A Variable Rate Solution to the Nonlinear Diffusivity Gas Equation

Barreto, Abelardo; Peres, Alvaro M.M.; Pires, Adolfo P.

doi:10.2118/145468-ms

Cited by 4 publications

(9 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…22 given above. This had been observed before by Peres et al (1990a) and Barreto Jr. et al (2013) for line-source wells. Because Eqs.…”

Section: The Finite Wellbore Radius Solutionsupporting

confidence: 80%

“…This solution structure of m D (r D , t D ) being equal to the slightly compressible liquid solution plus a corrective term for the c t variation had been point out earlier by Peres et al (1990a) and Barreto Jr. et al (2013) for line-source wells. Note that the corrective term depends upon the gas properties, that is, different gases lead to different values of this term.…”

Section: The Finite Wellbore Radius Solutionsupporting

confidence: 71%

“…pseudo-pressure solution which is to be compared to previous published pseudo-pressure line-source solution (Barreto Jr., 2011b;Barreto Jr. et al, 2013). For completeness, the latter solution is presented in this sub-section.…”

Section: The Line Source Solutionmentioning

confidence: 99%

“…Recently, Barreto Jr. (2011a, 2011b and Barreto Jr. et al (2012, 2013 presented a new solution procedure for Eq. 3 by Green's Function (GF).…”

Section: Introductionmentioning

confidence: 99%

“…Large class of gas well testing problems can be solved by this new approach. Thus far, this technique was successfully applied to a vertical near a sealing fault (Barreto Jr., 2011b), a hydraulic fractured vertical well (Barreto Jr. et al, 2012) and to variable gas rate (Barreto Jr. et al, 2013). In all cases, the vertical well was represented by line-sources.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Finite Wellbore Radius Solution for Gas Wells by Green's Functions

Sousa

Barreto

Peres

2014

Day 3 Fri, May 23, 2014

View full text Add to dashboard Cite

Even when written in terms of a pseudo-pressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, non-linear because the viscosity-compressibility product is pseudopressure dependent. However, several techniques and analysis procedures neglect such non-linearity. A new methodology for constructing solutions for gas reservoirs through the Green's Function (GF) technique was recently proposed in the literature. Such methodology handles the viscosity-compressibility product variation rigorously and it has been applied to solve several gas well tests problems successfully. In those problems the wellbore is always represented by a line source. This work extends a little further the theory by considering a finite wellbore radius boundary condition for a single vertical well producing at constant rate from an isotropic homogeneous and infinite gas reservoir. The presented solution does not consider non-Darcy flow effects and wellbore storage and skin as well. Results from our finite wellbore radius (f.w.r) solution are compared to a commercial finite difference reservoir simulator that shows a very close agreement. We also compared the f.w.r solution to the correspondent line source solution to study the difference between the two solutions. As expected, the pseudo-pressure solutions do not match at early times but they do agree at long times, which is exactly how f.w.r and line-source well solutions for slightly compressible fluids behave. It seems that, even for gas well problems, the wellbore can be satisfactorily represented by a line source without significant loss of generality. The line source assumption greatly simplifies the math and the computational effort. This aspect is especially attractive for complex non-linear gas well problems that remains to be solve by the GF approach.

show abstract

“…22 given above. This had been observed before by Peres et al (1990a) and Barreto Jr. et al (2013) for line-source wells. Because Eqs.…”

Section: The Finite Wellbore Radius Solutionsupporting

confidence: 80%

Section: The Finite Wellbore Radius Solutionsupporting

confidence: 71%

Section: The Line Source Solutionmentioning

confidence: 99%

“…Recently, Barreto Jr. (2011a, 2011b and Barreto Jr. et al (2012, 2013 presented a new solution procedure for Eq. 3 by Green's Function (GF).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finite Wellbore Radius Solution for Gas Wells by Green's Functions

Sousa

Barreto

Peres

2014

Day 3 Fri, May 23, 2014

View full text Add to dashboard Cite

show abstract

A New Rigorous Analytical Solution for a Vertical Fractured Well in Gas Reservoirs

2012

View full text Add to dashboard Cite

Significant gas reserves are found in low permeability reservoirs world wide. Economical flow rates are only achieved on these reservoirs by massive hydraulic fractures. Well testing is one of the most used techniques to reservoir management and monitoring, however pressure transient analysis can be a real challenge under these conditions. The gas hydraulic diffusivity equation is commonly linearized by means of pseudo pressure function m(p). The resulting partial differential equation having m(p) as dependent variable remains non-linear because the viscosity-compressibility product that multiplies the partial derivative of m(p) which respect to time varies with m(p). This type of differential equation is named quasi-linear. A wide spread assumption in the well testing literature is to consider that the viscosity-compressibility product remains approximately constant throughout the well test. This assumption is acceptable for small pressure drawdown only but rarely met in tight gas reservoirs. One can find in literature a few attempts to account for the viscositycompressibility product variation analytically; hence, there is still room for improvements in this research topic. This paper considers a vertical fractured well in a homogeneous isotropic infinite gas reservoir produced by constant flow rate. In this work the variation of the viscosity-compressibility product is considered as a non-linear source term. This approach leads to a closed form analytical solution based on Green's Functions. The solution is presented as an integraldifferential equation which must be evaluated numerically. A multidimensional numerical integration package was used to obtain the pseudo pressure results. This numerical scheme is capable of handling accurately large viscosity-compressibility variations.Results for a constant rate test for a vertical fractured well in an infinite isotropic reservoir show good agreement to a finite difference numerical simulator. It is shown that the behavior of dimensionless pseudo pressure and its log-time derivative is similar to the classical slightly compressible fluid fracture well solution. During the pseudo-radial flow regime the pseudo pressure solution is given by the correspondent liquid solution plus a small negative constant. This constant, however is rate sensitive. The solution technique presented in this paper may be successfully applied to harder gas well testing problems.

show abstract

Coupled Integro-Differential-Assymptotic Solution for Permeabiliy Loss/Restoration Monitoring in Pressure-Sensitive Oil Reservoirs

Fernandes

Braga

Souza

et al. 2022

Day 4 Thu, May 05, 2022

View full text Add to dashboard Cite

The Nonlinear Hydraulic Diffusivity Equation (NHDE) models the isothermal single-phase Darcian flow through porous media considering the variation in the properties of the rock and the fluid present inside its pores. Typically, the dimensionless solution of the Linear Hydraulic Diffusivity Equation (LHDE) pD(rD, tD) for constant permeability oil flow in porous media is computed through Laplace and Fourier transform or Boltzmann transformation. For the constant permeability approach, the dimensionless general solution in cylindrical coordinates is expressed by the transcendental function exponential integral Ei(rD,tD). This work develops analytically a new coupled perturbative-integro-differential model to solve the NHDE for oil flow in a permeability-pressure-sensitive reservoir with source. The general solution is computed combining a first order asymptotic series expansion, Green's functions (GF's) and a Volterra's second kind integro-differential formulation. A set of pore pressure and permeability values from two sandstones samples of an offshore reservoir from Brazil is obtained experimentally using the geomechanical elastic parameters e.g. the Young's modulus and Poisson's ratio, and uni-axial cell data. These data are used as an input in the computational code to run the proposed model and evaluate the reservoir permeability change. After these data input, the model runs and allowing to compute the instantaneous reservoir permeability values over the well-reservoir life cycle. The model calibration is performed using a porous media oil flow simulator and the results are accurate.

show abstract

A Variable Rate Solution to the Nonlinear Diffusivity Gas Equation

Cited by 4 publications

References 12 publications

Finite Wellbore Radius Solution for Gas Wells by Green's Functions

Finite Wellbore Radius Solution for Gas Wells by Green's Functions

A New Rigorous Analytical Solution for a Vertical Fractured Well in Gas Reservoirs

Coupled Integro-Differential-Assymptotic Solution for Permeabiliy Loss/Restoration Monitoring in Pressure-Sensitive Oil Reservoirs

Contact Info

Product

Resources

About