Pseudofunctions for Water Coning in a Three-Dimensional Reservoir Simulator

1977 ◽  
Vol 17 (04) ◽  
pp. 251-262 ◽  
Author(s):  
E.G. Woods ◽  
A.K. Khurana
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Three Dimensional ◽  
Production Performance ◽  
Vertical Cross Section ◽  
Two Dimensional ◽  
Well Performance ◽  
Reservoir Model ◽  
Reservoir Simulator ◽  
Static Conditions

Abstract Three-dimensional numerical models of bottom-water-drive reservoirs show delayed water breakthrough into individual wells when compared with observed well performance and individual-well coning models. This reservoir-model behavior results from masking of the well coning effect by volume-averaging pressure and saturation profiles around a well over a grid block with a large volume. The reservoir-simulator prediction of well performance can be improved by mathematically performance can be improved by mathematically transforming the production performance of a detailed well-coning model into a set of time-independent pseudorelative-permeability and capillary-pressure curves that then can be used in the reservoir model. A procedure for obtaining the required pseudofunctions is described and the results of their application in simple models and in a large reservoir-simulator model are shown. Introduction The prohibitive cost of numerical reservoir simulation with fine-grid definition models of large reservoirs has led to development of techniques whereby vertical saturation distribution and/or localized flow conditions in the vicinity of individual wells can be approximately accounted for in relatively coarse-grid models at an acceptable incremental cost. In particular, vertical cross-section models under capillary and gravity equilibrium have been used to derive pseudorelative permeabilities and capillary pressures for use in two-dimensional, areal models to simulate the average vertical distribution of flow without having to pay the computing price of a full three-dimensional model. Coats et al. described the use of the vertical equilibrium concept for developing pseudorelative-permeability and capillary-pressure pseudorelative-permeability and capillary-pressure functions for simulating the vertical dimension in a two-dimensional, areal simulator model This method assumes gravity-capillary equilibrium in the vertical direction. Also, Coats et al. developed a dimensionless parameter for estimating when these conditions are valid. Martin formed a mathematical basis for pseudofunctions by reducing the equations for pseudofunctions by reducing the equations for three-phase, three-dimensional, compressible flow to two-dimensional relations by partial integration of the equations of flow. Hearn extended the pseudorelative-permeability concept by adapting it pseudorelative-permeability concept by adapting it to stratified reservoirs where viscous rather than gravity and capillary forces dominate the vertical sweep efficiency. Hawthorne studied the effects of capillary pressure on pseudorelative permeability derived from the Hearn stratified model. Jacks et al. further enlarged thepseudorelative-perrneability concept by developing dynamic pseudorelative permeabilities. (Dynamic pseudos, denoting pseudos permeabilities. (Dynamic pseudos, denoting pseudos determined under flowing rather than static conditions, allow one to account for the interaction between viscous and gravity forces resulting from rate variation in the vertical plane.) Kyte and Berry generalized the work of Jacks et al. by introducing the concept of pseudocapillary pressures and improving dynamic pseudofunction calculations to include varying flow potential gradients. Emanual and Cook expanded the concept of vertical cross-section, pseudorelative permeabilities to include the vertical performance of individual wells. Their procedure combines the effect of coning and well pseudorelative permeabilities for use in a two-dimensional, areal model. Chappelear and Hirasaki used a different approach to including of coning effects in a two-dimensional, areal simulator by developing a functional relationship among water cut, average oil-column thickness, and total rate based on an analytical incompressible, steady-state model. The most sophisticated approach to representing well-coning effects in a reservoir simulator has been taken by Mrosovsky and Ridings and Akbar et al. They incorporated detailed numerical well models into the reservoir-model grid. SPEJ P. 251

New Pseudo Functions To Control Numerical Dispersion

1975 ◽  
Vol 15 (04) ◽  
pp. 269-276 ◽  
Author(s):  
J.R. Kyte ◽  
D.W. Berry
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Three Dimensional ◽  
Vertical Cross Section ◽  
Numerical Dispersion ◽  
Two Dimensional ◽  
Cross Sectional ◽  
One Dimensional ◽  
Sectional Model ◽  
The Cross

Abstract This paper presents an improved procedure for calculating dynamic pseudo junctions that may be used in two-dimensional, areal reservoir simulations to approximate three-dimensional reservoir behavior. Comparison of one-dimensional areal and two-dimensional vertical cross-sectional results for two example problems shows that the new pseudos accurately transfer problems shows that the new pseudos accurately transfer the effects of vertical variations in reservoir properties, fluid pressures, and saturations from the properties, fluid pressures, and saturations from the cross-sectional model to the areal model. The procedure for calculating dynamic pseudo-relative permeability accounts for differences in computing block lengths between the areal and cross-sectional models. Dynamic pseudo-capillary pressure transfers the effects of pseudo-capillary pressure transfers the effects of different pressure gradients in different layers of the cross-sectional model to the areal model. Introduction Jacks et al. have published procedures for calculating dynamic pseudo-relative permeabilities fro m vertical cross-section model runs. Their procedures for calculating pseudo functions are procedures for calculating pseudo functions are more widely applicable than other published approaches. They demonstrated that, in some cases, the derived pseudo functions could be used to simulate three-dimensional reservoir behavior using two-dimensional areal simulators. For our purposes, an areal simulator is characterized by purposes, an areal simulator is characterized by having only one computing block in the vertical dimension. The objectives of this paper are to present an improved procedure for calculating dynamic pseudo functions, including a dynamic pseudo-capillary pressure, and to demonstrate that the new procedure pressure, and to demonstrate that the new procedure generally is more applicable than any of the previously published approaches. The new pseudos previously published approaches. The new pseudos are similar to those derived by jacks et al. in that they are calculated from two-dimensional, vertical cross-section runs. They differ because (1) they account for differences in computing block lengths between the cross-sectional and areal models, and (2) they transfer the effects of different flow potentials in different layers of the cross-sectional potentials in different layers of the cross-sectional model to the areal model. Differences between cross-sectional and areal model block lengths are sometimes desirable to reduce data handling and computing costs for two-dimensional, areal model runs. For very large reservoirs, even when vertical calculations are eliminated by using pseudo functions, as many as 50,000 computing blocks might be required in the two-dimensional areal model to minimize important errors caused by numerical dispersion. The new pseudos, of course, cannot control numerical pseudos, of course, cannot control numerical dispersion in the cross-sectional runs. This is done by using a sufficiently large number of computing blocks along die length of the cross-section. The new pseudos then insure that no additional dispersion will occur in the areal model, regardless of the areal computing block lengths. Using this approach, the number of computing blocks in the two-dimensional areal model is reduced by a factor equal to the square of the ratio of the block lengths for the cross-sectional and areal models. The new pseudos do not prevent some loss in areal flow-pattern definition when the number of computing blocks in the two-dimensional areal model is reduced. A study of this problem and associated errors is beyond the scope of this paper. Our experience suggests that, for very large reservoirs with flank water injection, 1,000 or 2,000 blocks provide satisfactory definition. Many more blocks provide satisfactory definition. Many more blocks might be required for large reservoirs with much more intricate areal flow patterns. The next section presents comparative results for cross-sectional and one-dimensional areal models. These results demonstrate the reliability of the new pseudo functions and illustrate their advantages pseudo functions and illustrate their advantages over previously derived pseudos for certain situations. The relationship between two-dimensional, vertical cross-sectional and one-dimensional areal reservoir simulators has been published previously and will not be repeated here in any detail. Ideally, the pseudo functions should reproduce two-dimensional, vertical cross-sectional results when they are used in the corresponding one-dimensional areal model. SPEJ P. 269

The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions

1973 ◽  
Vol 13 (03) ◽  
pp. 175-185 ◽  
Author(s):  
Hugh H. Jacks ◽  
Owen J.E. Smith ◽  
C.C. Mattax
Keyword(s):  
Capillary Pressure ◽  
Cross Section ◽  
Relative Permeability ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Relative Permeabilities ◽  
Large Reservoir ◽  
Model Simulations ◽  
Wide Range ◽  
Section Model

Abstract Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs. Introduction Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions". DESCRIPTION OF PSEUDO-RELATIVE PERMEABILITY FUNCTIONS PERMEABILITY FUNCTIONS Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities. SPEJ P. 175

Two-Dimensional Radial Treatment of Wells Within a Three-Dimensional Reservoir Model

1974 ◽  
Vol 14 (02) ◽  
pp. 127-138 ◽  
Author(s):  
Ivan Mrosovsky ◽  
R.L. Ridings
Keyword(s):  
Pore Volume ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Well Performance ◽  
Reservoir Model ◽  
Fine Grid ◽  
Well Production ◽  
Radial Model ◽  
2D Model ◽  
Radial Well

Abstract A reservoir simulator is described in which two-dimensional cylindrical models are used for individual wells within the framework of a three-dimensional, rectangular-grid, reservoir model. This approach to simulation should give more realistic results than those incorporating simpler well production routines. Application should be production routines. Application should be particularly pertinent to cases in which gas and particularly pertinent to cases in which gas and water coning are important. Computation cost is increased by this sophisticated treatment of wells, but total running costs are nevertheless low enough that judicious use of the model is warranted. Introduction Changes in pressure and fluid saturations occurring in a reservoir on production may be described by certain well known differential equations. These equations can be solved by finite-difference techniques. To this end a reservoir may be divided, more or less regularly, into a number of blocks. We consider here a three-dimensional model. A rectangular grid is superimposed on the reservoir, which is further divided vertically into several layers. This kind of subdivision is generally adequate for treating most of a reservoir. However, difficulty arises in the neighborhood of a well. Fluid saturations and pressures typically exhibit steep gradients in the neighborhood of a producing well. Moreover, these properties have values near the well that are markedly different from those prevailing over most of the well's drainage area. To prevailing over most of the well's drainage area. To model this situation using finite-difference methods, a fine grid would be necessary in the vicinity of the well. To extend such a fine grid throughout a reservoir, with many wells, would result in an impracticably large number of blocks. Instead, it is customary to simplify and approximate the calculation of well performance. A common recourse is to use a radial flow formula such as(1) Here the external radius, re, is set so that the area of a circle of this radius should equal the area of the block in which the well is situated. Obviously, this formula takes no account of the radial variation of fluid saturations. In this paper a different approach to the calculation of well performance is described. Each well in the three-dimensional (3D) reservoir model is also assigned a two-dimensional (2D) radial model covering its immediate vicinity. These radial models are solved simultaneously with the 3D reservoir model. Thus, precision is concentrated where it is most needed - near the wells; yet the 3D reservoir model is not blown up unduly. The equations and method of solution for the 3D model are briefly sketched in the Appendix. The 2D radial models are basically the same as that described by Letkeman and Ridings. We now discuss some details of the radial well models. In particular we shall be concerned with the interface particular we shall be concerned with the interface between the 2D radial well models and the 3D reservoir model. RADIAL WELL MODEL To make the discussion simpler and more concrete we take, as an example, a reservoir model we have run. This 3D reservoir model had 15 layers. Consider a well situated in one vertical column of 15 blocks. The 2D radial model for this well also has 15 layers. Each layer in the 2D model is further split radially into six concentric blocks. The relationship between the two models is depicted in Fig. 1. The total pore volume in any layer of the 2D model must equal the pore volume of the corresponding block in the same pore volume of the corresponding block in the same layer of the 3D model; that is, the block through which the well passes. Consider now the calculation of one time step, say 30 days, for the 3D reservoir model. To make this calculation, we require a well production rate. Actually we need three rates: one each for oil, water, and gas. These rates are obtained by solving the 2D model over the same 30-day period. SPEJ P. 127

Microstructure Evolution of Epitaxial (Ba, Sr) TiO3/ (001) HgO Thin Films

1994 ◽  
Vol 361 ◽  
Author(s):  
V.A. Alyoshin ◽  
E.V. Sviridov ◽  
V.I.M. Hukhortov ◽  
I.H. Zakharchenko ◽  
V.P. Dudkevich
Keyword(s):  
Thin Films ◽  
Cross Section ◽  
Microstructure Evolution ◽  
Three Dimensional ◽  
Gas Pressure ◽  
Smooth Surfaces ◽  
Two Dimensional ◽  
Surface Versus ◽  
Deposition Conditions

ABSTRACTSurface and cross-section relief evolution of ferroelectric epitaxial (Ba,Sr)TiO3 films rf-sputtered on (001) HgO crystal cle-avage surface versus the oxygen worKing gas pressure P and subst-rate temperature T were studied. Specific features of both three-dimensional and two-dimensional epitaxy mechanisms corresponding to various deposition conditions were revealed. Difference between low and high P-T-value 3D epitaxy was established. The deposition of films with mirror-smooth surfaces and perfect interfaces is shown to be possible.

Plane turbulent jets in a bounded fluid layer

1992 ◽  
Vol 241 ◽  
pp. 587-614 ◽  
Author(s):  
T. Dracos ◽  
M. Giger ◽  
G. H. Jirka
Keyword(s):  
Experimental Investigation ◽  
Cross Section ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Turbulent Jets ◽  
Two Dimensional ◽  
Vortical Structures ◽  
Secondary Currents ◽  
Fluid Layers ◽  
Bounding Surfaces

An experimental investigation of plane turbulent jets in bounded fluid layers is presented. The development of the jet is regular up to a distance from the orifice of approximately twice the depth of the fluid layer. From there on to a distance of about ten times the depth, the flow is dominated by secondary currents. The velocity distribution over a cross-section of the jet becomes three-dimensional and the jet undergoes a constriction in the midplane and a widening near the bounding surfaces. Beyond a distance of approximately ten times the depth of the bounded fluid layer the secondary currents disappear and the jet starts to meander around its centreplane. Large vortical structures develop with axes perpendicular to the bounding surfaces of the fluid layer. With increasing distance the size of these structures increases by pairing. These features of the jet are associated with the development of quasi two-dimensional turbulence. It is shown that the secondary currents and the meandering do not significantly affect the spreading of the jet. The quasi-two-dimensional turbulence, however, developing in the meandering jet, significantly influences the mixing of entrained fluid.

scholarly journals Derivation of Equations for a Size Distribution of Spherical Particles in Non-Transparent Materials

2021 ◽  
Vol 5 (4) ◽  
pp. 53-60
Author(s):  
Daniel Gurgul ◽  
Andriy Burbelko ◽  
Tomasz Wiktor
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Size Distribution ◽  
Density Function ◽  
Cross Section ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Mathematical Formulas ◽  
Transparent Materials

This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF3) of the spherical radii (r3) of particles randomly placed in non-transparent materials. We have presented two attempts here, both of which are based on data collected from a random planar cross-section passed through space containing three-dimensional nodules. The first attempt uses a Probability Density Function (PDF2) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii (r2). These radii are produced by an intersection of the space by a random plane. In turn, the second solution also uses an experimentally obtained Probability Density Function (PDF1). But the form of PDF1 has been created on the basis of a set containing chord lengths collected from a cross-section.The most important finding presented in this paper is the conclusion that if the PDF1 has proportional scopes, the PDF3 must have a constant value in these scopes. This fact allows stating that there are no nodules in the sample space that have particular radii belonging to the proportional ranges the PDF1.

scholarly journals Membrane analogy for multi-material bars under torsion

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190124 ◽  
Author(s):  
Laura Galuppi ◽  
Gianni Royer-Carfagni
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Element Model ◽  
Elastic Problem ◽  
Two Dimensional ◽  
Torsion Problem ◽  
Linear Elastic ◽  
Physical Apparatus

Prandtl's membrane analogy for the torsion problem of prismatic homogeneous bars is extended to multi-material cross sections. The linear elastic problem is governed by the same equations describing the deformation of an inflated membrane, differently tensioned in regions that correspond to the domains hosting different materials in the bar cross section, in a way proportional to the inverse of the material shear modulus. Multi-connected cross sections correspond to materials with vanishing stiffness inside the holes, implying infinite tension in the corresponding portions of the membrane. To define the interface constrains that allow to apply such a state of prestress to the membrane, a physical apparatus is proposed, which can be numerically modelled with a two-dimensional mesh implementable in commercial finite-element model codes. This approach presents noteworthy advantages with respect to the three-dimensional modelling of the twisted bar.

scholarly journals Computational Study of Zigzag Spacer Design with Elliptical Cross-Section Filaments

2020 ◽  
Vol 307 ◽  
pp. 01047
Author(s):  
Gohar Shoukat ◽  
Farhan Ellahi ◽  
Muhammad Sajid ◽  
Emad Uddin
Keyword(s):  
Mass Transfer ◽  
Cross Section ◽  
Cross Sections ◽  
Computational Study ◽  
Flow Direction ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Two Dimensional ◽  
Permeate Flux ◽  
Elliptical Cross

The large energy consumption of membrane desalination process has encouraged researchers to explore different spacer designs using Computational Fluid Dynamics (CFD) for maximizing permeate per unit of energy consumed. In previous studies of zigzag spacer designs, the filaments are modeled as circular cross sections in a two-dimensional geometry under the assumption that the flow is oriented normal to the filaments. In this work, we consider the 45° orientation of the flow towards the three-dimensional zigzag spacer unit, which projects the circular cross section of the filament as elliptical in a simplified two-dimensional domain. OpenFOAM was used to simulate the mass transfer enhancement in a reverse-osmosis desalination unit employing spiral wound membranes lined with zigzag spacer filaments. Properties that impact the concentration polarization and hence permeate flux were analyzed in the domain with elliptical filaments as well as a domain with circular filaments to draw suitable comparisons. The range of variation in characteristic parameters across the domain between the two different configurations is determined. It was concluded that ignoring the elliptical projection of circular filaments to the flow direction, can introduce significant margin of error in the estimation of mass transfer coefficient.

Optimum profiles in two-dimensional Stokes flow

1995 ◽  
Vol 450 (1940) ◽  
pp. 603-622 ◽  
Keyword(s):  
Circular Cylinder ◽  
Cross Section ◽  
Stokes Flow ◽  
Unit Length ◽  
Three Dimensional ◽  
Variational Formula ◽  
Effective Radius ◽  
Two Dimensional ◽  
Equivalent Radius ◽  
Optimum Profile

We consider the problem of designing the section of a cylinder to minimize the drag per unit length it experiences when placed perpendicular to a uniform stream at low Reynolds number; we suppose the area of the cross-section to be given, and the flow to be two-dimensional. The relevant properties of a cylinder of general cross-section in a particular orientation can conveniently be expressed in terms of its equivalent radius; when the drag and flow at infinity are parallel, this equivalent radius is the radius of the circular cylinder giving rise to the same drag per unit length. We obtain a variational formula for this equivalent radius when the surface of the cylinder is perturbed; this shows that the optimum profile we seek must be such that the flow past it has a vorticity of constant magnitude at its surface, and this fact enables the optimum to be determined analytically. The efficacy of a particular section may be measured by its effective radius, this being the equivalent radius when the length scale is chosen to give the section an area π ; thus a circular cylinder has an effective radius of 1. The minimum possible effective radius, achieved by the optimum profile, is 0.88876. To illustrate some of the arguments we exploit in a more familiar setting, we also obtain a variational formula for the drag on a three-dimensional body in Stokes flow when its surface is perturbed.

Research on Buckling Mechanism of U Type Steel Support’s Pinnas Based on ANSYS

2012 ◽  
Vol 466-467 ◽  
pp. 1271-1274
Author(s):  
Jian Zhuang Liu ◽  
Jia Li ◽  
Li Gan ◽  
Fang Fang Du
Keyword(s):  
Stress Distribution ◽  
Cross Section ◽  
Coal Mine ◽  
Optimization Design ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Type Steel ◽  
Buckling Failure ◽  
Simulation Results ◽  
Three Dimensional Models

The prevenient mechanical research on U type steel support focuses on two dimensional analysis in the installing plan. Interested in the buckling failure of U type steel support’s pinnas in one coal mine of Huainan Group, the authors build three dimensional models to stimulating 25U and 29U support. The appealing simulation results verify ANSYS’s 3-D computational capabilities about complicated section support. The main objective of their investigation has been to obtain some knowledge of the mechanism, stress distribution, and load magnitude in the support’s deforming. Further, the deviating longitudinal load play an accelerating deformation role on its distorting, which should be paied more attention to cross-section optimization, design of support parameters, and installing way of steel lacing.

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