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Transport in Porous Media

, Volume 129, Issue 3, pp 779–810 | Cite as

An Analytical Solution and Nonlinear Regression Analysis for Sandface Temperature Transient Data in the Presence of a Near-Wellbore Damaged Zone

  • Filippo PaniniEmail author
  • Mustafa Onur
  • Dario Viberti
Article
  • 80 Downloads

Abstract

This study presents an approximate analytical solution for predicting drawdown temperature transient behaviors of a fully penetrating vertical well in a two-zone radial composite reservoir system. The inner zone may represent a damaged (skin) zone, and the outer (non-skin) zone represents an infinitely extended reservoir. The analytical solution is obtained by solving the decoupled isothermal pressure diffusivity equation and temperature equation for the inner and outer zones with the Boltzmann transformation. The convection, transient adiabatic expansion and Joule–Thomson heating effects are all accounted for in the solution. The developed solution compares well with the results of a thermal numerical simulator. The analytical solution is used as a forward model for estimating the parameters of interest by nonlinear regression built on a gradient-based maximum likelihood estimation (MLE) method. A methodology, based on semilog analyses of pressure and temperature data as well as log–log diagnostic plots of pressure- and temperature-derivative data, is proposed to obtain good initial guesses of parameters, which derive the MLE objective function to have reliable optimized estimates. The statistical measures such as estimated standard deviation of noise in pressure and temperature data, confidence intervals for parameters and correlation coefficients between parameter pairs are used to evaluate the goodness of fit and reliability of the estimated parameters from history matching pressure and/or temperature data sets. The results show that the rock, fluid and thermal properties of the skin zone and non-skin zone can be reliably estimated by regressing on temperature transient data jointly with pressure transient data in the presence of noise.

Keywords

Temperature transient analysis Nonlinear regression Analytical solution Boltzmann transformation Skin zone Radial composite reservoir model 

List of Symbols

\(\alpha \)

Thermal diffusivity constant (\(\hbox {m}^2\)/s)

\(\eta \)

Hydraulic diffusivity constant (\(\hbox {m}^2\)/s)

\(\gamma \)

Euler’s constant (0.577215)

\(\kappa \)

Thermal conductivity (J/m s K)

\(\lambda \)

Mobility (\(\hbox {m}^2\)/Pa s)

\(\mu \)

Viscosity of fluid (Pa s)

\(\phi \)

Porosity (fraction)

\(\rho \)

Density (kg/\(\hbox {m}^3\))

\(\sigma \)

Throttling coefficient (J/kg Pa)

\(\mathbf m \)

Model parameter vector

\(\mathbf m ^*\)

Optimized model parameter vector

\(\varepsilon _\mathrm{JT}\)

Joule–Thomson expansion coefficient (K/Pa)

\(\varphi \)

Adiabatic expansion coefficient (K/Pa)

B

Formation volume factor (\(\hbox {m}^3\)/\(\hbox {Sm}^3\))

c

Isothermal compressibility (\(\hbox {Pa}^{-1}\))

\(c_\mathrm{p}\)

Specific heat capacity (J/kg K)

\(c_\mathrm{pR}\)

Ratio of volumetric heat capacity of fluid to volumetric capacity of fluid-saturated porous medium

Ei

Exponential integral

h

Reservoir thickness (m)

k

Effective permeability (\(\hbox {m}^2\))

p

Pressure (Pa)

q

Volumetric flow rate at standard conditions (\(\hbox {Sm}^3\)/s)

r

Radius (m)

\(r_\mathrm{w}\)

Wellbore radius (m)

S

Skin factor (unitless)

s

Saturation (fraction)

T

Temperature (K)

t

Drawdown time (s)

\(u_\mathrm{c}\)

Velocity of convective heat transfer (m/s)

v

Darcy’s velocity (m/s)

z

Boltzmann transformation (unitless)

Subscripts

m

Component

O

Outer zone

o

Oil

S

Skin zone

s

Solid

T

Temperature

t

Total

w

Water

Notes

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Environment, Land and Infrastructure EngineeringPolitecnico di TorinoTurinItaly
  2. 2.McDougall School of Petroleum EngineeringUniversity of TulsaTulsaUSA

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