SPE Members Abstract A horizontal well usually yields a high rate through its long perforation interval, resulting in a large frictional pressure drop that is believed to reduce the well productivity. Unlike conventional horizontal well methods which consider the horizontal section only, the proposed model also includes the vertical section of the well and the surface facility from the well head to the GOSP. The hydraulics model was validated stage-by-stage by various known solutions, and then applied to a high rate well. It was found that the effect of frictional pressure drop on oil rate was not as pronounced as suggested by analytical methods, or some numerical simulation studies. In summary, the pressure drop in the vertical section of the horizontal well plays an especially important role in deciding how much fluid can be produced. Therefore, it is recommended to be an integral part of horizontal well models. Introduction In models which consider the horizontal section only, frictional pressure drop was shown to reduce the productivity of horizontal wells. Dikken presented an analytical horizontal well model that combined the fluid flow in the horizontal section of a well and the reservoir flow. He concluded that the flow inside horizontal wells is either transition or turbulent in most practical situations. Furthermore, for single phase turbulent flows, he found that appreciable reduction in drawdown occurred at positions farther away from the start of the section. In a water coning example, he demonstrated that little additional production results from extending a 300 m long well with a diameter of 11.4 cm. Novy generalized Dikken's model so that it can be applied to the recovery of gas. In the process, he found that the friction factor correlation used by Dikken gives friction factors that are too high for rough tubes. Novy performed extensive sensitivity studies and he concluded that if the ratio of well-bore pressure drop to drawdown at the producing end exceeds 10%, friction is apt to reduce productivity by 10% or more. In a simulation study where the frictional pressure drop was included in the horizontal well model, Seines et al. found that by varying the effective roughness of the well, thereby changing the pressure drop, they obtained a 10% difference in cumulative production after 1 year. The well was producing at a fixed flowing bottomhole pressure (FBHP). This paper presents the work in which a complete hydraulics model for horizontal wells was developed and validated. The model was then applied to a high flowrate case. Calculated results were compared with analytical predictions. Investigation of the discrepancy in results provided some insights about the modeling of horizontal wells. P. 407
This method reduces the numerical dispersion and the grid orientation effect simultaneously. Introduction Reservoir simulators have become one of the valuable tools which allow petroleum engineers to answer many important questions arising in the studies of petroleum recovery. They are the products of evolution over several decades and the technology is quite matured. There still exist some shortcomings, however. The most significant among them include:In an unfavorable mobility ratio piston-type displacement, the oil recovery is dependent upon the grid system employed,The error caused by the numerical dispersion inherent to low order numerical schemes can be significant in some cases. Much work has been done in the past to alleviate these problems. Since the literature is quite extensive, we will not attempt to review all of it here. The grid orientation effect was first reported by Todd et. al. and confirmed later by Coats et. al. Yanosik and McCracken proposed a nine-point scheme which has been proved to be effective in alleviating the grid orientation effect for some problems. The two point upstream weighting method, proposed by Todd et. al., reduces numerical dispersion. Although it also reduces the grid orientation effect for immiscible displacement, it still show serious grid orientation effect for unfavorable mobility-ratio piston-type displacement. Even though numerous other methods have been proposed in the past to address the grid orientation and the numerical dispersion problems, it appears to the best knowledge of the authors that the nine-point scheme of Yanosik and McCracken and its variations, and the two point upstream weighting method of Todd et. al. are probably the only methods which have been widely accepted by the petroleum industry. Unfortunately, however, none of these methods can simultaneously handle both grid orientation effect and numerical dispersion problems. The nine-point scheme is still of first order accuracy in mobility and hence has the numerical dispersion problem. The two point upstream weighting method, as was pointed by Yanosik and McCracken, can not reduce the grid orientation effect for piston-like displacement. We present in this paper a simple and new concept which can be effectively utilized to attack the grid orientation effect and numerical dispersion problems simultaneously. We also describe a feasible scheme to implement this concept. We will concentrate, for simplicity, on a two dimensional, two phase problem with uniform grid size, zero capillary pressure and each phase containing only one component. Extension to more complicated cases is straightforward. Basic Conditions Using Darcy's law and assuming the flow direction is from (i, j) to (i+1, j) (see Fig. 1), we obtain the flux of phase P across the block boundary BC as (1) The fluxes across three other block boundaries and for other phases are computed in a similar way. With the substitution of the finite difference approximation of equation (1) into the material balance equation, we have finite difference equations for pressure, saturation and, for compositional models, concentration. The resulting pressure difference equation can be represented in the following form: where n = 1,2, …, s are grid points adjacent to point (i, j). Since the pressure equation occurring in many reservoir simulations is either elliptic or near elliptic, the coefficients must satisfy the following conditions: Condition (1) - non-negativeness of the coefficients Condition (2) - diagonal dominance of the coefficient matrix with strict inequality for at least one point. Satisfaction of the above conditions is fundamental since they ensure that the difference equation satisfies a maximum principle. The resulting saturation equation on the other hand has the nature of a first order hyperbolic equation. It was pointed out by many investigators that the mobility should be monotonic for a numerical scheme to successfully produce a non-oscillating saturation front. Hence an additional condition we will require a numerical scheme to satisfy is: Condition (3) - the mobility should be monotonic between the nodal points.
SPE Member Abstract This paper presents a new iteration method for the numerical solution of pressure equations arising in petroleum reservoir engineering. It utilizes a newly developed preconditioner and generalized conjugate gradient acceleration with a new residual constraint procedure. The Alternating Interblock Direction (AID) preconditioner exploits a natural way of alternating sweeps in xyz-directions and is free from the dependence on the choice of the first internal direction which is common with other methods. The generalized conjugate gradient algorithm is ORTHOMIN. It is further accelerated by an alternating direction residual constraints (ADC) which could make the summation of the residuals be zero simultaneously in all three axes. The new method can be easily vectorized and parallelized. It can also be adapted to equations other than the pressure equation. Its speed and robustness are proved to be favorable to previously published methods by numerical experiments. Introduction In the core of a reservoir simulator exists the solution routines which are designed to solve a system of algebraic equations arising from the finite difference (or element) approximations of the non-linear partial differential equations. Let us denote such a system of equations by (1) The size of the coefficient matrix, A, is very large in the simulation of giant oil reservoirs such as the ones in Saudi Arabia. The use of the direct solvers for these cases is not practical due to the extensive CPU time and the space requirement and therefore the iterative solvers are employed. Saudi Arabia's reservoirs have some unique features. They includeThe number of grid dimensions in horizontal directions is much larger than the one in vertical direction.The grid sizes in the horizontal directions are much larger than the one in vertical direction. In this case, the ratio of the vertical direction transmissibility to the horizontal direction transmissibility is very large.If the horizontal grid size is refined to the same order as the vertical grid size, the ratio of the vertical direction transmissibility over the horizontal direction transmissibility becomes very small because of very low vertical direction permeability. The objective of this paper is to develop an iterative technique to be used in the simulation of such reservoirs. Since the numerical solution of the pressure equation poses more serious challenge (due to its elliptic nature) than the saturation equation or molar equation, we concentrate in this paper on the application of the new method to the solution of the pressure equation. THE ALGORITHM OF THE GENERALIZED CONJUGATE GRADIENT METHOD The most powerful iterative technique currently available to solve equation (1) appears to be the variation of the Generalized Conjugate Gradient Method. P. 71
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