Analytic solution to the non-linear diffusion equation for fluids of constant compressibility

Jelmert, Tom Aage; Vik, S.A.

doi:10.1016/0920-4105(95)00050-x

Cited by 20 publications

(10 citation statements)

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“…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%

“…While the presence of this factor clearly has no effect on the satisfaction of the differential equation, it is necessary, unless dimensionless variables are used throughout. A nonzero choice of b (Kikani and Pedrosa 1991;Yeung et al 1993;Jelmert and Vik 1996;Braeuning et al 1998) is used to ensure that the transformation variable vanishes at some reference pressure. Only Tong and Wang (2005) make the choice a = b = 0.…”

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: The Cole-hopf Transformationmentioning

confidence: 99%

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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“…To overcome these difficulties several suggestions and methods have been proposed such as that of Vik and Jelmert, who planned a solution using pressure logarithm transform for diffusivity equation included pressure squared term (Jelmert and Vik 1996). Xu-Long et al in their study found the exact solution for radial fluid flow along with pressure squared term.…”

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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“…When porosity, permeability and fluid density depend exponentially on pressure, the diffusivity equation reduces to a diffusion equation containing a squared gradient term. Many published articles have described analytical solutions for this equation through variable modifications (Chakrabarty et al, 1993a, b;Jelmert and Vik, 1996;Odeh and Babu, 1998;Wang and Dusseault, 1991), which are special cases of the Hopf-Cole transformation (Marshall, 2009). Applications in dual-porosity and fractal (Tong and Wang, 2005) media have also been described.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear flow model for well production in an underground formation

Guo

Nie²

2013

Nonlin. Processes Geophys.

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Fluid flow in underground formations is a nonlinear process. In this article we modelled the nonlinear transient flow behaviour of well production in an underground formation. Based on Darcy's law and material balance equations, we used quadratic pressure gradients to deduce diffusion equations and discuss the origins of nonlinear flow issues. By introducing an effective-well-radius approach that considers skin factor, we established a nonlinear flow model for both gas and liquid (oil or water). The liquid flow model was solved using a semi-analytical method, while the gas flow model was solved using numerical simulations because the diffusion equation of gas flow is a stealth function of pressure. For liquid flow, a series of standard log-log type curves of pressure transients were plotted and nonlinear transient flow characteristics were analyzed. Qualitative and quantitative analyses were used to compare the solutions of the linear and nonlinear models. The effect of nonlinearity upon pressure transients should not be ignored. For gas flow, pressure transients were simulated and compared with oil flow under the same formation and well conditions, resulting in the conclusion that, under the same volume rate production, oil wells demand larger pressure drops than gas wells. Comparisons between theoretical data and field data show that nonlinear models will describe fluid flow in underground formations realistically and accurately

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Analytic solution to the non-linear diffusion equation for fluids of constant compressibility

Cited by 20 publications

References 1 publication

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

Nonlinear flow model for well production in an underground formation

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