Numerical Three-Phase Model of the Two-Dimensional Steamflood Process

1970 ◽  
Vol 10 (04) ◽  
pp. 405-417 ◽  
Author(s):  
N.D. Shutler
Keyword(s):  
Fluid Flow ◽  
Cross Section ◽  
Oil Recovery ◽  
Vertical Cross Section ◽  
Recovery Efficiency ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Oil Water ◽  
Three Phases

Abstract This paper describes a numerical mathematical model that is a significant extension of a previously published one-dimensional model of the steamflood published one-dimensional model of the steamflood process. process. The model describes the simultaneous flow of the three phases - oil, water and gas - in two dimensions. Interphase mass transfer between water and gas phases is allowed, but the oil is assumed nonvolatile and the hydrocarbon gas insoluble in the liquid phases. The model allows two-dimensional heat convection within the reservoir and two-dimensional heat conduction in a vertical cross-section spanning the oil sand and adjacent strata. Example calculations are presented which, on comparison with experimental results, tend to validate the model. Steam overriding due to gravity effects is shown to significantly reduce oil recovery efficiency in a thick system while jailing to do so in a thinner system. A study of the effect of capillary pressure indicates that failure to scale capillary forces in laboratory models of thick sands may lead to optimistic recovery predictions, while properly scaled capillary forces may be sufficiently low as to play no important role in oil recovery. Calculations made with and without vertical permeability show that failure to account for vertical fluid flow can lead to predictions of pessimistic oil recovery efficiency. pessimistic oil recovery efficiency Introduction Mathematical tools of varying complexity have been used in studying the steamflood process. A "simplified" class of mathematical models has served primarily as aids in engineering design. A more comprehensive class of models has improved understanding of the nature of the process. The model described in this report is of the latter class, but it is more comprehensive than any previously published model. published model. All previously available calculations of the steamflood process are confined to one space dimension in their treatments of fluid flow. Thus all previous models necessary ignore all effects of gravity reservoir heterogeneity, and nonuniform initial fluid-phase distributions on fluid flow in a second dimension. This model, an extension of a previously published model accounts for heat and previously published model accounts for heat and fluid transfer in two space dimensions and, hence, can evaluate these effects on simultaneous horizontal and vertical flow. While the model can describe the areal performance of a steamflood (in which case the heat transfer is described in three dimensions), this aspect will not be considered in this paper. Rather, this paper will describe the model in its application to a vertical cross-section through the reservoir and will consider some preliminary investigations to demonstrate the importance of being able to simultaneously account for horizontal and vertical fluid flow. Mathematical details are given in appendices. MATHEMATICAL DESCRIPTION OF STEAMFLOODING Darcy's law provides expressions for the velocities of the three phases (oil, water and gas), which, when combined with oil, water and gas mass balances give the partial differential equations governing Now of the three phases within a reservoir sand: OIL PHASE ..(1) WATER PHASE ..(2) SPEJ P. 405

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

An Experimental Verification of a Two-Dimensional Technique for Computing Performance Of Gas-Drive Reservoirs

1963 ◽  
Vol 3 (01) ◽  
pp. 19-27 ◽  
Author(s):  
P.M. Blair ◽  
D.W. Peaceman
Keyword(s):  
Cross Section ◽  
Space Dimension ◽  
Vertical Cross Section ◽  
Low Flow ◽  
Two Dimensional ◽  
Gas Oil ◽  
One Dimensional ◽  
Flow Equations ◽  
The One ◽  
One Space Dimension

Abstract The shape and position of the gas-oil transition zone during downdip displacement of oil by gas has been calculated using flow equations which include the effects of gravity, relative permeability, capillary pressure and compressibility of the fluids. The calculations treat the problem in two space dimensions, and results are compared with data from a laboratory model tilted at 30 degrees and 60 degrees from the horizontal on displacements near and above the maximum rate at which gravity segregation prevents channeling of the gas along the top of the stratum. The good agreement between calculated and experimental results demonstrates the validity of the technique as well as that of the flow equations. Introduction Knowledge of the fluid distribution and movement in and oil reservoirs important in producing operations and estimation of reserves. The history of the oil industry has included steady progress in improving the accuracy of calculations which provide the required knowledge. The earliest method of calculating reservoir performance consisted of material-balance equations based on the assumption that all properties were uniform throughout a reservoir. For many reservoirs such a simple formulation is still the most useful. However, when large pressure and saturation gradients exist in a reservoir, the assumption of uniform values throughout may lead to significant error. To reduce these errors, Buckley and Leverett introduced a displacement equation which considers pressure and saturation gradients. Methods available at that time permitted solutions to the Buckley-Leverett equation in one space dimension; these solutions have been very useful in solving many problems related to the production of oil. However, the one-dimensional methods are not adequate for systems in which saturations vary in directions other than the direction of flow. An example of such a system is the case of gas displacing oil down a dipping stratum in which the gas-oil contact becomes significantly tilted. Of course, the Buckley-Leverett displacement method cannot predict the tilt of the gas-oil contact. Recent improvements of the one-dimensional Buckley-Leverett method achieve some success in predicting the tilt of the gas-oil contact at sufficiently low flow rates. However, at rates high enough that the viscous pressure gradient nearly equals or exceeds the gravity gradient, even these improved one-dimensional methods incorrectly predict the shape and velocity of the contact. Further progress in estimating such fluid movements in a reservoir appears to require consideration of the problem in more than one space dimension. The recent two-dimensional method of Douglas, Peaceman and Rachford appears adaptable to calculate changes with time of the saturation distribution in a vertical cross-section of a reservoir. The movement of saturation contours should represent the moving fluid contacts and include the effects of crossflow due to gravity, as well as variations in the rock and fluid properties. The nonlinear nature of the equations used in the method has prevented proof of the validity of the solutions. Douglas, Peaceman and Rachford made some comparisons with experiment but did not include cases in which gravity was important nor cases involving displacement by the nonwetting phase. Forthesereasons, atestof the two-dimensional method for a case in which these factors are included would be very desirable. The test selected was a comparison of calculated results with those from a carefully controlled laboratory experiment on a model with measured physical properties. The model selected was one in which gas displaced oil down a tilted, rectangular sand pack. The model can be thought of as representing a vertical cross-section taken parallel to the dip of a reservoir. The displacement thus simulates gas displacing oil downdip that might result from gas-cap expansion or gas injection. SPEJ P. 19^

Oil, Water, Slag Three Phases Separation Technology of Heavy Aging Oil

2014 ◽  
Vol 936 ◽  
pp. 1553-1555
Author(s):  
Meng Zheng
Keyword(s):  
Water Content ◽  
Field Test ◽  
Oil Recovery ◽  
Recovery Rate ◽  
Storage Capacity ◽  
Quality Requirements ◽  
Separation Technology ◽  
Oil Water ◽  
Export Quality ◽  
Three Phases

The technology was used for handling heavy aging oil by demulsifier and three phases horizontal scrow centrifuge. Through laboratory and field test, it showed that the water content of the processed aging oil dropped from 50% to 5% below, purity oil recovery rate reached more than 95%, meeting export quality requirements. The technology improved the effective storage capacity of flow station, is of great significance to the safe and steady operation of flow station.

Mine Impact Burial Prediction From One to Three Dimensions

2008 ◽  
Vol 62 (1) ◽  
Author(s):  
Peter C. Chu
Keyword(s):  
Three Dimensional ◽  
Time History ◽  
Three Dimensions ◽  
Vertical Position ◽  
Burial Depth ◽  
Prediction Errors ◽  
Two Dimensional ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Models

The Navy’s mine impact burial prediction model creates a time history of a cylindrical or a noncylindrical mine as it falls through air, water, and sediment. The output of the model is the predicted mine trajectory in air and water columns, burial depth/orientation in sediment, as well as height, area, and volume protruding. Model inputs consist of parameters of environment, mine characteristics, and initial release. This paper reviews near three decades’ effort on model development from one to three dimensions: (1) one-dimensional models predict the vertical position of the mine’s center of mass (COM) with the assumption of constant falling angle, (2) two-dimensional models predict the COM position in the (x,z) plane and the rotation around the y-axis, and (3) three-dimensional models predict the COM position in the (x,y,z) space and the rotation around the x-, y-, and z-axes. These models are verified using the data collected from mine impact burial experiments. The one-dimensional model only solves one momentum equation (in the z-direction). It cannot predict the mine trajectory and burial depth well. The two-dimensional model restricts the mine motion in the (x,z) plane (which requires motionless for the environmental fluids) and uses incorrect drag coefficients and inaccurate sediment dynamics. The prediction errors are large in the mine trajectory and burial depth prediction (six to ten times larger than the observed depth in sand bottom of the Monterey Bay). The three-dimensional model predicts the trajectory and burial depth relatively well for cylindrical, near-cylindrical mines, and operational mines such as Manta and Rockan mines.

Advancement Towards the Full-Field Implementation of Marmul Alkaline-Surfactant-Polymer in the Sultanate of Oman

2021 ◽  
Author(s):  
Dawood Al Mahrouqi ◽  
Hanaa Sulaimani ◽  
Rouhi Farajzadeh ◽  
Yi Svec ◽  
Samya Farsi ◽  
...  
Keyword(s):  
Oil Recovery ◽  
Phase 1 ◽  
Recovery Efficiency ◽  
Sultanate Of Oman ◽  
Water Separation ◽  
Full Field ◽  
Oil Water Separation ◽  
Oil Water ◽  
New Formulation ◽  
Alkaline Surfactant Polymer

Abstract In 2015-2016, the Alkaline-Surfactant-Polymer (ASP) flood Pilot in Marmul was successfully completed with ∼30% incremental oil recovery and no significant operational issues. In parallel to the ASP pilot, several laboratory studies were executed to identify an alternative and cost-efficient ASP formulation with simpler logistics. The studies resulted in a new formulation based on mono-ethanolamine (MEA) as alkali and a blend of commercially available and cheaper surfactants. To expediate the phased full field development, Phase-1 project was started in 2019 with the following main objectives are confirm high oil recovery efficiency of the new ASP formulation and ensure the scalability and further commercial maturation of ASP technology; de-risk the injectivity of new formulation; and de-risk oil-water separation in the presence of produced ASP chemicals. The Phase 1 project was executed in the same well pattern as the Pilot, but at a different reservoir unit that is more heterogeneous and has a smaller pore volume (PV) than those of the Pilot. This set-up allowed comparing the performance of ASP formulations and taking advantage of the existing surface facilities, thus reducing the project cost. The project was successfully finished in December 2020, and the following major conclusions were made: (1) with the estimated incremental recovery of around 15-18% and one of the producers exhibiting water cut reversal of more than 30%, the new ASP formulation is efficient and will be used in the follow-up phased commercial ASP projects; (2) the injectivity was sustained throughout the entire operations within the target rate and below the fracture pressure; (3) produced oil quality met the export requirements and a significant amount of oil-water separation data was collected. With confirmed high oil recovery efficiency for the cheaper and more convenient ASP formulation, the success of ASP flooding in the Phase-1 project paves the way for the subsequent commercial-scale ASP projects in the Sultanate of Oman.

Quantum entanglement in some topological insulator models

2019 ◽  
Vol 33 (24) ◽  
pp. 1950284 ◽  
Author(s):  
L. S. Lima
Keyword(s):  
Quantum Entanglement ◽  
Topological Insulator ◽  
Topological Charge ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Model Case ◽  
Topological Transition ◽  
One Dimensional ◽  
Zhang Model ◽  
The One

Quantum entanglement is studied in the neighborhood of a topological transition in some topological insulator models such as the two-dimensional Qi–Wu–Zhang model or Chern insulator. The system describes electrons hopping in two-dimensional chains. For the one-dimensional model case, there exist staggered hopping amplitudes. Our results show a strong effect of sudden variation of the topological charge Q in the neighborhood of phase transition on quantum entanglement for all the cases analyzed.

The Influence of Wood Specimen Surface Coating on Moisture Movement during Drying

Holzforschung ◽  
2002 ◽  
Vol 56 (5) ◽  
pp. 541-546 ◽  
Author(s):  
R. Baronas ◽  
F. Ivanauskas
Keyword(s):  
Surface Coating ◽  
Transverse Section ◽  
Wood Specimen ◽  
Two Dimensional ◽  
Drying Process ◽  
Dimensional Model ◽  
Wood Drying ◽  
Moisture Movement ◽  
One Dimensional ◽  
The One

Summary A model of wood drying under isothermal conditions taking into consideration coating of the surface of a specimen is presented in this paper in a two-dimensional formulation. The influence of the surface coating degree as well as geometrical shape of a wood specimen on the dynamics of drying is investigated. Exponentials, describing the dependence of the halfdrying time on the degree of coating of the edges, as well as on the ratio of the width to the thickness of the transverse section of specimens from the northern red oak (Quercus rubra), are presented for drying from above the fiber saturation point. This paper describes the conditions of usage of the two-dimensional moisture transfer model in contrast to the one-dimensional model for accurate prediction of the drying process taking into consideration the coating of edges of specimens having a rectangular transverse section. A measure of reliability of the one-dimensional model to predict the wood drying process of sawn boards is introduced in this paper.

scholarly journals A one-dimensional tearing mode equation for pedestal stability studies in tokamaks

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
J. W. Connor ◽  
R. J. Hastie ◽  
C. Marchetto ◽  
C. M. Roach
Keyword(s):  
Cross Section ◽  
Approximate Equation ◽  
Stability Studies ◽  
Dimensional Model ◽  
Tearing Mode ◽  
One Dimensional ◽  
Tearing Modes ◽  
The Stability ◽  
Poloidal Flux

Starting from expressions in Connor et al. (Phys. Fluids, vol. 31, 1988, p. 577), we derive a one-dimensional tearing equation similar to the approximate equation obtained by Hegna & Callen (Phys. Plasmas, vol. 1, 1994, p. 2308) and Nishimura et al. (Phys. Plasmas, vol. 5, 1998, p. 4292), but for more realistic toroidal equilibria. The intention is to use this approximation to explore the role of steep profiles, bootstrap currents and strong shaping in the vicinity of a separatrix, on the stability of tearing modes which are resonant in the H-mode pedestal region of finite aspect ratio, shaped cross-section tokamaks, e.g. the Joint European Torus (JET). We discuss how this one-dimensional model for tearing modes, which assumes a single poloidal harmonic for the perturbed poloidal flux, compares with a model that includes poloidal coupling Fitzpatrick et al. (Nucl. Fusion, vol. 33, 1993, p. 1533).

One-dimensional model of fourth order for rods with loading on lateral boundary: The case of rectangular cross section

2016 ◽  
Vol 22 (12) ◽  
pp. 2269-2287 ◽  
Author(s):  
Erick Pruchnicki
Keyword(s):  
Cross Section ◽  
Fourth Order ◽  
Displacement Field ◽  
Rectangular Cross Section ◽  
Transverse Dimension ◽  
Lateral Boundary ◽  
Dimensional Model ◽  
Equilibrium Equations ◽  
Lagrange Equations ◽  
One Dimensional

We propose deducing from three-dimensional elasticity a one dimensional model of a beam when the lateral boundary is not free of traction. Thus the simplification induced by the order of magnitude of transverse shearing and transverse normal stress must be removed. For the sake of simplicity we consider a beam with rectangular cross section. The displacement field in rods can be approximated by using a Taylor–Young expansion in transverse dimension of the rod and we truncate the potential energy at the fourth order. By considering exact equilibrium equations, the highest-order displacement field can be removed and the Euler–Lagrange equations are simplified.

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