Transport in Porous Media

, Volume 128, Issue 1, pp 45–73 | Cite as

Dynamic Temperature Analysis Under Variable Rate and Pressure Conditions for Transient and Boundary Dominated Flow

  • Yilin MaoEmail author
  • Mehdi Zeidouni


Current analytical approaches for temperature transient analysis heavily rely on the assumption of the constant rate production, which is often invalid for the extended period of oil production. This work addresses this issue by introducing novel analytical approaches to model the temperature signal under dynamic rate and pressure conditions. The introduced methods share underlying theories of superposition principle and production rate normalization from pressure transient analysis and include a newly derived analytical solution when these theories are not applicable. With adapting these methods, cases with complex production history are modeled using analog cases producing with a constant rate. The dynamic temperature analysis developed herein is validated using synthetic temperature data both graphically and by quantitative estimation of reservoir properties. The estimation outputs of these methods include permeability, drainage area, and damaged zone properties. Other features of existing temperature transient analysis, such as fluid property correction and monitoring well surveillance, are also extended to variable rate and pressure conditions in this paper. Two cases documented in the literature are addressed by dynamic temperature analysis for which decent reservoir characterization results are obtained. The dynamic temperature analysis proposed in this paper extends the scope of temperature transient analysis to complex production constraints and demonstrates convincing results for practical purposes.


Reservoir characterization Temperature transient analysis Variable rate and pressure Boundary dominated flow Rate transient analysis 

List of Symbols

\(\eta \)

Diffusivity for pressure transient (\(\hbox {L}^{2}\,\hbox {t}^{-1},\, \hbox {m}^2\,\hbox {s}^{-1})\)

\(\mu \)

Fluid viscosity \((\hbox {m\,L}^{-1}\,\mathrm{t^{-1}}\), Pa\(\cdot \) s)

\(\mu _{JT,f}, \mu _{JT,w}, \mu _{JT,m}\)

JT coefficient of the fluid phase, water phase, and saturated porous medium, respectively \((\hbox {L\,t}^2\,\mathrm{T/m, \hbox {K/Pa}})\)

\(\phi \)

Average porosity in the porous medium

\(\rho _f, \rho _s, \rho _w, \rho _m\)

Density of the fluid phase, solid phase, water phase, saturated porous medium, respectively (\(\hbox {m\,L}^{-3}, \hbox {kg/m}^3\))


Drainage area \((\hbox {L}^2, \hbox {m}^2)\)


Constant defined in Eq. 18

\(c_f, c_s, c_w, c_m\)

Specific heat capacity of the fluid phase, solid phase, water phase, and saturated porous medium, respectively \((\hbox {L}^2/\hbox {t}^2\hbox {T}, \hbox {J/(kg\,K)})\)


Exponential decline constant (1/t, 1/s)


Reservoir thickness (L, m)

\(k, k_s\)

Reservoir permeability and damaged zone permeability \((\hbox {L}^2, \hbox {m}^2)\)


Rock conductivity \((\hbox {ML\,t}^{-3}\,\hbox {T}^{-1}, \hbox {W/(mK)})\)


Relative permeability


Slopes used in Eqs. 23, 2629 for TTA characterization procedures


Pressure \((\hbox {m\,L}^{-1}\,\hbox {t}^{-2}, \hbox {Pa})\)


Cumulative production rate \((\hbox {L}^3, \hbox {m}^3)\)

\(q, q_i\)

Downhole production rate and initial production rate \((\hbox {L}^3\,\hbox {t}^{-1}, \hbox {m}^3/\hbox {s})\)

\(Q_A, Q_s, Q_e, Q_p, Q_{JT}\)

Intercepts used in Eqs. 24 and 30 for TTA characterization procedures, superposition cumulative production function defined in Eq. 6, and the start and end of the quasi-linear behavior used in Eq. 25\((\hbox {L}^3, \hbox {m}^3)\)


Radius (L, m)

\(r_D, t_D, T_D\)

Dimensionless radius, time, and temperature, respectively

\(r_s, r_e, r_{sf}\)

Near wellbore damaged zone radius, outer closed boundary radius and the radius of sand-face, respectively (L, m)


Near wellbore damage skin factor


Saturation of the residual water


Temperature (T, K)


Time (t, s)


Initial reservoir temperature, respectively (T, K)


Darcy velocity (\(\hbox {L\,t}^{-1}\), \(\hbox {m}\,\hbox {s}^{-1}\))


Total compressibility of the fluid saturated porous media \((\hbox {m}^{-1}\hbox {L\,t}^2, \hbox {1/Pa})\)



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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Craft and Hawkins Department of Petroleum EngineeringLouisiana State UniversityBaton RougeUSA

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