-uniformity of the steady flow across the far-field is considered. The calculated unsteady modes are used to decompose the unsteady flow at the far-field into modes. The direction of each mode is determined, and incoming modes are prescribed and outgoing modes are extrapolated. The results of 2D and 3D inviscid linearised flow simulations using the new boundary condition are presented.

Partial Integration of Equations of Multiphase Flow

1968 ◽  
Vol 8 (04) ◽  
pp. 370-380 ◽  
Author(s):  
John C. Martin
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Flow Equations ◽  
Vertical Equilibrium ◽  
The Third ◽  
Dimensional Flow ◽  
Three Phase ◽  
Capillary Pressures ◽  
Third Dimension ◽  
Partial Integration

Abstract Equations for three-phase, three-dimensional, compressible flow (including capillarity) are reduced to two-dimensional relations by a partial integration. This reduction allows three-dimensional flow problems to be treated with mathematics for only two spatial dimensions. The results can be used to formulate flow equations for two-dimensional reservoir simulators i-n which the effects of capillarity and fluid segregation in the third dimension are represented. Such reservoir simulators would retain many of the advantages of two-dimensional simulators while simulating three-dimensional effects. The principal restriction of the method is that the thickness of the reservoir should be small, compared to the distance across the reservoir. Introduction In recent years, computers have been used to calculate performances of many reservoirs. Most of the detailed calculations, however, are based on finite difference solutions of the flow equations, and present day computers are seldom able to handle a sufficient number of cells to produce entirely satisfactory solutions, even for produce entirely satisfactory solutions, even for reservoirs represented by two-dimensional arrays of cells. The simulation becomes much worse when one wishes to approximate the reservoir by a three-dimensional array. A great saving in computation or a more detailed solution can be obtained for many reservoirs by using the partial integration of the equations of flow, presented in this paper. The integration reduces the three-dimensional equations to two-dimensional relations; ant for studies of two-dimensional flows in vertical cross-sections, the equations are reduced to one-dimensional relations. Most reservoir performance calculations currently are based on one- or two-dimensional flow relations. In some cases flow in the third dimension is approximated by assuming a particular type of vertical saturation distribution, such as gravity segregation. The relations developed in this paper approach those for segregated flow as the capillary pressures approach zero, and they approach those for uniformly distributed saturations as the capillary pressures are increased. For this analysis, the ratio of the reservoir's thickness to the maximum distance across it must be small. The capillary pressures between the oil and water should also be small compared to the maximum pressure difference in the reservoirs. It is requirement is met by most reservoirs. It is assumed that the capillary-pressure curves are well defined, whether or not hysteresis effects are included. Also, the reservoir must have sufficient vertical permeability to allow the fluids to segregate. The results presented here provide a firm theoretical foundation to Coats' et al. assumption of vertical equilibrium and extend the relations to three-phase flow. Coats' assumption of vertical equilibrium, which he verified by calculations and experiment, is developed here mathematically from basic flow equations. Discussion SATURATION AND PRESSURE DISTRIBUTIONS Appendix A presents a mathematical analysis of fluid flow in reservoirs where the ratio of thickness to maximum distance across the reservoir is small. The results indicate:that the fluids along any line perpendicular to such a reservoir's upper surface are in antic capillary equilibrium (vertical equilibrium);that, to a first approximation, the fluid pressures and properties are functions of only areal position in the reservoir and time; andthat hydrostatic pressure gradients exist along any line perpendicular to the reservoir's upper surface. The results might be expected after studying several physical considerations. First, no flow is allowed normal to the upper and lower reservoir boundaries, which are relatively close together. SPEJ P. 370

Reservoir Simulation of the Empire Abo Field: The Use of Pseudos in a Multilayered System

1979 ◽  
Vol 19 (05) ◽  
pp. 279-288 ◽  
Author(s):  
J.E. Killough ◽  
H.P. Foster
Keyword(s):  
Three Dimensional ◽  
Computer Time ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Relative Permeabilities ◽  
Three Dimensional Model ◽  
Layered Model ◽  
Multilayered System ◽  
Capillary Pressures ◽  
Recovery Schemes

Abstract A three-layered model with vertical equlibrium pseudo-relative permeabilities and pseudo-capillary pseudo-relative permeabilities and pseudo-capillary pressures was used to match the results from a pressures was used to match the results from a 22-layered model of a portion of the Empire Abo field. Results of the two models were almost identical in both history match and prediction phases. Directionally dependent pseudo-relative permeabilities were introduced to match the drainage mechanism of the reservoir. Both horizontal displacement and drainage mechanisms can be modeled simultaneously in a multilayered system using these pseudo functions.The excellent comparison between the finely gridded model and the pseudo model allowed the simulation of alternative recovery schemes using the pseudo model with the same confidence as the pseudo model with the same confidence as the original three-dimensional model. In particular, a comparison between pressure maintenance with carbon dioxide and nitrogen was simulated, with a substantial savings in both manpower and computer time. Development of this pseudo model makes the simulation of the entire Empire Abo field feasible. Introduction For several years vertical equilibrium pseudo-relative permeabilities and pseudo-capillary pressures have permeabilities and pseudo-capillary pressures have been used throughout the industry to collapse a three-dimensional model with many layers to a two-dimensional areal model with a single layer. The obvious advantages of pseudo models are reduced computer and manpower costs. The justification for the use of pseudo models has been the comparison of results from one-dimensional models using pseudos with those from two-dimensional, cross-sectional models. Few comparisons have been made between an areal model using pseudos and a three-dimensional simulation.The Empire Abo field offered an excellent example for such a comparison. The field had been simulated in three dimensions, using 22 layers. An analysis of the simulation results showed that conditions were such that vertical equilibrium existed in most columns of the model. By applying pseudos to the three-dimensional model, it appeared possible to reduce the number of layers and associated computer time substantially.In this paper, the techniques for obtaining a match of the three-dimensional models with a pseudo model are described. The original three-dimensional model is described along with a history of the Empire Abo field. Coning correlations and directionally dependent pseudo-relative permeabilities also are described. Results between the pseudo model and the three-dimensional model also are compared. Finally, the use of the pseudo model for predicting alternative recovery schemes is discussed. Field History The Empire Abo field is located in southeastern New Mexico, about 8 miles (13 km) southeast of Artesia, NM (Fig. 1). The Abo reservoir was discovered by Amoco Production Co. in Nov. 1957. Subsequent development and unitization of the Abo reservoir was documented by Christianson. The producing horizon is a carbonate reef, basal Permian (lower Leonard) in age. The productive reef is about 12 1/2 miles (20.12 km) long and 1 1/2 miles (2.47 km) wide (Fig. 2), covering about 11,339 surface acres (45.9 x 10(6) m2). The reef dips gently from southwest to northeast at about 1 degree. JPT p. 279

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

A linearity test for thick specimens imaged by HVEM over a 180° angular range

1991 ◽  
Vol 49 ◽  
pp. 552-553
Author(s):  
Yu Liu
Keyword(s):  
Phase Contrast ◽  
Three Dimensional ◽  
Angular Range ◽  
Two Dimensional ◽  
Back Projection ◽  
Line Integral ◽  
Exponential Law ◽  
Transmission Electron ◽  
Absorption Law ◽  
Linearity Test

The image obtained in a transmission electron microscope is the two-dimensional projection of a three-dimensional (3D) object. The 3D reconstruction of the object can be calculated from a series of projections by back-projection, but this algorithm assumes that the image is linearly related to a line integral of the object function. However, there are two kinds of contrast in electron microscopy, scattering and phase contrast, of which only the latter is linear with the optical density (OD) in the micrograph. Therefore the OD can be used as a measure of the projection only for thin specimens where phase contrast dominates the image. For thick specimens, where scattering contrast predominates, an exponential absorption law holds, and a logarithm of OD must be used. However, for large thicknesses, the simple exponential law might break down due to multiple and inelastic scattering.

An Image Reconstruction System Applied to High Voltage Microscopy

1975 ◽  
Vol 33 ◽  
pp. 292-293
Author(s):  
D. E. Johnson
Keyword(s):  
High Voltage ◽  
Prior Information ◽  
Block Diagram ◽  
Three Dimensional ◽  
Real Space ◽  
Two Dimensional ◽  
Efficient Manner ◽  
Fourier Methods ◽  
Tilt Series ◽  
Methods Of Reconstruction

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

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