Transport in Porous Media

, Volume 129, Issue 3, pp 901–940 | Cite as

Pressure Transient Behavior of High-Fracture-Density Reservoirs (Dual-Porosity Models)

  • Mojdeh RasoulzadehEmail author
  • Fikri J. Kuchuk


Naturally fractured reservoirs (NFRs) contain primary (macro), secondary, and tertiary (minor) fractures. These reservoirs are heterogeneous, where fracture properties such as length, conductivity, aperture, and orientation significantly vary spatially. In general, fractures in these reservoirs are very conductive whether they are in a discrete or continuous fracture network. The conventional dual-porosity-type models (Warren and Root in SPE J 3(3):245–255, 1963) are widely used to model NFRs either in conventional or unconventional formations. In shale, the naturally fractured shale is often considered as a dual-porosity model and is coupled with discrete network of hydraulic fractures. Expressing flow by dual-porosity model in the complex pore structure of NFRs is an oversimplification which has demonstrated several deficiencies and inconsistencies with field transient data and field applications. We show that the dual-porosity models are applicable to specific sets of NFRs and under restrictive conditions. In several NFRs, fractures create a network, communicate hydraulically with each other globally, and provide overall conductivity for the reservoir. The matrix provides the overall storage capacity, but it should be conductive enough to pass the fluid towards the fractures. Given the contrast between the conductivity of the fractures and the matrix, several flow regimes may form at microscale. For low contrasts, the conductivities of the matrix and fractures are close enough that the medium behaves as a homogenous medium. For higher contrasts, matrix is not conductive enough to be a part of the overall flow process in the reservoir, so it behaves as a source of fluid to fractures. For very high contrasts, the matrix is almost not conductive, so it cannot pass the fluid to the fracture. In this case, the overall flow will be governed by the flow in the fracture network. Other intermediate cases may also occur. The pressure behavior of continuously fractured reservoirs is investigated for various contrasts in matrix and fracture properties. The homogenized model is obtained by using the two-scale homogenization technique. The homogenization is performed by starting from the microscale flow equations in the matrix and fracture coupled via a set of no-jump boundary conditions on the interface of fracture and matrix. This approach captures the details of the flow both within and between the porous matrix and fracture. A general equivalent macroscopic model is proposed; the macroscopic homogenized model is the average of the microscale equations of flow in fracture and matrix. The equivalent porosity and the equivalent permeability of the averaged medium are derived. The equivalent permeability depends on both the fracture’s permeability and the geometry of each block of nonhomogeneity. The memory effect in the medium is included in the averaged model through an integro-differential term. Memory effect represents the difference between the response time of matrix and fracture to the same pressure drop. Using the homogenization technique, several averaged models for fractured medium can be obtained, depending on the ratio of the fracture to matrix diffusivity, the fracture-to-matrix volume ratio, and the number of scales of nonhomogeneity in the medium. For certain contrasts in matrix and fracture properties, such as the ratio of matrix to fracture permeability and the fracture aperture, the general model reduces to Warren and Root’s model. In general, Warren and Root’s model does not describe every fractured reservoir. The impact of the presence of additional fracture network, such as secondary and tertiary fractures on the equivalent model, and pressure transient behavior is highlighted.


Dual porosity Multi-scale media Two-scale homogenization Warren and Root model Naturally fractured media 



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AlabamaTuscaloosaUSA

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