Use of Analytical Solutions in Improvement of Simulator Accuracy
Authors: Jasti J.K. , Penmatcha V.R. , Babu D.K. 
Published: December 2007

Abstract
Analytical solutions previously developed for wellbore design and scoping analysis, were utilized in this study to benchmark and modify the existing wellbore models in reservoir simulator. The analytical solutions were developed for single phase oil or gas flow problems, and they are currently being used for well performance studies covering a wide range of well configurations: vertical, horizontal, deviated, and arbitrary well geometry (branched wells and undulating wells); fractured completions in single or multi-layered reservoirs. In this paper, some of the analytical solutions are utilized for benchmarking the existing well models in the reservoir simulator. Alternative well models are proposed for improving the accuracy of simulator predictions. Two methods of calculating gridblock-wellbore-connection factors are described. Results show that the new methods impact the simulator accuracy significantly for 3 dimensional flow cases and for highly deviated wells.

Introduction
Flowing bottomhole pressure and flow rate are the two important parameters computed by reservoir simulators. It is essential that these quantities are estimated accurately, especially in view of increased application of complex well trajectories for exploiting oil and gas reservoirs. A relationship between well block pressure and wellbore pressure is utilized in the reservoir simulators to determine the wellbore pressure if the flow rate is specified, or to determine the well block flow rate, if the wellbore pressure is given. Accurate evaluation of connection factors', defining the relationship between the simulator-generated well block pressure and the wellbore flowing pressure, therefore becomes an important item. Peaceman, through his landmark publications established a mathematical relationship between well block pressure and wellbore pressure for a fully penetrating vertical well. Babu et al. extended this work for the case of a fully penetrating horizontal well in a slab like drainage area. This topic continues to generate interest and attention by various investigators: Kim's work on non square grid blocks; Morita et al.'s work on fine mesh and finite element 3D simulators; Peaceman's work on off-centered and multiple wells in a single grid block; Lin's work on partially penetrating vertical wells, heterogeneous media, and non-uniform grids; Ding's work on double layer potentials with transmissibility adjustments; Sharpe and Ramesh's publication on orthogonal grid generation, non-uniform grids and 3D flow aspects; Chen et al.'s work on productivity index calculation in reservoir simulators; etc.
In this paper, we present results from our attempts to increase simulator accuracy when solving problems connected with the production from a system of wells. The wells can be partially penetrating (allowing for 3D flow around the wellbore), and having arbitrary trajectories. A sequence of analytical and numerical procedures is outlined here to better estimate the relationship between the wellbore and well block pressures.

Methodology
The rate-pressure relationship can be summarized using a wellbore-to-gridblock connection factor CFi as:
(1)
Explicit formulas for the connection factors for homogenous media with uniform grids can be found in previous publications by Peaceman for vertical wells, and Babu et al. for horizontal wells. Whereas the previous formulas covered essentially two dimensional flow, the present work is applicable to fully three dimensional problems, and is designed to handle arbitrary well trajectories in three dimensional space.
Two methods are presented for computing the connection factors. The first method applies when a uniform grid is employed in the simulator. In this approach, the connection factors are determined from direct analytical solutions. For locally refined grids, and for arbitrarily structured grids, a second method is developed. Both methods are applicable for homogenous and anisotropic media with either one well or multiple wells being active. The reservoir region is assumed to be rectangular box-shaped, with all six faces being closed to flow. The connection factors are determined in a two step procedure : in the first step an analytical single phase solution is used to determine production rates qi and wellbore pressures pwf,i; the second step involves computation of the pblock, i.
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