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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1989 SEG Annual Meeting, October 29–November 2, 1989

Paper Number: SEG-1989-1068

... ABSTRACT No preview is available for this paper. finite difference scheme 4th-order scheme grid initial

**boundary****value****problem**calculation difference operator equation discretization cohen operator propagation seg annual international meeting wave equation reservoir...Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*17 (05): 345–352.

Paper Number: SPE-5726-PA

Published: 01 October 1977

... partial differential equation Upstream Oil & Gas

**boundary**condition basis function Galerkin Method differential equation finite-difference method Generalized Collocation Methods for Time-Dependent, Nonlinear**Boundary**-**Value****Problems**ABSTRACT R. F. SINCOVEC* MEMBER SPE~AIME This paper briefly...
Abstract

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*11 (04): 374–388.

Paper Number: SPE-2806-PA

Published: 01 December 1971

...W.E. Culham; Richard S. Varga This paper presents and examines in detail extensions to the Galerkin method oil solution that make it numerically superior to conventional methods used to solve a certain class of time-dependent, nonlinear

**boundary****value****problems**. This class of**problems**includes...
Abstract

This paper presents and examines in detail extensions to the Galerkin method oil solution that make it numerically superior to conventional methods used to solve a certain class of time-dependent, nonlinear boundary value problems. This class of problems includes the equation that describes the flow of a fully compressible fluid in a porous medium. The Galerkin method with several different piecewise polynomial subspaces and a non-Galerkin piecewise polynomial subspaces and a non-Galerkin method specifically employing cubic spline functions are used to approximate the solution of a nonlinear parabolic equation with one spatial variable. With a parabolic equation with one spatial variable. With a known analytic solution of the problem, the accuracies of these approximations are determined and compared with conventional finite-difference approximations. Specially, the various methods are compared on the basis of the amount of computer time necessary to achieve a given accuracy, as well as with respect to the order oil convergence and computer core storage required. These tests indicate that the higher-order Galerkin methods require the least amount of computer time for a given range of accuracy. Introduction The purpose of this paper is to outline in detail the application of the Galerkin method, employing piecewise polynomials, to solve nonlinear piecewise polynomials, to solve nonlinear boundary-value problems and compare the computational efficiency of the Galerkin method with more conventional numerical methods. Numerical methods compared with the Galerkin technique include a non-Galerkin method that utilizes cubic spline interpolation and the conventional finite-difference methods. Four conventional time approximations were also studied in conjunction with the above mentioned space discretization methods. In an earlier paper, Price and Varga showed theoretically that higher-order approximations to certain semilinear convection-diffusion equations were possible by means of Galerkin techniques, but complete numerical results for such approximations were not given. Also, in a paper that introduced the Galerkin method to the petroleum industry, Price et al. demonstrated that higher-order approximations were far superior numerically to the conventional methods used to solve certain linear convection-diffusion type problems. Jennings, Douglas and Dupont and Douglas et al. have considered the application of Galerkin methods to various nonlinear problems, but again complete numerical results, problems, but again complete numerical results, including comprehensive comparisons with existing numerical methods, were not given. Thus, in addition to presenting some new and computationally efficient Galerkin formulations for nonlinear problems and numerically demonstrating their problems and numerically demonstrating their higher order accuracies, it was also desirable to test these. methods to determine if they also exhibited the same superiority in regard to computational efficiency as was demonstrated for the Galerkin methods applied to linear problems. If so, then the Galerkin technique could prove to be an important advancement toward developing faster numerical models for field application. To test and compare each method of solution, a problem involving the nonlinear gas-flow equation problem involving the nonlinear gas-flow equation in one spatial variable with a specific volumetric source term was chosen, for which a closed-form or analytic solution was known. Using this particular problem and its analytic solution, it was possible problem and its analytic solution, it was possible to determine numerically the order of convergence of each method, to compare each method on the basis of computer time expended to obtain a given accuracy, and to compare each method with respect to computer core storage required. In addition, the experimental data were used to define "consistent quadrature" and "consistent interpolation" schemes for the Galerkin methods. Finally, it was possible to formulate conclusions regarding the computational efficiency of the four time approximations investigated. SPEJ P. 374

Journal Articles

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*9 (02): 204–220.

Paper Number: SPE-2034-PA

Published: 01 June 1969

... numerical solution multiphase flow approximation dx dy pressure gradient Fluid Dynamics

**boundary****value****problem**inequality matrix unit square subspace Galerkin Method basis element equation Upstream Oil & Gas block ri basis function varga exact solution Galerkin Methods...
Abstract

This paper presents a new technique for solving some of the partial differential equations that are commonly used in simulating reservoir performance. The results of applying this technique to a simple problem show that one obtains accurate pressure problem show that one obtains accurate pressure values near wells, as well as accurate pressure gradients, which can be explicitly calculated. The method is completely rigorous in that convergence of the discrete numerical solution to the continuous solution for both pressure and pressure gradient is established. High-order, piecewise-polynomial approximations are used near the wells where pressure gradients are steep, while low-order, pressure gradients are steep, while low-order, piecewise-polynomial approximations are used piecewise-polynomial approximations are used elsewhere to reduce greatly the calculation time. This combination is shown to give a uniformly good approximation to the solution. These approximations, obtained by using a Galerkin process with suitable Hermite subs paces, are shown to be theoretically and numerically superior to the usual approximations obtained from standard finite-difference techniques. Not only are much greater accuracies obtained, but computer times are also greatly reduced. The application of this technique to multiphase flow problems (e.g., single well coning problems) would have considerable practical interest, but such extensions of this technique with full mathematical rigor have not been made as yet. However, the numerical methods presented here are general, and in principle extend to multidimensional, multiphase flow. Moreover, the preliminary results given in this paper are sufficiently encouraging that we feel the effort in attempting these extensions is justified. Introduction The problem of obtaining accurate pressure distributions and pressure gradients around wells is of considerable importance in the numerical simulation of reservoir performance. The most common approach to solving this problem is to use finite difference techniques (see McCarty and Barfield or Peaceman and Rachford). This approach, however, has many disadvantages, the major one being that many grid points are generally necessary for accurately describing the pressure distribution and the pressure gradient around wells. This need for a fine grid results in large computer times and often in prohibitively high costs. Besides investigating the method of finite differences, some authors, such as Welge and Weber and Roper, Merchant and Duvall, have considered a combination of analytical and numerical techniques with some success. These approaches, however, are all nonrigorous and quite -often cannot even be applied. In this paper, we present a numerical formulation of high-order present a numerical formulation of high-order accuracy, based on the Galerkin method, for solving this problem. We treat here only the partial differential equation that describes steady-state, single-phase flow. However, the methods presented are general and in principle extend to multidimensional, multiphase principle extend to multidimensional, multiphase flow. Specifically we treat special cases of the problem in two dimensions described by: problem in two dimensions described by: (1) (2) where G is a rectangle (with sides parallel to the coordinate axis) with boundary G, / n denotes the outward normal, and and beta are non-negative constants such that + beta greater than 0. SPEJ p. 204

Proceedings Papers

Paper presented at the 11th ISRM Congress, July 9–13, 2007

Paper Number: ISRM-11CONGRESS-2007-173

... of the strains. Since strains have the differential relations with displacements as shown in equations 4 and 5, the strain changes can be estimated from the measured displacements. 3

**BOUNDARY**CONDITIONS Generally speaking, a**boundary****value****problem**can be divided into "stress**boundary****value****problem**...
Abstract

ABSTRACT This paper presents an analytical closed-form elasto-plastic solution for displacement around a circular tunnel. The solution is based on strain-related parameters such as initial strain, strain boundary conditions and a strength criterion in termas strains. Engineering approaches based on such strain-related solutions are advantageous for applications in the so called observational method. This paper will demonstrate that the size of the eventual plastic zone and the strain-related parameters can be estimated directly by evaluations of measured displacements. These parameters are important to evaluate the observed tunnel behaviors compared with the expected behaviors. 1 INTRODUCTION Displacements are often monitored during rock excavations, not only for monitoring the stability, but also for re-evaluating the design. For instance, displacement monitoring is an essential part of the New Austrian Tunnelling Method (NATM). A design approach known as "the observational method", in which the design is reviewed by measurements during construction, has been adopted in the Eurocodes. When using the observational method, the response time of the instrumentations and the procedures for analysing the results shall be sufficiently rapid in relation to the possible evolution of the construction work. At present, back-calculations based on stress analysis are often performed. But this is a time-consuming procedure. Sakurai (1981) developed a strain evaluation method to determine the strain distribution in an underground opening directly from the measured displacement.However the method has not been widely used in rock engineering, partially because of lack of a general strength criterion expressed in strains. In order to utilise the approaches of strain analyses in rock engineering, a general and simple criterion for rock strength in term of strains has been proposed by Chang (2006), based on the compilation of test data from the literature. 2 STRAINS ASSOCIATED WITH TUNNELLING Before a tunnel excavation, the rock is considered as being constrained by initial strains, denoted by { ε o }. During the course of tunnelling, the strains will be changed from the initial state { ε o } to a new state { ε } in the strain space. Based on analyses of test data Chang (2006) has pointed out that under triaxial loading the rock exhibits volumetric contraction i.e d ε v /d ε 1 > 0 under elastic condition; whereas rock exhibits plastic behaviour with volumetric expansion i.e. d ε v /d ε 1 < 0 when the strains reach the strain yielding surface defined by equation 1. This is an important statement, by which the extents of yielding zone around a tunnel can be detected by measuring the changes of the strains. Since strains have the differential relations with displacements as shown in equations 4 and 5, the strain changes can be estimated from the measured displacements. 3 BOUNDARY CONDITIONS Generally speaking, a boundary value problem can be divided into "stress boundary value problem" or "displacement boundary. value problem", as illustrated in Figure 2. When stresses on all boundaries are specified, the problem is classified as a "stress boundary value problem"; whereas if the displacements or strains on all boundaries are specified, the problem is classified as a "displacement boundary value problem".

Proceedings Papers

Paper presented at the The 27th International Ocean and Polar Engineering Conference, June 25–30, 2017

Paper Number: ISOPE-I-17-702

... of modern mathematicians Hall [Li ZC, etc 2013], and American scholar Lang [Ramachandran PA 2002] also discussed Lie-group and Lie algebra. As the development of group theory is maturing, matures, many physical

**problems**are solved, such as coupling**problem**,**boundary****value****problem**and ordinary differential...
Abstract

ABSTRACT An equal norm multiple scale Trefftz method (MSTM) associated with the Lie-group scheme GL(n, R) in Euclidean space is developed to describe two dimensional nonlinear sloshing behaviors. In this paper, the explicit and implicit GL(n, R) in the Euclidean space are used for time integration and the results in terms of computational efficiency and accuracy are very good. The MSTM combined with the vector regularization method (VRM) is adopted the first time to eliminate the phenomena of higher-order numerical oscillation and noisy dissipation. The proposed method in this paper can overcome the boundary noisy disturbance and improve the stability and accuracy of the sloshing problems. Numerical scheme is developed and verified by benchmark tests. Different shapes of the fluid tank are simulated with various excitation frequencies. The occurring waves are successfully modeled and the results will be discussed later in detail. Comparisons of the results with other methods shows that the proposed method in this paper indeed does a better job on both accuracy and running time. INTRODUCTION In recent years, the application of SPH method by Monaghan [Kim Y, 2001], Ma[Zhang T, etc. 2016], Jan [Monaghan JJ, 1994], respectively, the application of SPH method, MLPG[Zhang T, etc. 2016], respectively, the simulations of sloshing behaviors exist singular problems of integrals and slow in convergence due to their finite element and boundary element methods. Local Radial Basis Function Collocation Method (LRBFCM) has also been used effectively to simulate the sloshing phenomenon, such as Fan and other scholars [Vaughan GL, etc. 2008]. In this paper, the Trefftz method is used to solve the sloshing problem. Trefftz method [Ma QW, 2005] was first proposed in 1926. Trefftz method using T-complete function as a base function to meet the problem of governing equations. In 2004, Kita et al. [Ali A, etc. 2005] first applied the Trefftz method to the simulation of the sloshing problem. When applying the Trefftz method with no singular sources, sufficient constraint equations should be established to increase the boundary discrete points in order to improve the simulation accuracy. But the high order base functions will cause numerical instability. Liu [Liu CS, etc. 2009] proposed modified Trefftz method (MTM) to introduce the feature length in the T-complete base function to improve the phenomenon of numerical instability, Chen [Chen YW, 2009, 2010, 2012] and other scholars using the modified Trefftz method And Geometric Multiple Scale Trefftz Method (MSTM) to solve the sloshing problem. The concept of dissipation factor and control volume is revised to improve the accuracy of the solution. In 2016, MTM and VRM were proposed to overcome Border interference. In the part of solving the initial value problem of sloshing problem, this paper will use the preserving group algorithm to carry on the operation, and the preserving group algorithm is generalized through the group concept. Some scholars of modern mathematicians Hall [Li ZC, etc 2013], and American scholar Lang [Ramachandran PA 2002] also discussed Lie-group and Lie algebra. As the development of group theory is maturing, matures, many physical problems are solved, such as coupling problem, boundary value problem and ordinary differential equation problem. Liu [Chen CS, etc 2006] introduced nonlinear dynamic system into augmented dynamic system in 2001 (Minkowski space) under the deduction of Lorentz group. And in 2013 [Jin BA, 2004] derived Lie-group differential algebraic equation method for solving the above problems. Lie-group differential algebraic equation method is a nonlinear differential algebraic equation, adding generalized linear group structure, so that it is converted to initial value problem solving. Through the developed method, simple formulae are derived to deal with complex problems. In this paper, we will use the explicit and implicit Lie-groups method in the Euclidean space for time integration. In this paper, the computational efficiency and accuracy are very good when the Lie method is used in the Euclidean space. The weighting factor calculated by the group preserving algorithm is introduced into the boundary value problem to correct the numerical error of the boundary value problem.

Proceedings Papers

Paper presented at the The Third International Offshore and Polar Engineering Conference, June 6–11, 1993

Paper Number: ISOPE-I-93-296

... of hydrodynamically loaded cable has been developed by Chiou and IISOPE Member Leonard (1991) in which the

**problem**is formulated as a two point**boundary****value****problem**. The**boundary****value****problem**is then transformed into an iterative set of quasi-linearized**boundary****value****problems**, which is then decomposed (Atkinson...
Abstract

ABSTRACT: The equations of motion for small tethered buoys floating in a nonlinear wave field have been developed. The coupling between rotational and translation3.I degrees of freedom is included in the equations and a three-dimensional response is assumed. The floating buoy is treated as one boundary condition of the governing differential equations for the mooring line coupled buoy-mooring problem. INTRODUCTION In this paper the coupling effects of rotational degrees of freedom of tethered floating buoys with the governing equations of the tether are considered. The cable algorithm is described in the following section. The equations of motion for tethered floating buoys in terms of the six degrees of freedom in translation and rotation, which constitute the boundary conditions for one end of the tether, are developed. An algorithm for quasi-linearization of those boundary conditions, which are used in determining the tether motions and buoy rotations for the coupled nonlinear system, is developed and. presented in a subsequent section. Validation of the methodology is provided in the final section. Buoys and their moorings are considered in this work to be classified as small bodies for which the relative-motion Morison equation may be adopted (Sarpkaya and Isaacson, 1981). A coupled analysis is needed for this ocean structure, since the motion of the buoy affects the motion of the mooring and visa versa (Berteaux, 1976). CABLE ALGORITHM An iterative algorithm of dynamic analysis of hydrodynamically loaded cable has been developed by Chiou and IISOPE Member Leonard (1991) in which the problem is formulated as a two point boundary value problem. The boundary value problem is then transformed into an iterative set of quasi-linearized boundary value problems, which is then decomposed (Atkinson, 1989) into a set of initial value problems so that spatial integration may be performed along the cable (Sun et al., 1993).

Proceedings Papers

Publisher: Offshore Technology Conference

Paper presented at the Offshore Technology Conference, April 21–23, 1970

Paper Number: OTC-1281-MS

... at two q.ifferent points along thecable These

**boundary****value****problems**are commonly solved for the equilib- rium position of the cable by integrating a family of initial**value****problems**and then se- lecting the desired solution. An iterative solution technique is pre- sented by which**boundary****value**...
Abstract

ABSTRACT In the design of systems towed or moored in a fluid by a flexible cable, boundary conditions are often specified at two different points along the cable. These boundary value problems are commonly solved for the equilibrium position of the cable by integrating a family of initial value problems and then selecting the desired solution. An iterative solution technique is presented by which boundary value problems in two and three dimensions can be solved directly. The differential equations of cable equilibrium are derived for a three-dimensional velocity field. The hydrodynamic loading functions are expressed in general forms so that the most appropriate hydrodynamic force model for a given cable application can be used, or solutions for different models can readily be compared. The differential equations are solved by a digital computer program which incorporates the solution technique for boundary value problems. The force components at the cable terminations necessary to achieve the given boundary conditions are obtained from the solution; for example, the hydrodynamic forces which a towed body must exert on its towing cable to maintain a specified position can be determined. Examples of solutions to two and three-dimensional problems with space variable velocity components are given. INTRODUCTION Flexible mooring and towing cables have had wide application as components of hydrodynamic and aerodynamic systems. Clearly, an understanding of the behavior of these cables is imperative for intelligent design of the complete system. The objective of this paper is to present different solution techniques which will provide a more direct solution to certain flexible cable problem s than was previously possible. The two-dimensional cable problem has been studied by a number of investigators. The most significant difference in the various investigations is the mathematical model used for the hydrodynamic forces acting on the cable, i.e., the hydrodynamic loading functions. Pode 1 provided a comprehensive report on the subject, and his solutions are still commonly used. He assumed a sine-squared relation for the normal component of hydrodynamic force and a constant for the tangential force component. The solutions were put in the form of integral "cable functions" which were evaluated numerically and tabulated. Knowing the tension and slope at one point on the cable, the conditions at other points along the cable can be found using Pode' s functions. Whicker 2 added to Pode' s work by developing cable functions using a modified hydrodynamic force model applicable to faired as well as round cables. The most complete reference to date on the solution of tow cable problems by the use of cable functions has been published by Eames 3 . He provides an extensive discussion on the form of the hydrodynamic loading functions as related to the physical conditions of the problem. Cable s are classified according to the relative significance of the various terms of his proposed loading models. This reference is particularly valuable for aiding in the selection of a hydrodynamic force model.

Proceedings Papers

Paper presented at the 6th ISRM Congress, August 30–September 3, 1987

Paper Number: ISRM-6CONGRESS-1987-214

... ABSTRACT: Conditions which are decisive for the stability of deep underground excavations are investigated in a numerical study. First a nonassociated elasto-plastic constitutive relation for brittle materials is derived. It is shown that an originally load controlled

**boundary****value****problem**...
Abstract

ABSTRACT: Conditions which are decisive for the stability of deep underground excavations are investigated in a numerical study. First a nonassociated elasto-plastic constitutive relation for brittle materials is derived. It is shown that an originally load controlled boundary value problem is equivalent to a displacement Controlled boundary value problem. As an application circular - and rectangular cavities in an isotropic stress field are investigated. RESUME: Les conditions decicives pour la stabilite des cavites très profondes ont ete etudiees par les etudes numeriques. D'abord on decrive une loi des conditions elasto-plastiques non associatives pour des materiaux cassants. On demontre comment un probleme aux conditions aux limites originalement controlle par la charge est equivalent à problème aux conditions aux limites controlle par le deplacement. Comme application on etude des cavites circulaires et rectangulaires dans un champ de tension isotropique. ZUSAMMENFASSUNG: Anhand numerischer Studien wird ueberprueft, unter welchen Bedingungen tiefliegende Hohlraume in sprödem Gestein versagen können. Zunachst wird ein nichtassoziiertes elasto-plastisches Stoffgesetz fuer sprödes Gestein hergeleitet. Es wird gezeigt, wie urspruenglich lastparametergeregelte Randwertprobleme in aquivalenter Weise verschiebungsgeregelt formuliert werden können. Als Anwendungen werden kreisförmige und rechteckige Untergrundhohlraume in einem isotropen Spannungsfeld behandelt. INTRODUCTION: One is frequently confronted with the problem of stability of deep underground excavations in brittle rock in connection with geophysical explorations or energy production. In this paper constitutive relationships are derived for the desription of the mechanical behaviour of strongly coherent rocks such as granite, sandstone etc. On the basis of this constitutive model the stability of circular and rectangular cavities is studied by means of the finite element method. It was pointed out by EGGER (1973) in an analytical study that the most important aspect of the material behaviour with respect to the stability of cavities is the law that describes how the material looses coherence. In the present paper this property will be considered in an elasto-plastic constitutive relation which allows the representation of frictional hardening and simultaneous cohesion softening. In finite element solutions in softening material behaviour one is confronted with two major problems: Load control in the incremental solution is not possible because the load-displacement relation is not monotoneous in general. This means in a tunnel problem that the intensity of the support pressure cannot be used as a control parameter in a finite element solution because in softening material the support pressure first decreases to a certain threshold level and has to be increased to maintain stability then. By applying the so called arc-length algorithm for Solution of nonlinear systems of equations this problem can be overcome (RIKS, 1979; CRISFIELD, 1985; de BORST, VERMEER, 1984). In arc-length algorithm the increments in externally prescribed loads or displacements do not have to be definded by the user. An increase or decrease of the loads is determined by the algorithm in such a way that e.g. a certain norm of the incremental displacements do not exceed a specified value. Alternatively, as will be done here, the surface deformation which is conjugate in energy to the surface pressure can be used as a control parameter (provided this surface deformation is monotoneously increasing). In an tunnel problem this means that the volume change of the tunnel is prescribed instead of the support pressure. The finite element solution becomes strongly mesh dependent. One of the reasons for this is the tendency of the displacement gradients to localize in thin shear bands. However, if the classical continuum theory is underlain the thickness of the shear bands is undetermined. To remedy this desease, it has been proposed to make the softening modulus dependent on the element size (PIETRUSCZAK, MROZ, 1981; WILLAH et al. 1984, 1986). A physically more evident approach is to underlay a generalized continuum theory as e.g. the Cosserat theory. In such a continuum the thickness of shear bands is defined by certain material parameters depending on the dimension of length (Grain diameter etc.) (BESDO 1985; MUHLHAUS 1986a, b; MUHLHAUS, VARDOULAKIS, 1986). In this article we content with the classical continuum description. It will be shown that in the case of the boundary value problems considered here the essential properties of the solution are mesh independent. In the following section a constitutive relation is defined. Subsequently it is shown that an originally load controlled boundary value problem can be made displacement controlled. In the fourth section the stability of the surface of a rectangular cavity in an infinite plate is considered. The in-plane principle stresses at infinity are assumed to be equal. To get an idea of the influence of the shape of the cavity on the stability, the case of a circular cavity is also considered.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 26–29, 2004

Paper Number: SPE-89919-MS

... injection results in a Riemann

**problem**for this hyperbolic system. The displacement of oil by a polymer slug with water drive is described by an initial and**boundary****value****problem**with piecewise constant initial data and results in wave interactions. The Riemann**problem**for the displacement of oil by hot...
Abstract

Abstract Compositional models describe chemical flooding and gas-based IOR processes. It is possible to show from the known analytical solutions of multicomponent polymer/surfactant flood, that the concentration "part" of the solution is completely defined by adsorption isotherms and does not depend on relative permeability and phase viscosities. Analytical models for 1-D displacement of oil by gas have been developed during last 15 years. It was observed from semi-analytical and numerical experiments that several thermodynamic features of the process (MMP, key tie lines, etc) are independent of transport properties. In this paper, we introduce potential coordinates that allow for splitting of the compositional model into a thermodynamics auxiliary system and one transport equation. The number of auxiliary equations is less than the number of equations in the compositional model by one. Explicit projection and lifting procedures are derived. The splitting is valid for either self-similar continuous injection problems or for non-self-similar slug injection problems. Analytical models for displacement of oil by a polymer slug and for oil displacement by rich gas slug with lean gas drive were developed by the splitting procedure. With respect to 3-D flows, splitting takes place only for the case of constant total mobility. Introduction The chemical methods of enhanced oil recovery include injection of aqueous solutions of several chemical components (polymers, surfactants, salts, etc.) that affect the flow of each phase in porous media 1 . One-dimensional displacement of oil by multicomponent chemical solutions taking into account adsorption is described by an (n+1)×(n+1) hyperbolic system of conservation laws, where n is the number of components in the aqueous displacing phase. Continuous polymer injection results in a Riemann problem for this hyperbolic system. The displacement of oil by a polymer slug with water drive is described by an initial and boundary value problem with piecewise constant initial data and results in wave interactions. The Riemann problem for the displacement of oil by hot water 2 is mathematically equivalent to one-component polymer flooding (n=1) for a convex sorption isotherm 3 . A graphical procedure for the solution of this problem has already been developed 4,5 . Several Riemann solutions for the case n=2 have been found 6 , where the i-th adsorbed concentration depends only on the concentration of the i-th component in the aqueous phase. The solution of the (n+1)×(n+1) system for two-phase n-component displacement was studied in different papers 7–10 . The particular case of one-phase n-component flow leads to an (n)×(n) hyperbolic system (called the associated system), which was used for solving the Riemann problem. The projection of the elementary waves of a two-phase system onto the associated one-phase system was constructed. The lifting procedure developed allows the calculation of any Riemann solution for the two-phase system once the associated one-phase solution is known. The theory developed is based on the fact that the Riemann problem solutions depend on a single parameter, x/t. Thus, the theory cannot be extended to non-self-similar Cauchy/initial-boundary value problems. Non-self-similar problems with wave interactions have been solved for n=1 11 and n=2 12 . Gas based methods of enhanced oil recovery include injection of different gases (methane, rich hydrocarbon gases, carbon dioxide, nitrogen and various combinations) in order to improve displacement by mass exchange between oleic and gas phases 1 . One dimensional displacement of oil by gas (solvent injection) is described by an (n-1)×(n-1) hyperbolic system of conservation laws, where n is the number of components13–16. Continuous gas injection results in a Riemann problem for this hyperbolic system. Displacement of oil by a gas slug with another gas drive is described by the initial and boundary value problem with piece-wise initial data 12 .

Proceedings Papers

Paper presented at the The Nineteenth International Offshore and Polar Engineering Conference, July 21–26, 2009

Paper Number: ISOPE-I-09-049

... In most of the wave theories, a

**boundary****value****problem**consisting of a partial differential equation and certain**boundary**conditions describing the various**boundaries**is solved in an approximate way. The complete**boundary****value****problem**has not been solved, even in the simple case of constant...
Abstract

In most of the wave theories, a boundary value problem consisting of a partial differential equation and certain boundary conditions describing the various boundaries is solved in an approximate way. The complete boundary value problem has not been solved, even in the simple case of constant water depth. The theory developed in this paper is limited to a flat bottom having a constant water depth. The waves are assumed regular. Water is considered as incompressible and inviscid as with most of the wave theories. Therefore the continuity equation which gives rise to the basic differential equation of wave motion is Laplace equation. In this paper, an ordinary differential equation which is equivalent to Laplace equation with the non-linear boundary conditions, is introduced for developing a 3-dimensional gravity wave theory. General solutions satisfying Laplace equation, bottom boundary condition and non-linear kinematic boundary condition on free surface are developed. By applying the general solutions to non-linear dynamic boundary condition on free surface, an ordinary differential equation is derived. The ordinary differential equation gives rise to eigen value problem. The eigen values of the equation are dispersion relationship. To verify the validity of the ordinary differential equation, it is proved that the general solutions developed in this paper include Dean''s solution which was developed using stream function theory. INTRODUCTION Numerous water wave theories have been developed which are applicable to different environments dependent upon the specific environmental parameters, e.g., water depth, wave height and wave period. Airy wave theory, Stokes wave theory, Cnoidal wave theory, Solitary wave theory and Dean's numerical stream function theory are commonly used in the design of offshore structure(Chakrabarti, 1987; DNV, 2007). For high Ursell numbers the wave length of the Cnoidal wave goes to infinity and the wave is a solitary wave(DNV, 2007).

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 26–29, 2004

Paper Number: SPE-89935-MS

... of a well produced with constant rate. In this work we present a new technique to evaluate single well productivity indices both for constant pressure and for constant rate conditions. The approach is based on the solution of two steady state

**boundary****value****problems**with constant pressure prescribed...
Abstract

Abstract The existing methods for evaluating the well productivity index are based on solution of transient problems. One approach is to consider the single well problem in infinite domain and subsequently apply the method of images. This puts restrictions on the geometry of the well and of the drainage volume. Another approach is to solve the transient problem in the bounded domain for late times. While in this case restrictions on the well geometry are less severe, the shape of the drainage volume is still limited to the simplest ones. In addition, for the constant rate case highly accurate wellbore pressures, for the constant pressure case highly accurate wellbore rates are required and that puts an extreme computational burden on the semi-analytical or numerical methods involved. Even with the most powerful methods and hardware available, the calculation of the productivity index of directionally drilled and partially penetrating wells, especially in more complex drainage volumes is a formidable task. An additional problem is that in general, the productivity index for a well produced under constant pressure condition is different, although very close, from the productivity index of a well produced with constant rate. In this work we present a new technique to evaluate single well productivity indices both for constant pressure and for constant rate conditions. The approach is based on the solution of two steady state boundary value problems with constant pressure prescribed on the wellbore. The two productivity indices, (for constant rate and constant wellbore pressure, respectively) are then computed as integral characteristics of the solutions of the corresponding time dependent boundary value problem. The two productivity indices are computed independently. The method can be applied to any geometry of the reservoir (both 2-D and 3-D, regular or irregular), and any direction and length of penetration of the well. Designer wells (with a freely prescribed path) can be also considered. Introduction We consider a bounded reservoir with no flow outer conditions. The fluid is single phase, slightly compressible. A well producing with either constant pressure or constant rate is characterized by the productivity index defined as [19]: Equation 1 where q(t) is the production rate, p w (t) is the flowing bottomhole pressure and p a (t) is the average reservoir pressure. We are particularly interested in the stabilized (late time) value of the PI. For the constant production rate stabilization means that the difference of average and wellbore pressure (the denominator) becomes time invariant. This flow regime is called pseudo-steady state (PSS). In the case of constant wellbore pressure, both the numerator and denominator keep changing with time, but their ratio stabilizes, leading to the flow regime called boundary-dominated (BD).

Proceedings Papers

Paper presented at the The Eighth International Offshore and Polar Engineering Conference, May 24–29, 1998

Paper Number: ISOPE-I-98-241

... ABSTRACT In this paper, the fully nonlinear computations are done in the time-domain, using an Euler-Lagrange method. At each time step, the resulting mixed

**boundary****value****problem**is solved using desingularized isolated sources. On the free-surface, the kinematic condition is used to time step...
Abstract

ABSTRACT In this paper, the fully nonlinear computations are done in the time-domain, using an Euler-Lagrange method. At each time step, the resulting mixed boundary value problem is solved using desingularized isolated sources. On the free-surface, the kinematic condition is used to time step the free surface elevation and the dynamic condition is used to march the potential. The waves generated by a source-sink pair and a ellipsoid moving below a free surface are considered. INTRODUCTION At the present time, most computations for ship seakeeping or for the diffraction-radiation motions of platforms are done in the frequency domain. To date, the majority of research has assumed that the water can be considered as incompressible and inviscid and that the flow around the body remains irrotational. In this case, the Laplace equation is valid everywhere in the fluid domain and the hydrodynamic forces acting on the body are determined as the solution to a boundary value problem. This problems can be solved by panel methods using either Rankine singularities (aerodynamic) or Kelvin ones satisfying a linearized free-surface boundary condition with use of Green's function. In the first case, unknown singularities have to be distributed on the body and also on a part of the free surface to be determined; furthermore the radiation condition is difficult to satisfy and computational difficulties can be observed at the truncated boundary of the free-surface. In the second case, if problems due to the radiation condition and to the boundary reflections are suppressed, high computational times are required due to the complicated form of the Green's function (Inglis and Price, 1981; Guevel and Bougis, 1982; Wu and Eatock Taylor, 1987; Squire and Wilson, 1992 and Iwashita and Okhusu, 1992 or more recently Ba and Guilbaud, 1995).

Proceedings Papers

Paper presented at the ISRM International Symposium, September 12–16, 1988

Paper Number: ISRM-IS-1988-066

... in the transformed field, since the

**boundaryes**there are all rectangular, and the computation in the transformed field thus is 536 regardless physical ofon a square grid the shape of the**boundaries**. Several transformed region configurationsmay be obtained rut the discution of this matter is out of the scope...
Abstract

ABSTRACT: A code for numerical generation of boundary-fitted curvilinear coordinate systems on fields containing any number of arbitrary two dimensional bodies have been developed. Both algebraic generation systems and partial differential equations procedures have been used. For this paper many grids for several situations with differents purpose are generated and the best grid configuration is interactively obtained. 1 INTRODUCION A numerically-generated grid is understood here to be the organized set of points formed by the intersections Of the lines of a boundary- Conforming curvilinear coordinate system. The procedure for the generation of curvilinear coordinate systems used here is the numerical solution of partial elliptic differential equation. Those of poisson and/or laplace. An algebric interpolation has been used to provide the first approximation to the numerical solution of the partial differential equation. In the section which follows. the conception of the curvilinear coordinate system and the grid generation system are discussed. The next two sections discover about the equations of the generation system and the implementation procedure. Finally. a sample solution showing some of the implemented program's facilities is presented. The extension of the program capabilitiescan be seen else where. 2 BASIC CONCEPTS Here the generation of the curvilinear coordinate system may be treated as follows: With the curvilinear coordinates specified on the boundaries. e.g., ~ (x.y) and T) (x.y).on the boundary B and A (this specification amounting to a constant value for either ~ or T) on each segment of B and A, with a specified monotonic variation of the other coordinate over the segment), generate the values, ~(x,y) and n(x,y), in the field bounded by Band A. This is thus a boundary value problem on the physical field with the curvilinear coordinate (s,n) as the dependent variables and the cartesian coordinates (x.y) as the independent variables, with boundary conditions specified on curved Figure 1: Physical Region. The problem may be simplified for computation, however, by first transforming so that the physical cartesian coordinates (x,y) become the dependent variables, with the curvilinear coordinates (s, n) as the independent variables. Since a constant value of one curvilinear coordinate, with monotonic variation 'of the other, has been specified on each boundary segment, it follows that these boundary segments in the physical field will correspond to vertical or horizontal lines in the transformed field. Also, since the range of variation of the curvilinear coordinate varying along a boundary segment has been made the same over opposing segments, it follows that the transformed field will be composed of retangular blocks. The boundary value problem in the transformed field then involves generating the values of the physacal.cartesian coordinates x (s, n) and y (E;" n), in the transformed field from the specified boundary values of x I["n) and y ([" n) on the retangular boundary of the transformed field, the boundary being formed of segments of constant S or n, i.e vertical or horizontal lines. Within = constant on a boundary segment, and the increments in S taken to be uniformly unity, this boundary value specification is implemented numerically by distributing the points as desired along the boundary segment and then assigning the values of the cartesian coordinates of each successive point as boundary values at the equally spaced boundary points on the botton (or top) of the transformed field.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE California Regional Meeting, March 27–29, 1985

Paper Number: SPE-13626-MS

... SPE Member Abstract A general theory of Green's functions for solving

**boundary**-initial**value****problems**governed by two linear pressure diffusion equations coupled through a fluid transfer term which depends on the pressure histories, is presented. The governing pressure histories, is presented...
Abstract

SPE Member Abstract A general theory of Green's functions for solving boundary-initial value problems governed by two linear pressure diffusion equations coupled through a fluid transfer term which depends on the pressure histories, is presented. The governing pressure histories, is presented. The governing equations, developed previously by Wijesinghe and Culham [1984], describe the flow of fluid in naturally fractured reservoirs with arbitrary fracture connectivity simultaneously taking into account both fracture and pore system permeabilities and unsteady interporosity fluid permeabilities and unsteady interporosity fluid transfer. In this study, the relationship between the Green's functions (ie. generalized source solutions) and the adjoint Green's functions is determined, and the boundary conditions which must be imposed on the adjoint Green's functions to obtain explicit solutions for the fracture and pore pressures, are presented. The fundamental Green's pressures, are presented. The fundamental Green's functions, which characterize the basic properties of the two coupled differential equations, are derived and examples of their application to reservoir fluid flow problems are given. As in the case of the classical diffusion equation for fluid flow in single porosity formations, the Green's, functions based on the present theory enable solutions to complex fluid flow problems in naturally fractured reservoirs to be systematically constructed from simpler source solutions. I. Introduction The Green's function method is a powerful and elegant technique for solving boundary value problems governed by linear partial differential problems governed by linear partial differential equations. It has been extensively utilized in diverse branches of physics and engineering and is well known in classical potential theory, in diffusional transport theory, in elastodynamic wave propagation and in theories of integral and partial differential equations for investigating the existence and uniqueness Of solutions. Although it has had a rich and varied history since its first introduction by G. Green in 1828, it has been viewed more as an abstract mathematical device than as a practical tool for solving routine problems. The Green's function method is not only a powerful analytical tool for studying the mathematical characteristics of the solutions to boundary value problems, but also is a means of constructing problems, but also is a means of constructing solutions to difficult problems from simpler, more basic, solutions. Furthermore, it provides a systematic procedure for reducing complex boundary value problems to integral representations which are much more amenable to numerical solution than the corresponding differential equations and associated boundary conditions. The development of boundary element techniques based on boundary integral representations using fundamental Green's functions and the advent of powerful digital computers and efficient matrix equation solution algorithms has brought the method within the realm of routine application to practical problems. These developments have resulted in problems. These developments have resulted in renewed interest in this classical technique. The application of the Green's function method to petroleum reservoir engineering problems was rare prior to the publication of the landmark paper by Gringarten and Ramey. They introduced to the petroleum industry the basic theory underlying the Green's function method, and the method of constructing solutions to complex problems from simpler solutions through problems from simpler solutions through superposition of solutions and the use of Newman's product method. Also, they presented a product method. Also, they presented a comprehensive list of Green's functions and source solution. for a wide variety of reservoir geometries and boundary conditions. Since then, many papers employing source solutions have been published on single-well and interference pressure published on single-well and interference pressure tests with different boundary conditions and reservoir configurations. P. 363

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, October 18–20, 2004

Paper Number: SPE-88461-MS

...) hyperbolic system of conservation laws, where n is the number of components in the displacing phase. Continuous polymer injection results in a Riemann

**problem**for this hyperbolic system. The displacement of oil by a polymer slug with water drive is described by an initial and**boundary****value****problem**...
Abstract

Abstract Compositional models with adsorption describe polymer/surfactant flooding with varying salinity. It is shown that the concentration "part" of the solution for 1-D multicomponent polymer/surfactant flooding is completely defined by adsorption isotherms and is independent of relative permeabilities and phase viscosities. We introduce potential coordinates allowing for the compositional model splitting into a thermodynamics auxiliary system and one transport equation. The number of auxiliary equations is less than the number of equations of the compositional model by one. Explicit projection and lifting procedures are derived. The splitting is valid for either self-similar continuous injection problems or non-self-similar slug injection problems. With respect to 3-D flows, splitting takes place only for the case of constant total mobility. For the general case of the total mobility variation, mixing between fluids that enter different streamlines occurs, and splitting does not happen any more. Introduction Enhanced Oil Recovery (EOR) methods include injection of different fluids into reservoirs to improve oil displacement. The EOR methods may be classified into the following kinds: chemical methods, solvents methods and thermal methods. The chemical fluids most commonly injected are polymers, surfactants, micellar solutions, etc 1 . One-dimensional displacement of oil by an aqueous solution containing several chemicals species considering adsorption is described by an (n+1)×(n+1) hyperbolic system of conservation laws, where n is the number of components in the displacing phase. Continuous polymer injection results in a Riemann problem for this hyperbolic system. The displacement of oil by a polymer slug with water drive is described by an initial and boundary value problem with piecewise constant initial data and results in wave interactions 2 . The Riemann problem for the displacement of oil by hot water 3 is mathematically equivalent to one-component polymer flooding (n=1) for a convex sorption isotherm 4 . Several Riemann solutions for the case n=2 have already been found 1–6 and a graphical procedure for the solution of this problem was developed 5,6 . The Riemann solution for n-component polymer flooding was found for the case where the i-th adsorbed concentration depends only on the concentration of the i-th component in the aqueous phase 6,7 . Exact solutions for non-self-similar slug problems were also published 2,8 . The Riemann solution of the (n+1)×(n+1) system for two-phase n-component displacement was studied in several papers 9–12 for any arbitrary shape of sorption isotherms. The particular case of one-phase n-component flow leads to an (n)×(n) hyperbolic system (called the associated system), which was used for solving the Riemann problem. The projection and lifting procedures developed allow the calculation of any Riemann solution for the two-phase system once the associated one-phase solution is known. The theory developed is based on the fact that the Riemann problem solutions depend on a single parameter, x/t. Thus, the theory cannot be extended to non-self-similar Cauchy/initial-boundary value problems. A potential function associated with the conservation of aqueous phase and used as an independent variable instead of time was introduced 13 and the change of independent variables reduced the number of equations by one. The reduced (auxiliary) system contains only thermodynamic functions whereas the original system contains both thermodynamic functions and transport properties. It is shown that the "concentration part" of the solution of the initial-boundary value problem for the (n+1)×(n+1) system satisfies the auxiliary (n)×(n) system. The equations for the projection and lifting of elementary waves allow the construction of the solution from the auxiliary system solution. The exact solution for displacement of oil by polymer slug with water drive was obtained using the technique developed 13 .

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 1–4, 2000

Paper Number: SPE-62976-MS

... a tomographic method), the permeability can be directly obtained from the solution of a non-linear

**boundary**-**value****problem**. In this paper we extend this approach to the case when the porous medium is anisotropic. When the principal axes of anisotropy are known and fixed, a procedure is proposed, in which...
Abstract

Abstract In a recent publication 1 we proposed a direct method for the inversion of the permeability field of an isotropic porous medium based on the analysis of the displacement of a passive tracer. By monitoring the displacement front at successive time intervals (for example, using a tomographic method), the permeability can be directly obtained from the solution of a non-linear boundary-value problem. In this paper we extend this approach to the case when the porous medium is anisotropic. When the principal axes of anisotropy are known and fixed, a procedure is proposed, in which the tracer is injected two (or three) consecutive times along the two (or three) principal directions (for the case of a 2-D (or 3-D) problem, respectively). It is shown that the diagonal components can be obtained from the solution of two (or three) coupled boundary-value problems involving the experimentally obtainedfields of arrival times. Numerical examples show that the method works well when the permeability variation is not very sharp (for example, for correlated distributions). When thepermeability tensor is full and the principal axes vary in space, we propose a procedure involving the injection in three different directions (for the case of a 2-D problem). In principle, the components of the permeability tensor canbe determined from the solution of three coupled boundary-value problems. However, the inversion method encounters significant numerical problems. For the case of small off-diagonal components, a practical procedure is proposed to decouple the problems in the inversion method for both 2-D and 3-D. Introduction Permeability heterogeneity is an important feature of natural porous media, as it affects significantly flow and fluid displacement properties. Two main approaches exist for its identification, one based on pressure transients and another on production data. The first approach makes use of the diffusion equation, which governs transient single-phase flow, the second requires the solution of convective flow equations. The classical approach for identifying permeability is based on the inversion of pressure data from well tests 2,3 . Typically, this approach gives information on the average permeability value around the testing well. Spatial heterogeneity can be roughly estimated through interpolation among estimated permeabilities at various wells and the application of geostatistics. Promising methods for the estimation of key statistical properties, for the case of small fluctuations of the logarithm of the permeability, were suggested by Oliver 4,5 . Under the assumption of a stationary field, Yortsos and Al-Afaleg 6 proposed a multiple-well pressure transient method that leads to the estimation of the correlation function (semi-variogram) of the heterogeneity.

Proceedings Papers

Paper presented at the The Sixteenth International Offshore and Polar Engineering Conference, May 28–June 2, 2006

Paper Number: ISOPE-I-06-288

...) is employed to solve the

**boundary****value****problem**at each time step. The fourth-order predictor-corrector Adams-Bashforth- Moulton (ABM4) scheme is used for the time-stepping integration of instantaneous free surface**boundary**conditions. A damping layer near the end-wall of wave tank is added to absorb...
Abstract

ABSTRACT An approach for two-dimensional (2D) numerical wave tank (NWT) in the time domain is developed within the frame of potential flow, in which the fully nonlinear (or exact) free surface boundary conditions are satisfied. In this approach, the desingularized boundary element method (BEM) is employed to solve the boundary value problem at each time step. The fourth-order predictor-corrector Adams-Bashforth- Moulton (ABM4) scheme is used for the time-stepping integration of instantaneous free surface boundary conditions. A damping layer near the end-wall of wave tank is added to absorb the outgoing waves. The saw-tooth instability is overcome via a five-point Chebyshev smoothing scheme. The present model is applied to study the diffraction problem of both regular and irregular incident waves by a horizontal cylinder in detail. INTRODUCTION The concept of Numerical Wave Tank (NWT) has received considerable attention in the past decades, whose objective is to reproduce physical wave basins as closely as possible. The Numerical Wave Tank Group (NWTG) of the International Society of Offshore and Polar Engineering (ISOPE) was initiated by Prof. C.H. Kim at the 5th ISOPE conference in Hague (1995). Thanks to the great contributions by many participating researchers and workshops on NWT, some progress on NWT has been made in the past decades. Kim et al. (1999) and Tanizawa (2000) presented comprehensive reviews of NWTs. One of the most popular and successful approaches in the simulation of fully nonlinear NWT is the mixed Eulerian-Lagrangian (MEL) formulation, which was originally developed by Longuet-Higgins and Cokelet (1976). The first is to solve the mixed boundary value problem for the velocity potential in an Eulerian frame. Next, the time-stepping integration for the free surface boundary conditions is performed in a Lagrangian manner in order to track the instantaneous free surface.

Journal Articles

*Int. J. Offshore Polar Eng.*29 (01): 42–52.

Paper Number: ISOPE-19-29-1-042

Published: 01 March 2019

... is first adopted to eliminate the higher-order numerical oscillation phenomena and noisy dissipation in the

**boundary****value****problem**. Then, the weighting factors of initial and**boundary****value****problems**are introduced into the linear system to prevent the elevation from vanishing without iterative...
Abstract

This paper presents an equal-norm multiple-scale Trefftz method (MSTM) associated with the group-preserving schemes (GPS) to tackle some difficulties in nonlinear sloshing behaviors. The MSTM combined with the vector regularization method is first adopted to eliminate the higher-order numerical oscillation phenomena and noisy dissipation in the boundary value problem. Then, the weighting factors of initial and boundary value problems are introduced into the linear system to prevent the elevation from vanishing without iterative computational controlled volume. More important, the explicit scheme, based on the GL (n, R), and the implicit scheme can be combined to reduce iteration number and increase computational efficiency. A comparison of the results shows that the proposed approach is better than previously reported methods. Introduction Sloshing of liquid in tanks has received considerable attention from many researchers in related engineering fields. The problem arises because excessive sloshing of the confined liquid can strongly damage the structure or the loads induced by sloshing, which may seriously modify the dynamics of the vehicle that supports the tanks—for example, fuel sloshing in liquid propellant launch vehicles (Lu et al., 2015), oil oscillations in large storage tanks as a result of long-period strong ground motions (Hashimoto et al., 2017), and sloshing in nuclear fuel pools owing to earthquakes (Eswaran and Reddy, 2016). Besides, sloshing effects in the ballast tanks of a ship may cause it to experience large rolling moments and eventually capsize because of loss of dynamic stability (Krata, 2013; Sanapala et al., 2018). Also, if the forcing frequency coincides with the natural sloshing frequency, the high dynamic pressures, by reason of resonance, may damage the tank walls. Thus, accurate prediction of sloshing behaviors in tanks driven by external forces is very critical for successful structural design and reducing impacts on vehicle maneuvering.

Proceedings Papers

Paper presented at the The Twenty-first International Offshore and Polar Engineering Conference, June 19–24, 2011

Paper Number: ISOPE-I-11-114

... equation. The flow

**problem**can be formulated as an initial-**boundary****value****problem**. At each time step, the**boundary****value****problem**for the velocity potential is solved using a desingularized**boundary**integral method. A timestepping approach is used, in which the kinematic and dynamic**boundary**conditions...
Abstract

ABSTRACT: This paper describes a simple, fast and effective numerical tool that can be employed to predict the onset of liquid sloshing in tanks and assess the severity of the hydrodynamic pressures applied on the tank walls. A potential flow model with linear boundary conditions was chosen to give an initial screening tool that provides reasonable accuracy and computational speed. The liquid in the tank is assumed incompressible and inviscid, and the flow is assumed irrotational. The flow can be then described by a velocity potential Ф( t , x , y , z ) which is governed by the Laplace equation. The flow problem can be formulated as an initial-boundary value problem. At each time step, the boundary value problem for the velocity potential is solved using a desingularized boundary integral method. A timestepping approach is used, in which the kinematic and dynamic boundary conditions on the liquid free surface are integrated in time to update the surface elevation and the velocity potential for the next time instant. Three-dimensional problems are studied. INTRODUCTION Sloshing of liquid cargo in a partially-filled tank can cause large pressures on the tank walls. In the case of wave periods close to the liquid natural periods, sloshing can become violent and may lead to structural failures. The underlying physics of the problem are very complex and not fully understood. A recent paper by Yung et al (2010) describes all the relevant parameters of the problem as well as the learning from an extensive model testing campaign. Advanced CFD methods (such as volume of fluid methods, level-set methods and SPH methods) have been proposed and developed in recent years for simulating liquid sloshing and providing better prediction of the phenomena (Yang and Lohner 2005, Lohner et al 2007, Pakozdi and Graczyk 2009, and Cao et al 2010).

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