Use of Analytical Solutions To Improve Simulator Accuracy
- J.K. Jasti (Mobil E&P Technical Center) | V.R. Penmatcha (Stanford U.) | D.K. Babu (Mobil E&P Technical Center)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 1999
- Document Type
- Journal Paper
- 47 - 56
- 1999. Society of Petroleum Engineers
- 1.2 Wellbore Design, 4.1.5 Processing Equipment, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.5 Reservoir Simulation, 4.1.2 Separation and Treating, 5.1.1 Exploration, Development, Structural Geology
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Analytical solutions previously developed for wellbore design and scoping analysis were utilized in this study to benchmark and modify the existing wellbore models in the 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, and fractured completions in single or multilayered 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 three-dimensional flow cases and for highly deviated wells.
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 wellblock 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 wellblock flow rate, if the wellbore pressure is given. Accurate evaluation of "connection factors," defining the relationship between the simulator-generated wellblock pressure and the wellbore flowing pressure, therefore becomes an important item. Peaceman, through his landmark publications,1,2 established a mathematical relationship between wellblock pressure and wellbore pressure for a fully penetrating vertical well. Babu3 et al. extended this work for the case of a fully penetrating horizontal well in a slablike drainage area. This topic continues to generate interest and attention by various investigators:4-11 Kim's work on nonsquare gridblocks; the work of Morita et al. on fine mesh and finite element three-dimensional (3D) simulators; Peaceman's work on off-centered and multiple wells in a single gridblock; Lin's work on partially penetrating vertical wells, heterogeneous media, and nonuniform grids; Ding's work on double layer potentials with transmissibility adjustments; Sharpe and Ramesh's publication on orthogonal grid generation, nonuniform grids and 3D flow aspects; the work of Chen et al. 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 wellblock pressures.
The rate-pressure relationship can be summarized using a wellbore-to-gridblock connection factor Fci as
Explicit formulates for the connection factors for homogenous media with uniform grids can be found in previous publications by Peaceman1,2 for vertical wells, and Babu3 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.
Step 1: Determination of Flow Rates and to Wellbore Pressures.
All expressions are derived as solutions to the general diffusion equation
For purposes of completeness, we indicate briefly a procedure to solve Eq. (2). The methods used are standard, and combine analytical and numerical techniques to obtain the solution, as indicated in Refs. 12-17.
The pressure change at location (x i, yi, zi) due to unit rate of production at location (xj , yj, zj) at time t is represented by C ij(t). The production at location (xj , yj, zj) could be due to any of the following types of completions: a line sink (representing horizontal, vertical, or inclined wells), or a planar surface sink (representing hydraulic fractures).
where p(( xi, yi, z j );(xj, yj, zi);t) denotes the pressure being observed at time t, location (xi , yi, zi) due to unit rate of production at location (xj, yj, zj).
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