Pressure Transient Model of a Vertically Fractured Well in a Fractal Reservoir
- R.A. Beier (Conoco Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Formation Evaluation
- Publication Date
- June 1994
- Document Type
- Journal Paper
- 122 - 128
- 1994. Society of Petroleum Engineers
- 5.4.2 Gas Injection Methods, 5.6.1 Open hole/cased hole log analysis, 4.3.4 Scale, 5.4 Enhanced Recovery, 5.6.4 Drillstem/Well Testing, 5.8.7 Carbonate Reservoir, 3 Production and Well Operations, 1.6.9 Coring, Fishing, 5.8.6 Naturally Fractured Reservoir
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In many cases, field pressure transient data are matched with homogeneous reservoir models to get effective reservoir properties, even though log and core data show the subject reservoir is heterogeneous. This paper presents field data and a model that demonstrate the above approach does not always work. A pressure transient model is developed for a heterogeneous reservoir with a fractal structure, The model treats a well with a vertical fracture and matches field data that previously could not be analyzed.
Conventional pressure transient models have been developed with the assumption that reservoirs are areally homogeneous. Yet, core, log, and outcrop data indicate this assumption is not justified in many cases. Still, homogeneous models are applied to obtain an effective permeability corresponding to a fictitious homogeneous reservoir. This approach seems reasonable if the correlation length of the permeability variation is small compared to the permeability variation is small compared to the interwell scale and the permeability variation is sufficiently small. Mishra et al. formed a heterogeneity index that contains both the spatial correlation and the variation. One would suspect the homogeneous model might eventually fail to match field data as the permeability variation and correlation length scale both increase. The field data given in the present report confirm that homogeneous pressure transient models do not always apply. Instead, a model developed for a heterogeneous reservoir with a fractal structure matches some recent field tests in the Crayburg/San Andres Formations in southeastern New Mexico.
The model considers a heterogeneous reservoir that contains both permeable and impermeable rock. The resulting permeable network is assumed to have a fractal structure. The fractal structure imposes heterogeneities at all length scales. The permeability distribution is bimodal, since at any permeability distribution is bimodal, since at any location, the permeability is either zero or a fixed finite value. The model treats a well with a vertical fracture in an infinite reservoir. The model is an extension of a previous fractal model that handles a finite circular wellbore. Chang and Yortsos have previously applied a fractal model to naturally fractured reservoirs.
FRACTAL PERMEABLE NETWORK
Core samples can give some indication of spatial permeability patterns. The thin section in Fig. 1 permeability patterns. The thin section in Fig. 1 was taken from a San Andres Formation (southeastern New Mexico) core that was impregnated by blue-dye epoxy. The dark color identifies the network of more permeable rock invaded by the epoxy. For the sake of discussion consider the dark pattern as representative of the permeable network at the scale of a well pattern. The uninvaded (lighter colored rock) Is treated as impermeable. Imagine a well at the upper left hand corner. Choose a nearby point In the dark area. A nearly straight path can be found that stays within the permeable dark area back to the well. Now choose points that are successively further from the imaginary well but still in the dark area. As the distance from the wellbore increases, the path back to the well will more likely encounter larger impermeable areas to wind around. For more distant points the path becomes more tortuous, This causes the effective hydraulic diffusivity to decrease away from the wellbore.
From the work of Cefen et al. we expect the hydraulic diffusivity, n, for a fractal permeable network to scale as
where O is a parameter related to the topology of the network (O < O). Here r is the distance from the wellbore. In practice Eq. 1 should apply over a finite range of length scales where and are the lower and upper cutoff lengths, respectively. Then, for , conventional averaging techniques such as arithmetic and/or geometric averaging might apply. These cutoff lengths have not been established for any oil or gas reservoirs.
The hydraulic diffusivity contains both porosity and permeability. The porosity of an areal network permeability. The porosity of an areal network scales as
where d is the fractal dimension.
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