A new fully implicit formulation for compositional simulators is presented. Rather than solving for pressure, saturations and phase compositions, the new formulation solves for pressure, overall concentrations and K-values. This change of primary unknowns to be solved improves numerical stability by yielding a more diagonally dominant Jacobian. The use of K-values as primary unknowns also allows the composition constraints to be solved separately from the other equations. The solution of the constraint equations is very efficient because they can be reformulated as monotonic functions*. The primary unknowns are further classified into reduced-unknowns and pivotal-unknowns. The latters are eliminated from the mass conservation equations using the fugacity equalities and constraints. The pivotal-unknowns are selected according to the sensitivity of the equations to the unknowns. This selection conforms to the partial pivoting strategy of Gaussian elimination and enhances numerical stability. Finally, a partial solution method is introduced to eliminate unnecessary calculations of those equations which have met the convergence criteria. For a simple system, it is shown that the new formulation results in a more diagonally dominant Jacobian matrix. Numerical experiments involving one dimensional multicontact-miscibile (MCM) and immiscible problems, a two dimensional MCM problem and a three dimensional MCM problem are conducted to illustrate the capability and efficiency of the new formulation. Results indicate that the number of iterations per time step required by the new formulation is only about 40% – 60% of an existing fully implicit scheme. It is almost as low as that required by an IMPES scheme.

Matching historical performance of a reservoir is a time-consuming exercise with uncertain outcome. The current practice of manual matching is subjective and goodness of the match owes largely to the experiences of the team members involved in a study and to the quality and quantity of various input and observed data. This paper describes approaches taken to speed up the history matching effort during the course of replicating multiyear performance of two West African reservoirs. The study entails validation of production data, and matching of single-well and full-field performances with two automated-history-matching (AHM) approaches based on the Gauss-Newton algorithm. These finite-difference results obtained with AHM are contrasted with those obtained from the traditional manual approach. Results show that both AHM approaches shorten length of a study, and preserve objectivity of a history match. The use of grid coarsening results in impressive speed gain that can quickly provide clues whether adjustments of certain parameters are justified for some wells. That multiple parameter adjustments can be made within specified bounds with a few iterations in a 24-hour run is a profound improvement in our speed of learning of key reservoir flow behavior. This speed, in turn, guides the study in a path that leads to rapid conclusion in a fraction of time taken in a traditional study. Introduction History matching, by nature, is an ill-posed inverse problem. To complicate the picture, manual history matching (MHM) is a subjective, labor-intensive process. That is because MHM involves one-factor-at-a-time adjustment of static reservoir parameters. Practically speaking, the quality of a MHM exercise is driven largely by the experience of the engineers and the budget, among other intangibles. Consequently, the reliability of forecasted results is often very short-lived, and the business decisions made have large uncertainty. Yet, flow simulation remains the most potent tool to search for infill potentials and/or instituting other recovery processes in producing fields. Equally important, acquisition of real-time data overwhelms MHM and rapid turnaround for timely intervention becomes infeasible. Therefore, increasing demands are being made to develop tools that speed up the history-matching process and interject objectivity at the same time. Attempts have been made since the mid 1960's 1,2 to automate history matching. Solving the single-phase flow problem for transient-pressure tests3–5 has had good success. However, the same is not true while tackling 3D, three-phase flow problems. Difficulties arise owing to highly nonlinear nature of the problem, and large computational requirements needed to solve a sizable field problem. Although significant computational and solver efficiencies have been gained over the past four decades, ever-increasing size of geostatistical earth models have continued to challenge the computational-speed issue. Historically, several approaches have emerged to do automated matching for three-phase, full-field problems. The common thread to all of them is rooted in minimizing an objective function, which typically includes a sum of production rate mismatch terms squared. Some of these approaches include the use of steepest descent,1 Gauss-Newton for single-phase2,6–8 and multiphase9,10 flow, and optimal control theory for single-phase,6,11 two-phase,8,12,13 and three-phase14 flow problems. Successful implementation15 and use16 of the gradient-based approach has also been reported. In general, these nonlinear regression methods require evaluation of sensitivity coefficients. The sensitivity coefficients are partial derivatives of the time-dependent reservoir parameters, such as pressure and saturation, with respect to static reservoir parameters, such as permeability or azimuth of a geostatistical variogram. Numerical differentiation is a typical way of obtaining these coefficients at each iteration of regression for each estimate of a parameter. Therein lays the difficulty of gaining computational efficiency for a sizable 3D, three-phase flow problem with many parameters. One way to circumvent this issue pivots around evaluating the coefficients analytically.17

Thi s paper presents a mult i component surface tension correlation based on scaling theory. In addition to particular exponents employed. the correlation contains two new features: (1) A corresponding-states equation is derived for a correlation coefficient. commonly referred to as a parachor.As a result. the hydrocarbon pseudocomponent parachors can be calculated through this equation. once their pseudocritical properties are estimated.(2) An approach is proposed to calcu-1 ate the parachors of mi xtures. In contrast to the conventional approach. which calculates the mixture parachor via molar mixing of component parachors, this approach first obtains the pseudocrtical properties of the mixture and then employs the correspondi ng-states equat i on to cal cul ate the mixture parachor.

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A new mathematical model is developed for analyzing single-well pulse test data from a reservoir consisting of two permeable zones separated by a less permeable zone. The model considers well-bore storage and skin effects and allows the three zones to have different rock properties. The line-source assumption used in developing the analytic solution is validated via comparison of results with another model also developed in this work. This latter model assumes identical rock properties for the two permeable zones but allows the wellbore radius to be finite. The numerical Laplace inversion methods available in the literature are evaluated with respect to some simple functions and the analytic Laplace space solution of the problem of interest. It is found that the Fourier-series and Zakian's methods can serve as alternatives to the well-known Stehfest algorithm. The effects of wellbore storage and skin on single-well pulse testing of layered reservoirs are investigated. Consistent with the findings in the literature for multiwell pulse testing, it is shown that the influence of wellbore storage and skin is to reduce the response amplitude and to increase the time lag of the pulse response. Among the wellbore storage and skin factors involved in a single well pulse testing, the skin associated with the observation interval is least influential on the pressure response. Using field data, it is shown that the mathematical model presented can be used, in conjunction with an optimization subroutine, to estimated the parameters of interests. It is further shown numerically that neglecting the effects of wellbore storage and skin could result in an underestimation of the formation permeabilities of interest. The problem of non-uniqueness of solution for parameter estimation using optimization techniques is also discussed for the subject problem.