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Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

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Abstract

Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical simulation. (1) In this paper, the effects of parameters including fracture conductivity, coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant, fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic fracturing design.

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Abbreviations

\(C_\mathrm{fD}\) :

Dimensionless fracture conductivity

\(t_\mathrm{D}\) :

Dimensionless time

\(C_\mathrm{D}\) :

Dimensionless wellbore storage coefficient

\(p_\mathrm{wD}\) :

Dimensionless well bottom pressure

\(p_\mathrm{D}\) :

Dimensionless pseudo pressure

\(dp_\mathrm{D}\) :

Dimensionless pseudo pressure derivative

\(p_\mathrm{fD}\) :

Dimensionless pseudo fracture pressure

\(S_\mathrm{k}\) :

Skin factor

\(x_{\mathrm{D}j}\) :

Midpoint of the \(j\) segment

\(\theta \) :

Fracture asymmetry factor

\(\lambda \) :

Tranfer coefficient of CBM from matrix to cleat

\(\omega \) :

Storage coefficient of cleat

\(\Xi \) :

Rate constant

\(s\) :

Time variable in Laplace domain, dimensionless

\(\tilde{p}_\mathrm{D}\) :

The dimensionless pseudo pressure \(p_\mathrm{D}\) in Laplace domain

\(\tilde{p}_\mathrm{wD}\) :

Bottom pressure \(p_\mathrm{wD}\) in Laplace domain

\(\tilde{p}_\mathrm{fD}\) :

Dimensionless pseudo fracture pressure \(p_\mathrm{fD}\) in Laplace domain

\(\tilde{q}(u)\) :

Fracture rate \(q(x,t)\) in Laplace domain

\(\tilde{q}_\mathrm{fD}\) :

Dimensionless fracture rate \(q_\mathrm{fD}\) in Laplace domain

\(c_\mathrm{t}\) :

Total compressibility, 1/psi

\(k\) :

Effective permeability, mD

\(p\) :

Bottomhole pressure, psi

\(p_\mathrm{ic}\) :

Initial formation pressure, psi

\(q\) :

Rate of per unit fracture length from formation, MMscf/d

\(\mu \) :

Fluid viscosity, cp

\(h\) :

Formation thickness, ft

\(\phi \) :

Porosity, fraction

\(r\) :

Reservoir radius, ft

\(r_\mathrm{e}\) :

Equivalent drainage radius, ft

\(t\) :

Time variable, h

\(h\) :

Formation thickness, ft

\(\tau \) :

Temperature, \(^{\mathrm{o}}\)R

\(Z\) :

Gas compressibility factor, fraction

\(x_\mathrm{f}\) :

Fracture half length, ft

\(w\) :

Width of the fracture, ft

\(x^{\prime }\) :

Integral variable

\(C\) :

Volumetric gas concentration in the micropores, scf/ft\(^{3}\)

\(D\) :

Diffusion coefficient, \(\text{ ft}^{2}\)/h

\(T\) :

Sorption time constant, h

\(V_\mathrm{L}\) :

Total sorption capacity, \(\text{ scf/ft}^{3}\)

\(K_{0}(x)\) :

Modified Bessel function (2nd kind, zero order)

\(K_{1}(x)\) :

Modified Bessel function (2nd kind, first order)

\(I_{0}(x)\) :

Modified Bessel function (1st kind, zero order)

\(I_{1}(x)\) :

Modified Bessel function (1st kind, first order)

\(a\) :

Macropore property

\(i\) :

Micropore property

\(f\) :

Fracture property

\(D\) :

Dimensionless

\(g\) :

Gas property

\(sc\) :

Standard condition

\(ic\) :

Initial condition

\(w\) :

Wellbore property

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Acknowledgments

This article was supported by Important National Science and Technology Specific Projects of the twelfth five Years Plan Period (Grant No.2011ZX05013-002) and the National Basic Research Program of China (Grant No.2011ZX05009-004).

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Authors and Affiliations

  1. School of Energy Resources, China University of Geosciences, Beijing, 100083, People’s Republic of China

    Lei Wang, Xiaodong Wang, Junqian Li & Jiahang Wang

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  1. Lei Wang
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Appendices

Appendix A

$$\begin{aligned} \alpha =2.637\times 10^{-4} \quad \beta =1.422\times 10^{6} \end{aligned}$$

Appendix B

See Table 4.

Table 4 SI metric conversion factors
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Wang, L., Wang, X., Li, J. et al. Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs. Transp Porous Med 97, 353–372 (2013). https://doi.org/10.1007/s11242-013-0128-z

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