2.2.1. Trends of Change in the Submergence Depth
According to Formulas (5) and (8) of the models, oil wells can be divided into three types: the full-pumping type, the transition type, and the nonfull-pumping type. For the maximum pumping speed
vMax of the pumping unit, after a long period of pumping, when
, the oil well is the full-pumping type; when
, the oil well is the nonfull-pumping type; otherwise, when
, the oil well is the transition type. These three well types correspond to two well states: full-pumping and nonfull-pumping (see
Table 1 and
Figure 1). When the submergence depth
, the oil pumping state of the oil well is the full-pumping state; otherwise, when the submergence depth
is the nonfull-pumping state. Among them, the oil well of the full-pumping type can only be in the full-pumping state, and the oil well of the nonfull-pumping type can only be in the nonfull-pumping state. In the transition-type oil well, the state of the oil well may be in the state of full pumping, or it may be in the state of nonfull pumping, and the well’s state may also transition between the two, depending on the well’s seepage and pumping conditions.
If the oil well is of the full-pumping type, for the full-pumping state model of Formula (5), . During the pumping process, when , the submergence depth is the initial submergence depth h0. At this time, . In the model, the submergence depth h changes with time t, and the comprehensive seepage parameter Cf changes with the oil-storage condition of the reservoir during the oil-pumping process. Normally, Cf changes very slowly, and its change is negligible over a considerable period. Therefore, Cf can generally be treated as a constant. However, after a long period of pumping, when the remaining oil storage in the reservoir decreases by a certain amount and the variation in Cf reaches a certain level, it needs to be recalculated to match the current state of the oil well to the greatest extent. As the pumping process continues, the time t increases gradually. If the reservoir has sufficient oil storage, the submergence depth h will be higher than the pump stroke height hp; that is, . The oil well will be in the full-pumping state for a long time.
If the oil well is of the nonfull-pumping type, for the nonfull-pumping state model of Formula (8), during the pumping process, the initial submergence depth of the oil well , and the change in the parameter Cn in the model is mainly related to the pumping speed v; thus, the change in the submergence depth h will be more sensitive to v. If the oil storage speed of the oil well is not less than the current oil pumping speed, after a long period, the oil pumping process reaches a certain time t, and the submergence depth h will gradually increase but will not be greater than hp. If the oil well storage speed is always lower than the current pumping speed, the submergence depth h will gradually decrease with increasing time t. When h is low to a certain extent, it is necessary to stop pumping.
If the oil well is of the transition type, it may be in a full-pumping state or a nonfull-pumping state, which is determined by the oil well’s permeability and pumping speed. The current state of the oil well needs to be determined according to the relationship between h and hp. During the pumping process, when , the oil well is in a full-pumping state, corresponding to the model of the full-pumping state in Formula (5). For the transition-type oil well in the full-pumping state, if the oil storage speed of the oil well is less than the current pumping speed, the submergence depth h will gradually decrease with increasing time t. After reaching a certain time, , the oil well will transition from the full-pumping state to the nonfull-pumping state. At this time, the corresponding Formula (8) is the nonfull-pumping state model. For the transition-type oil well in the state of nonfull pumping, if the oil storage speed of the well is not less than the current pumping speed, after a long pumping time, the submergence depth h will gradually increase until , and the well will again transition from the nonfull-pumping state to the full-pumping state. When the pumping state changes, that is, from the full-pumping state to the nonfull-pumping state, or from the nonfull-pumping state to the full-pumping state, its initial submergence depth h0 is replaced by hp, and the time t starts counting from 0.
Regardless of whether a well is in the full or nonfull-pumping state, for a given pumping speed, its submergence depth is in one of three phases: the rising phase, the holding phase, or the falling phase. For the full-pumping model, these three phases (see
Figure 2) can be expressed as follows:
When
, the submergence depth is in the rising phase, and when
, the submergence depth is in the holding phase; otherwise, the submergence depth is in the falling phase. Similarly, for the nonfull-pumping model, the three phases (see
Figure 3) can be expressed as follows:
The trend of change in the submergence depth is determined by the comprehensive seepage parameter Cf of the oil well and the pumping speed v of the pumping unit. Especially for the nonfull-pumping model, because the oil well has fewer liquid reserves, the variation trend of submergence depth highly depends on the pumping speed.
2.2.2. Pumping State Transition
During the pumping process, the oil well of the full-pumping type is in the full-pumping state, and the oil well of the nonfull-pumping type is in the nonfull-pumping state. However, for the oil well of the transition type, the state of the oil well may change during the pumping process, from the full-pumping state to the nonfull-pumping state or from the nonfull-pumping state to the full-pumping state, depending on the well’s comprehensive seepage parameter Cf and pumping speed v.
Figure 4 shows a schematic diagram of the transition of an oil well submergence depth from the full-pumping state to the nonfull-pumping state. Combining Formulas (5) and (9), it can be found that this state transition should satisfy the following requirements:
The denominator of Formula (11) is ; otherwise, the submergence depth of the full-pumping state model in Formula (5) will have a constant value of h0, and the pumping state will not change.
It is assumed that
; then, an oil well transitioning from the nonfull-pumping state to the full-pumping state should satisfy the following conditions:
Similarly, the denominator is
; otherwise, the submergence depth of the nonfull-pumping state model in Formula (8) will have a constant value of
h0, and the pumping state of the oil well will not change. The submergence depth of the oil well from the nonfull-pumping state to the full-pumping state is shown in
Figure 5.
Regarding the parameters hE and Cf of the oil well model in the full-pumping state (Formula (5)), when the oil well transitions into the nonfull-pumping state, the current submergence depth satisfies ; the parameters hE and Cf do not change, and at this time, the initial submergence depth h0 is replaced by hp. The state model Formula (5) is replaced by Formula (8), and the time restarts. The transition of an oil well from the nonfull-pumping state to the full-pumping state is similar to the transition process described above.
The models (Formulas (5) and (8)) of the oil well’s pumping submergence depth can also be applied to include only the submergence depth variation of pure oil storage or the pure oil pumping process. In fact, the pure oil storage submergence depth model and the pure oil pumping submergence depth model are special cases of oil well pumping submergence depth models. According to the oil well pumping submergence depth constructed above, the oil well pure storage model and the oil well pure pumping model can be obtained:
(1) When the oil well stops pumping; that is,
, the pure oil storage model [
12] is as follows:
Obviously, Formula (13) is a special case of Formulas (5) and (8) in the case of .
(2) Under the assumption that the permeability coefficient of the oil well is
, the pure oil pumping model [
16] is as follows:
Similarly, Formula (14) is a special case of Formulas (4) and (7) in the case of . In reality, oil wells that satisfy Formula (14) hardly exist, and they are suitable for the liquid pumping process of a straight vessel.