Recent studies have shown field data exhibiting a negative half slope trend in the pressure derivative that cannot be explained as spherical flow. In one case the well was located in an elongated fluvial reservoir bounded on one end by an aquifer acting as a constant pressure boundary. In another case the well was also in an elongated reservoir, this time crossed by a highly conductive fault. None has shown a rigorous derivation for the analytical equations for this flow regime.
This study derives flow regime equations for two new flow regimes encountered by a well near a constant pressure boundary. When there is no evidence of another nearby boundary, the pressure derivative trends are radial flow until the constant pressure boundary is encountered, and after that a straight trend with negative unit slope is observed. This flow regime is named here as dipolar flow. When the well is in an elongated reservoir near a constant pressure boundary perpendicular to the elongated direction, possible flow regimes include radial, dipolar flow, linear flow, and a flow regime with negative half slope, which is named here as dipole linear flow.
Normally falling derivative behavior due to a constant pressure boundary is assumed to signal the end of any useful parameter estimation, but the new dipole flow regimes are sensitive to permeability and to the distances to the constant pressure boundary and to boundaries defining the elongated reservoir. This study shows how to use the flow regimes to determine distances to closed and constant pressure boundaries, and to identify bedding plane permeability anisotropy (kx from well to constant pressure boundary, ky parallel to constant pressure boundary plane). The new flow regimes are present in standard single fault and rectangle models for pressure transient behavior, but they have never been rigorously derived or described.
Introduction
The plot of the log of the pressure change and its derivative with respect to superposition time as a function of the log of elapsed time was first introduced by Bourdet et al.1 as an aid to type-curve matching. Referring to the Bourdet plot as the log-log diagnostic plot, Ehlig-Economides2,3 summarized relationships between pressure derivative responses and flow geometries described as "flow regimes". In the past 20 years, flow regime analysis has been accepted by industry and used widely in commercial interpretation software.
A commonly encountered flow regime, a negative half slope trend in the derivative is known as an indication of spherical or hemispherical flow. Ehlig-Economides et al.4 studied in detail this flow regime, which is associated with the limited entry completion.
However, some field data exhibit a negative half slope in the pressure derivative but cannot be interpreted as spherical or hemispherical flow. Two cases can be found in literature. One case is given by Escobar,5 who studied an elongated reservoir with a constant pressure boundary normal to the elongation. However, the author mistakenly named the negative half sloping flow regime parabolic flow because he observed a parabolic shape for the isobars created by a numerical simulator. In this work, we will show the correct model does not give parabolically shaped isobars. The other case was presented by Al-Ghamdi et al.6 This time a highly conductive fault serves as the constant pressure boundary in an elongated reservoir. This paper showed field examples illustrating the same negative 1/2 slope trend and provided a model to match the data, but it did not provide a flow regime equation for the behavior.
In both cases the well was located in an elongated reservoir which is bounded on one end by a constant pressure boundary. The schematic plan view is given in Fig. 1. Here the term "elongated reservoir" means a long, narrow reservoir which could have a stratigraphic origin such as fluvial deposition, or a structural origin such as parallel sealing faults.5, 7–9
In addition, unlike the rapid drop in the pressure derivative seen when a well is surrounded by a circular or square constant pressure boundary, when the well is near a single lateral linear constant pressure boundary, the pressure derivative drops with a slope of -1. Physical examples of such a constant pressure boundary include a downdip aquifer10 or an updip gas cap, or a well near a finite conductivity fault.11 A diagram of a well near a single constant pressure boundary is shown in Fig. 2. This figure also illustrates that a well near a constant pressure boundary is equivalent to the well plus an image well with a rate of opposite sign at a distance twice that between the well and the constant pressure boundary.