This paper summarizes some research that was conducted to construct finite-element models for reservoir flow problems. The models are based on Galerkin's method, but the method is applied in an unorthodox manner to simplify calculation of coefficients and to improve stability. Specifically, techniques of compatibility relaxation, capacity lumping, and upstream mobility weighting are used, and schemes are obtained that seem to combine the simplicity and high stability of conventional finite-difference models with the generality and modeling flexibility of finite-element methods.The development of a model for single-phase gas flow and a two-phase oil/water model is described. Numerical examples are included to illustrate the usefulness of finite elements. In particular, the triangular element with linear interpolation is shown to be an attractive alternative to the standard five-point, finite-difference approximation for two-dimensional analysis. Introduction During the past decades, finite-element methods have been developed to a high level of sophistication and have gained wide popularity within several branches of engineering science. In some fields, such methods have replaced to some extent the older finite-difference methods in engineering practice because they have been regarded as a more convenient tool for numerical analysis. An increasing interest in finite-element methods, or variational methods in general, also may be noticed in the field of numerical reservoir simulation, but so far no definitive breakthrough has occurred in this field.One reason for this probably is the complexity of reservoir flow problems. Reservoir flow equations in most cases are nonlinear, and for multiphase flow, they are usually found on the borderline between parabolic and hyperbolic equations. For such parabolic and hyperbolic equations. For such problems, the dissimilarities between problems, the dissimilarities between finite-difference and finite-element methods are much more pronounced than for linear problems of the elliptic type. This means that the finite-element method may not be looked upon as easily as an extension or generalization of finite-difference methods. Second, one can question whether all the advantages that are gained in other instances by using finite elements may be realized at all.Applications of variational methods to single-phase flow problems, or diffusion-type problems in general, have been studied extensively. The merits of finite elements for such problems are apparently well established, at least as far as linear problems are concerned.The literature on variational methods in multiphase flow is comparatively sparse, and so far the results are inconclusive regarding the relative advantages of variational methods and finite- difference methods in this field. In summary, variational methods offer the potential advantages of (1) easy implementation of higher-order approximations, (2) a more proper treatment of variable coefficients, and (3) greater modeling flexibility.Previously, attention was focused on Aspects 1 and 2. Several authors used cubic Hermitian basis functions. SPEJ P. 333

Ttiia paper wea eelected for presentation by an SPE Program Committee followingreview of informationcontained in an abatractaubmkfed by the q Utf80@, contents of the paper, aa preeanted, have not been reviewed by the society of Pelroleum Engineare and are subject 10 correctionby the q uthor(a),The matarial, ae preeented, does not necesaerily reftectq ny positionof the Sociity of Petroleum Engineers, itsofficere, or members. Papers -tad qt SPE m.etinga are aubjocf to Publioatitmreview by Editorial Commifteee 01 the Society of Pelrofaom Engineers. Permieeion to copy la mtdotad to an e5afreot ot not more then 300 words. Ittuatratlonamay ne4 be c@@ The qbafraaf should oonfainoonapkw eoknMa@nam of where and by whom the paper ia preeentest,Write PubfloelionaManqer, SPE, P.O. Sox SSSSSS, Riohardaors, TX 7~, Tafax, 7S0SSSSPEDAL. AbstractDistorted grids or "corner point geometry" have received increased attention as a means of enhancing the modeliing flexibility of finite difference type reservoir simulators. Non-rectangular grids permit better alignment with reservoir boundaries and faults and may lead to fewer grid blocks and therefore less computational efforts in simulation of complex reservoirs.This paper shows that caution should be used as regards the transmissibility calculations for distorted grids. Specifically, it is demonstrated by simulation experiments that methods which try to maintain (for 2-D problems) the five-point connectivity of standard finite differences may lead to serious errors, A generalized nine-point scheme is constructed and shown to be more appropriate for distorted grids, For instance, the scheme seems to preserve the accuracy of a square nine-point scheme if the distortion is kept within certain limits, Although based on finite element techniques, the scheme is given a finite difference type form, and can therefore easily be implemented in more generalpurpose finite difference type reservoir simulators,

The Gullfaks Field is a large oilfield in the Norwegian Sector of the North Sea. The field is compartmentalized by a dense and complex fault pattern, and most of the oil is contained in poorly consolidated but high-quality reservoir sands. Production started in 1986, and waterflooding is the main recovery method. Water-alternating-gas (WAG) injection -or supplementary injection of gas in existing water injection wells -has been identified as a possible method for increasing and accelerating oil recovery from Gullfaks. To verify the potential of this technique, a WAG pilot was initiated in 1991. The target area for the gas was primarily one fault block where major water breakthrough had occurred. In this paper, relevant field observations are presented, and the effects of the WAG pilot injection on flow performance and recoveries are discussed on the basis of the field observations and a detailed, history-matched 3D simulation model for the pilot area. In the present case, the gas migrates rapidly towards the top of the formation and accumulates in secondary gas caps. Improved immiscible displacement of attic oil is the main benefit of gas injection in this case.

Modeling of reservoir processes like gas miscible t100ding may require consideration of phase equilibrium between multiple liquid phases. Under certain conditions two hydrocarbon liquid phases may form; one may also want to account for mass transfer between the hydrocarbon and the aqueous phases.This paper describes a refined successive substitution (SS) method for calculating multi phase t1ash equilibrium. The phase behavior procedure proceeds in a stepwise manner, imd additional phases are introduced by a special testing scheme based on phase fugacities. This is to avoid trivial solutions and to ensure continuity across phase boundaries.The method has been tested on various three-and fourphase systems; and examples of application show that the method performs well. Calculation Steps of Multiphase FlashThe basis for our approach to the multi phase t1ash equilibrium problem is the SS method, which consists of the following steps.1. Assume equilibrium K-values. 2. Calculate the phase distribution and compositions corresponding to the given K-values.3. Calculate component fugacities in each phase and check for equality.4. If equality is not achieved, correct the K-values on the basis of the fugacities and repeat from Step 2.We assume that fugacities are given from a cubic EOS (Redlich-Kwong, Peng-Robinson), but the problem of selecting suitable parameters, especially for lumped and/or heavy components, is considered beyond the scope of this paper.If the initial K-value estimates are sufficient, simultaneous handling of all phases is probably the most efficient method. Frequently this is not the case, however, and the method then easily becomes unstable and leads to trivial solutions. We have found it advantageous, therefore, to develop a more stepwise approach. Existence of the different phases is tested explicitly, and the sets of equilibrium constants are developed phase by phase before all phases are handled simultaneously.The calculation steps involved for any assumed

An element model was used to investigate the effect of using the existing crestal gas injectors in the Statfjord Fm at the Statfjord Field for additional updip water injection. The element was extracted from a history matched full field model and refined. The simulations gave accelerated production and increased oil recovery of approximately 5% with updip water injection, primarily because of better pressure maintenance, better sweep of lower parts of the formation and less problems with gas production constraints. Sensitivities on selected parameters did not indicate anything in disfavour of updip water injection. Full field simulations have confirmed the potential of updip water injection, and the scheme is found to fit very well into the overall Statfjord Field strategy.