Analytical solutions for radial pressure distribution including the effects of the quadratic‐gradient term

Chakrabarty, Chayan; Ali, Sajjad; Tortike, W.S.

doi:10.1029/92wr02892

Cited by 33 publications

(20 citation statements)

References 11 publications

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“…1.7 depends not only on β l and β K , but also on the pressure gradient. In the purely radial flow problem considered by Odeh and Babu (1988), Chakrabarty et al (1993); Jelmert and Vik (1996); Braeuning et al (1998), and Tong and Wang (2005), very large pressure gradients can be expected in the vicinity of a narrow wellbore. In view of this, the range of compressibility for which significant nonlinearity can be expected is better investigated with reference to one-dimensional flow between parallel planes, since in this geometry the pressure gradient is bounded.…”

Section: One-dimensional Dirichlet Problemsmentioning

confidence: 98%

“…1.7 are special cases of the CHT, corresponding to various choices of the integration constants a and b. Most treatments (Odeh and Babu 1988;Chakrabarty et al 1993;Jelmert and Vik 1996;Braeuning et al 1998) omit the pre-exponential factor of 1/β in Eq. 2.7, which is equivalent to setting a = ln β.…”

Section: The Cole-hopf Transformationmentioning

confidence: 99%

“…1.8 subject to various boundary conditions. In contrast, many treatments of the radial flow problem, motivated by the interpretation of well-drawdown transients, have removed the squared gradient term by a change of variable (Odeh and Babu 1988;Chakrabarty et al 1993;Jelmert and Vik 1996;Braeuning et al 1998;Tong and Wang 2005). While nonlinear effects will obviously become significant if β l + β K is sufficiently large, there are no published studies determining the numerical values for which solutions of Eq.…”

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confidence: 93%

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Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

Marshall

2008

Transp Porous Med

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In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole-Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pruß equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.

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Section: One-dimensional Dirichlet Problemsmentioning

confidence: 98%

Section: The Cole-hopf Transformationmentioning

confidence: 99%

mentioning

confidence: 93%

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Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

Marshall

2008

Transp Porous Med

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“…Several procedures have been proposed to solve the pressure diffusivity equation. Take Chakrabarty et al (1993) as an example, who provided a quantitative analysis of the effects of neglecting the quadratic gradient term on solving the diffusion equation governing the transient state (Chakrabarty et al 1993). It should be noted that among the flow regimes in reservoir, the transient flow is the most significant state upon which such important characteristics as permeability, reservoir capacity, and skin factor can be determined using well test analysis (Van Everdingen 1953;Lee 1992).…”

Section: Introductionmentioning

confidence: 99%

Parametric analysis of diffusivity equation in oil reservoirs

Azin

Mohamadi‐Baghmolaei

Sakhaei

2016

J Petrol Explor Prod Technol

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The governing equation of fluid flow in an oil reservoir is generally non-linear PDE which is simplified as linear for engineering proposes. In this work, a comprehensive numerical model is employed to study the role of non-linear term in reservoir engineering problems. The pervasive sensitivity analysis is performed on rock and fluid properties, and it is shown that rock permeability and fluid viscosity are the most significant parameters which influence the pressure profile. Moreover, the critical reservoir radius was determined in four different cases including heavy and light oil along with high and low permeable rocks, in which the pressure difference of linear and non-linear diffusivity equation is significant. Results of this study give an insight into proper simplification of diffusivity equation and provide an engineering-based decision strategy for different reservoir properties having certain rock and fluid properties in oil reservoirs. The results show the significance of depletion time, production rate, and reservoir radius in calculation of pressure drop.

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“…Wang Y [4] established the nonlinear flow model in poroelastic media. Chakrabarty C [5] derived the mathematical model with nonlinear diffusion equation and made the quantitative analysis of the quadratic term. Braeuning S [6] established the nonlinear radial flow model of the variable-rate well-test.…”

Section: Introductionmentioning

confidence: 99%

Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics

Nie¹,

Ding²

2010

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For bottom water reservoir and the reservoir with a thick oil formation, there exists partial penetration completion well and when the well products the oil flow in the porous media takes on spherical percolation. The nonlinear spherical flow equation with the quadratic gradient term is deduced in detail based on the mass conservation principle, and then it is found that the linear percolation is the approximation and simplification of nonlinear percolation. The nonlinear spherical percolation physical and mathematical model under different external boundaries is established, considering the effect of wellbore storage. By variable substitution, the flow equation is linearized, then the Laplace space analytic solution under different external boundaries is obtained and the real space solution is also gotten by use of the numerical inversion, so the pressure and the pressure derivative bi-logarithmic nonlinear spherical percolation type curves are drawn up at last. The characteristics of the nonlinear spherical percolation are analyzed, and it is found that the new nonlinear percolation type curves are evidently different from linear percolation type curves in shape and characteristics, the pressure curve and pressure derivative curve of nonlinear percolation deviate from those of linear percolation. The theoretical offset of the pressure and the pressure derivative between the linear and the nonlinear solution are analyzed, and it is also found that the influence of the quadratic pressure gradient is very distinct, especially for the low permeability and heavy oil reservoirs. The influence of the non-linear term upon the spreading of pressure is very distinct on the process of percolation, and the nonlinear percolation law stands for the actual oil percolation law in reservoir, therefore the research on nonlinear percolation theory should be strengthened and reinforced.

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Analytical solutions for radial pressure distribution including the effects of the quadratic‐gradient term

Cited by 33 publications

References 11 publications

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

Parametric analysis of diffusivity equation in oil reservoirs

Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics

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