# Pressure transient response of stochastically heterogeneous fractured reservoirs

## Abstract

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblatt*et al.*'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters*λ* and*Ω*, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters*λ* and*Ω*, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir*λ* and*Ω*, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.

## Key words

Fractured reservoirs double-porosity models## Nomenclature

*a*coefficient in the sink/source function

*c*compressibility

*E*{ }ensemble average

*G*(,)Green function

*h*thickness of reservoir

*k*permeability

*¯k*trend component of

*k*- \(\hat k\)
fluctuated component of

*k**L*characteristic length

*n*_{i}unit normal vector

*p*pressure

- \(\hat p\)
Laplace transform of

*p**r*_{0}autocorrelation distance

*r*_{w}well radius

*q*rate of flow

*u*flux

*t*time

**x**Cartesian coordinates

*α*geometrical parameter

- γ
*i* part of the boundary

*λ*(

*ar*_{w}^{2}*k*_{1}/*k*_{2}) dimensionless parameter*Ω**Φ*_{2}*c*_{2}/(*Φ*_{1}*c*_{1}+*Φ*_{2}*c*_{2}) dimensionless parameter*Μ*dynamic viscosity

- \(\sigma _{\upsilon _k }\)
standard deviation governing the variation of\(\hat k_2\)

- \(\sigma _{\upsilon _\phi }\)
standard deviation governing the variation of\(\hat \phi _2\)

*ρ*autocorrelation function

*Φ*porosity

- \(\bar \phi\)
trend component of

*Φ*- \(\hat \phi\)
fluctuated component of

*Φ*- ∇
nabla operator

## Subscripts

- 1
medium of blocks (matrix)

- 2
medium of fracture

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## References

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