Semi-Analytical Well Model of Horizontal Wells With Multiple Hydraulic Fractures
- J. Wan (Stanford U.) | K. Aziz (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2002
- Document Type
- Journal Paper
- 437 - 445
- 2002. Society of Petroleum Engineers
- 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.6.4 Drillstem/Well Testing, 5.5 Reservoir Simulation, 3 Production and Well Operations, 5.8.6 Naturally Fractured Reservoir, 2.5.4 Multistage Fracturing, 5.1.5 Geologic Modeling, 4.1.2 Separation and Treating
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A well model is required to relate the well rate to the well pressure and the wellblock pressure while modeling wells in reservoir simulators. The well index is calculated automatically by simulators for conventional wells, but it is generally calculated by the user and supplied in the input while modeling nonconventional wells when the default procedure available in commercial simulators is not adequate.
In this paper we describe a new semi-analytical solution for horizontal wells with multiple fractures. The fractures can be rotated at any horizontal angle to the well, and they need not fully penetrate the formation in the vertical direction. This solution for hydraulically fractured wells is obtained by applying Fourier analysis to a 2D solution; therefore, this solution is easy to obtain when the 2D solution is available. The analytical solution provides the well pressure that can be combined with a numerically computed gridblock pressure to obtain the well index (WI). Results of our rigorous solution are compared with empirical approaches currently in use for calculating the well index and the productivity index. Examples are given when horizontal wells with fractures are modeled using our new approach and conventional methods. These results demonstrate the need for computing the correct WI.
The technology of fracturing horizontal wells is being widely used by the petroleum industry. Therefore one will ask the questions: Are the currently available models adequate for predicting performance of horizontal wells with fractures? If not, how can we develop more appropriate models?
For the modeling of wells in reservoir simulators, a well model is required to relate the well flow rate to the well pressure and the wellblock pressure. Peaceman1,2 established a mathematical relationship between the wellblock pressure and wellbore pressure for a fully penetrating vertical well under the condition of 2D flow. This relationship is valid for isolated wells under steady-state or pseudosteady conditions. Peaceman's well model is usually the default in general reservoir simulators. Because of the variety of nonconventional wells used by industry, such as horizontal and fractured wells, and because of the variety of grid systems employed, much research is being conducted on improving simulator accuracy. Some of the more important works are summarized here.
Babu et al.3 extended Peaceman's work to the case of a uniform flux horizontal well in a slab-like drainage area. Penmatcha et al.4,5 have extended Babu and Odeh's model to the case of infinite well conductivity. Jasti et al.6 present results to increase simulator accuracy when solving the problem of a system of partially penetrating wells having arbitrary trajectories. Ding7 used the concept of layer potentials and applied transmissibility adjustments. Maizeret8 developed an analytical solution to investigate well indices for nonconventional wells. In 1998, Wan et al.9 compared several well models, including explicit well models, and showed the importance of correct well indices.
Our focus here is on horizontal wells with single or multiple fractures. Basically, there are three ways to model hydraulic fractures:
Refine the grids, so that the true geometry of the fractures is modeled. In this method, fractures are represented by very fine gridblocks; properties of the fractures, such as permeability, are assigned to those fracture blocks. Schulte10 evaluated the transient pressure behavior of a fractured vertical well in this way. However, the size of the fracture blocks was so small that numerical instabilities were encountered. To overcome this problem, Roberts et al.11 increased the fracture grid width and decreased the fracture permeability.
Modify the transmissibilities of the blocks that contain the fractures. Work has been done in independently generating fracture transmissibilities with the transmissibilities of reservoir blocks adjacent to the fracture plane.12,13 Therefore, the productivity of the fracture is included while the negligible volume of the fracture is excluded. These authors also apply various coupling methods to deal with the interfaces between the fracture model and the reservoir model. Hegre14 computed correct transmissibilities for a coarse grid, which gives the same long-term pressure behavior as a fine-grid model. The transmissibility adjustments are based on results from a model, which is fine gridded in the near wellbore/ fracture area with explicit modeling of the fractures.
Modify the effective wellbore radius or well index. Cinco- Ley and Samaniego15 showed that the effective wellbore radius is one-fourth of the total fracture length for an infinite conductivity vertical fracture, so this effective wellbore radius can be used instead of rw in the productivity index formulas. Hegre and Larsen16 showed that the effective wellbore radius for a horizontal well intercepted by either a longitudinal fracture or multiple transverse fractures can be analytically determined. They used the rew calculated in this manner in the well index formula (see Eq. 15) instead of rw.
The approach of modifying the effective wellbore radius is used in this study. A semi-analytical approach is used to calculate the well index.6,8 First we develop the analytical solution for a horizontal well with multiple fractures in a brick-shaped drainage volume. With the wellbore pressure from this analytical solution and the wellblock pressure from a single-phase numerical simulator, the effective wellbore radius is calculated.
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