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ECMOR X  10th European Conference on the Mathematics of Oil Recovery
 Conference date: 04 Sep 2006  07 Sep 2006
 Location: Amsterdam, Netherlands
 ISBN: 9789073781474
 Published: 04 September 2006
41  60 of 78 results


A Sequential Splitting Strategy for CO2 Storage Modelling
Authors L. Trenty, A. Michel, E. Tillier and Y. Le GalloResearch and development methodologies for the storage of CO2 in geological formation are in developing over the last 10 years. In this context, numerical simulators are the practical tools to understand the physical processes involved by acid gas injection and evaluate the long term stability of the storage.
CO2 storage models can be seen as a mix between two types of models: a reservoir model coupling multiphase flow in porous media with local phase equilibria and an hydrogeochemical model coupling transport in aqueous phase with local chemical equilibria and kinetic reaction laws.
In a recent paper, Nghiem et al. [1] proposed a fullycoupled method to solve this problem extending the local PVTunknowns elimination strategy to geochemical problems. This paper presents a different approach which combine the two models in a single computation using a sequential splitting method. The main advantage of this approach is the ability to choose a specific numerical scheme for each model.
From a mathematical point of view, this method can be seen as a two stage time integration scheme. At each stage we compute an approximate solution of the main problem using some predicted terms estimated from previous stages. This scheme is a sequential splitting strategy so it may induce some numerical errors relevant for the physical coherence of the model. To correct this error without using any iterative method, we propose to use a correction stage.
This method has been implemented in a research computer code. The code is applied to model a fieldscale CO2 storage in an heterogeneous saline aquifer.
[1] Long Nghiem, Peter Sammon, Jim Grabenstetter and Hiroshi Ohkuma, ”Modeling CO2 Storage in Aquifers with a FullyCoupled Geochemical EOS Compositional Simulator”, SPE 89474, April 004.



Unstructured Higher Resolution Convective Schemes for Flow in Porous Media
Authors S. Lamine and M. G. EdwardsNovel high resolution schemes for convective flow approximation are developed and coupled with general continuous fulltensor Darcy flux approximations. This development leads to new locally conservative formulations for multiphase flow in porous media.
The higher order schemes are designed for fluid transport on unstructured grids and are constructed using slope limiters such that they are stable with a maximum principle that ensures solutions are free of spurious oscillations. The schemes are developed to handle unstructured meshes with variable grid spacing and different formulations are compared.
Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The new formulation is compared with standard reservoir simulation schemes and controlvolume finite element methods. The new schemes are shown to effectively reduce numerical diffusion while resolving flow induced by rapid changes in permeability, leading to superior resolution of concentration and saturation fronts compared to other schemes.



Symmetric Positive Definite Subcell CVD Schemes for CellCentred Quadrilateral Grids
Authors M. G. Edwards and M. PalA new family of locally conservative subcell fluxcontinuous schemes is presented for cellcentred approximation of the generaltensor pressure equation on quadrilateral grids. The local position of flux continuity quadrature point defines the scheme. This work continues the development of symmetric positive definite subcell ControlVolume Distributed (CVD) schemes first introduced in [1, 2], where a piecewise constant general geometrypermeability tensor approximation is introduced over each subcell of a controlvolume. Physicalspace fluxcontinuous schemes possess nonsymmetric matrices for general quadrilateral cells, or indeed any general cell type. The subcell tensor approximation ensures that a fluxcontinuous finite volume scheme is obtained with a symmetric positive definite (SPD) discretization matrix on any grid, structured or unstructured [1, 2, 3].
By definition, tensor approximation at the subcell level leads to a finer scale tensor approximation compared to the cell level, and consequently can be expected to yield a superior SPD approximation to that of earlier cellcentred cellwise constant SPD tensor schemes. A numerical convergence study confirms that the SPD subcell tensor approximation reduces solution errors when compared to the SPD cellwise constant tensor schemes. In addition, constructing the subcell tensor approximation using controlvolume face geometry yields the best results consistent with design of the schemes. Comparisons are also made with the physical space schemes. A particular quadrature point is found to give the best numerical convergence of the subcell schemes for the cases tested.



Algebraic Multigrid Based Preconditionners for Oil Reservoir Simulation
Authors P. Bonneau, Y. Achdou, R. Masson and P. QuandalleAlgebraic MultiGrid (AMG) methods have become attractive methods when used as preconditioners for Krylov subspace linear solvers.
We present here two preconditioning strategies using AMG methods for the solving of large ill conditioned systems arising from the discretization and linearization of fluid mass conservation laws.
The first strategy is a "combinative" preconditioner combining an ILU(0) step on all types of unknowns(pressure, saturation) and a more specific step on pressure unknowns with an AMG method.
The second strategy is performed in a single step with a Block Aggregation AMG. The Block Aggregation AMG method has been developed to cope with matrices araising from the discretization and the linearization of PDE systems. It is also known for its low computational cost and its simple implementation in comparison to classical AMG. This simplicity has to be paid: Aggregation AMG is less efficient than classical AMG, which led us to enhance it with an ILU(0) smoother.
One difficulty of PDE systems is the coupling existing between the different types of unknowns. A widespread idea, in reservoir simulation, proposes to perform a "local decoupling" by scaling the original matrix of the system by the inverse of the block diagonal matrix built with the diagonal blocs of the original matrix.
We will see that the scaling/decoupling of the original system is not well suited for AMG based preconditioners, resulting in bad performances of the linear solver. As a consequence, we will abandon the scaling/decoupling of the original matrix when using an AMG preconditioner.
The two AMG based preconditioners and a simple ILU(0) preconditioner will be compared on "black oil" simulations on fields with highly heterogeneous permeabilities.



Multiscale Mortar Mixed Finite Elements for Multiphase Flow in Porous Media
Authors M. F. Wheeler, G. Pencheva, S. Thomas and I. YotovWe present a multiscale mortar mixed finite element method for multiphase flow in porous media. The method is based on a domain decomposition (or coarse grid). Mass balance equations in matching fine grids of scale h, while continuity of fluxes is imposed via mortar finite elements on a coarse scale H. Higher order mortar spaces on appropriately chosen coarse interface grids are used to provide optimal fine scale convergence. For example, for the lowest order RaviartThomas mixed method or cellcentered finite differences, a choice of H = O(sqrt h) and quadratic mortars gives O(h) convergence for both the pressure and the velocity. The nonlinear algebraic system in a fully implicit discretization is solved via a non overlapping domain decomposition algorithm, which reduces the global problem to an interface problem for one pressure and one saturation. Computational experiments for oilwater displacement in highly heterogeneous media illustrate the efficiency of the multiscale mortar formulation versus the singlescale
approach.



Control Volume Discretisation on NonMatching Meshes in 3D
Authors E. Øian, B. O. Heimsund, G. T. Eigestad and I. AavatsmarkComplex geometric and geological features of realistic reservoirs motivate the need for flexible and robust flux discretisation for forecasting of fluid flow in porous media. Geomodelling tools allow faults in the stratigraphy and local grid refinements which typically result in nonmatching meshes with hanging nodes. Flow simulations based on cell centred control volume discretisations can be performed when connectivities are established and transmissibilities are computed across the nonmatching interfaces. Ideally, the interface couplings should be in a form that allows the use of robust flux discretisation techniques. Existing approaches for such problems include generating prismatic ghost cells along the nonmatching interfaces or deriving specialised stencils for possible mesh configurations.
We study an alternative technique that deals with nonmatching meshes by combining a conversion to a topologically conforming mesh with a general Omethod multi point flux approximation. Hangingnodes are removed by adding them to the mesh interfaces, creating general polyhedral mesh cells. Then the general pointbased algorithm for setting up Omethod interaction regions is used.
Since, in three dimensions, this approach can lead to a mismatch between the number of degrees of freedom and the number of continuity conditions, the mesh must be created with care. Due to the advantages from both an implementation perspective and potential good quality in the solution, it is nevertheless worthwhile to examine the properties of this approach. We present guidelines to ensure that the resulting conforming mesh is consistent with the Omethod and also how to apply a reduced flux stencil for arbitrary polyhedral meshes.



Nonlinear TwoPoint Flux Approximations for Simulating Subsurface Flows with FullTensor Anisotropy
Authors B. T. Mallison, Y. Chen and L. J. DurlofskyFulltensor anisotropy effects are often encountered in subsurface flow due to either grid nonorthogonality or permeability anisotropy. A multipoint flux approximation (MPFA) is generally needed to accurately simulate flow for such systems, though the resulting discrete system is more complex and may be less robust than that resulting from a twopoint flux approximation (TPFA). In this paper, we present and apply a different approach, nonlinear twopoint flux approximation (NTPFA), for modeling systems with fulltensor effects and grid nonorthogonality. NTPFA incorporates global flow into the determination of the twopoint flux transmissibilities, which is analogous to the use of twopoint flux transmissibility in global and localglobal upscaling procedures. For a given flow scenario, the global MPFA solution can be used to compute the twopoint flux transmissibility, leading to a global NTPFA scheme. To avoid solving the full global MPFA system, we have also developed a localglobal NTPFA procedure, in which the global flow is approximated by local MPFA solutions iteratively coupled with a global TPFA solution. The NTPFA methods described here are applied to 2D parallelogram grids, heterogeneous fulltensor permeability fields, and a 3D multiblock model with grid nonorthogonality. Results from both NTPFA schemes are generally in close agreement with the MPFA solution and provide considerable improvement over the standard TPFA scheme.



Fast Sequential Implicit Porous Media Flow Simulations Using Multiscale Finite Elements and Reordering of Cells for Solution of Nonlinear Transport Equation
Authors J. E. Aarnes, S. Krogstad, K. A. Lie and J. R. NatvigIt is demonstrated previously in the literature that multiscale methods can used to provide accurate highresolution velocity fields at a low computational cost. However, to achieve enhanced accuracy in flow simulations compared with a standard approach, the multiscale method must be accompanied by a transport solver that can account for the finescale structures of the velocity fields. In this paper, we use the standard implicit singlepoint upwind (SPU) finitevolume method for computing transport. This method requires that a nonlinear system is solved at each timestep. However, if we assume (as in streamline methods) that capillary forces can be disregarded and that gravity can be treated by operator splitting, and reordering the cells in an optimal way, the nonlinear systems in each implicit advective step can be solved on a cellbycell (or blockbyblock) basis. This approach makes the standard SPU method at least as fast as a streamline method, even on geocellular models with multimillion cells, and alleviates many limitations that streamline methods have. In particular, the method is mass conservative, compressibility can be handled in a straightforward manner, and pressure can be updated frequently without severely influencing the computational efficiency. By combining this transport solver with a multiscale pressure solver, we obtain a very efficient solution method capable of direct simulation of geocellular models with multimillion cells within an acceptable timeframe on a single desktop computer.



A Compact MPFA Method with Improved Robustness
Authors I. Aavatsmark, G. T. Eigestad and J. M. NordbottenMPFA methods were introduced to solve control volume formulations on general grids. While these methods are general in the sense that they may be applied to any grid, their convergence properties vary.
An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper, conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations which lead to smaller flux stencils, are desirable for grids with high aspect ratio or severe skewness and for media with strong anisotropy or strong heterogeneity.
We introduce a new MPFA method for quadrilateral grids termed the Lmethod. The methodology is valid for general media. For homogeneous media and uniform grids, this method has fourpoint flux stencils and sevenpoint cell stencils in two dimensions. The reduced stencil appears as a consequence of adapting the method to the closest
neighboring cells.
We have tested the convergence and monotonicity properties for this method, and compared it with the Omethod. For moderate grids the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the Omethods is lost, while the Lmethod converges with a reduced convergence rate.
The Lmethod has a somewhat larger monotonicity range than the Omethods, but the dominant difference is that when monotonicity is lost, the Omethods may give large oscillations, while the oscillations with the Lmethod are small or absent.



A New Stability Criterion for the IMPSAT Formulation of Compositional Fluid Flow
Authors J. Haukås, I. Aavatsmark, M. Espedal and E. ReisoA new stability criterion for the IMPSAT formulation of compositional fluid flow has been developed. The new criterion may allow for significantly larger stable timesteps than the conventional IMPSAT stability criterion.
The IMPSAT formulation implies that interblock flow terms are evaluated with pressure and saturations from the current time level (implicitly), but with additional variables from the previous
time level (explicitly). The new criterion is associated with explicit treatment of variables that may be interpreted as complementary to volumes, referred to as isochoric variables. Analysis shows that the isochoric variables represent the part of the system that may change even though pressure and volumes are kept fixed. The interpretation leads to the identification of two mutually orthogonal subspaces of the computational space, a volume space and an isochoric space. The stability criterion is based on projections onto the isochoric space, and requires that the hyperbolic, isochoric part of the flow must not exceed the corresponding part of the mass present in a gridblock.
Whereas the conventional IMPSAT stability criterion puts restrictions on the entire flow of each component, the new criterion only limits the isochoric part of the flow. In many cases, larger timesteps are therefore allowed. The relative improvement may be in the range of 50150 %, as shown in numerical examples, and the predicted maximum stable timesteps are reasonably close to the observed maximum stable timesteps. However, for cases where there is little or no saturation change between the hydrocarbon phases, e.g., for retrograde gas condensate cases or single hydrocarbon phase cases, the improvement is insignificant.



Grid Optimization for Improved Monotonicity of MPFA Solutions on Unstructured Grids
Authors M. J. Mlacnik and L. J. DurlofskyUnstructured grids are useful for resolving key geological or flow features in reservoir simulation. Accurate finite volume discretization of the reservoir flow equations on such grids generally requires the use of multipoint flux approximation (MPFA). MPFA can be applied to heterogeneous, anisotropic systems on generally unstructured grids, though it may suffer from loss of monotonicity of the inverse of the resulting linear operator at moderate to high permeability anisotropy ratios. As a consequence, the resulting pressure solution can show errors in the form of spurious oscillations. In recent work, we developed new algorithms for 2D and 3D systems that address this loss of monotonicity from a grid optimization perspective. Given an underlying finescale heterogeneous permeability field, the approaches can be generalized (via iteration) to couple permeability upscaling with the unstructured grid optimization.
In our previous work, solutions of the singlephase incompressible pressure equation were considered for 2D and 3D models. In this paper we describe our grid optimization procedures, consider new example cases, and apply the approach to a twophase flow problem. We demonstrate that the overall procedure is capable of providing accurate solutions (for both singlephase and twophase flows) that are free of spurious pressure oscillations. An upscaling example demonstrates results in reasonably close agreement with the corresponding finegrid solution. Although not presented here, the method can be coupled with a flowbased grid generation procedure to provide an overall gridding capability that resolves key flow regions while minimizing or eliminating spurious pressure oscillations.



A SemiAnalytical Model for Productivity Testing of Complex Well Configurations
Authors P. A. Fokker, G. K. Brouwer, F. Verga and D. FerreroThis paper presents a semianalytical method for the modeling of productivity testing of vertical, horizontal or multilateral wells. The method, which is applicable to both oil and gas reservoirs, automatically accounts for well interference. The use of analytical expressions ensures proper handling of transient shorttime behavior and semisteadystate longtime behavior, both close to the well and further into the reservoir. Calculation times are still very limited, in the order of a few minutes down to a few seconds when there are vertical wells only. This makes the tool suitable for well testing evaluation.
The approach is based on an earlier derived productivity prediction tool, in which the steadystate equations were solved. It has now been extended to solve the timedependent diffusion equation and it is thus more rigorous than the extension to timedependent behavior using solutions to the Laplace equation and moving pressure boundaries, which was presented recently. In our current method, the equations have first been transformed using the Laplace transformation. The expressions for the producing wells are combined with auxiliary sources outside the reservoir. The core of the semianalytic method involves an adjustment of the positions and strengths of these sources in order to approximate the boundary conditions at the reservoir boundaries. The solution that is obtained is transformed back into the time domain using a Stehfest algorithm.
The new approach has been validated with numerical tools, including both reservoir simulators and welltest interpretation software. Validations were performed with artificial cases using both singlewell and multiplewell production tests. The results of these tests were excellent.



Application of SFunction for Well Test Analysis
By T. ShimamotoOnedimensional radial flow equation, of which permeability and porosity are in inversely proportional to the radius, is equivalent to the equation of linear flow. In other words, if we take the radial variations of permeability and porosity into consideration, we can represent the linear flow by onedimensional radial flow equation. To expand this idea and to apply it for various types of inner and outer boundary shapes, we defined the Sfunction as "The functions of reservoir properties in radial direction which should be used for a radial composite model in order to represent the pressure transient behaviors at the well location for given inner and/or outer boundary shapes". We found that the approximate Sfunction can be easily estimated from complex velocity potentials for simple boundary shapes(SPE100174).
In this paper, the Sfunctional analyses are expanded to freeform boundary shapes. For this purpose, we have to derive the complex velocity potential for arbitrary boundary shapes. The solutions for these kinds of problems are known as “SchwarzChristoffel conformal mapping”. In decades, the development of the computer technologies enables us to calculate the conformal mapping easily.
The typical SchwarzChristoffel conformal mapping is a formula that will transform the complex velocity potential in a canonical domain into a polygon of the physical domain. This theory can be extended to the problem that sinks and /or sources (singular points) exist. Thus, we can obtain the complex velocity potential and at last we can calculate the pressure behavior for arbitrary shaped boundary problems. Here we will mainly discuss a “strip type transform” for the bent channel connected to fan system and about a “disk type transform” for the several hydraulic plane fractures which have different directions each other.



Deflation Accelerated Preconditioned Conjugate Gradient Method in Finite Element Methods in Oil Reservoirs
Authors F. J. Vermolen, P. L. J. Zitha and C. VuikOil reservoirs generally contain several layers with highly varying permeabilities. Hence modeling oil and water flows often involves solving partial differential equations with very large contrasts in the coefficients. Since the injection and production wells are very small compared to dimensions of the reservoir, the injection wells and production wells are modelled by the use of deltafunctions appearing in the right hand side of the partial differential equation for the pressure. In the presentation we consider a reservoir that consists of several layers with extreme contrasts of the permeability at the interfaces between the adjacent layers. Further, the finite element mesh is refined in the vicinity of the production and injection wells.
The finite element discretization of the above equation gives a stiffness matrix with extremely varying coefficients and hence the spectrum consists of large eigenvalues and eigenvalues that are almost zero, which gives a very high condition number. Hence a very bad convergence behavior for an iterative solver such as the conjugate
gradient method results. A preconditioner, like ILU, removes almost all the small eigenvalues, however, some small eigenvalues due to the large ratio of the coefficients at the interfaces persist. These small eigenvalues are removed by deflation based on a set of vectors that approximate the span of the corresponding eigenvectors.
Herewith the speed of convergence is successfully enhanced and the computational cost are reduced significantly. By the use of a proper choice for the deflation vectors, we show that the speed of convergence of our method does not depend on either the value of the contrasts in the coefficients or on the number of layers with varying coefficients. Further, the method is scalable in a parallel computing environment.



Scalability and Load Balancing Problems in Parallel Reservoir Simulation
Authors T. Guignon, J. M. Gratien, J. F. Magras, P. Q. Quandalle and O. R. RicoisScalability and Load Balancing Problems in Parallel Reservoir Simulation
New parallel reservoir simulator software designed for Linux clusters enable to overcome hardware limitation and to simulate models with large amount of data. Reservoir Engineering industry is very interested in using ever growing dataset with more and more complex physics and detailed models. The key issue still remains running simulations in an acceptable CPU time. As, the trend in hardware technologies is not to improve drastically the performance of individual CPUs but to facilitate the aggregation of computation facilities (with high bandwidth network, multicore architectures, ...), the challenge is to improve the efficiency of reservoir simulation software on a large number of processors.
New numerical difficulties and performance problems appear when the number of cells and the number of processors are growing. As a matter of fact, the architecture of Linux clusters is very sensible to memory distribution and load balancing:
 the cost of parallel solver algorithm is usually sensible to the size of the reservoir model (lack of scalability) and the consequences on CPU performance can no more be neglected;
 the domain decomposition algorithms used to distribute data between processors have a great influence on the computing load balancing between processors;
 using adaptive numerical schemes with dynamic space criteria (AIM schemes, flash algorithms based on the thermodynamical state of each cell) is a source of unbalance that cannot statically be resolved;
 simulation result storages on irregular data structures, such as unstructured grids, multilateral smart wells and perforated cells, lead to store an important amount of information during the simulation. With the variety of IO subsystems found on Linux clusters the simulator must be able to adapt its IO strategy to the underlaying IO library/file system and hardware.
In this paper, we present different approaches to overcome these kinds of problems. We discuss technical choices such like :
 advanced scalable linear solver algorithm ;
 load balancing issue with different domain decomposition strategies ;
 mesh partitioner strategy and parallel solver performance management;
 flexible IO strategy from simple file system to more complex parallel file system or database.
We have developed and benchmarked these different solutions on published reference large scale problems and actual case studies with several tens millions of cells. We analyze the results and discuss the efficiency of each solution to overcome the scalability difficulties and performance limitations due to load unbalance.



NonEqulibrium TwoVelocity Effects in GasCondensate Flow through Porous Media
Authors I. Panfilova, M. Panfilov and S. OladyshkinThe flow of gascondensate mixture in porous media is characterized by three properties which determine a significant nonequilibrium in transfers between gas and liquid: 1) a capillary and gravity coagulation of small liquid drops with forming large aggregates, 2) a high difference in diffusion coefficients for liquid and gas, which leads to a delay in establishing the liquid aggregate composition, and 3) a high difference between the liquid and gas mobility, which causes the dependence of the nonequilibrium parameters on the relative phase velocity. The objective of this study is to construct a closed nonequilibrium compositional flow model and to reveal various regimes of the nonequilibrium behaviour.
The analysis was based on separating in time the capillarygravity coagulation and the phase exchange. The coagulation was studied using the method of pore network modelling. An isolated liquid drop limited by two meniscus inside a pore can move in direction of the resulting capillary or gravity force. The drop motion is limited by percolation conditions saying that the gas from the neighbouring pores can not be displaced if it has no cluster connection to the medium exit. As the result we obtained the relations between the correlation length of the pore radii field or the permeability field and the scale of the liquid aggregate.
At the next step we constructed a set of the averaged compositional flow models taking into account the phase nonequilibrium. The averaging was performed by twoscale asymptotic expansion method. Three regimes of the nonequilibrium were revealed. At a small relative phase velocity the macroscale exchange is diffusion limited, being described by a nonlocal integrodifferential operator. Its kernel is calculated as the result of solution to a cell problem. A growth of the relative phase velocity determines an increasing influence of the rotational flow inside liquid aggregates on mass transfer between the phases. Such a diffusionrotation regime is described in terms of the double relaxation model, presenting a secondorder nonlinear kinetic differential equation. The diffusion and rotation relaxation times are calculated as the result of solution to a problem of gas flow around a macroscale liquid aggregate in a porous medium with mass transfer. The fluid flow was described by the Brinkman equations which allow the rotations, in contrast to the Darcy law. The “slip regime” arises at very high relative phase velocities, when the nonequilibrium degree becomes to decrease, however the system tends to a new equilibrium state in which the time of contact between liquid and gas appears to be much lower than the time of mass exchanges. The generalized kinetic model of the slip regime is obtained by homogenization technique. A number of examples are simulated for gascondensate flow in the vicinity of a well, where the nonequilibrium degree appears to be the most significant.
The influence of the capillary number, the Forchheimer effect and the velocitydependent relative permeability on the nonequilibrium was analysed.
We develop generalized relations which describes uniformly all the three nonequilibrium regimes, which may be used as a plugin to the existing PVT or hybrid thermodynamichydrodynamic software, with providing a new option to simulate the nonequilibrium behaviour of multicomponent gascondensate mixtures.
The research is financed by the Schlumberger Moscow Research Center.



Splitting the Thermodynamlics and Hydrodynamics in Compositional GasLiquid Flow through Porous Reservoirs
Authors S. Oladyshkin and M. PanfilovFor twophase compositional flow with mass transfer in porous media, it is shown that the existing steadystate solutions are unstable for gasliquid systems, which are characterized by a high relative phase mobility. Instead of them the process model allows stable “semistationary” solutions which correspond to a limit model when the liquidgas mobility ratio tends to zero. In a semistationary model the pressure and phase concentration fields are quasistationary, while the liquid saturation, in conditions of a continuous condensation/evaporation, is strongly nonstationary and has no a stationary limit in time. Within the framework of the semistationary model we have obtained a full splitting between hydrodynamics and thermodynamics, thanks to the fact that N2 equations describing the phase concentration transport allow an explicit first integration along streamlines. This enables to transform them into differential equations of thermodynamic type (out of space and time) with differentiating over pressure. Being added to the usual equilibrium and EOS equations, such a system constitutes a new closed thermodynamic model which depends on the pressure only. This model describes the thermodynamic behaviour in an open system which corresponds to gasliquid flow along each streamline. The remaining two flow equations determine the pressure and the saturation fields, with coefficients dependent on thermodynamics. To determine them it is sufficient to calculate the thermodynamic system one time, in contrast to the full compositional model where the thermodynamics is calculated at each space and time point.
The obtained hydrodynamic model allows obtaining the analytical solutions along streamlines, by using the singular perturbation method with respect to the relative phase mobility parameter. The irregular character of the asymptotic expansions is determined by the fact that the formal limit solution corresponds to an immobile liquid, which means a full pore plugging by liquid and breaking of gas movement. The full analytical solutions to the non selfsimilar multicomponent flow problems towards a well were constructed with using the matching asymptotic expansion method. This result was used to propose a new method of gascondensate well representation in reservoir simulations. The comparison of the solutions obtained for the split thermodynamic and hydrodynamic models with numerical ECLIPSEbased fullcompositional simulations has shown a very good agreement. The suggested method was used to develop a new kind of the streamline compositional simulator. We illustrate some examples of its functioning.
The research was financed by the Schlumberger Abingdon Technology Center.



The distillation mechanism in steam displacement of oil
Authors D. Marchesin and J. BruiningDuring steam drive recovery of oil a thin zone of high volatile oil concentration is formed between the upstream hot steam region and the downstream cold liquid region containing water and the oil at its initial composition. Distillable components, coinjected or stripped from the oil remaining in the upstream steam region, are transported and build up this region. The existence of this zone is responsible for the high recovery of oil from the steam zone. This paper investigates the growth and the stability of this zone, for the case when relatively small amounts of volatile oil are present. We show that this zone is described by ODE's typical of traveling waves. The growth and stability are illustrated numerically. Our results suggest that steam drive recovery can be improved by the preinjection of a small slug of volatile oil.



Efficient Integration of Stiff Kinetics in Reactive Compositional and Thermal Porous Media Processes
Authors M. R. Kristensen, M. Gerritsen, P. G. Thomsen, M. L. Michelsen and E. H. StenbyWe propose the use of implicit onestep ESDIRK (Explicit Singly Diagonal Implicit RungeKutta) methods for integration of the stiff kinetics in reactive, compositional and thermal processes that are solved using operatorsplitting type approaches. To facilitate the algorithmic development we construct a virtual kinetic cell model. The model serves both as a tool for the development and testing of tailored solvers and as a testbed for studying the interactions between chemical kinetics and phase behavior. As case study, two chemical kinetics models with 6 and 14 components, respectively, are implemented for insitu combustion, a thermal oil recovery process. Through benchmark studies using the 14 component reaction model the new ESDIRK solvers
are shown to improve computational speed when compared to the widely used multistep BDF methods DASSL and LSODE.
Phase changes are known to cause convergence problems for the integration method. We propose an algorithm for detection and location of phase changes based on discrete event system theory. Experiments show that the algorithm improves the robustness of the integration process near phase oundaries by lowering the number convergence and error test failures by more than 50% compared to direct integration without the new algorithm.



Subsidence Simulations for Depleting Reservoirs – Considerations on the Role of Hydromechanical Nonlinearities
Authors G. Musso, D. Selvaggi and S. ManticaThe paper aims at investigating some issues related to the theoretical and numerical modelization of subsidence phenomena induced by hydrocarbon withdrawal. First a brief review of the implications of facing the problem as coupled / uncoupled is provided accordingly to the theory of mechanics of porous media as formulated by Biot. On the basis of these considerations, the classical nucleus of strain solution given by Geertsma is coupled to the hydrodinamic problem of a single phase fluid draining towards a well. The approach implemented has the advantage of providing a series of subsidence profiles evolving with time, together with the well production. The problem is handled both introducing or not the possibility of hydromechanical nonlinearities in the behaviour of the reservoir rock. When accounted, the mechanical nonlinearity is accomplished by assuming that the rock deforms accordingly to classical logarithmic relationships for oedometric stress paths, as foreseen by critical state models such as Cam Clay. On its turn the hydraulic nonlinearity is accounted for by implementing a literature relationship valid for Mexico Gulf sands, postulating that an exponential relationship exists between porosity and permeability.
