trust region based solver for nonlinear transport in heterogeneous porous media we describe a new nonlinear solver for immiscible two phase transport in porous media where viscous buoyancy and capillary forces are significant the flux fractional flow function f is a nonlinear function of saturation and typically has inflection points and can be non monotonic the non convexity and non monotonicity of f are major sources of difficulty for nonlinear solvers of coupled multiphase flow and transport in natural porous media we describe a modified newton algorithm that employs trust regions of the flux function to guide the newton iterations the flux function is divided into saturation trust regions delineated by the inflection unit flux and end points the updates are performed such that two successive iterations cannot cross any trust region boundary if a crossing is detected the saturation value is chopped back to the appropriate trust region boundary the proposed trust region newton solver which is demonstrated across the parameter space of viscous buoyancy and capillary effects is a significant extension of the inflection point strategy of jenny et al jcp 2009 5 for viscous dominated flows we analyze the discrete nonlinear transport equation obtained using finite volume discretization with phase based upstream weighting then we prove convergence of the trust region newton method irrespective of the timestep size for single cell problems numerical results across the full range of the parameter space of viscous gravity and capillary forces indicate that our trust region scheme is unconditionally convergent for 1d transport that is for a given choice of timestep size the unique discrete solution is found independently of the initial guess for problems dominated by buoyancy and capillarity the trust region newton solver overcomes the often severe limits on timestep size associated with existing methods to validate the effectiveness of the new nonlinear solver for large reservoir models with strong heterogeneity we compare it with state of the art nonlinear solvers for two phase flow and transport using the spe 10 model compared with existing nonlinear solvers our trust region solver results in superior convergence performance and achieves reduction in the total newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost

a finite volume method for a two phase multicomponent polymer flooding multicomponent polymer flooding used in enhanced oil recovery is governed by a system of coupled non strictly hyperbolic conservation laws in the presence of gravity the flux functions need not be monotone and hence designing godunov type upwind schemes is difficult and computationally expensive to overcome this difficulty we use the basic idea of discontinuous flux to reduce the coupled system into an uncoupled system of scalar conservation laws with discontinuous coefficients for these scalar equations we use the dflu flux developed in 5 to construct a second order scheme the scheme is shown to satisfy a maximum principle and the performance of the scheme is shown on both one and two dimensional test problems

implicit hybrid upwinding for two phase flow in heterogeneous porous media with buoyancy and capillarity abstract we consider the numerical solution of the partial differential equations governing multiphase flow in porous media for highly nonlinear problems the temporal discretization of choice is often the unconditionally stable fully implicit method however the nonlinear systems often solved with newton s method are difficult to solve thus the computational cost is strongly dependent on the nonlinear convergence rate and enhancing this convergence property is key to speed up subsurface flow simulation we focus on the case of spatially discontinuous capillary pressure between rock regions to efficiently and accurately simulate the flow dynamics in heterogeneous porous media the flux computation combines implicit hybrid upwinding with transmission conditions between different rock regions this leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence we extend our previous results hamon et al 2016 18 by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions the generalized scheme is supported by an analysis of its mathematical properties our multidimensional numerical examples which range from buoyancy driven flow with capillary barriers to viscous dominated flow demonstrate that the implicit hybrid upwinding scheme improves the accuracy compared to the standard phase based upwinding scheme while leading to significant reductions in the number of nonlinear iterations in multiple dimensions

on a multicommodity stochastic flow network with unreliable nodes subject to budget constraint from the quality management and decision making view point reliability and unreliability are important indices to measure the quality level for a stochastic flow network in a multicommodity stochastic flow network with unreliable nodes the branches and nodes all have several possible capacities and may fail different types of the commodity which are transmitted through the same network simultaneously compete the capacities of branches and nodes in this paper we first define the system capacity as a vector for a multicommodity stochastic flow network with unreliable nodes then we design a performance index which is the probability that the upper bound of the system capacity is a given pattern subject to the budget constraint it can be applied to evaluate the quality level for such a network a simple approach based on minimal cuts is thus presented to evaluate the performance index

canonical forms and parameter identification problems in perspective systems we introduce canonical forms for perspective dynamical systems under the action of a perspective group and illustrate their application to parameter identification with the aid of a single charged coupled device camera we show that the parameters in the canonical form can be identified uniquely using an extended kalman filter

numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure abstract in this paper we propose and analyze the convergence of a tpfa two points flux approximation finite volume scheme to approximate the two incompressible phase flow with dynamic capillary pressure in porous media the fully implicit scheme is based on nonstandard approximation on mobilities and capillary pressure on the dual mesh we derive a discrete variational formulation and we present a new result of convergence in a two or three dimensional porous medium in comparison with static capillary pressure the non equilibrium capillary model requires more powerful techniques especially the discrete energy estimates are not standard

calculation of minimal capacity vectors through k minimal paths under budget and time constraints abstract reducing the transmission time is an important issue for a flow network to transmit a given amount of data from the source to the sink the quickest path problem thus arises to find a single path with minimum transmission time more specifically the capacity of each arc is assumed to be deterministic however in many real life networks such as computer networks and telecommunication networks the capacity of each arc is stochastic due to failure maintenance etc hence the minimum transmission time is not a fixed number such a network is named a stochastic flow network in order to reduce the transmission time the network allows the data to be transmitted through k minimal paths simultaneously including the cost attribute this paper evaluates the probability that d units of data can be transmitted under both time threshold t and budget b such a probability is called the system reliability an efficient algorithm is proposed to generate all of lower boundary points for d t b the minimal capacity vectors satisfying the demand time and budget requirements the system reliability can then be computed in terms of such points moreover the optimal combination of k minimal paths with highest system reliability can be obtained

vertex approximate gradient discretization preserving positivity for two phase darcy flows in heterogeneous porous media abstract in this article a new nodal discretization is proposed for two phase darcy flows in heterogeneous porous media the scheme combines the vertex approximate gradient vag scheme for the approximation of the gradient fluxes with an hybrid upwind hu approximation of the mobility terms in the saturation equation the discretization in space incorporates naturally nodal interface degrees of freedom d o f allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves it is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3d large discrete fracture matrix dfm models

a posteriori error estimates stopping criteria and adaptivity for multiphase compositional darcy flows in porous media in this paper we derive a posteriori error estimates for the compositional model of multiphase darcy flow in porous media consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations we show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators we then decompose the estimators into the space time linearization and algebraic error components this allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error moreover the spatial and temporal error components can be balanced by time step and space mesh adaptation our analysis applies to a broad class of standard numerical methods and is independent of the linearization and of the iterative algebraic solvers employed we exemplify it for the two point finite volume method with fully implicit euler time stepping the newton linearization and the gmres algebraic solver numerical results on two real life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria already on fixed meshes without any noticeable loss of precision

a combined finite volume nonconforming finite element scheme for compressible two phase flow in porous media we propose and analyze a combined finite volume nonconforming finite element scheme on general meshes to simulate the two compressible phase flow in porous media the diffusion term which can be anisotropic and heterogeneous is discretized by piecewise linear nonconforming triangular finite elements the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh the relative permeability of each phase is decentred according the sign of the velocity at the dual interface this technique also ensures the validity of the discrete maximum principle for the saturation under a non restrictive shape regularity of the space mesh and the positiveness of all transmissibilities next a priori estimates on the pressures and a function of the saturation that denote capillary terms are established these stabilities results lead to some compactness arguments based on the use of the kolmogorov compactness theorem and allow us to derive the convergence of a subsequence of the sequence of approximate solutions to a weak solution of the continuous equations provided the mesh size tends to zero the proof is given for the complete system when the density of the each phase depends on its own pressure

a realization theory for perspective systems with applications to parameter estimation problems in machine vision in this paper we introduce and study a linear dynamical system with a perspective observation function the study of these systems has been shown to be useful in motion and shape estimation problems in machine vision we introduce the notion of perspective observability and obtain under a special case a necessary and sufficient condition that would guarantee observability of the initial condition of the linear dynamical system up to a one parameter magnitude scaling subsequently a new realization theory is introduced which is useful for studying linear systems with perspective observation our main result is to show that parameters can be recovered up to orbits of a suitable perspective group a new rescaling algorithm is described to identify parameters up to orbits of the perspective system the identification problem is further analyzed in detail for various problems that are of interest in machine vision

limit boundary conditions for finite volume approximations of some physical problems in the industrial context finite volume schemes are used to compute an approximation of the solution of a system of equations set on a certain domain when this domain is bounded some numerical boundary conditions have to be implemented in order to complete the computation of the finite volume scheme this is a tricky step in the elaboration of the scheme which is still not mastered in fact at a closer sight it appears that there is a deep interaction between the understanding of the physical phenomena at the boundary of the domain and the implementation of the numerical boundary conditions unfortunately this link is not always completely intelligible and a reason for this lack of clarity is the fact that whereas the the continuous equation satisfied by the limit of the numerical solution is known the boundary conditions satisfied by this very limit are not well understood the purpose of this paper is to clarify this point in three industrial situations of one dimensional two phase flows

nonlinear analysis of multiphase transport in porous media in the presence of viscous buoyancy and capillary forces nonlinear convergence problems in numerical reservoir simulation can lead to unacceptably large computational time and are often the main impediment to performing simulation studies of large scale problems we analyze the nonlinearity of the discrete transport mass conservation equation for immiscible incompressible two phase flow in porous media in the presence of viscous buoyancy and capillary forces although simulation problems are multi dimensional with large numbers of cells and variables we find that the essence of the nonlinear behavior can be understood by studying the discretized numerical flux function for the interface between two cells the numerical flux is expressed in terms of the saturations of the two cells discontinuities in the first order derivative of the flux function referred to as kinks and inflection lines are identified as the cause of convergence difficulty these critical features kinks and inflections change the curvature of the numerical flux function abruptly and can lead to overshoots oscillations or divergence in newton iterations based on our understanding of the nonlinearity a nonlinear solver is developed referred to as the numerical trust region ntr solver the solver is able to guide the newton iterations safely and efficiently through the different saturation trust regions delineated by the kinks and inflections specifically overshoots and oscillations that often lead to convergence failure are avoided numerical examples demonstrate that our ntr solver has superior convergence performance compared with existing methods in particular convergence is achieved for a wide range of timestep sizes and courant friedrichs lewy cfl numbers spanning several orders of magnitude in addition a discretization scheme is proposed for handling heterogeneities in capillary pressure saturation relationship the scheme has less degree of nonlinearity compared with the standard single point phase based upstream weighting scheme leading to an improved nonlinear convergence performance especially when used together with our ntr solver our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by jenny et al 2009 6 and wang and tchelepi 2013 7 significantly the nonlinearity of multiphase transport in porous media is analyzed the analysis considers viscous buoyancy and capillary forces the critical features that cause nonlinear convergence failures are identified a nonlinear solver that converges without oscillations and divergence is developed the solver converges faster than existing methods and allows large timesteps

internet packet routing application of a k quickest path algorithm this paper describes a study on the application of an algorithm to rank the k quickest paths to the routing of data packets in internet networks for this purpose an experimental framework was developed by considering two types of random generated networks to simulate values of the ip packet sizes a truncated pareto distribution was defined having in mind to reflect a key feature of internet traffic namely its self similar stochastic nature results concerning the average cpu times of the algorithm for the different sets of experiments will be presented and discussed

a perspective theory for motion and shape estimation in machine vision in this paper we consider the problem of motion and shape estimation of a moving body with the aid of a monocular camera we show that the estimation problem reduces to a specific parameter estimation of a perspective dynamical system surprisingly the above reduction is independent of whether the data measured is the brightness pattern which the object produces on the image plane or whether the data observed are points lines or curves on the image plane produced as a result of discontinuities in the brightness pattern many cases of the perspective parameter estimation problem have been analyzed in this paper these cases include a fairly complete analysis of a planar textured surface undergoing a rigid flow and an affine flow these two cases have been analyzed for orthographic pseudo orthographic and image centered projections the basic procedure introduced for parameter estimation is to subdivide the problem into two modules one for spatial averaging and the other for time averaging the estimation procedure is carried out with the aid of a new realization theory for perspective systems introduced for systems described in discrete time and in continuous time finally much of our analysis has been substantiated by computer simulation of specific algorithms developed in order to explicitly compute the parameters detailed simulation that would answer the perspective realizability question is a subject of future research

the dflu flux for systems of conservation laws the dflu numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space we show how this flux can be used to solve a certain class of systems of conservation laws such as systems modeling polymer flooding in oil reservoir engineering furthermore these results are extended to the case where the flux function is discontinuous in the space variable such a situation arises for example while dealing with oil reservoirs which are heterogeneous numerical experiments are presented to illustrate the efficiency of this new scheme compared to other standard schemes like upstream mobility lax friedrichs and force schemes

upstream differencing for multiphase flow in reservoir simulation upstream weighting for multiphase flow in reservoir simulation is analyzed the associated numerical flux is shown to be well defined monotone lipschitz continuous and consistent in the case of a two phase flow the corresponding numerical flux is compared to that of godunov and engquist and osher finally simple way to obtain a higher order scheme is outlined

homogeneous dynamical systems theory we study controlled homogeneous dynamical systems and derive conditions under which the system is perspective controllable we also derive conditions under which the system is observable in the presence of a control over the complex base field in the absence of any control input we derive a necessary and sufficient condition for observability of a homogeneous dynamical system over the real base field the observability criterion obtained generalizes a well known popov belevitch hautus rank criterion to check the observability of a linear dynamical system finally we introduce rational exponential interpolation problems as an important step toward solving the problem of realizing homogeneous dynamical systems with minimum state dimensions

a multiscale restriction smoothed basis method for high contrast porous media represented on unstructured grids a wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of poisson type equations modeling flow in porous media we present a new multiscale restricted smoothed basis msrsb method that is designed to be applicable to both rectilinear grids and unstructured grids like many other multiscale methods msrsb relies on a coarse partition of the underlying fine grid and a set of local prolongation operators multiscale basis functions that map unknowns associated with the fine grid cells to unknowns associated with blocks in the coarse partition these mappings are constructed by restricted smoothing starting from a constant a localized iterative scheme is applied directly to the fine scale discretization to compute prolongation operators that are consistent with the local properties of the differential operators the resulting method has three main advantages first of all both the coarse and the fine grid can have general polyhedral geometry and unstructured topology this means that partitions and good prolongation operators can easily be constructed for complex models involving high media contrasts and unstructured cell connections introduced by faults pinch outs erosion local grid refinement etc in particular the coarse partition can be adapted to geological or flow field properties represented on cells or faces to improve accuracy secondly the method is accurate and robust when compared to existing multiscale methods and does not need expensive recomputation of local basis functions to account for transient behavior dynamic mobility changes are incorporated by continuing to iterate a few extra steps on existing basis functions this way the cost of updating the prolongation operators becomes proportional to the amount of change in fluid mobility and one reduces the need for expensive tolerance based updates finally since the msrsb method is formulated on top of a cell centered conservative finite volume method it is applicable to any flow model in which one can isolate a pressure equation herein we only discuss single and two phase incompressible models compressible flow e g as modeled by the black oil equations is discussed in a separate paper

convergence of implicit monotone schemes with applications in multiphase flow in porous media phase based upstreaming which is a commonly used spatial discretization for multiphase flow in reservoir simulation naturally gives rise to implicit monotone schemes when implicit time stepping is used the nonlinear gauss seidel and jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations thus for one dimensional problems we obtain an alternate constructive proof that such schemes are well defined and converge to the entropy solution of the conservation law for arbitrary cfl numbers the accuracy of phase based upstream solutions is studied for various spatial and temporal grids revealing the importance of unconditional stability when nonuniform grids and or variable porosity is involved

implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy abstract numerical simulation of coupled multiphase flow and transport in porous media is used in many scientific and industrial applications such as groundwater management and oil recovery to solve the governing partial differential equations we focus on the fully implicit backward euler finite volume method fim in this numerical scheme large algebraic systems must be solved at each time step using newton s method this can be quite expensive especially for highly nonlinear problems with buoyancy in this work we present a new first order approximation of the flux in the fim leading to improved nonlinear convergence rate and radius this approximation is based on implicit hybrid upwinding and relies on separate treatments of the viscous and buoyancy parts of the numerical flux moreover in the viscous part the total velocity discretization is key as it controls the coupling between the elliptic flow problem and the hyperbolic multiphase transport problem to achieve robustness and efficiency we propose a differentiable total velocity discretization that adapts to the balance of forces at each interface we analyze the theoretical properties of the resulting numerical scheme and prove that saturations remain between physical bounds challenging two and three phase test cases illustrate that our numerical scheme brings a significant reduction in the number of nonlinear iterations compared to a finite volume discretization relying on the widely used phase per phase upstream weighting our numerical scheme therefore reduces the simulation cost while providing similar accuracy