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Improved numerical inverse Laplace transformation to improve the accuracy of type curve for analyzing well-testing data

  • Short Communication - Hydrology
  • Published:
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Abstract

Analyzing well-testing data by the type-curve matching is a modern well-testing analysis method and is widely used in the petroleum and gas industry. By improving accuracy of type curve, we can get more accurate results from analyzing well-testing data, which provide a scientific base for development of oil, gas and water resources. By solving percolation equations, we can obtain type curves. The Laplace transformation methods are often used to solve them. In this paper, we improve the accuracy of type curve by improving the numerical inverse Laplace transformation (NILT) based on infinite series. We combine the NILT based on infinite series with Levin convergence acceleration and determine necessary parameters through numerical experiments to improve accuracy and speed. To verify this method, we compare the improved method with the Stehfest method using some functions such as trigonometric function. Type curves for analysis of well-testing data for the homogeneous reservoir with elastic outer boundary and a dual porosity reservoir are plotted and compared by using the improved numerical inversion and the Stehfest numerical inversion, respectively. These results show that type curves plotted by the improved method are less in vibration and fluctuation than ones plotted by the Stehfest method.

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Abbreviations

a, k, σ, N :

Constant

B :

Formation volume factor, m3/m3

C :

Wellbore storage, m3/Pa

C D :

Dimensionless wellbore storage, dimensionless

C t :

Total compressibility, Pa−1

C m t, C f t :

Total compressibility of matrix and fracture systems, respectively, Pa−1

F(s):

Function in Laplace domain

f(t):

Function in time domain

\(\tilde{f}\left( t \right)\) :

Approximation of the exact \(f\left( t \right)\)

h :

Thickness of reservoir, m

i :

Imaginary unit

Im:

Imaginary part

K :

Permeability, m2

n :

Number of series

P :

Pressure, Pa

P D :

Dimensionless pressure, dimensionless

P i :

Initial pressure in reservoir, Pa

P fD :

Dimensionless pressure in fracture, dimensionless

P mD :

Dimensionless pressure in matrix, dimensionless

P wf :

Wellbore pressure, Pa

\(\bar{P}_{D}\) :

Dimensionless pressure in Laplace space, dimensionless

q :

Production rate, m3/s

r :

Radial distance, m

R :

Boundary radius, m

r w :

Wellbore radius, m

r D :

Dimensionless radius, dimensionless

R D :

Dimensionless boundary radius, dimensionless

S :

Skin, dimensionless

S n :

Series

t :

Time, s

t D :

Dimensionless time, dimensionless

z :

Laplace variable

\(\beta = \text{Re} (z)\) :

numerical inverse Laplace transformation (NILT)

λ :

Interporosity flow parameter, dimensionless

π :

3.141592653589793…

µ :

Viscosity of fluid, Pa·s

\(\varepsilon _{\Gamma }^{{P_{D} }}\) :

Elastic coefficient, dimensionless

ϕ :

Porosity, fraction

ϕ m , ϕ f :

Fporosity of matrix and fracture, respectively (fraction), dimensionless

ω :

Storage coefficient, dimensionless

\(\Delta ^{k}\) -k st :

Order finite difference

D :

Difference

f :

Fracture

m :

Matrix

\(^-\) :

Laplace transformation

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Acknowledgements

We gratefully acknowledge Dr. Yun for her valuable suggestions and discussions.

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Authors and Affiliations

  1. Faculty of Mining Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

    Song Chol Kim, Yong Il Song & Chol Gwang Han

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  1. Song Chol Kim
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Contributions

Song Chol Kim took part in design, methodology, writing—original draft preparation. Yong Il Song involved in resources, writing—review and editing. Chol Gwang Han participated in comparison study.

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Correspondence to Song Chol Kim.

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Communicated by Michael Nones, Ph.D. (CO-EDITOR-IN-CHIEF).

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Kim, S.C., Song, Y.I. & Han, C.G. Improved numerical inverse Laplace transformation to improve the accuracy of type curve for analyzing well-testing data. Acta Geophys. 69, 919–930 (2021). https://doi.org/10.1007/s11600-021-00585-7

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