The DAN method for calculation of vapour-liquid equilibria using an equation of state; Bubble and dew point calculations

1985 ◽  
Vol 50 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Josef P. Novák ◽  
Vlastimil Růžička ◽  
Anatol Malijevský ◽  
Jaroslav Matouš ◽  
Jan Linek
Keyword(s):  
Equation Of State ◽  
Equilibrium Point ◽  
Critical Point ◽  
Dew Point ◽  
Computational Technique ◽  
Equilibrium Conditions ◽  
Close Vicinity ◽  
Newton Raphson ◽  
Raphson Method ◽  
Phase Envelope

A modification of the computational technique for calculating bubble and dew points using an equation of state has been proposed. The procedure consists in the Double Application of the Newton-Raphson method (DAN) to the set of equilibrium conditions. The algorithm is very effective as it provides both values of equilibrium variables and a very qualified first estimate of the next equilibrium point. This enables to proceed along the phase envelope rather quickly and to achieve convergence within a few iterations except in the close vicinity of the critical point.

The DAN method for calculation of vapour-liquid equilibria using an equation of state; Flash calculation

1985 ◽  
Vol 50 (1) ◽  
pp. 23-32 ◽  
Author(s):  
Josef P. Novák ◽  
Vlastimil Růžička ◽  
Anatol Malijevský ◽  
Jaroslav Matouš ◽  
Jan Linek
Keyword(s):  
Equation Of State ◽  
Computational Technique ◽  
Equilibrium Conditions ◽  
Newton Raphson ◽  
Flash Calculation ◽  
Raphson Method ◽  
Number Of Iterations

A modification of the computational technique for flash calculations using an equation of state has been developed. The procedure consists in the double application of the Newton-Raphson method (DAN) to the set of equilibrium conditions. The algorithm is designed to minimize the number of iterations. It is, therefore, especially useful in successive calculations, where a family of solutions at slightly changing conditions is desired.

Critical Point and Saturation Pressure Calculations for Multipoint Systems (includes associated paper 8871 )

1980 ◽  
Vol 20 (01) ◽  
pp. 15-24 ◽  
Author(s):  
L.E. Baker ◽  
K.D. Luks
Keyword(s):  
Equation Of State ◽  
Critical Point ◽  
Critical Region ◽  
Equations Of State ◽  
Saturation Pressure ◽  
Direct Calculation ◽  
Iterative Solution ◽  
Dew Point ◽  
K Value ◽  
Bubble Point

Abstract Calculation of fluid properties and phase equilibria isimportant as a general reservoir engineering tool andfor simulation of the carbon dioxide or rich gasmultiple-contact-miscibility (MCM) mechanisms. Of particular interest in such simulations is thenear-critical region, through which the compositionalpath must go in an MCM process.This paper describes two mathematical techniquesthat enhance the utility of an equation of state forphase equilibrium calculations. The first is animproved method of estimating starting parameters(pressure and phase compositions) for the iterativesaturation pressure (bubble-point or dew-point)solution of the equation of state. Techniquespreviously have been presented for carrying out thisiterative solution; however, the previously describedprocedure for obtaining initial parameter values wasnot satisfactory in all cases. The improved methodutilizes the equation of state to estimate theparameter values. Since the same equation then isused to calculate the saturation pressure, the methodis self-consistent and results in improved reliability.The second development is the use of the equationof state to calculate directly the critical point of afluid mixture, based on the rigorous thermodynamic criteria set forth by Gibbs. The paper presents aniterative method for solving the highly nonlinearequations. Methods of obtaining initial estimates ofthe critical temperature and pressure also arepresented.The techniques described are illustrated withreference to a modified version of the Redlich-Kwong equation of state (R-K EOS); however, theyare applicable to other equations of state. They havebeen used successfully for a wide variety of reservoirfluid systems, from a simple binary to complexreservoir oils. Introduction MCM processes such as CO2 or rich gas miscibledisplacements (conducted at pressures below thecontact-miscible pressure) traverse a compositional path that goes through the near-critical region. Thishas been described in several papers. Simulationof an MCM process requires the use of an equation of state to describe the liquid- and vapor-phasesaturations and compositions. Fussell and Yanosikdescribed an MVNR (minimum-variableNewton-Raphson)method for solution of a version of theR-K EOS. They discussed some of the difficulties ofobtaining solutions to the equation of state in the near-critical region and showed that the MVNRmethod gave improved results.Experience with the MVNR method hasdemonstrated a need for an improved estimate ofinitial iteration parameters (pressure, phasecompositions)for an iterative solution of saturation(dew-point and bubble-point) pressures. It waslearned that the semitheoretical K-value correlationused for initial estimates usually gave satisfactoryresults when the fluid system contained significantamounts of heavy components (C7+) but was oftenunsatisfactory for fluid systems containing only lightcomponents. This type of system is exemplified bythe fluids in a dry gas-rich gas mixing zone or bymixtures rich in CH4, CO2, or N2.Experience also has demonstrated a need for direct calculation of the critical point. While the MVNRsolution technique discussed by Fussell and Yanosikexhibits improved convergence in the near-critical region, it often is difficult to obtain convergedsolutions of the equation of state at compositionswithin a few percent of the critical composition. SPEJ P. 15＾

scholarly journals A new procedure for computing the factor of safety using the Morgenstern-Price method

2001 ◽  
Vol 38 (4) ◽  
pp. 882-888 ◽  
Author(s):  
D Y Zhu ◽  
C F Lee ◽  
Q H Qian ◽  
Z S Zou ◽  
F Sun
Keyword(s):  
Factor Of Safety ◽  
Recurrence Relations ◽  
Limit Equilibrium ◽  
Solution Process ◽  
Stability Factor ◽  
Equilibrium Conditions ◽  
Scaling Parameter ◽  
Newton Raphson ◽  
Safety Limit ◽  
Raphson Method

By employing the same assumption regarding interslice forces as that used in the Morgenstern-Price method, two concise recurrence relations between interslice forces and interslice moments are derived which satisfy both force and moment equilibrium conditions. The Newton-Raphson method is used for determining the factor of safety and the associated scaling parameter of the interslice force function. Algebraic derivatives required in the solution process are evolved in a recursive manner which can be easily implemented in a computer program. The choices of initial values of safety factor and scaling parameter are suggested. The procedure proposed in this paper proves to be efficient and solutions converge rapidly.Key words: slope, stability, factor of safety, limit equilibrium method.

Axisymmetric flow near the critical point on the moving boundary

2010 ◽  
Vol 7 ◽  
pp. 182-190
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh. Nasibullaeva
Keyword(s):  
Fluid Flow ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Axial Velocity ◽  
Moving Boundary ◽  
Axisymmetric Flow ◽  
Boundary Motion ◽  
Newton Raphson ◽  
Raphson Method

In this paper the investigation of the axisymmetric flow of a liquid with a boundary perpendicular to the flow is considered. Analytical equations are derived for the radial and axial velocity and pressure components of fluid flow in a pipe of finite length with a movable right boundary, and boundary conditions on the moving boundary are also defined. A numerical solution of the problem on a finite-difference grid by the iterative Newton-Raphson method for various velocities of the boundary motion is obtained.

A Generalized Compositional Approach for Reservoir Simulation

1983 ◽  
Vol 23 (05) ◽  
pp. 727-742 ◽  
Author(s):  
Larry C. Young ◽  
Robert E. Stephenson
Keyword(s):  
Enhanced Recovery ◽  
Gas Condensate ◽  
Reservoir Modeling ◽  
Pressure Equation ◽  
Compositional Modeling ◽  
Fluid Property ◽  
Black Oil ◽  
Fluid Properties ◽  
Newton Raphson ◽  
Raphson Method

A procedure for solving compositional model equations is described. The procedure is based on the Newton Raphson iteration method. The equations and unknowns in the algorithm are ordered in such a way that different fluid property correlations can be accommodated leadily. Three different correlations have been implemented with the method. These include simplified correlations as well as a Redlich-Kwong equation of state (EOS). The example problems considered area conventional waterflood problem,displacement of oil by CO, andthe displacement of a gas condensate by nitrogen. These examples illustrate the utility of the different fluid-property correlations. The computing times reported are at least as low as for other methods that are specialized for a narrower class of problems. Introduction Black-oil models are used to study conventional recovery techniques in reservoirs for which fluid properties can be expressed as a function of pressure and bubble-point pressure. Compositional models are used when either the pressure. Compositional models are used when either the in-place or injected fluid causes fluid properties to be dependent on composition also. Examples of problems generally requiring compositional models are primary production or injection processes (such as primary production or injection processes (such as nitrogen injection) into gas condensate and volatile oil reservoirs and (2) enhanced recovery from oil reservoirs by CO or enriched gas injection. With deeper drilling, the frequency of gas condensate and volatile oil reservoir discoveries is increasing. The drive to increase domestic oil production has increased the importance of enhanced recovery by gas injection. These two factors suggest an increased need for compositional reservoir modeling. Conventional reservoir modeling is also likely to remain important for some time. In the past, two separate simulators have been developed and maintained for studying these two classes of problems. This result was dictated by the fact that compositional models have generally required substantially greater computing time than black-oil models. This paper describes a compositional modeling approach paper describes a compositional modeling approach useful for simulating both black-oil and compositional problems. The approach is based on the use of explicit problems. The approach is based on the use of explicit flow coefficients. For compositional modeling, two basic methods of solution have been proposed. We call these methods "Newton-Raphson" and "non-Newton-Raphson" methods. These methods differ in the manner in which a pressure equation is formed. In the Newton-Raphson method the iterative technique specifies how the pressure equation is formed. In the non-Newton-Raphson method, the composition dependence of certain ten-ns is neglected to form the pressure equation. With the non-Newton-Raphson pressure equation. With the non-Newton-Raphson methods, three to eight iterations have been reported per time step. Our experience with the Newton-Raphson method indicates that one to three iterations per tune step normally is sufficient. In the present study a Newton-Raphson iteration sequence is used. The calculations are organized in a manner which is both efficient and for which different fluid property descriptions can be accommodated readily. Early compositional simulators were based on K-values that were expressed as a function of pressure and convergence pressure. A number of potential difficulties are inherent in this approach. More recently, cubic equations of state such as the Redlich-Kwong, or Peng-Robinson appear to be more popular for the correlation Peng-Robinson appear to be more popular for the correlation of fluid properties. SPEJ p. 727

scholarly journals Correction to: Efficient analysis of welding thermal conduction using the Newton–Raphson method, implicit method, and their combination

2020 ◽  
Author(s):  
Zhongyuan Feng ◽  
Ninshu Ma ◽  
Wangnan Li ◽  
Kunio Narasaki ◽  
Fenggui Lu
Keyword(s):  
Thermal Conduction ◽  
Implicit Method ◽  
Newton Raphson ◽  
Raphson Method

A Correction to this paper has been published: https://doi.org/10.1007/s00170-020-06437-w

scholarly journals Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 47
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz ◽  
U. Iturrarán-Viveros ◽  
R. Caballero-Cruz
Keyword(s):  
Fractional Derivative ◽  
Iterative Methods ◽  
Fractional Integral ◽  
Order Of Convergence ◽  
Fractional Operators ◽  
Speed Of Convergence ◽  
Newton Raphson ◽  
Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

scholarly journals Comparing Different Approaches for Solving Large Scale Power-Flow Problems With the Newton-Raphson Method

IEEE Access ◽  
2021 ◽  
Vol 9 ◽  
pp. 56604-56615
Author(s):  
Manolo D'orto ◽  
Svante Sjoblom ◽  
Lung Sheng Chien ◽  
Lilit Axner ◽  
Jing Gong
Keyword(s):  
Power Flow ◽  
Large Scale ◽  
Flow Problems ◽  
Newton Raphson ◽  
Raphson Method
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