RU2549663C1 - Method for determining heat conductivity coefficients of rocks, heat transfer through tubing strings and casing string and length of circulation system of well - Google Patents

Abstract

FIELD: oil and gas industry.

SUBSTANCE: stopped well is chosen; its flushing is performed, and with that, temperature at the circulation system outlet is recorded. With that, pumping of hot liquid (heat carrier) is performed through annular space, with that, at its inlet the liquid temperature varies as per a periodic law and is recorded, and heat conductivity coefficient λ_"п" and coefficients of heat transfer through tubing strings k₁ and casing string k₂ are calculated as per mathematical formulas.

EFFECT: improving measurement accuracy of an average integral value of heat conductivity of mine rocks as to a well log and determining coefficients of heat transfer through the tubing strings and through the casing string, length of the circulation system of the well.

Description

The present invention relates to the oil and gas industry and relates to the determination of thermal properties (heat and thermal diffusivity) of rocks that make up the well section and the formation as a whole.

A common method for removing paraffin deposits on the inner surface of tubing (tubing) that reduces the live section of the tubing string and reduces its throughput in the production of highly viscous paraffinic oils is the thermal method of dewaxing wells by pumping coolants, for example, hot oil.

To determine the heat and thermal diffusivity of rocks, a method was developed using a mobile concentrated non-contact source of thermal energy and an installation based on it ("Thermophysical properties of rocks", Babaev VV, Budymka VF, Sergeeva TA, etc. . - M .: Nedra, 1987, p. 37-38).

The noted installation consists of a movable platform on which the reference sample and the studied core of an optical emitter with a power of 0.5-5.0 W, which serves as a source of thermal energy, a non-contact temperature sensor, which is a low-temperature radiometer with a sensitivity of 0.10C, a self-recording potentiometer KSP- 4 and power supply.

In the experiment, a mobile platform with rock samples located on it moves at a constant speed from 0.3 to 1.0 cm / s. In this case, the surface of the samples is heated at a given point (along the line) by a laser beam and after a certain time the temperature of this surface is detected by a non-contact sensor.

Significant disadvantages of the noted method and installation include the lack of modeling of thermobaric conditions of occurrence of rocks. In particular, experience shows that the coefficients of heat and thermal diffusivity are significantly dependent on temperature, which does not provide reliable measurements at temperatures exceeding room temperature (20 ° C).

In this regard, it is of interest to determine the thermal properties of rocks in conditions of their natural occurrence. The work (Nikolaev S.A., Nikolaeva N.G., Salamatin A.N. “Thermal Physics of Rocks.” - Kazan: KSU Publishing House, 1987, pp. 66-73) presents a borehole method for determining the thermal diffusivity of formation rocks based on the restoration of temperature in the well after it is stopped.

Around the existing well, the natural temperature field, as a rule, is disturbed due to heat exchange of the fluid flow moving in it with the environment. The process of temperature recovery, which begins after the latter has stopped, depends on many factors, including the history of the operation of the well, its design, the thermal characteristics of its elements and the surrounding massif.

The disadvantages of this technique are the complexity and duration of the experiment and the lack of measurement of the coefficient of thermal conductivity of rocks.

In addition to the above, both analogs described above do not allow determining the heat transfer coefficients from the coolant in the tubing to the flow in the annulus in the borehole circulation system.

Closest to the essence of the present invention is a modified method (methods A and B) for determining the thermal and thermal diffusivity of rocks surrounding a well by the rate of restoration of the temperature of the drilling fluid after a violation of the established heat field (B. Yakovlev, “Forecasting the oil and gas potential of mineral resources according to geothermal data ", M: Nedra, 1996, p. 47-56).

Method A takes into account the effect on the restoration of violated temperatures in the well of the cement ring in the annulus. In this case, a homogeneous formation is revealed, opened by a vertical cased well filled with liquid. The reservoir has the same temperature t∞, the same temperature and the filling fluid (long idle well). After a violation of a uniform temperature field (for example, flushing a well), it is restored at a rate that depends on the intensity of flushing, the design of the well, the thermal properties of the annular cement and surrounding rocks.

The methodology for conducting downhole experiments in order to obtain initial data for determining heat and thermal diffusivity is as follows:

- a long idle well with sufficiently thick homogeneous formations is selected;

- the steady-state natural temperature is recorded along the wellbore (geothermogram);

- the well is flushed (in which it is necessary to completely replace the fluid filling it in 1-2 hours and the temperature of the outflowing fluid should be recorded throughout this time);

- before the end of the washing, thermal measurement is carried out along the wellbore in the studied depth interval;

- subsequent temperature measurements in the same interval are made after 1, 4, 8, 24 hours of downtime after flushing.

The features of the method for determining thermal properties using the Laplace integral transform are: taking into account in the mathematical model the influence of the thermal properties of the cement ring on the temperature field of the well and the possibility of using numerical iterative methods to minimize the residuals realized by a computer.

Method B (a simplified statistical method), the implementation of which is also possible with the use of a computer, is based on solving the problem of cooling a cylinder of infinite length, finite diameter in an unlimited homogeneous medium, when heat transfer in it from the coolant occurs according to the law of heat conduction.

In solving the problem, it is assumed that within the studied rock layers, lithologically homogeneous, the change in the thermal field is insignificant; the influence of interfaces (underlying and overlapping layers) takes place only in the intervals directly adjacent to these sections; the effect of convection due to the pronounced viscosity of the clay solution is small; the diameter of the well within the studied section interval does not change.

The disadvantage of methods A and B is the low accuracy of determining the average integral values of the thermal and thermal diffusivity of the rocks along the well section.

Comparison of the results of determining the thermal conductivity λ and thermal diffusivity a of rocks by the rate of temperature recovery with the data of laboratory definitions, conducted by B.A. Yakovlev, shows that in methods A and B, the values of heat and thermal diffusivity are overestimated. When determining thermal properties by a statistical method (method B), rocks according to λ and a are less differentiated (especially according to a ).

The principal disadvantage of the prototype is the lack of measurement of heat transfer coefficients from the coolant in the tubing to the flow in the annulus and the length of the circulation system of the well.

The aim of the invention is to improve the accuracy of measuring the average integral value of the thermal conductivity of the rocks along the well section and determining the heat transfer coefficients through the tubing and through the casing, as well as the length of the circulation system of the well.

This goal is achieved by the fact that in accordance with the known method (prototype), a stopped well is selected, it is flushed, the temperature of the outgoing flushing fluid is recorded, while in contrast to the prototype, hot fluid (coolant) is injected through the annulus, at the temperature inlet this fluid is changed according to a periodic law, and recorded, and the thermal conductivity λ _n and tubing through the heat transfer coefficients k ₁ and k ₂ the casing, and the length of loop h translational wellbore system are calculated using the following formulas:

where a , b and Bi are found from the solution of the system of equations:

Legend:

And _e , ϕ _e is the amplitude and phase shift found by processing the measured temperature at the outlet of the circulation system;

p _e , q _e , - parameters characterizing the average component of the temperature at the outlet of the circulation system and found by processing the measured temperature;

t ₁ , t _n is the dimensionless time of the beginning and end of the interval in which the measured temperature is processed at the outlet of the circulation system;

h is the length of the circulation system, m;

T _m - average temperature at the entrance to the annulus, ° C;

B is the temperature of the neutral layer, ° C;

θ _m1 (t ₁ , 0), θ _m1 (t _n , 0) are dimensionless temperatures at the outlet of the tubing at moments t ₁ , t _n defined by the formulas

Where

and the function φ (t) can be calculated by the formula

η, ξ, are the parameters defined by the expressions:

η = Re [ψ ₁ (iw, 0)], ξ = Im [ψ ₁ (iw, 0)],

where i is the imaginary unit; Re, Im - real and imaginary parts of the function

K ₀ (z), K ₁ (z) - Bessel functions of an imaginary argument of the second kind of zero and first order, as = iw;

;w = ω · τ ₀ , ω is the frequency of temperature fluctuations, τ ₀ is the characteristic process time in hours, indicating the time during which the thermal disturbance propagates around the well by $${\overline{r}}_2$$

;

v ₁ , v ₂ , - flow rates in the tubing and in the annulus, m / h;

G = f _1,2 ρv _1,2 - mass flow rate, kg / h;

f ₁ , f ₂ - the cross-sectional area of the tubing and annulus, respectively, m ² ;

c _p - heat capacity kJ / (kg * K);

ρ is the density, kg / m ³ ;

r ₁ , r ₂ - internal radii of the tubing and casing, m;

- толщина слоя парафина на стенке НКТ, м; $${\overline{r}}_{\mathrm{one}}=r_{\mathrm{one}}-{\delta }_p,{\delta }_p$$

- the thickness of the paraffin layer on the tubing wall, m;

k ₁ , k ₂ - heat transfer coefficients from the coolant in the tubing to the fluid in the annulus and from the coolant in the annulus to the rocks surrounding the well, respectively, W / (m ² * K);

- средняя по длине скважины толщина обсадной колонны, м; $${\overline{r}}_2=r_2-{\delta }_m,{\delta }_m$$

- average casing string thickness, m;

λ _n is the coefficient of thermal conductivity of the rocks surrounding the well, W / (m ² * K);

/λ_n; Bi - dimensionless Biot criterion, Bi = k ₂ $${\overline{r}}_2$$

/ λ _n ;

 / 2 „;

K _G - dimensionless parameter of the form K _G = Gh / (T _m -B).

The invention can be described as follows. Before flushing, the well is stopped and maintained until the temperature field in it takes the form of a geothermal distribution. Next, flushing the well through the annulus begins. An essential point of washing is that the temperature at the inlet to it is a function of time and varies according to a periodic law. In this case, a regular heat transfer regime occurs, in which the temperature field in the well ceases to depend on the initial data and is determined by periodic fluctuations in the input temperature. When such a regime occurs in the heat treatment of a well, the temperature at the exit from it contains a lot of useful information about heat transfer processes in the heat carrier stream.

The temperature field in the case of a regular heat transfer mode is formed as a superposition of two fields, due to the average temperature at the entrance to the well and induced by periodic temperature fluctuations. The temperature in the well, due to the average temperature at the inlet, is calculated by the formulas:

Here, the functions U _1,2 , and Z _{1,2 are} defined by the following expressions:

, $$U_{\mathrm{one}}(t,x)=({\lambda }_{\mathrm{one}}{e}^{{\lambda }_{2}x}-{\lambda }_2{e}^{{\lambda }_2-{\lambda }_{\mathrm{one}}(\mathrm{one}-x)})/({\lambda }_{\mathrm{one}}{e}^{{\lambda }_2-{\lambda }_{\mathrm{one}}}-{\lambda }_2)$$

,

, $$U_2(t,x)=({\lambda }_{\mathrm{one}}{e}^{{\lambda }_2-{\lambda }_{\mathrm{one}}(\mathrm{one}-x)}-{\lambda }_2{e}^{{\lambda }_{2}x})/({\lambda }_{\mathrm{one}}{e}^{{\lambda }_2-{\lambda }_{\mathrm{one}}}-{\lambda }_2)$$

,

,

,

,

,

where λ _1,2 , and the function φ (t) can be calculated by formulas (4), (5).

The relationship of dimensional and dimensionless variables is carried out by the following expressions

where τ is the washing time, hour;

z - coordinate of the well section, counted from the wellhead, m;

T _1,2 - temperature in the tubing and annulus, respectively;

T ₀ = Гz + В, ° С.

If the temperature at the entrance to the annulus of the well varies with time according to a periodic law, then it can always be represented in the following form

where A _f is the amplitude, and ω is the frequency of temperature fluctuations at the entrance to the circulation system, or, using (11), in dimensionless form

Using the theory of functions of a complex variable, one can obtain formulas that determine the periodic component of the temperature field in the well:

where, as before, s = iw, i is the imaginary unit; Im - means the imaginary part of the corresponding complex functions:

in which the quantities ψ ₁ , ψ ₂ are calculated by the formulas:

определяются выражениями (6), (7), (8).Values µ _1,2 , $$\overline{\varphi }(s)$$

are determined by expressions (6), (7), (8).

Thus, with a periodic change in temperature at the entrance to the annulus of the well over time, a regular heat transfer regime occurs. In this case, the temperature field can be calculated by the following formulas:

The functions θ _m1 (t, x), θ _m2 (t, x), θ _p1 (t, x), θ _p2 (t, x) are found from expressions (9), (10), (14). Expressions (14) - (16) are original and are given for the first time.

Computational experiments conducted using formulas (16) allow us to draw the following conclusions:

- the amplitude of the temperature fluctuations at the outlet of the tubing differs significantly from the amplitude of the fluctuations in the input temperature. This fact is explained by the intense heat exchange of the well with the surrounding rock mass;

- there is a phase shift in temperature fluctuations at the entrance to the annulus and at the exit of the tubing;

- the average temperature weakly depends on time and in a fairly wide range can be approximated by a linear law. In particular, if the oscillation period is much shorter than the well flushing duration, then in the time interval having the order of the oscillation period, a linear approximation is quite justified.

The first two facts indicate that if we measure the outlet temperature and determine the amplitude of the fluctuations of the outlet temperature and the phase shift from the obtained data, then comparing these values with the obtained theoretical expressions leads to two equations for determining any parameters, in particular, parameters a and b associated with heat transfer coefficients. The third fact suggests that if during the injection process at the exit from the tubing of the well we measure the temperature and select its average component at the current moment, comparing it with θ _m1 (t, x) we can obtain the third equation for calculating the parameter Bi, depending on thermal conductivity of rocks.

Processing experimental data

From the foregoing, it follows that in order to determine the thermal properties, it is necessary to know the average component of the temperature, the amplitude of the oscillations, and the phase shift. The calculation of these parameters is carried out according to the following algorithm. Let T _ei , τ _i be the measured temperature values and the corresponding time instants, 1≤i≤n, where n is the number of measurements. It was previously noted that the temperature at the wellhead (both inlet and outlet) is approximated fairly well by the expression

where p, q, A, ϕ are parameters requiring determination from the analysis of experimental data.

To find the quantities p, q, A, ϕ, the least squares method is used. Its essence is that we consider a functional of the form

. $$J(p,q,A,\phi )={\displaystyle \sum _{i=\mathrm{one}}^n(p+q\cdot t_i+A\sin }[w\cdot t_i+\phi ]-{\theta }_{\mathrm{uh}i}{)}^2$$

.

Here t _i , θ _ei is the dimensionless time and the corresponding dimensionless value measured at the wellhead temperature. The values of p, q, A, ϕ are found from the minimum condition for the functional in question:

Algorithms that solve problem (18) are implemented in modern computer packages. For example, in the package “Mathematics” this problem is solved using one operator:

FindFit [data, p + q * t + A * Sin [w * t + ϕ], {p, q, A, ϕ}, t},

where data is the name of the array with experimental data.

The proposed experimental data processing algorithm allows us to accurately determine both the average component of the temperature field and the periodic one with a small number of measurements. In this case, the amplitude of the temperature fluctuations and the phase shift are found.

Calculation formulas for determining thermal properties.

The thermal properties of the well are estimated based on an analysis of the temperature at the outlet of the tubing. Given equalities (9), (10), (11), (14), (16), for the temperature at the outlet of the tubing, we can write:

where i is the imaginary unit.

The function ψ ₁ (iw, 0) is a complex function, i.e.

ψ ₁ (iw, 0) = η + iξ, η = Re [ψ ₁ (iw, 0)], ξ = Im [ψ ₁ (iw, 0)].

Therefore, equality (19) can be rewritten in the form

where χ = arctan (ξ / η).

Comparing expression (20) with the results of processing measurements of temperature at the outlet of the tubing, we obtain a system of four equations:

Index "e", as before, indicates the parameters obtained on the basis of experimental data processing.

An analysis of the resulting system shows that the first three equations can be used to determine a , b, Bi, and the fourth equation gives an estimate of the length of the circulation system. This is due to the fact that the phase shift is practically independent of Bi and little depends on the parameters a , b in a wide range of their changes. The phase shift is determined by the frequency of the temperature fluctuation and the parameter K _1t + K _2t , which characterizes the passage of the coolant through the entire channel of its movement. So, for 0 < a <1.0 <b <4, the phase shift is well described by the equations

From the fourth equation of the above system, formulas for estimating the length of the circulation system of the well (2) are found, and the first three equations form a system (3) that determines the parameters a , b, Bi. Knowing the values of a, b, Bi and using formulas (1), we can find the heat transfer coefficients through the tubing and through the casing of the well k ₁ , k _2, respectively, as well as the thermal conductivity of the rocks λ _n .

By measuring the temperature at the entrance to the well and exit from the tubing, it is possible to determine their average component, the amplitude of the temperature fluctuations and phase shift. According to experimental data, heat transfer coefficients through the tubing and through the casing are calculated, as well as the coefficient of thermal conductivity surrounding the well bore and the length of the circulation system of the well.

The results obtained, in particular, make it possible to more reliably determine the temperature distribution along the wellbore and, as a result, increase the efficiency of the process of thermal dewaxing of the well, which, ultimately, will increase its current production rate.

Thus, during the heat treatment of the well in the mode of periodically measuring the temperature at the entrance to the annulus, a regular heat transfer regime occurs over time. In this case, the temperature field is independent of the initial data and is a superposition of the average temperature and the periodic component.

Claims (1)

The method of determining the coefficients of thermal conductivity of rocks, heat transfer through tubing and casing, as well as the length of the circulation system of the well, which consists in selecting a stopped well, flushing it and recording the temperature at the outlet of the circulation system, characterized in that the injection hot liquid (coolant) is produced through the annulus, while at the entrance to it the temperature of the liquid changes according to the periodic law and is recorded, and ffitsient thermal conductivity λ _n and the coefficients of heat transfer through the tubing k ₁ and k ₂ the casing calculated by the formulas

where a , b and Bi are found from the solution of the equations of the system

and the length of the circulation system is determined by the formulas

where the following conventions are used:
And _e , ϕ _e is the amplitude and phase shift found by processing the measured temperature at the outlet of the circulation system;
p _e , q _e - parameters characterizing the average component of the temperature at the outlet of the circulation system and found by processing the measured temperature;
t ₁ , t _n is the dimensionless time of the beginning and end of the interval in which the measured temperature is processed at the outlet of the circulation system;
h is the length of the circulation system, m;
T _m - average temperature at the entrance to the annulus, ° C;
B is the temperature of the neutral layer, ° C;
θ _m1 (t ₁ , 0), θ _m1 (t _n , 0) are dimensionless temperatures at the outlet of the tubing at moments t ₁ , t _n defined by the formulas

Where

, ϖ = a + 0.5bφ (t), and the function φ (t) can be calculated by the formula

η, ξ - parameters defined by the expressions:
η = Re [ψ ₁ (iw, 0)], ξ = Im [ψ ₁ (iw, 0)],
where i is the imaginary unit; Re, Im - real and imaginary parts of the function

$$\overline{\varphi }(s)=\frac{\mathrm{one}}{s}\cdot \frac{\sqrt{s}{K}_{\mathrm{one}}(\sqrt{s})}{\sqrt{s}{K}_{\mathrm{one}}(\sqrt{s})+Bi{K}_0(\sqrt{s})}$$

,
K ₀ (z), K ₁ (z) are the Bessel functions of the imaginary argument of the second kind of zero and first order, and s = iw;
w = ω · τ ₀ , ω is the frequency of temperature fluctuations, τ ₀ is the characteristic process time in hours, indicating the time during which the thermal disturbance propagates around the well by $${\overline{r}}_2$$

;
K _t1 = τ ₁ / τ ₀ , K _t2 = τ ₂ / τ ₀ , τ ₁ = h / v ₁ , τ ₂ = h / v ₂ , v ₁ , v ₂ are the flow rates in the tubing and in the annulus, m /hour;
G = f _{1, 2} ρv _{1, 2} - mass flow rate, kg / h;
f ₁ , f ₂ - the cross-sectional area of the tubing and annulus, respectively, m ² ;
C _p is the specific heat kJ / (kg * K);
ρ is the density, kg / m ³ ;
r ₁ , r ₂ - internal radii of the tubing and casing, m;
$${\overline{r}}_{\mathrm{one}}=r_{\mathrm{one}}-{\delta }_p$$

, δ _p is the thickness of the paraffin layer on the tubing wall, m;
k ₁ , k ₂ - heat transfer coefficients from the coolant in the tubing to the fluid in the annulus and from the coolant in the annulus to the rocks surrounding the well, respectively, W / (m ² * K);
$${\overline{r}}_2=r_2-{\delta }_m$$

, δ _m is the average casing string thickness, m;
λ _n is the coefficient of thermal conductivity of the rocks surrounding the well, W / (m ² * K);
Bi - dimensionless Biot criterion, $$Bi=k_2{\overline{r}}_2/{\lambda }_n$$

;
K _G - dimensionless parameter of the form K _G = Gh / (T _m -B).