Summary
This paper discusses pressure responses from a formation with two communicating layers in which a fully penetrated high permeability layer is adjacent to a low-permeability layer. An analytical reservoir model is presented for well-test analysis of the layered systems, with the bottom of the low-permeability layer being a constant-pressure boundary. The strength of the support from the low-permeability layer is characterized with two parameters: layer bond constant and storage capacity.
Introduction
The log-log plot of pressure derivative vs. time is called a diagnostic plot in well-test analysis. Special slope values of the derivative curve usually are used for identification of reservoir and boundary models. These slopes include 0-slope, 1/4-slope, 1/2-slope, and unity slope. In many cases, however, the derivative curves do not exhibit slopes of these special values, and it is believed that some nonspecial slopes also reflect certain flow patterns in the reservoirs. Layered, thick reservoirs are one such example.1
In a layered reservoir, it is common practice to perforate a high-permeability section intentionally (adjacent sections are known to be less permeable) or unintentionally (adjacent sections are believed to be impermeable). It is expected that the flow in the perforated high-permeability layer will be partially fed by fluids in the adjacent layers. Warren and Root2 classified this type of layered reservoir as one of the dual-porosity systems in which the storage effect of the low-permeability layer is considered while the crossflow between layers is neglected. They presented a model based on the mathematical concept of superposition of the two media, as introduced previously by Barenblatt et al.3
This paper discusses the pressure response from a formation with two communicating layers. The flow in the two-layer system is referred to as Linearly Supported Radial Flow (LSRF) in this study. The reservoir model is depicted in Fig. 1. The LSRF may exist in the drainage area of a vertical well where radial (normally horizontal) flow prevails in a high-permeability layer and linear (normally vertical) flow into the high-permeability layer dominates in a low-permeability layer. The LSRF also may exist in the drainage area of a horizontal well after pseudoradial flow in the high-permeability layer is reached. Two LSRF systems were investigated:an LSRF system with a no-flow boundary at the opposite side of (not adjacent to) the high-permeability layer, andan LSRF system with a constant boundary pressure at the opposite side of (not adjacent to) the high-permeability layer.
Model Description
LSRF With No-Flow Boundary at Bottom.
An LSRF system with a no-flow boundary at the bottom of the low-permeability layer was investigated with a finite-element-based numerical simulator. The simulator was fully tested and commercially available in the market. Model configuration and input data are summarized in Table 1. The model well flowed 1,000 hours at a constant flow rate of 1,000 STB/D. A diagnostic plot of the generated response is shown in Fig. 2. It is seen from the figure that the radial-flow derivative is V-shaped in a certain time period. This is an expected signature of dual-porosity systems. It is concluded that the radial-flow derivative curve is similar to the derivative curve of single-layer double-porosity reservoirs. The signature of the pressure-derivative responses cannot be used for further diagnostic purposes. Other information from fracture/void detections is required.
LSRF With Constant-Pressure Boundary at Bottom.
Pressure response from an LSRF system with a constant-pressure boundary at the bottom of the low-permeability layer was also investigated with the numerical simulator. Model configuration and input data were kept the same as those in Table 1. The model well flowed 300 hours. A diagnostic plot of the generated response is shown in Fig. 3. It is seen from the figure that pressure derivative drops sharply in the later time. This is an expected signature of reservoirs with bottomwater or gas-cap gas drive. One may use a bottomwater- drive reservoir model to determine horizontal and vertical permeabilities in the perforated layer. However, one cannot be sure whether the derived vertical permeability is the permeability of the perforated layer or the low-permeability layer. Also, one cannot characterize the strength of the waterdrive based on the pressure-transient data.
To retrieve true reservoir properties and characterize the strength of the waterdrive based on pressure-transient data, an analytical reservoir model was derived in this study. The mathematical formulation of the model is shown in the Appendix. When U.S. field units are used, the resultant constant-rate solution for oil takes the following form:Equation 1
where pd = p-pwf.
The constants B and C are defined asEquations 2 and 3
Noticing that the derivative of Ei (t) is given byEquation 4
the diagnostic derivative of pressure for radial flow becomesEquation 5
Taking the 10-based logarithm of this equation givesEquation 6
This equation indicates that the diagnostic derivative currently used in well-test-analysis practice for radial-flow identification is not a constant during the LSRF (i.e., the radial-flow pressure derivative curve will not have a plateau but will decrease with time). This rate of increase depends on B and C if no other boundary effect exists. Therefore, constants B and C can be used to characterize the strength of the supporting layer.
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