Evaluation of reservoir parameters through well-test and decline-curve
analysis is a current practice used to estimate formation parameters and to
forecast production decline identifying different flow regimes, respectively.
From practical experience, it has been observed that certain cases exhibit
different wellbore pressure and production behavior from those presented in
previous studies. The reason for this difference is not understood completely,
but it can be found in the distribution of fractures within a naturally
fractured reservoir (NFR). Currently, most of these reservoirs are studied by
means of Euclidean models, which implicitly assume a uniform distribution of
fractures and that all fractures are interconnected. However, evidence from
outcrops, well logging, production-behavior studies, and the dynamic behavior
observed in these systems, in general, indicate the above assumptions are not
representative of these systems. Thus, the fractal theory can contribute to
explain the above. The objective of this paper is to investigate the
production-decline behavior in an NFR exhibiting single and double porosity
with fractal networks of fractures. The diffusion equations used in this work
are a fractal-continuity expression presented in previous studies in the
literature and a more recent generalization of this equation, which includes a
temporal fractional derivative. The second objective is to present a combined
analysis methodology, which uses transient-well-test and
boundary-dominated-decline production data to characterize an NFR exhibiting
fractures, depending on scale. Several analytical solutions for different
diffusion equations in fractal systems are presented in Laplace space for both
constant-wellbore-pressure and pressure-variable-rate inner-boundary
conditions. Both single- and dual-porosity systems are considered. For the case
of single-porosity reservoirs, analytical solutions for different diffusion
equations in fractal systems are presented. For the dual-porosity case, an
approximate analytical solution, which uses a pseudosteady-state
matrix-to-fractal fracture-transfer function, is introduced. This solution is
compared with a finite-difference solution, and good agreement is found for
both rate and cumulative production. Short- and long-time approximations are
used to obtain practical procedures in time for determining some fractal
parameters. Thus, this paper demonstrates the importance of analyzing both
transient and boundary-dominated flow-rate data for a single-well situation to
fully characterize an NFR exhibiting fractal geometry.
Synthetic and field examples are presented to illustrate the methodology
proposed in this work and to demonstrate that the fractal formulation
consistently explains the peculiar behavior observed in some real
Evaluation of reservoir parameters through decline-curve analysis has become
a common current practice (Fetkovich 1980; Fetkovich et al. 1987). The main
objectives of the application of decline analysis are to estimate formation
parameters and to forecast production decline by identifying different flow
Different solutions have been proposed during both transient
(Ehlig-Economides and Ramey 1981; Uraiet and Raghavan 1980) and
boundary-dominated (Fetkovich 1980; Fetkovich et al. 1987; Ehlig-Economides and
Ramey 1981; Arps 1945) flow periods. Both single- and double-porosity (Da Prat
et al. 1981; Sageev et al. 1985) systems have been addressed. During the
boundary-dominated-flow period, in homogeneous systems, there is a single
production decline, but for NFRs in which the matrix participates, there are
two decline periods, with an intermediate constant-flow period (Da Prat et al.
1981; Sageev et al. 1985).
Carbonate reservoirs contain more than 60% of the world’s remaining oil.
Yet, the very nature of the rock makes these reservoirs unpredictable.
Formations are heterogeneous, with irregular flow paths and circulation traps.
In spite of this complexity, at present, all studies on
constant-bottomhole-pressure tests found in petroleum literature assume
Euclidean or standard geometry is applicable to both single-porosity reservoirs
and NFRs (Fetkovich 1980; Fetkovich et al. 1987; Ehlig-Economides and Ramey
1981; Uraiet and Raghavan 1980; Arps 1945; Da Prat et al. 1981; Sageev et al.
1985), even though real reservoirs exhibit a higher level of complexity.
Specifically, natural fractures are heterogeneities that are present in
carbonate reservoirs on a wide range of spatial scales. It is well known that
flow distribution within the reservoir is controlled mostly by the distribution
of fractures (i.e., geometrical complexity). There could be regions in the
reservoir with clusters of fractures and others without the presence of
fractures. The presence of fractures at different scales represents a relevant
element of uncertainty in the construction of a reservoir model. Thus, highly
heterogeneous media constitute the basic components of an NFR, so Euclidean
flow models have appeared powerless in some of these cases. Alternatively,
fractal theory provides a method to describe the complex network of fractures
(Sahimi and Yortsos 1970).
The power-law behavior of fracture-size distributions, characteristic of
fractal systems, has been found by Laubach and Gale (2006) and Ortega et al.
(2006). Distributions of attributes such as length, height, or aperture can
frequently be expressed as power laws. Scaling analysis is important because it
enables us to infer fracture attributes such as fracture strike, number of
fracture sets, and fracture intensity for larger fractures from the analysis of
microfractures found in oriented sidewall cores. This approach offers a method
to overcome fracture-sampling limitations, with microfractures as proxies for
related macrofractures in the same rock volume (Laubach and Gale 2006; Ortega
et al. 2006).
The first fractal model applied to pressure-transient analysis was presented
by Chang and Yortsos (1990). Their model describes an NFR that has, at
different scales, poor fracture connectivity and disorderly spatial
distribution in a proper fashion. Acuña et al. (1995) applied this model and
found the wellbore pressure is a power-law function of time. Flamenco-Lopez and
Camacho-Velazquez (2003) demonstrated that to characterize a NFR fully with a
fractal geometry, it is necessary to analyze both transient- and
pseudosteady-state-flow well pressure tests or to determine the fractal-model
parameters from porosity well logs or another type of source.
Regarding the generation of fracture networks, Acuña et al. (1995) used a
mathematical method for this purpose, while Philip et al. (2005) used a
fracture-mechanics-based crack-growth simulator, instead of a purely stochastic
method, for the same objective.
In spite of all the work done on decline-curve analysis, the problem of
fully characterizing an NFR exhibiting fractal geometry by means of production
data has not been addressed in the literature. Thus, the purpose of this work
is to present analytical solutions during both transient- and
boundary-dominated-flow periods and to show that it is possible to characterize
an NFR having a fractal network of fractures with production-decline data.
© 2008. Society of Petroleum Engineers
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- Original manuscript received:
14 June 2006
- Meeting paper published:
31 August 2006
- Revised manuscript received:
21 January 2008
- Manuscript approved:
18 February 2008
- Version of record:
20 June 2008