Research on working condition diagnosis method of downhole oil-water separation system with sucker-rod pump

Abstract

Compared with the rod-pumping oil production system of the conventional pumping unit, the structure of the down-hole oil-water separation system from the rod pump is more complicated, and the down-hole working conditions are worse. Existing technology no longer meets the needs of its downhole condition diagnosis. It seriously affected the production output and economic benefits of the oil field. Because of this, a fault pre-diagnosis method for the downhole oil-water separation system from the rod pump was proposed. Through the theoretical calculation of oil well production and pump efficiency. After field investigation and collection of many suspension point indicators diagrammed for comparison and analysis, the injection pump’s failure probability was 90% above. At this time, the extraction pumps work normally. Therefore, a reverse calculation method is proposed: establish and solve the static/dynamic load model of the production pump, and obtain the load-displacement function of the production pump. The load-displacement function of the suspension point derived from the suspension point indicator diagram is subtracted from the calculated load-displacement function of the production pump to obtain the load-displacement function of the injection pump so that the working condition diagnosis can be performed according to the injection pump work diagram. In the application of 18 wells within the field, the judgment result of injection pump failure and the result of pump inspection coincided with 91.3%, which verified the correctness of the calculation model. This method provides necessary theoretical guidance for the field application and working condition diagnosis of rod pump downhole oil-water separation system and is of great significance for improving the comprehensive benefit of oilfield production.

Introduction

Rod pump downhole oil-water separation technology is a very effective way to stabilize oil and control water for the development of ultra-high water-cut oil wells. It achieves a combination of production and injection in a single wellbore, effectively reducing the water content of the produced fluid, resulting in significant cost savings and environmental benefits. The overall benefit of the oil field has significantly improved. ¹ However, the downhole part of this system is composed of an injection pump and a production pump. Compared with the conventional rod pump pumping system, its structure is more complicated and the downhole working conditions are more severe. The existing technology is no longer suitable for the downhole fault diagnosis of the system. If the diagnosis and treatment are not timely, the production and economic benefits of the oil field will be affected to varying degrees. ² Therefore, timely analysis of the working conditions of the system, accurate and reliable fault diagnosis results, and corresponding measures are of great significance to improve the comprehensive benefits of oilfield production.

In general, the working condition of an oil well can be initially diagnosed by a suspension point indicator diagram, but if the working condition of the well is to be accurately diagnosed, it is better to use a pump power diagram. Numerous scholars have studied conventional down-hole fault diagnosis methods for beam-pumping units. ³ There are roughly five periods in the development of diagnostic techniques: Sun ⁴ proposed the “five-finger test” analysis method relies on experienced workers to perceive the vibration of the polished rod with their hands, and then judge the working state of the underground according to their own experience. Although easy to implement, its accuracy was low, and it was gradually replaced by other methods. Li ⁵ proposed the surface indicator diagram method mainly uses a dynamometer to measure the suspension point diagram and matches it with a standard dynamometer diagram to obtain the downhole working status. However, this method suffers from the following problems: there are too many assumed conditions, and some of the actual measured dynamometer diagrams are too singular in shape, and it is difficult to select the closest standard diagram, thus limiting its range of application. Qiu ⁶ proposed the downhole dynamometer method uses a downhole power instrument directly downhole to measure the dynamometer diagram of the oil pump. The method eliminates many of the uncertain factors and makes the test more accurate. However, all the downhole devices need to be taken out of the ground to install the downhole dynamic instrument, and the installation cost is steep, which is not conducive to practical promotion. Gibbs ⁷ and Li ⁸ proposed the computer diagnosis method combines the mathematical model with the damping wave equation and the surface dynamometer diagram to obtain the corresponding downhole pump dynamometer diagram, and uses the dynamometer diagram to implement corresponding comparative analysis. Compared with the downhole dynamometer method, it not only saves the cost, but also improves the accuracy. However, this method still needs to rely on the skills and work experience of the operator, making it difficult to promote its use. Huang ⁹ proposed the “Artificial Intelligence Diagnosis” proposed by using computer-aided diagnosis and intelligent techniques to analyze faults in the entire pumping unit, because of its advantages of simple operation and high precision, it has gradually become the current research field and has extensive development space. However, it also has certain inherent shortcomings. Since the down-hole pump group of the same well injection and production system works at a depth of approximately 1000 m underground, it is difficult to directly detect its working state parameters, and the method of using a down-hole dynamic instrument is also infeasible due to factors such as inconvenient installation and elevated cost. A reasonable fault diagnosis model and drawing a pump power diagram are the core techniques for fault diagnosis. ¹⁰

Therefore, based on the actual working conditions at the site, starting from the Gibbs model, ¹¹ in this paper, we propose a reverse calculation method to establish and solve the dynamic load model and additionally combine the actual measurements of the suspended point indicator diagram to draw the output and injection pump indicator diagrams. The fault map of the oil pumping system is used for fault diagnosis, which better solves the problem. And further drawing the production pump with the help of Visual Basic programming language. And based on the structure of the rod pump downhole oil-water separation system, a calculation model for the static load of the production pump is established. Meanwhile, based on the unconventional structure of the rod pump downhole oil-water separation system, a reverse calculation method is proposed, and the dynamometer diagrams of the production pump and the injection pump are drawn respectively. The dynamometer diagrams of the pump and injection pump create the conditions for the next step in the construction of a fault diagnosis system.

Structure and working principle of downhole oil-water separation system with the rod pump

Structure of downhole oil-water separation system with rod pump

The rod-pump downhole oil-water separation system consists mainly of a conventional beam pumping unit, a production pump, a sealing piston, a bridge packer, an injection pump, and an oil-water separation system, as shown in Figure 1.

Figure 1.

Schematic of injection/production string system in the same well.

The working principle of the injection production pump system

As shown in Figure 2, during the downstroke, the plunger of the injection pump goes down with the sucker rod, the volume of Pump Inner Ring Cavity 2 increases, Lower Valve 2 opens, Upper Valve 2 closes, and the water separated by the separator enters the inner ring cavity of the pump. At the same time, the production pump plunger is pulled down and the separated oil enters Pump Inner Ring Cavity 1 of the production pump through the bridge-packer channel. Therefore, the injection pump and the production pump simultaneously complete the process of water injection and oil pumping.

Figure 2.

Working principle of the downstroke.

As shown in Figure 3, during the upstroke, when the pump plunger is driven by the rod, the volume of the Pump Inner Ring Cavity 2 is reduced, Lower Valve 2 is closed, and Upper Valve 2 is opened. Liquid from Pump Inner Ring Cavity 2 is reinjected into the injection layer. At the same time, the production pump plunger goes up synchronously and the liquid in Pump Inner Ring Cavity 1 is lifted to the surface, so the process of oil recovery and injection into the formation is simultaneously completed.

Figure 3.

Working principle of upstroke.

Fault prediction method

The rod pump downhole oil-water separation system is different from the conventional rod pump system in composition and structure. There are two pumps for injection and production in the downhole working at the same time. Conditions, another known indicator diagram of the production pump or injection pump is required as a supplementary boundary condition. However, it is difficult to directly and accurately test the pump diagram at a distance of several thousand meters downhole in the existing technology. Therefore, this paper is the first to perform a preliminary pre-diagnosis of the rod-pump downhole oil-water separation system: we calculate the well fluid production and pump efficiency using the corresponding data from the dynamometer maps measured in the field, and propose a fault pre-diagnosis method.

Theoretical calculation of production pump fluid production

The liquid production rate of the oil well in the rod pump downhole oil-water separation system is the liquid production rate of the production pump. The loss of liquid production of the production pump mainly includes three parts: ① The loss caused by the gas compression in the annulus of the large and minor plungers; ② The stroke loss caused by the elastic elongation of the sucker rod; ③ The loss caused by the leakage of the production pump.

Compression loss of gas in the ring air of large and little plungers

Calculation of the gas-liquid ratio at the bottom of the well

The gas-liquid satisfies the law of mass conservation during the entire transportation process. At different pressures, liquids are incompressible while gases are compressible. Set the oil-gas ratio at the wellhead at the atmospheric pressure (0.1 MPa) as $\chi$ , and the water cut as $\gamma$ . Then there is a relationship:

$$V_g:V_o=\chi :1$$

(1)

$$V_o:V_w=(1-\gamma ):\gamma$$

(2)

Combining the above formulas, we get:

$$V_g:V_o:V_w=\chi ·(1-\gamma ):(1-\gamma ):\gamma$$

(3)

Then, within a unit volume, the percentage of gas, oil, and water in the total volume at the wellhead is:

$${\kappa }_g=\frac{\chi ·(1-\gamma )}{\chi ·(1-\gamma ):(1-\gamma ):\gamma }$$

(4)

$${\kappa }_o=\frac{1-\gamma }{\chi ·(1-\gamma ):(1-\gamma ):\gamma }$$

(5)

$${\kappa }_w=\frac{\gamma }{\chi ·(1-\gamma ):(1-\gamma ):\gamma }$$

(6)

Where $V_g$ is the volume of gas, ${\mathrm{m}}^3$ ; $V_o$ is the volume of oil, ${\mathrm{m}}^3$ ; $V_w$ is the volume of water, ${\mathrm{m}}^3$ ; ${\kappa }_g$ is the percentage of gas produced at the wellhead to the total volume; ${\kappa }_o$ is percentage of the total volume of oil produced at the wellhead; ${\kappa }_w$ is percentage of wellhead produced water in total volume.

According to the previous assumption, oil and water are incompressible; that is, the volume of oil and water at the wellhead and bottom of the well remains the same, and only the volume of gas changes, so the percentage of the three phases of oil, gas, and water at the bottom of the well changes.

Set the surface temperature as $T_0$ , bottom hole temperature as $T_1$ , and submergence as $h$ . The mass-specific gravity of the annular air of the large and small plungers remains constant and satisfies the ideal gas law:

$$\mathrm{PV}=\mathrm{nRT}$$

(7)

where $R$ is gas constant (proportional constant); $P$ is ideal gas pressure, unit is Pa; $V$ is gas volume, ${\mathrm{m}}^3$ ; $n$ is the amount of gas substance, mol; $T$ is temperature, K.

When the pumping unit is in the bottom dead center position, the total pressure on the valve of the little plunger pump is:

$$P_1=\rho \mathrm{gH}+P_0$$

(8)

where $\rho$ is Liquid density in tubing, $\mathrm{kg}/{\mathrm{m}}^3$ ; $P_1$ is small plunger pump valve upper pressure, Pa; $H$ is pump’s depth, m; $P_0$ is Wellhead pressure, Pa.

According to the gas equation of state, there are:

$$P_0{V}_{g1}=\mathrm{nR}{T}_0$$

(9)

$$P_1{V}_{g2}=\mathrm{nR}{T}_1$$

(10)

where:

$$V_{g1}=1·{\kappa }_{g1}$$

(11)

Then the volume of the bottom hole gas is:

$$V_{g2}=V_{g1}·\frac{P_0}{P_1}·\frac{T_1}{T_0}$$

(12)

The volume percentage of the bottom-hole gas is:

$${\kappa }_g^{\prime }=\frac{V_{g2}}{V_{g2}+1·{\kappa }_0+1·{\kappa }_w}$$

(13)

where $V_{g1},V_{g2}$ is well-head and bottom-hole gas volume, ${\mathrm{m}}^3$ ; ${\kappa }_g^{\prime }$ is bottom hole gas volume percentage.

Gas compression moment in the annular space of large and little plunger in the upstroke

When the plunger of the production pump is at the bottom dead center, the pressure in the annular space is the submerged pressure (defined as P₂). During the up-stroke, the volume of the annular space between the large and small plungers decreases, and the pressure increases, when the pressure in the annular space of the large and small plunger is brought to the upper-pressure P₁ of the pump valve of the small plunger, which can then be pushed open to discharge the liquid into the oil pipe.

According to the PV equation:

• (a) At the bottom dead center position, when the plunger is about to move upward:

$$P_2{V}_1=\mathrm{nR}{T}_1$$

(14)

• (b) The plunger runs upwards and when the upper valve is pushed open:

$$P_1{V}_2=\mathrm{nR}{T}_1$$

(15)

where $P_2$ is sinking pressure; $V_1$ is the gas volume of the plunger at the bottom dead center position, $V_1=V·\kappa \prime$ , ${\mathrm{m}}^3$ ; $V$ is the volume of the annulus at the bottom dead center position of the plunger, ${\mathrm{m}}^3$ ; $V_2$ is The plunger goes up until the valve ball of the small plunger is pushed away, the remaining volume in the annulus of the large and small plunger, ${\mathrm{m}}^3$ .

Simultaneous equations (14) and (15), we can get:

$$V_2=\frac{P_2}{P_1}\times V_1$$

(16)

During this period, the volume of the annulus of the large and small plungers decreases ( $V_1-V_2$ ), and the distance that the plungers travel upwards (compression distance of the annulus of the large and small plungers) is:

$${\lambda }_1=(V_1-V_2)/(F_b-F_s)$$

(17)

where $F_b$ is large plunger area, ${\mathrm{m}}^2$ ; $F_s$ is small plunger area, ${\mathrm{m}}^2$ .

Stroke loss of sucker rod elastic deformation

The loss of stroke caused by the difference in load between the upper and lower stroke is:

$$\lambda =\frac{\mathrm{\Delta }P·L}{E{f}_0}$$

(18)

Where $\mathrm{\Delta }P$ is up and down stroke load difference, N; $L$ is pump’s depth, m; $E$ is modulus of elasticity of sucker rod, $2.1\times {10}^{11}$  Pa; $f_0$ is sucker rod cross-sectional area, ${\mathrm{m}}^2$ .

In a rod pump downhole oil-water separation system, the large plunger of the production pump is stretched under force during the upstroke, which reduces the amount of discharged liquid, but the small plunger remains under force during the downstroke. That is, the sucker rod is still stretched, but the elongation is small. Thus, the stroke loss is the difference between the elongation due to the force on the large plunger and the elongation due to the force on the small plunger.

Calculation of pump leakage

Different from ordinary oil well pumps, the production pump in a downhole oil-water separation system consists of two plungers, an upper and a lower. If the pressure in the annular space is the same as the discharge pressure during the upstroke, the small plunger will not leak. If the pressure in the annular space is greater than that of the oil. Without pressure in the ring cavity, the large plunger leaks. During the downstroke, the pressure in the annular space is equal to the discharge pressure; the small plunger leaks; the pressure in the annular space is equal to the pressure in the annular space of the oil sleeve, and the large plunger does not leak.

The leakage caused by the pressure difference is:

$$q_3=\frac{\pi d{\delta }_0^3\mathrm{\Delta }p}{12\mu l}(1+\frac{3e^2}{2{\delta }_0^2})$$

(19)

where $q_3$ is gap leakage rate of oil well pump, ${\mathrm{m}}^3/\mathrm{s}$ ; $\mu$ is liquid dynamic viscosity, Pa s; ${\delta }_o$ is concentric clearance of plunger pump barrel, m; $e$ is eccentric distance, m (take 0.04 mm); $d$ is Plunger diameter, m; $\mathrm{\Delta }p$ is the difference between the liquid column pressure in the tubing and the submerged pressure, Pa; $l$ is plunger’s length, m.

Calculation of wellhead fluid production and pump efficiency

Based on the above derivation, the calculated value of the actual fluid production at the wellhead can be obtained as follows:

$$Q_{\mathrm{AP}}=(S-\lambda -{\lambda }_1)·(F_L-F_{\mathrm{S}})·(1-\kappa {\prime }_{\mathrm{g}})-q_3$$

(20)

The theoretical liquid production rate at the wellhead is:

$$Q_{\mathrm{TP}}=(F_{\mathrm{L}}-F_{\mathrm{S}})\mathrm{Sn}$$

(21)

The theoretical pump efficiency is:

$${\eta }_{\mathrm{TE}}=\frac{Q_{\mathrm{AP}}}{Q_{\mathrm{TP}}}\times 100\mathrm{\%}$$

(22)

According to the dynamometer diagram collected on-site, the actual pump efficiency can be calculated as:

$${\eta }_{\mathrm{AP}}=\frac{Q_{\mathrm{AD}}}{Q_{\mathrm{AP}}}\times 100\mathrm{\%}$$

(23)

where $Q_{\mathrm{AP}}$ is calculated value of actual fluid production at the wellhead, ${\mathrm{m}}^3/\mathrm{d}$ ; $Q_{\mathrm{TP}}$ is theoretical fluid production at the wellhead, ${\mathrm{m}}^3/\mathrm{d}$ ; $Q_{\mathrm{AD}}$ is the actual liquid production reflected by the indicator diagram, ${\mathrm{m}}^3/\mathrm{d}$ ; ${\eta }_{\mathrm{TE}}$ is theoretical pump efficiency; ${\eta }_{\mathrm{AP}}$ is actual pump efficiency.

Statistical analysis of fluid production

The dynamometer test reports collected from the site are classified into two categories: normal and faulty. Using Visual Basic software, the above formulas were compiled into a fluid production calculation program, into which the parameters in the dynamometer plots were further fed to obtain the corresponding wellhead fluid production and pump efficiencies. Some of the data records are shown in Tables 1 and 2.

Table 1.

Analysis data of wellhead fluid production under normal working conditions.

Number	Stroke (m)	Stroke times (min)	Sinking degree (m)	Pump depth (m)	Moisture content (%)
1	5.07	3.2	605.51	740.6	89
2	4.98	3.5	402.56	740.6	86.5
3	4.45	2.7	563.3	744.7	96.9
4	4.5	2.8	514.68	744.7	93.4
5	4.57	3.5	560.79	738.1	85
6	4.65	3.5	560.79	738.1	85.3
7	4.68	2.8	703.11	738.1	97.5
8	4.58	2.3	724.45	738.1	97.3
	$Q_{\mathrm{AP}}$ (m3/d)	$Q_{\mathrm{TP}}$ (m3/d)	$Q_{\mathrm{AD}}$ (m3/d)	${\eta }_{\mathrm{TE}}$ (%)	${\eta }_{\mathrm{AP}}$ (%)	$\frac{{\eta }_{\mathrm{TE}}-{\eta }_{\mathrm{AP}}}{{\eta }_{\mathrm{TE}}}$ (%)
1	19.46	30.28	17.84	64.26	58.92	8.31
2	15.19	32.53	17.01	46.70	52.29	11.97
3	17.00	22.42	18.21	75.83	81.21	7.10
4	15.48	23.52	16.31	65.85	69.36	5.33
5	16.35	29.85	15.51	54.78	51.96	5.16
6	16.84	30.37	17.51	55.44	57.65	3.99
7	19.80	24.46	23.02	80.98	94.13	16.24
8	15.71	19.66	13.17	79.91	66.99	16.17

Table 2.

Analysis data of wellhead fluid production under failure conditions.

Number	Stroke (m)	Stroke times (min)	Sinking degree (m)	Pump depth (m)	Moisture content (%)
1	4.76	3.1	465.98	85.5	740.6
2	5.12	5.3	749.31	76.5	749.3
3	5.37	5.2	749.31	96.9	749.3
4	5.47	4.2	89.26	95.7	972.9
5	5.56	4.2	68.1	96.9	972.9
6	5.55	5.7	329.85	96.9	1036
7	5.49	6.3	37.01	95.7	1036
8	4.92	4.2	413.3	95.1	1122
	$Q_{\mathrm{AP}}$ (m3/d)	$Q_{\mathrm{TP}}$ (m3/d)	$Q_{\mathrm{AD}}$ (m3/d)	${\eta }_{\mathrm{TE}}$ (%)	${\eta }_{\mathrm{AP}}$ (%)	$\frac{{\eta }_{\mathrm{TE}}-{\eta }_{\mathrm{AP}}}{{\eta }_{\mathrm{TE}}}$ (%)
1	13.42	27.54	28.48	48.75	107.7	112.7
2	29.84	50.64	57.04	58.92	112.6	91.14
3	43.52	52.11	60.08	83.52	115.3	98.86
4	18.29	42.88	74.96	42.67	174.8	309.8
5	21.05	43.58	84.75	48.31	194.5	302.5
6	41.96	59.04	115.3	71.06	195.4	174.9
7	21.89	64.55	101.4	33.90	157.1	363.4
8	24.77	38.57	46.32	64.21	120.1	87.03

It can be seen from Table 1 that under normal working conditions, the calculated wellhead liquid production rate is not much different from the measured wellhead liquid production rate, that is, the theoretical pump efficiency is close to the actual pump efficiency value, and the difference between the two is less than 20 percentage points; It can be seen from Table 2 that under fault conditions, there is a large difference between the calculated wellhead liquid production rate and the measured wellhead liquid production rate, and the percentage point difference between the theoretical pump efficiency and the actual pump efficiency reaches more than 90, and some data even exceed 300, which shows that the well has failed. At the same time, there is still a phenomenon where the actual pump efficiency is greater than 1, whereas in a normal working environment the actual pump efficiency of a production pump should be less than 1 regardless of whether it is faulty or not. Therefore, this phenomenon indicates that the fault occurred at the injection pump. After on-site investigation, a large number of suspension point indicator diagrams, and calculation and analysis, it was found that the failure of the injection pump accounted for more than 90% of the total failures, while the production pump was working normally.

After on-site investigations, extensive suspension point indicator diagrams, and computational analysis, it was found that the injection pump failures accounted for more than 90% of the total failures, while the production pump was working normally.

Backward calculation method

Based on the actual situation on site, and assuming that only the production pump is considered and that the production pump is considered to be working normally, a reverse calculation is proposed: ① Establish and solve the static load model of the production pump; ② Calculate the dynamic load of the production pump, including inertial load, vibration load, and friction load, in which the Gibbs model is used for the vibration load; ③ The load-displacement function at the suspension point is derived from the dynamometer diagram of the suspension point, and the calculated load-displacement function of the extraction pump is subtracted to obtain the load-displacement function of the injection pump. Therefore, fault diagnosis can be carried out according to the power diagram of the injection pump.

Static load model of the production pump

Assuming that only the production pump is considered, and the sucker rod and the production pump are taken as the research objects, as shown in Figure 4, the static load of the production pump on the up and down strokes obtained from the force balance equation as follows:

$$\begin{array}{c}\multicolumn{1}{c}{P_{\mathrm{SU}}=\frac{\pi }{4}(D_{\mathrm{LP}}^2-d_{\mathrm{R}}^2)(p_{\mathrm{D}}+p_0)-\frac{\pi }{4}(D_{\mathrm{LP}}^2-d_{\mathrm{R}}^2)(p_{\mathrm{I}}+p_C)}\\ +G_1+G_R\hfill \end{array}$$

(24)

$$\begin{array}{c}\multicolumn{1}{c}{P_{\mathrm{SD}}=\frac{\pi }{4}(D_{\mathrm{SP}}^2-d_{\mathrm{R}}^2)(p_{\mathrm{D}}+p_0)-\frac{\pi }{4}(D_{\mathrm{SP}}^2-d_{\mathrm{R}}^2)(p_{\mathrm{I}}+p_C)}\\ +G_1+G_{\mathrm{R}}\hfill \end{array}$$

(25)

Where $P_{\mathrm{SU}}$ , $P_{\mathrm{SD}}$ is the static load of the up and down strokes of the extraction pump, N; $D_{\mathrm{LP}}$ , $D_{\mathrm{SP}}$ is production pump large and small plunger diameter, m; $d_R$ is sucker rod diameter, m; $p_C$ is casing pressure, Pa; $p_0$ is oil pressure, Pa; $p_{\mathrm{I}}$ , $p_{\mathrm{D}}$ is production pump suction inlet pressure, discharge pressure, Pa; $p_{\mathrm{A}}$ is production pump annular pressure, Pa; $G_1$ is the sum of the plunger weights of the extraction pumps, N; $G_{\mathrm{R}}$ is sucker rod weight, N.

Figure 4.

Mechanics model of production pump.

Production pump dynamic load model

In addition to static loads, production pumps are also affected by dynamic loads such as inertia, vibration, and friction. The running speed of the plunger is related to the stroke and the number of strokes, and the dynamic loads will also change accordingly. ¹²

Inertia load

The inertial load is caused by the variable velocity motion, which is the joint action of the rod and the fluid mass in the upstroke, and the rod’s weight in the downstroke. If the influence of the pole column and lifting oil elasticity is ignored, the law of motion of the liquid column and each point of the rod can be considered to be consistent with the suspension point.

On the upstroke:

$$P_{\mathrm{UL}}=P_{\mathrm{RL}}+P_{\mathrm{FL}}=\frac{G_R}{g}\times a_c+\frac{G_{\mathrm{F}}}{g}\times a_c\times \epsilon$$

(26)

$$\epsilon =\frac{F-f_{\mathrm{P}}}{F_{\mathrm{T}}-f_{\mathrm{P}}}$$

(27)

On the downstroke:

$$P_{\mathrm{DR}}=P_{\mathrm{RL}}=\frac{G_R}{g}\times a_c$$

(28)

where $a_c$ is suspension point acceleration, $\mathrm{m}/{\mathrm{s}}^2$ ; $g$ is acceleration of gravity, $\mathrm{m}/{\mathrm{s}}^2$ ; $\epsilon$ is Coefficient of reduction of liquid column acceleration when oil pipe cross-sectional area expands; $F$ is Oil well pump plunger cross-sectional area, ${\mathrm{m}}^2$ ; $f_{\mathrm{P}}$ is sucker rod cross-sectional area, ${\mathrm{m}}^2$ ; $F_{\mathrm{T}}$ is oil pipe cross-sectional area, ${\mathrm{m}}^2$ .

Vibration load

There are three main reasons for the vibration: ① The sucker rod on the upper part of the production pump is an elastic body; ② The reciprocating motion of the sucker rod; ③ The loading and unloading of the liquid load. In this paper, the Gibbs model is used to analyze the vibration load ¹³ :

$$c^2\frac{{\partial }^{2}u}{\partial t^2}=\frac{{\partial }^{2}u}{\partial t^2}+v\frac{\partial u}{\partial t}$$

(29)

where $c$ is propagation speed of the sound wave in sucker rod, $\mathrm{m}/\mathrm{s}$ ; $u$ is elastic displacement of any section of the pole, m; $x$ is the distance from the suspension point to any section of the column, m; $t$ is time since start, s; $v$ is damping coefficient, $0.2\le v\le 2$ .

After numerical calculation, the vibration load $F_v$ is:

$$F_v=-E{f}_{\mathrm{P}}{\frac{\partial u}{\partial x}|}_{x=0}=E{f}_{\mathrm{P}}\psi \frac{v_{B,D}}{c}{K}_v$$

(30)

$$K_v=(-1{)}^k{e}^{-\frac{v}{2}t}(\frac{c}{L}t-2k)$$

(31)

$$t=\frac{\theta -{\theta }_{B,D}}{6n}$$

(32)

where $k$ is coefficient, depending on the value of $\frac{c}{L}t$ ; $L$ is pole length, m; $\theta$ is crank angle, rad; ${\theta }_{B,D}$ is crank angle at the end of static deformation, rad.

Friction load

Frictional loads are mainly: ① Frictional force between pump barrel and plunger; ② Frictional load between wellhead polished rod and packing; ③ Frictional force between the tubing and liquid column; ④ Frictional force between liquid column and sucker rod string. See the application literature ¹⁴ for the calculation method.

Simulation analysis process

Figure 5 is the calculation flow chart of the simulation analysis. First, a MATLAB program is written to extract the data of the actual suspension point indicator diagram, and the load-displacement function is obtained after interpolation and fitting processing, which is used as the input of the next step program. The coefficients of the vibration model are obtained through numerical iterative calculation, and the load-displacement function of the production pump is calculated in combination with the method described above, and the load-displacement function of the injection pump is further calculated. Based on the movement law of the ground pumping unit and the above calculation results, the dynamometer diagrams of the two pumps were drawn by using Visual Basic programming. Besides referring to the fault maps of the conventional oil pumping systems, pattern recognition technology is applied to identify the dynamometer diagram of the injection pumps to determine whether a certain type of fault has occurred in the injection pump.

Figure 5.

Calculation flow chart.

Application cases

The production data information of 18 oil wells using the rod pump downhole oil-water separation system was collected, and the above method was used to draw the indicator diagrams of the injection and production pumps, and the fault of the injection pump was judged and compared with the actual pump inspection results. ¹⁵ The statistical data results are shown in Table 3:

Table 3.

Diagnostic system test results.

Number	Fault type	Number of samples	diagnostic result
1	Leakage of upper pump valve	80	77 dynamometer diagrams were correct, 3 dynamometer diagrams were diagnosed as no fault
2	Broken pole	103	89 dynamometer diagrams were correct, 9 dynamometer diagrams were diagnosed as lower valve leakage, 5 dynamometer diagrams were diagnosed as continuous pumping and spraying
3	Insufficient fluid supply	140	127 dynamometer diagrams were correct, and 13 dynamometer diagrams were diagnosed as air influence
4	Both praying and pumping	160	145 dynamometer diagrams were correct, 15 dynamometer diagrams were diagnosed as broken pole
5	Bump pump	136	All dynamometer diagrams were correct
6	Air influence	80	67 dynamometer diagrams were correct, 11 dynamometer diagrams were diagnosed as insufficient fluid supply, and 2 dynamometer diagrams were diagnosed as leakage of the lower pump valve
7	Leakage of lower pump valve	35	28 dynamometer diagrams were correct, 5 dynamometer diagrams were diagnosed as broken pole, and 2 dynamometer diagrams were diagnosed as both spraying and pumping
8	Simultaneous leakage of upper and lower pump valves	36	34 dynamometer diagrams were correct and 2 dynamometer diagrams were diagnosed as leakage of the upper pump valve

Statistics show that, among the 770 dynamometer diagrams, a total of 703 were correctly diagnosed, with an agreement rate of 91.3%.

Application case 1

A 1# well adopts a φ83/φ44 injection pump, a φ70/φ57 production pump, and a φ38 sealed piston rod pump downhole oil-water separation system for production. The model of the pumping unit is CYJ14-5.5-89, and the stroke is 4.4 m. Stroke times 2.4min⁻¹, pump depth 744. 7 m, water content 0.964, oil, and casing pressure 0.21, 0.25 MPa. The dynamometer diagram of the measured suspension point is shown in Figure 6, and the dynamometer diagram of the production pump and injection pump obtained through calculation is shown in Figure 7. The dynamometer diagram of the injection pump shown in Figure 7(b) is consistent with the typical dynamometer diagram of the pump working normally, ¹⁶ indicating that the downhole oil-water separation system of the rod pump is working normally. This result is consistent with the actual production situation.

Figure 6.

Dynamometer diagram of 1# measured suspension point.

Figure 7.

Simulation dynamometer diagram of 1# downhole pump: (a) dynamometer diagram of production pump and (b) dynamometer diagram of injection pump.

Application case 2

The pump specifications used in a 2# well are the same as those of the 1# well. The pumping unit model is CYJ14-6-89, the stroke is 4.93 m, the stroke frequency is 2.8/min⁻¹, the pump depth is 785.59 m, the water content is 0.86, and the oil and casing the pressure is 0.49, 0.36 MPa. The measured dynamometer diagram of the suspended point is shown in Figure 8, and the calculated dynamometer diagram of the production pump and injection pump is shown in Figure 8. Figure 9(b) is compared with the pump dynamometer map of typical faults ¹⁶ ; it can be judged that the injection pump at this time has a down-touch pump fault. This result is consistent with the pump inspection result.

Figure 8.

Dynamometer diagram of 2# measured suspended point.

Figure 9.

Simulation dynamometer diagram of 2# downhole pump: (a) dynamometer diagram of production pump and (b) dynamometer diagram of indicator pump.

Conclusions

Based on the structure of the downhole an oil-water separation system with a rod pump, the static load calculation model of a production pump is established. Based on the unconventional structure of the rod pump downhole oil-water separation system, this paper proposes a reverse calculation method. The mechanical model is solved according to the Gibbs model assuming that the production pump is normal. Collect the suspended point dynamometer diagrams of 18 wells where the rod pump downhole oil-water separation system is applied in the field, use the method described in this paper to draw the downhole pump dynamism diagram, and judge the fault of the injection pump. The judgment result is 91.3% consistent with the pump inspection result, verifying the correctness of the calculated model.