The Permeability Variogram from Pressure Transients of Multiple Wells: Theory and 1-D Application
- Yanis C. Yortsos | Nabeel Al-Afaleg
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 328 - 337
- 1997. Society of Petroleum Engineers
- 5.6.4 Drillstem/Well Testing, 1.6.9 Coring, Fishing, 5.6.1 Open hole/cased hole log analysis, 5.1 Reservoir Characterisation, 5.6.5 Tracers, 5.1.1 Exploration, Development, Structural Geology, 5.6.3 Pressure Transient Testing, 4.3.4 Scale, 5.1.5 Geologic Modeling
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This paper presents a new approach for the estimation of the large-scale correlation function of reservoir heterogeneity from the analysis of the pressure transient response of multiple wells. The approach is based on the theory of small fluctuations of the logarithm of the permeability and models the response of the "ensemble average pressure", obtained by averaging the pressure response of multiple well tests. A non-local diffusivity equation for the ensemble-average pressure, which incorporates directly the permeability correlation function, is obtained, from the analysis of which the permeability semi-variogram of the reservoir can be constructed in principle. We consider a particular application to 1-D geometries, where a type-curve is also derived for the case of an exponential correlation function. Numerical simulations results support the applicability of the method.
The stochastic representation of subsurface reservoirs requires reliable estimates of the correlation structure of reservoir attributes. Basing the correlation structure on core or well log derived data requires a substantial number of probing points, each of which has a limited radius of investigation. Single- or multiple-well pressure transient tests offer a substantially larger investigation capability, while requiring fewer probing points. At present, however, a theoretical foundation to relate the pressure transient responses to heterogeneity characteristics, in general, and to the nature of correlation structure, in particular, is missing.
The application of well-testing to idealized homogeneous reservoirs has been extensive and it is described in many classical works. The classical approach provides a first approximation to of the flow properties, such as the average permeability. With the increasing recognition of the heterogeneous nature of oil reservoirs, however, many attempts have recently been made to extend the usefulness of well testing to the interpretation of the heterogeneity of reservoir properties.
The origin of most of the works in this area can be traced to Oliver,1 who evaluated the next, beyond the homogeneous, term in the asymptotic expansion of the well pressure response for a system with a weakly varying permeability. Oliver1 showed that the problem effectively reduces to that of flow in a reservoir with an effective permeability which is radially dependent. Oliver's solution is expressed in terms of a composite integral, involving the product of the unknown radially-averaged permeability fluctuation with a known kernel (similar to an integral transform), the inversion of which could in principle yield the radially-averaged permeability fluctuations. In a subsequent paper, Oliver2 argued against the existence of a unique solution when matching this solution with a discrete well response. He, then, proposed an approximation based on the method of Backus and Gilbert,3 which provides a smooth approximation to the true solution. Feitosa et al.4 proposed a different approach, based on an Inverse Solution Algorithm (ISA), which although not guaranteeing uniqueness in all cases, gives practically useful results and appears to circumvent some of the problems.
Subsequent work built on the same line of reasoning. Oliver5 extended his approach to problems involving storativity (in addition to transmissibility) fluctuations, and proceeded with the development of an analogous equation for interference testing. Yadavalle et al.6,7 proceeded one step further, and used the results of Feitosa et al.4 to infer a semi-variogram based on the inverted data of the radially-averaged fluctuations.
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