Determination of Fractal Parameters of Fracture Networks Using Pressure-Transient Data
- F. Flamenco-Lopez (Pemex E&P) | R. Camacho-Velazquez (Pemex E&P)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- February 2003
- Document Type
- Journal Paper
- 39 - 47
- 2003. Society of Petroleum Engineers
- 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 5.1.1 Exploration, Development, Structural Geology, 4.1.2 Separation and Treating, 5.6.4 Drillstem/Well Testing, 5.8.6 Naturally Fractured Reservoir, 5.5 Reservoir Simulation, 5.6.1 Open hole/cased hole log analysis, 5.6.3 Pressure Transient Testing, 5.4.2 Gas Injection Methods, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation
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The objective of this paper is to investigate the transient pressure behavior of naturally fractured reservoirs with fractal characteristics. This work is based on the findings of previous studies, which have shown that the networks of fractures in some reservoirs are fractals. Thus, using this assumption, an approximate analytical solution for dual-porosity systems is derived in the Laplace space.
The approximate solution presented in this work uses a pseudosteady- state matrix-to-fractal fracture transfer function. This solution is compared with a finite-element solution, and good agreement is found. Short- and long-time approximations are used to obtain procedures in time to determine some fractal parameters. These approximations are compared to the appropriate expressions when an unsteady-state matrix-to-fractal fracture transfer function is used.
Synthetic and field examples are presented to illustrate the methodology proposed in this work.
This paper also presents an analytical solution for the pseudosteady- state flow period and demonstrates the importance of analyzing both transient and pseudosteady-state flow-pressure data for a single-well situation to fully characterize a naturally fractured reservoir with a fractal geometry.
Reservoir heterogeneity influences fluid-flow trajectories. These heterogeneities are present on a wide range of scales. Fractal theory provides a method to describe the network of fractures in a rock and to connect heterogeneities at smaller scales to those at larger scales and vice versa. The simplest fractal models assume a power-law scaling procedure in which the exponent is related to the fractal dimension, which is the main tool of fractal geometry. The dimension provides a description of how much space the set of fractures fills; it contains a great deal of information about the geometrical properties of the fracture set. Thus, this fractal parameter describes variations over a range of scales; if it can be estimated, reservoir properties such as porosity and permeability can be defined properly.
Chang and Yortsos1 applied a fractal model to pressuretransient analysis. This model describes a naturally fractured system with different scales, poor fracture connectivity, and disorderly spatial distribution in a proper fashion. Acuña et al.2 applied this model to analyze pressure transient tests, and they (like Chang and Yortsos) discovered that the change in wellbore pressure is a power-law function of time, in which the fraction or spectral dimension can be obtained. This parameter is a function of both the fractal dimension and the conductivity index. The latter is related to the topology of the fracture network. Both Refs. 1 and 2 point out that to determine the values of fractal parameters, additional information regarding the transient test is needed.
Olarewaju3 also examined the pressure-transient response of naturally fractured reservoirs by using a fractal model, but instead of assuming a pseudosteady-state transfer function between matrix and fracture systems as Refs. 1 and 2 did, he used a transient interporosity flow assumption.
Beier4 extended the fractal model of Chang and Yortsos1 to consider a hydraulic fractured well. He also observed a power-law behavior during the linear and radial flow periods.
In spite of all the work done on pressure-transient analysis using fractal geometry, it is not possible to fully characterize a naturally fractured reservoir with fractal geometry using well-test data. As Chang and Yortsos1 mentioned in their seminal work, the determination of the four parameters of their fractal model (one of these being the fractal dimension) from transient well-test information is similar to the problem of determining porosity and permeability for the Euclidean geometry case from a single-well transient test. It is also well known that most porosity well logs have been found to have a fractal character; in fact, the fractal dimension can be obtained by using techniques like rescaled range, variogram, and spectral methods.5-9 Thus, to determine the fractal model parameters, we either use information from porosity well logs, or we try to attain this information from another type of test.
The purpose of this work is to present an analytical solution during the pseudosteady-state flow period and to show that it is possible, using both this solution and the transient response, to obtain the values of all four parameters of the fractal model.
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