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Impact of pressure-dependent diffusivity on transient pressure analysis of a dry Coalbed Methane (CBM) wells: A new approach

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Abstract

Coalbed Methane (CBM) reservoirs demonstrate sensitivity to in-situ stress conditions. The depletion of reservoir pressure due to production leads to changing stress conditions within the reservoir. The matrix shrinkage phenomenon due to the coal matrix gas desorption results in altering the pore volume of the cleat network. Consequently, the porosity and permeability of the coal cleat system change as pressure depletes. This phenomenon contradicts the assumption made in the derivation of the diffusivity equation. The inherent assumption of reservoir properties produces inaccurate results for CBM reservoirs during conventional pressure transient analysis under the stress-driven permeability effects. This study deals with the issues related to pressure-dependent diffusivity in dry CBM reservoirs. In this paper, the authors attempt to present a new equation that incorporates pressure-dependent cleat porosity and permeability in the transient state flow equation for dry CBM reservoirs. The concept of stress-dependent pseudo pressure (SDPP) and pseudo-time has been leveraged to derive a new solution to the diffusivity equation under transient flow conditions. The outcome of this study finds its application in accurately estimating the cleat permeability and skin in a dry CBM well that flows with reservoir conditions of pressure-dependent permeability.

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Abbreviations

\({A}_{m}\) :

Matrix shrinkage compressibility (atm–1)

\({B}_{g}\) :

Gas formation volume factor (rcc/scc)

\({C}_{f}\) :

Pore volume compressibility (atm–1)

\({C}_{g}\) :

Gas compressibility (atm–1)

\({C}_{s}\) :

Sorption compressibility (atm–1)

\({C}_{t}\) :

Total compressibility (atm–1)

h :

Reservoir thickness (cm)

\({k}_{i}\) :

Initial cleat permeability (Darcy)

\(k\) :

Cleat permeability at pressure P (Darcy)

K :

Bulk modulus (atm)

\(M\left(P\right)\) :

Stress dependent pseudo pressure function (atm2/cp)

\(M\left({P}_{r,t}\right)\) :

Stress dependent pseudo pressure at a distance (r) from the well at any time t (atm2/cp)

\(M\left(\overline{{P }_{{r}_{inv},t}}\right)\) :

Average pseudo pressure (SDPP) over radius of investigation (atm2/cp)

\(M(\overline{P})\) :

Volume averaged stress dependent pseudo pressure (atm2/cp)

M :

Constrained axial modulus (atm)

P :

Pressure at any point within reservoir (atm)

\({P}_{sc}\) :

Standard condition pressure (atm)

\(Pb\) :

Reference pressure (atm)

\({P}_{i}\) :

Initial reservoir pressure (atm)

\(\overline{P}\) :

Average reservoir pressure (atm)

\({P}_{wf}\) :

Flowing bottom hole pressure (atm)

\({P}_{L}\) :

Langmuir pressure (atm)

\(q\) :

Gas flow rate at reservoir condition (cc/sec)

\({q}_{sc}\) :

Standard condition gas flow rate (cc/sec)

\({r}_{e}\) :

Outer boundary radius (cm)

\({r}_{inv}\) :

Radius of investigation (cm)

\({r}_{w}\) :

Wellbore radius (cm)

\({r}_{w}^{\prime}\) :

Effective wellbore radius (cm)

S :

Skin (Dimensionless)

\({t}_{np}\) :

Normalised pseudo time (sec)

\({T}_{sc}\) :

Temperature at standard condition (°R)

T :

Reservoir temperature (°R)

V :

Volume of cylinder representing reservoir (cc)

\({V}_{L}\) :

Langmuir volume (cc/g)

Z :

Real gas compressibility factor (Dimensionless)

\(\alpha \) :

Constant of proportionality (k and \(\varphi \) relation) (Darcy)

\({\varepsilon }_{L}\) :

Volumetric strain at infinite pressure (Dimensionless)

\(\mu \) :

Gas viscosity (cp)

\(\rho \) :

Gas density at reservoir pressure P (g/cc)

\({\rho }_{b}\) :

Coal bulk density (g/cc)

\({\rho }_{sc}\) :

Gas density at standard condition (g/cc)

ν:

Poisson’s ratio (Dimensionless)

\(\varphi \) :

Cleat porosity at reservoir pressure P (Dimensionless)

\({\varphi }_{i}\) :

Initial fracture porosity (Dimensionless)

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Acknowledgements

We express our sincere gratitude to Computer Modelling Group (CMG) for supporting the academic and research activities at Indian Institute of Technology (Indian School of Mines), Dhanbad by donating the reservoir simulation software for research studies. We have used the CMG-GEM numerical simulator to validate the results of the analytical model derived in this paper.

Author information

Authors and Affiliations

  1. Department of Petroleum Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, India

    Rajeev Upadhyay & Vinay Kumar Rajak

  2. Department of Applied Geophysics, Indian Institute of Technology (Indian School of Mines), Dhanbad, India

    Saurabh Datta Gupta

Authors
  1. Rajeev Upadhyay
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  2. Saurabh Datta Gupta
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  3. Vinay Kumar Rajak
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Contributions

Rajeev Upadhyay: Conceptualisation, coding, data compilation, processing, and manuscript writing. Saurabh Datta Gupta: Conceptualisation, data compilation and manuscript writing, problem statement, and result validation. Vinay Kumar Rajak: Data compilation, overall validation, and manuscript writing.

Corresponding author

Correspondence to Saurabh Datta Gupta.

Additional information

Communicated by Arkoprovo Biswas

Appendices

Appendix A: Derivation of equation for transient state flow condition in CBM well flowing under the conditions of pressure-dependent cleat permeability and porosity

Equation (A1) is given by

$$\frac{1}{r}\frac{\partial }{\partial r}\left(\alpha r\frac{\partial M(P)}{\partial r}\right)=\frac{\mu {C}_{t}}{{\varphi }^{n-1}}\left(\frac{\partial M(P)}{\partial t}\right).$$
(A1)

Let us define normalised pseudo time as:

$${t}_{np}={\mu }_{i}{C}_{ti}\int_{0}^{t}\frac{{\varphi }^{n-1}}{\mu {C}_{t}}\partial t.$$
(A2)

Differentiating both sides,

$$\frac{\partial {t}_{np}}{\partial t}= \frac{{{\mu }_{i}{C}_{ti}\varphi }^{n-1}}{\mu {C}_{t}}.$$
(A3)

Let us assume

$$s= \frac{{r}^{2}}{2{t}_{np}}$$
(A4)
$$\frac{\partial s}{\partial r}= \frac{r}{{t}_{np}}$$
(A5)

and

$$\frac{\partial s}{\partial {t}_{np}}= -\frac{{r}^{2}}{2{t}_{np}^{2}}.$$
(A6)

Equation (A1) can be written as:

$$\frac{1}{r}\frac{\partial }{\partial s}\left(\alpha r\frac{\partial M(P)}{\partial s}\frac{\partial s}{\partial r}\right)\frac{\partial s}{\partial r}=\frac{\mu {C}_{t}}{{\varphi }^{n-1}}\left(\frac{\partial M(P)}{\partial {t}_{np}}\frac{\partial {t}_{np}}{\partial t}\right).$$
(A7)

Substituting the values of \(\frac{\partial s}{\partial r} \text{ and } \frac{\partial {t}_{np}}{\partial t}\),

$$\frac{1}{r}\frac{\partial }{\partial s}\left(\alpha r\frac{\partial M\left(P\right)}{\partial s} \frac{r}{{t}_{np}}\right) \frac{r}{{t}_{np}}=\frac{\mu {C}_{t}}{{\varphi }^{n-1}}\left(\frac{\partial M\left(P\right)}{\partial {t}_{np}}\frac{{{\mu }_{i}{C}_{ti}\varphi }^{n-1}}{\mu {C}_{t}}\right),$$
(A8)
$$\frac{\partial }{\partial s}\left(\alpha \frac{\partial M\left(P\right)}{\partial s} \frac{{r}^{2}}{{t}_{np}}\right) \frac{1}{{t}_{np}}={\mu }_{i}{C}_{ti}\frac{\partial M\left(P\right)}{\partial s} \frac{\partial s}{\partial {t}_{np}} ,$$
(A9)
$$\frac{\partial }{\partial s}\left(2\alpha s\frac{\partial M\left(P\right)}{\partial s} \right) \frac{1}{{t}_{np}}=-\frac{{r}^{2}{\mu }_{i}{C}_{ti}}{2{t}_{np}^{2}} \frac{\partial M\left(P\right)}{\partial s},$$
(A10)
$$\frac{\partial }{\partial s}\left(2\alpha s\frac{\partial M\left(P\right)}{\partial s} \right)=-s {\mu }_{i}{C}_{ti}\frac{\partial M\left(P\right)}{\partial s},$$
(A11)
$$2\alpha s\frac{\partial }{\partial s}\left(\frac{\partial M\left(P\right)}{\partial s}\right)+ 2\alpha \frac{\partial M\left(P\right)}{\partial s}= -s {\mu }_{i}{C}_{ti}\frac{\partial M\left(P\right)}{\partial s}.$$
(A12)

Let \(\frac{\partial M\left(P\right)}{\partial s}=P^{\prime}\)

$$2\alpha s\frac{\partial P^{\prime}}{\partial s}+ 2\alpha {P}^{{\prime}}= -s{\mu }_{i}{C}_{ti} {P}^{{\prime}},$$
(A13)
$$\frac{\partial P^{\prime}}{P^{\prime}}= -\frac{{\mu }_{i}{C}_{ti}}{2\alpha } \partial s-\frac{\partial s}{s}.$$
(A14)

Integrating both sides, we get:

$${{sP}}^{{\prime}}={C}_{1}{e}^{\frac{-s{\mu }_{i}{C}_{ti}}{2\alpha }}.$$
(A15)

\({C}_{1}\) can be evaluated using the line source boundary condition

$$\underset{r\to 0}{\text{lim}}\left(r\frac{\partial M\left(P\right)}{\partial r}\right)=\frac{{\varphi }^{n}}{\mu }\frac{P}{Z}\frac{q\mu }{2\pi kh}$$
(A16)
$$\underset{r\to 0}{\text{lim}}\left(r\frac{\partial M\left(P\right)}{\partial s}\frac{\partial s}{\partial r}\right)=\frac{{\varphi }^{n}}{\mu }\frac{P}{Z}\frac{q\mu }{2\pi kh}$$
(A17)

From equations (A4 and A5),

$$r\frac{\partial s}{\partial r}=2s.$$

Equation (A16) becomes,

$$2s\frac{\partial M\left(P\right)}{\partial s}= \frac{{\varphi }^{n}}{\mu }\frac{P}{Z}\frac{q\mu }{2\pi kh},$$
(A18)
$$s{P}^{{\prime}}=\frac{{\varphi }^{n}}{\mu }\frac{P}{Z} \frac{q\mu }{4\pi kh}.$$
(A19)

From equation (A4), as r \(\to 0, s\to 0.\) As \(s\to 0\), equation (A15) reduces to

$${{sP}}^{{\prime}}={C}_{1}.$$
(A20)

Combining equations (A19 and A20), we get:

$${C}_{1}=\frac{{\varphi }^{n}}{\mu }\frac{P}{Z} \frac{q\mu }{4\pi kh}.$$
(A21)

Substituting for \({C}_{1}\) in equation (A15), we get:

$${P}^{{\prime}}=\frac{{\varphi }^{n}}{\mu }\frac{P}{Z}\frac{q\mu }{4\pi kh}\frac{{e}^{\frac{-s{\mu }_{i}{C}_{ti}}{2\alpha }}}{s},$$
(A22)
$$\frac{\partial M\left(P\right)}{\partial s}=\frac{{\varphi }^{n}}{\mu }\frac{P}{Z}\frac{q\mu }{4\pi kh}\frac{{e}^{\frac{-s{\mu }_{i}{C}_{ti}}{2\alpha }}}{s}.$$
(A23)

Applying the equation of state for real gas,

$$\frac{Pq}{Z}=\left(\frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\right).$$
(A24)

Substituting \(\frac{Pq}{Z}\) from equation (A24),

$$\frac{\partial M\left(P\right)}{\partial s}=\frac{{{\varphi }^{n}P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi kh}\frac{{e}^{\frac{-s{\mu }_{i}{C}_{ti}}{2\alpha }}}{s}.$$
(A25)

Using nth exponent porosity-permeability relationship (\(k= \alpha {\varphi }^{n})\), equation (A25) can be expressed as:

$$\frac{\partial M\left(P\right)}{\partial s}=\frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi \alpha h}\frac{{e}^{\frac{-s{\mu }_{i}{C}_{ti}}{2\alpha }}}{s}.$$
(A26)

Integrating both sides

$$\int_{P_i}^{P} \partial M\left(P\right)= \frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi \alpha h} \int_{\propto}^{{\frac{{r}^{2}}{2{t}_{np}}}}\frac{{e}^{\frac{-s{\mu }_{i}{C}_{ti}}{2\alpha }}}{s} \partial s.$$
(A27)

Let \(\frac{s{\mu }_{i}{C}_{ti}}{2\alpha }= \lambda ; \partial s=\frac{2\alpha }{{\mu }_{i}{C}_{ti}}\partial \lambda \)

$$\int_{P_i}^{P} \partial M\left(P\right)= \frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi \alpha h} \int_{\propto}^{{\frac{{r}^{2}{\mu }_{i}{C}_{ti}}{4\alpha {t}_{np}}}} \frac{{e}^{-\lambda }}{\lambda } \partial \lambda ,$$
(A28)
$$M\left({P}_{r,t}\right)= M\left({P}_{i}\right)- \frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi \alpha h} \int_{{\frac{{r}^{2}{\mu }_{i}{C}_{ti}}{4\alpha {t}_{np}}}}^{\propto}\frac{{e}^{-\lambda }}{\lambda } \partial \lambda ,$$
(A29)
$$M\left({P}_{r,t}\right)= M\left({P}_{i}\right)- \frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi \alpha h} E_i\left(\frac{{\mu }_{i}{C}_{ti}{r}^{2}}{4\alpha {t}_{np}}\right).$$
(A30)

We know that Ei(x) ≈ −ln\((\gamma{x} )\) for x < 0.01, where Ei(x) is the exponential integral and \(\gamma \) is the exponential of Euler’s constant (0.5772), i.e., \(\gamma \) = Exp(0.5772).

$$M\left({P}_{r,t}\right)= M\left({P}_{i}\right)- \frac{{P}_{sc}{q}_{sc}T}{{T}_{sc}}\frac{1}{4\pi \alpha h}\text{ln}\left(\frac{4\alpha {t}_{np}}{\gamma {{\mu }_{i}{C}_{ti}r}^{2}}\right).$$
(A31)

Appendix B: Plot of \({\varvec{M\left({P}_{wf}\right)}}\) \({\varvec{ vs. }}\, {\textbf{ln}}({\varvec{t}}_{\varvec{np}})\)

This appendix contains the plot of \(M\left({P}_{wf}\right)\) vs. ln(tnp) for various cases listed in table 2 (see figures B1, B2, B3, B4).

Figure B1
figure 7

Plot of \(M\left({P}_{wf}\right)\) vs. \(\text{ln}({t}_{np})\) – Case 1. The digression from straight line during early time is ascribed to time-dependent desorption in simulator model (CMG-GEM). The transient flow equation derived in this paper corresponds to an instantaneous desorption model. Conversely, the CMG-GEM simulator considers time-dependent desorption. To minimise the gap with analytical model, the simulator model has been assigned small sorption time (1 day). Though minor, the time-dependent desorption in CMG-GEM simulator causes digression from straight line during early time.

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Figure B2
figure 8

Plot of \(M\left({P}_{wf}\right)\) vs. \(\text{ln}({t}_{np})\) – Case 2. Details same as figure B1.

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Figure B3
figure 9

Plot of \(M\left({P}_{wf}\right)\) vs. \(\text{ln}({t}_{np})\) – Case 3. Details same as figure B1.

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Figure B4
figure 10

Plot of \(M\left({P}_{wf}\right)\) vs. \(\text{ln}({t}_{np})\) – Case 4. Details same as figure B1.

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Upadhyay, R., Datta Gupta, S. & Rajak, V.K. Impact of pressure-dependent diffusivity on transient pressure analysis of a dry Coalbed Methane (CBM) wells: A new approach. J Earth Syst Sci 132, 34 (2023). https://doi.org/10.1007/s12040-022-02040-7

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