Systems and methods for real-time monitoring of downhole pump conditions
Systems and methods for improved monitoring of downhole pump conditions may provide real-time monitoring, high accuracy, and low noise when monitoring downhole pump conditions. Systems for monitoring pump conditions may be coupled to any suitable sucker rod pump, and may gather desired data from the pump. The desired data may be gathered at several points-in-time during a pump stroke to provide real-time monitoring. A wave equation corresponding to the behavior of the downhole pump may be solved when the desired data is received to provide real-time monitor. In some embodiments, the wave equation may be solved by separating it into static and dynamic solutions. In some embodiments, the dynamic solution of the wave equation may be solved utilizing an integral-based method.
1. A method for monitoring downhole pump conditions in real-time, the method comprising:
coupling a load sensor and a position sensor to a rod pump provided at a surface of a well;
gathering surface load data and position data from the load and position sensors of the rod pump;
estimating downhole load and downhole position in real-time throughout a pump stroke utilizing the surface load data, the surface position data, and a nonhomogenous viscous damped wave equation, wherein the downhole load and the downhole position is determined by
estimating a static downhole position and a static downhole load utilizing a static solution σ(x) of the nonhomogenous viscous damped wave equation,
estimating a dynamic downhole load and a dynamic downhole position utilizing a dynamic solution γ(x, t) transformed into a function of complex frequency that is integrated over all frequencies (ω) and time (t), and
determining a total solution ψ(x, t) from the static and the dynamic solutions, wherein the downhole load is determined from the static downhole load and the dynamic downhole load, and the downhole position is determined from the static downhole position and the static downhole position; and
plotting the downhole load and the downhole position in real-time to provide a plot of the downhole position v. the downhole load.
2. The method of claim 1 , wherein the dynamic solution γ(x, t) of the nonhomogenous viscous damped wave equation is applicable to a non-periodic data set of the surface load data and position data.
3. The method of claim 1 , wherein a rod of the rod pump has multiple tapers.
4. The method of claim 1 , wherein the estimating step involving the dynamic solution γ(x, t) determines the dynamic downhole position L at a time t from
γ
(
L
,
t
)
=
1
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
ⅆ
ω
ⅆ
ξ
+
1
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
1
κ
sin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where ω represents frequency, ƒ(ξ) represents the surface position as a function of time, F(ξ) represents the surface load as a function of time, κ represents
1
a
(
ic
+
ω
)
ω
,
α represents the propagation velocity of a wave in a rod material, and c represents a semi-empirical dampening constant; and
the dynamic downhole load from
EA
∂
γ
∂
x
(
L
,
t
)
=
EA
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
κ
sin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
-
EA
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where E represent the Young's modulus of a rod string, and A represent a cross-sectional area of the rod string.
5. The method of claim 1 , wherein the estimating step involving the dynamic solution γ(x, t) determines the dynamic downhole position L at a time t from
γ
(
L
,
t
)
=
1
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
+
1
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
1
κ
sin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where ω represents frequency, ƒ(ξ) represents the surface position as a function of time, F(ξ) represents the surface load as a function of time, κ represents
1
a
(
ic
+
ω
)
ω
,
α represents the propagation velocity of a wave in a rod material, and c represents a semi-empirical dampening constant.
6. The method of claim 1 , wherein the estimating step involving the dynamic solution γ(x, t) determines the dynamic downhole load from
EA
∂
γ
∂
x
(
L
,
t
)
=
EA
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
κsin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
-
EA
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where E represent the Young's modulus of a rod string, and A represent a cross-sectional area of the rod string.
7. A system for monitoring downhole pump conditions in real-time, the system comprising:
a rod pump providing a horsehead and sucker rod coupled to the horsehead, wherein the rod pump is position at a surface to pump fluids from a well;
a prime mover coupled to the rod pump, wherein the prime mover drives the horsehead;
a position sensor coupled to the rod pump at the surface, wherein the position sensor measures surface position data of the sucker rod;
a load sensor coupled to the rod pump at the surface, wherein the load sensor measures surface load data of the sucker rod;
a processor receiving the surface load and surface position data, wherein the processor estimates downhole position and downhole load in real-time throughout a pump stroke utilizing the surface load data, the surface position data, and a nonhomogenous viscous damped wave equation, and the downhole position and the downhole load are estimated by
estimating a static downhole position and a static downhole load utilizing a static solution σ(x) of the nonhomogenous viscous damped wave equation,
estimating a dynamic downhole load and a dynamic downhole position utilizing a dynamic solution γ(x, t) transformed into a function of complex frequency that is integrated over all frequencies (ω) and time (t), and
determining a total solution ψ(x, t) from the static and the dynamic solutions, wherein a total downhole load is determined from the static downhole load and the dynamic downhole load, and a total downhole position is determined from the static downhole position and the static downhole position; and
a display for plotting the downhole load and the downhole position in real-time to provide a plot of the downhole position v. the downhole load.
8. The system of claim 7 , wherein the dynamic solution γ(x, t) of the nonhomogenous viscous damped wave equation is applicable to a non-periodic data set of the surface load data and position data.
9. The system of claim 7 , wherein the sucker rod of the rod pump has multiple tapers.
10. The system of claim 7 , wherein the estimating step involving the dynamic solution γ(x, t) determines the dynamic downhole position L at a time t from
γ
(
L
,
t
)
=
1
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
+
1
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
1
κ
sin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where ω represents frequency, ƒ(ξ) represents the surface position as a function of time, F(ξ) represents the surface load as a function of time, κ represents
1
a
(
ic
+
ω
)
ω
,
α represents the propagation velocity of a wave in a rod material, and c represents a semi-empirical dampening constant; and
the dynamic downhole load from
EA
∂
γ
∂
x
(
L
,
t
)
=
EA
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
κ
sin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
-
EA
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where E represent the Young's modulus of a rod string, and A represent a cross-sectional area of the rod string.
11. The system of claim 7 , wherein the estimating step involving the dynamic solution γ(x, t) determines the dynamic downhole position L at a time t from
γ
(
L
,
t
)
=
1
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
+
1
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
1
κ
sin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where ω represents frequency, ƒ(ξ) represents the surface position as a function of time, F(ξ) represents the surface load as a function of time, κ represents
1
a
(
ic
+
ω
)
ω
,
α represents the propagation velocity of a wave in a rod material, and c represents a semi-empirical dampening constant.
12. The system of claim 7 , wherein the estimating step involving the dynamic solution γ(x, t) determines the dynamic downhole load from
EA
∂
γ
∂
x
(
L
,
t
)
=
EA
2
π
∫
-
∞
∞
f
(
ξ
)
∫
-
∞
∞
κsin
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
-
EA
2
π
∫
-
∞
∞
F
(
ξ
)
∫
-
∞
∞
cos
(
κ
L
)
e
i
ω
(
ξ
-
t
)
d
ω
d
ξ
,
where E represent the Young's modulus of a rod string, and A represent a cross-sectional area of the rod string.