Abstract. We propose streamline-based simulation as a possible alternative to particle tracking for modeling solute transport. Like particle tracking, the pressure field is computed on an underlying grid using conventional techniques. The flow velocity at cell edges is then computed using Darcy's law, and this information is used to trace streamlines throughout the domain. In particle tracking, mass is transported by moving particles along streamlines. In the method we describe, a one-dimensional conservation equation is solved numerically along each streamline. If the flow field changes, the solute concentration is mapped onto the underlying grid and the streamlines are recomputed. The concentration is then mapped back onto the new streamlines, and the simulation proceeds as before. The method is suited for modeling advectively dominated multispecies transport in heterogeneous aquifers. We illustrate the streamline approach with synthetic example problems in fully saturated, confined aquifers: conservative, sorbing, and decaying tracer; a four-component radioactive decay chain; and saltwater intrusion where the flow field changes with time. Where possible, we compare the results with analytical solutions and results from particle tracking codes. It is natural to consider the use of streamline-based methods to model solute transport. Streamline methods have many similarities with particle tracking. In both cases the flow field is computed on a underlying grid using conventional numerical techniques, and contaminant mass is moved along streamlines. In particle tracking this mass is moved explicitly as a collection of particles, whereas in a streamline method the appropriate one-dimensional conservation equation is solved numerically along each streamline. Unlike conventional finite difference or In this paper we extend the streamline method to study a variety of contaminant transport problems. Where possible, the results are compared with other numerical codes and analytical solutions. First, to introduce this work, we provide a discussion of traditional approaches to model solute transport. Second, we describe the streamline method. Then, we apply this method to a variety of simple example problems in fully saturated confined aquifers: conservative, sorbing, and decaying tracer flow; a four component radioactive decay chain; and saltwater intrusion where the flow field changes with time. In conclusion, we propose streamline-based simulation as a possible alternative to particle tracking. Numerical Methods for Modeling Solute TransportTransport models can be categorized as Eulerian, Lagrangian, or mixed Eulerian-Lagrangian. In the Eulerian approach the transport equation is solved on a fixed spatial grid, so that concentrations are associated with fixed points or volume elements in space [Bear, 1972]. Both finite difference and finite element methods are Eulerian. These methods handle dispersion-dominated transport accurately and efficiently, but in advection-dominated transport problems, Eulerian methods suffer ...

A new 3-D three phase compositional reservoir simulator based on extension of the streamline method has been developed. This paper will focus on the new methods developed for compositional streamline simulation as well as the advantages and disadvantages of this strategy compared with more traditional approaches. Comparisons with a commercially available finite difference simulator will both validate the method and illustrate the cases in which this method is useful to the reservoir engineer. Introduction Streamline methods have been used as a tool for numerical approximation of the mathematical model for fluid flow since the 1800's (Helmholtz28 and later Muskat45) and have been applied in reservoir engineering since the 1950's and 1960's19,29–31,21,57. The reason behind using the approach has been both the needs for solving the governing equations accurately and achieving reasonable computational efficiency. Streamline methods continued to be explored through the 1970's by LeBlanc and Caudle37, Martin et al40,41 and Pitts et al50 and 1980's by Lake et al36,70, Cox11, Bratvedt et al8 and Wingard et al71, but the focus of reservoir simulation was on developing finite difference simulators. In the 1990's streamline methods have emerged as an alternative to finite difference simulation for large, heterogeneous models that are difficult for traditional simulators to model adequately. These efforts are described in numerous papers notably by Renard56, Batycky et al1–4, Peddibhotla et al48,49, Thiele et al64–66, Ingebrigtsen33, Ponting52 and in an overview paper by King and Datta-Gupta35. The application of the method has been described by numerous other authors10,12–17,26,34,53–55,69. Use of the streamline simulator used for the work in this paper has also been extensively described24,39,42,47,58,61,62,68. Several similar approaches such as the method of characteristics18,27,38,43,44, particle tracking20,63 and front-tracking25,7 have also been used in reservoir simulation. Conventional finite difference methods suffer from two drawbacks, numerical smearing and loss of computational efficiency for models with a large numbers of grid cells. Large models (105 –106 cells) are routinely generated in order to accurately represent geologically heterogeneous, multi-well problems.. Finite difference methods based on an IMPES approach suffer from the time step length limiting CFL condition, so as the number of cells increases the maximum time step length get shorter for a given model. For a large number of cells the shortness of the time step can render the total CPU time for a simulation impractical. Fully implicit finite difference simulators can take longer time steps but require the inversion of a much larger matrix than the IMPES approach. This is an even larger issue with compositional simulation where a large number of components will make the matrix very large. Also the non-linearity of the governing equations might require a limitation on the time step length again making very large models impractical to run. This can be improved by using an adaptive implicit aproach73. Conventional streamline methods are based on an IMPES method. In these methods the pressure is solved implicitly and then streamlines are computed based on this pressure solution. In this way the 3D domain is decomposed into many one-dimensional streamlines along which fluid flow calculations are done. This method assumes that the pressure is constant throughout the movement of fluids. One weakness with this concept is the lack of connection between the changing pressure field and the movement of fluids. This can cause instabilities and limitations on the time step length.

Because of increased speed and accuracy, 3D streamline simulation now can be used in a wide range of reservoirs. Large reservoirs with hundreds or thousands of wells, several hundred thousand grid blocks and an extensive production history always have been a challenge for finite-difference simulation. The size and complexity of these reservoirs generally have limited simulation to sections or patterns. The streamline technique enables simulation of these reservoirs by reducing the 3D domain to a series of 1D streamlines along which the fluid flow computations are performed, offering computational benefits orders of magnitude greater. Additionally, increased accuracy is achieved by maintaining the sharp flood fronts from the displacement processes and reducing grid orientation effects. The streamline simulation results substantially have more value as a reservoir management tool when used in conjunction with traditional reservoir engineering techniques such as standard finite-difference simulators. Several case studies that highlight a variety of situations where streamline methods proved highly beneficial are presented in this paper. These studies will help the practicing reservoir engineer decide whether to apply streamline methods and the optimal timing for the application. Introduction Streamline and streamtube methods have been used in fluid flow computations for many years31,48. Early applications for hydrocarbon reservoir simulation were reported by Fay and Pratts22 in the 1950's, Higgins et al32–34 in the early 1960's, and Pitts and Crawford53, LeBlanc and Caudle41 and Martin and Wegner et al43,44 in the 1970's. Streamline/streamtube methods numerically solve the complex fluid flow models for multiphase flow in porous media with a reasonable balance between the computational efficiency and the physics modeled. As computers became more powerful, attention turned toward developing simulators based on finite-difference (FD) methods60, including more physical effects. However, computer models of reservoirs have grown in complexity and geological models with tens of millions of grid cells can now be created. Conventional finite-difference methods suffer from two drawbacks, numerical smearing and computational efficiency for models with a large numbers of grid cells. For models with a large number of wells, the number of cells required to achieve acceptable accuracy between wells can be prohibitive. Also, accurate modeling of geological heterogeneities can require a very large number of cells. A finite-difference method based on an IMPES approach suffers from the time-step length limiting Courant-Friedrichs-Lewy (CFL) condition. As the number of cells grows higher, the maximum time-step length gets shorter for a given model. For a very large number of cells, the shortness of the time step can render the total CPU time impractical for a simulation. The advantage of the fully implicit approach is stability of the solutions. The time-step length is only limited by the nonlinearities; however, these can be strong and, in practice, keep the time-step length relatively short. A disadvantage of the fully implicit approach is the tendency to smear the solution (numerical dispersion) even more than the IMPES approach.

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