Transport in Porous Media

, Volume 117, Issue 1, pp 69–102 | Cite as

Addressing the Influence of a Heterogeneous Matrix on Well Performance in Fractured Rocks

  • R. RaghavanEmail author
  • C. Chen


We address the influences of heterogeneity and complex geology of the matrix of fractured rocks on transient flow. Fractional constitutive flux laws reflect the stochastic framework that we consider. We model transient diffusion in both the matrix and fracture systems in terms of a continuous time random walk. Our procedure is particularly suited to address a complex geology that may exist on a number of scales and which may include dead ends and discontinuities. The performance of a horizontal well produced through arbitrarily located, multiple hydraulic fractures with distinct properties (length, width, permeability) forms the basis for our thesis. The pressure distribution in a rectangular drainage region where the well may be placed arbitrarily is expressed in terms of the Laplace transformation. The required solutions are obtained numerically. The focus of our work is on long-term behaviors for production at a constant pressure. In addition to numerical solutions, asymptotic results that provide information on the structure of the solutions are presented. Agreement between the asymptotic and numerical solutions is excellent. We show that long-term responses are governed by two distinct, two-parameter Mittag–Leffler functions and are an outcome of the complexities we desire to model in both the matrix and fracture systems. As a consequence, power-law behaviors that reflect the heterogeneity inherent in the system define long-time expectations. We show that the new solutions we derive do reduce to those of classical diffusion; that is, results corresponding to classical diffusion are a subset of the new results obtained here. Our results are particularly suited to model transient flow in shale reservoirs.


Multiporosity Rate decline Pressure behavior Fractured rocks Hydraulic fractures Fractured horizontal wells Fractional diffusion Anomalous diffusion Mittag–Leffler function Power-law decline 

List of symbols


Drainage area (\(\hbox {L}^2\))


See Eq. (78)

\( \tilde{C}\)

See Eq. (84)


Compressibility (\(\mathrm{L T}^2/\mathrm{M}\))


Shape factor


Distance between the outer fractures (L)

\(E_{\alpha , \beta }( z) \)

Mittag–Leffler function; see (Eq. 57)


Function defined in Eq. (24)


Thickness (L)

\(K_{\nu }(z)\)

Modified Bessel function of the second kind of order \(\nu \)


Permeability (\(\hbox {L}^2\))

\(k_\mathrm{F} \)

Permeability of hydraulic fracture (\(\hbox {L}^2\))


See Eq. (3)

\(I_{\nu }(z)\)

Modified Bessel function of the first kind of order \(\nu \)

\(L_\mathrm{f} \)

One half of the length of hydraulic fracture (L)

\({{{\mathrm{\mathcal {L}}}}}\)

Laplace transform

\(\ell \)

Reference length (L)


Number of fractures


Pressure (\(\hbox {M}/\hbox {L}/\hbox {T}^2\))


Rate (\(\mathrm{L}^3/T\))


Time (T)


Laplace variable (1/T)


Coordinates (L)


Width (L)

\(\alpha \)

Exponent; see Eq. (2)

\(\Gamma (x)\)

Gamma function

\(\gamma \)

Euler’s constant (\(0.5772 \ldots \) )

\(\sigma \)

Pseudoskin factor; single fracture

\(\eta \)

Diffusivity; various

\(\tilde{\eta }\)

‘Diffusivity’; see Eq. (16) (\(\mathrm{L}^{2}/\mathrm{T}^\alpha \))

\(\vartheta \)

See Eq. (1)

\(\lambda \)

Mobility (L/T/M)

\(\mu \)

Viscosity (M/L/T)

\(\nu \)

Exponent; see Eq. (82)

\(\phi \)

Porosity (\(\hbox {L}^3/\hbox {L}^3\))







Natural fracture system


Hydraulic fracture










Laplace transform


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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.R. Raghavan, IncTulsaUSA
  2. 2.Kappa EngineeringHoustonUSA

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