# Calculation Model of Bottom Hole Flowing Pressure of Double-Layer Combined Production in Coalbed Methane Wells

Authors: Peng Zhang; Xiangchun Wang; Yan Zhang; Chao Wang; Lihui Zheng; Maozong Gan; Yamin Liao
Journal: International Journal of Energy Research
 (2023)

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## Abstract

Bottom hole flowing pressure (BHFP) is the key factor to determine a reasonable working system and achieve long-term stable production of coalbed methane (CBM) wells. However, there is no special BHFP model for double-layer combined production (DLCP). Generally, the constant mass model (CMM) for single-layer production is applied to treat the double reservoirs as a whole, ignoring the changes of fluid mass in each section and the acceleration pressure drop in the reservoir section. The calculation results have great errors, and the BHFP of the lower reservoir is used to adjust the production system of the two reservoirs, which does not meet the requirements of the upper reservoir. In this paper, the expression of acceleration pressure drop assumed to be zero in CMM is decomposed and derived, the relationship between acceleration pressure drop and unit length radial flow is established, and then the pressure drop formula of reservoir section with radial inflow is obtained. The reservoir is divided into several sublayers, and the pressure drop equation for each sublayer is established. According to the water flow and gas flow in the reservoir and nonreservoir sections, the corresponding velocity equations of water phase and gas phase are derived. The above equations are combined to establish the variable mass model (VMM) for DLCP with three stages. The field data are substituted into the VMM and the CMM, and the accuracy of the new model is verified. The results show that in the stages of double-layer water production and double-layer gas production, the errors of the two models are less than 5%, while in the stage of gas-water coproduction, the error of the VMM is 2.75%-6.58%, and the error of the CMM is 7.15%-15.18%. The VMM is more accurate. In addition, in the stages of water production and gas-water coproduction during DLCP, the BHFP of the two reservoirs differs greatly, with a maximum difference of 49.1%. Therefore, the two reservoirs need to adjust the production rule according to their respective BHFP. To sum up, the VMM can accurately give the BHFP of each reservoir, which is more realistic. It also solves the problem that one BHFP cannot accurately adjust the production rule of the two reservoirs, so as to provide technical support for the formulation of optimal production rule and the realization of high production.


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## Body

## 1. Introduction

Coalbed methane (CBM) is an efficient and clean unconventional natural gas resource [1, 2]. The discovered CBM reserves in the world account for more than 30% of the world’s total natural gas reserves, while China’s CBM reserves account for about 41% of China’s total natural gas resources [3, 4]. The efficient development of CBM is conducive to easing energy tension and reducing carbon emissions. CBM is mainly adsorbed gas, and its production goes through the process of desorption, diffusion, and seepage [5, 6]. Therefore, a reasonable production rule can maximize the stable production time and improve the final output. The core of CBM production rule is to control the bottom hole flowing pressure (BHFP) [7, 8]. When the BHFP is too high, the time to reach the desorption pressure is too long, and the economic benefit is low. When the BHFP is too low, it will form an excessive production pressure difference, which is prone to speed sensitivity and stress sensitivity, resulting in the blockage of the pores and the closure of the cleats by the discharge of pulverized coal to different degrees, resulting in the reduction of permeability, and then the pressure drop is difficult to further penetrate to deeper formation, resulting in the inability of CBM to continue desorption. It can be seen that the BHFP is the basis for formulating a reasonable production system [9–11] and is the key parameter for the dynamic evaluation of gas wells [12, 13]. Therefore, the production rule can be adjusted in real time to maintain long-term stable production by obtaining the BHFP regularly and accurately.At present, the methods to obtain BHFP include measurement method, well test method, empirical equation method, and theoretical calculation method [14–17]. Measurement and well test method are expensive and difficult to implement frequently. The empirical equation method is difficult to be widely used because of its harsh application conditions and large errors. Relatively speaking, the theoretical calculation method is relatively mature and has few restrictions; so, it is widely used [18, 19]. The theoretical calculation method generally adopts the average temperature-average deviation coefficient method [20] and Cullender-Smith method [21] or the improved Cullender-Smith method [22] to calculate the annulus pure gas section pressure, adopts Chen Jialang-yue Xiang’an method [23], the Podio Modified S-Curve method [24], and Hasan-Kabir analytical method [25] to calculate the pressure in gas-liquid mixing section, and then combines different methods to obtain the BHFP. Some scholars predict the shut-in bottom hole pressure from the wellhead pressure by considering the transient behavior of hydrostatic pressure loss during the shut-in period [26].However, the theoretical calculation methods are all aimed at single-layer production, and there is no special method for double-layer combined production (DLCP) at present [27, 28]. When it is necessary to obtain the BHFP of DLCP, the two reservoirs are generally regarded as a whole as a large single layer, and the theoretical calculation method of single layer is adopted for approximate calculation [27, 29, 30]. In the calculation, the fluid velocity of the whole well is calculated according to the total water production and total gas production of the two layers [31, 32], ignoring the fluid mass changes at different depths from the upper reservoir to the bottom of the well. In order to compare with the model to be established in this paper, it is called constant mass model (CMM) here. The approximate treatment may produce large errors. Taking the gas phase velocity of gas-water coproduction stage as an example, the gas flow through the section between the two layers is the gas production of the lower layer, while the gas flow through the section above the upper reservoir is the sum of the gas production of the two layers. The change of gas flow will change the original two-phase flow pattern [33] and then change the pressure drop. It will also produce an acceleration pressure drop that cannot be ignored, which is different from the single-layer production. It is necessary to establish a corresponding variable mass model (VMM) according to the flow process in which the mass of different sections changes during DLCP. In addition, the BHFP of the lower reservoir is taken as the common BHFP value of the two reservoirs, so as to adjust the production rule of the two reservoirs [34–36].It can be seen that there are two problems in the current method. On the one hand, CMM does not meet the characteristics of DLCP for no consideration of the acceleration pressure drop of the reservoir section and the fluid mass change of each section. On the other hand, the calculation result of CMM is only the BHFP of the lower reservoir. When the BHFP of the two reservoirs was greatly different, it will lead to large errors in adjusting the production system of the upper reservoir.In order to finely describe the changes of fluid velocity through the two reservoirs, interlayer section, and sections above the dynamic liquid level (DLL) and below the DLL during DLCP, the whole well is divided into five sections based on the DLL and the roof and floor of the two reservoirs. The pressure drop formula for the reservoir section, considering the acceleration pressure drop, is derived based on the single-layer CMM, while the nonreservoir section still uses the pressure drop formula of CMM. Then, according to water and gas flowing through each section, the corresponding water and gas velocities equations are established, respectively. The VMM for DLCP with three stages is established. Finally, the iterative solution process of the model is given.


## 2. Methods

In this section, the whole production process is divided into three stages firstly. On the basis of the CMM of single-layer production in each stage, the relationship between the acceleration pressure drop and the radial inflow rate per unit length is established. Then, the pressure drop equation and velocity equations of gas and water for five well sections are established, respectively, and then the VMM of BHFP for DLCP is obtain.The three stages are as follows: (1) both reservoir I (upper reservoir) and reservoir II (lower reservoir) produce water, called double-layer water production stage; (2) gas and water are produced simultaneously from the wellhead, called gas-water coproduction stage; and (3) both reservoir I and reservoir II produce gas, called double-layer gas production stage.The following assumptions are made when establishing the model: (1) the rod pump is at least below the reservoir I, (2) the tubing and casing are concentric and vertical, (3) when a reservoir is exposed, that is, when the DLL is lower than the reservoir, it is considered that this reservoir only produces gas and no water [37], (4) the volume coefficient of water is 1, and (5) the two reservoirs are intercommunicated only in the well.


### 2.1. Establishment of BHFP Model in Double-Layer Water Production Stage

#### 2.1.1. CMM of Single-Layer Water Production

When a single layer only produces water, there is a static gas column above the DLL, and the BHFP is composed of wellhead casing pressure, static gas column pressure drop, and dynamic liquid column pressure drop.(1) Static Gas Column Pressure Drop Model. Wellhead casing pressure can be obtained directly, and static gas column pressure PAO can be calculated directly by equation (1) [38].
(1)P=P0+P0Z01+hMT0M/Mg−1,(2)Mg=103RMwg,where P is the downhole pressure, MPa; P0 is the wellhead casing pressure, MPa; h is the DLL depth, m; M is the geothermal heating rate, m/K; Mg is the self-heating rate of gas, m/K, R is the general gas constant, 8.314 J/(mol•K), MW is the molar mass of the gas, g/mol; g is the gravitational acceleration, m/s2; T0 is the wellhead thermodynamic temperature, K; and Z0 is the compressibility coefficient of wellhead gas, dimensionless.(2) Dynamic Liquid Column Pressure Drop Model. In the CBM drainage stage, in order to reach the gas-water production stage as soon as possible, it is necessary to reduce the DLL to decrease the BHFP; so, the water pump displacement is required to be greater than the sum of the formation water production and the reduction of down-hole fluid accumulation. Therefore, the flow direction of the water in the annulus is downward, and the water is subject to upward friction, which is different from the conventional upward flow of gas and water.Referring to reference [39], the microsegment dh is selected, the direction is positive downward, and the basic model of single-layer water production is established through the momentum conservation equation, as shown in equation (3).
(3)−AdP+ρwgdhA−fwρwvw22πD1+D2dh=dρwAvw2.In equation (3), the three terms on the left are the pressure difference between the two ends of the microsegment, gravity, and friction, and the right side is the momentum change, which is equivalent to the acceleration pressure drop. In the general CMM, it is considered that the mass of the whole well is constant, the flow rate is constant, and the right side is taken as 0. The model is shown in equation (4)
(4)−dPdh=λwρwvw22D1−D2−ρwg,where
(5)λw=1.14−2lgeD1−D2+21.25Rew0.9−2,Rew=1000D1−D2ρwvwμw,vw=Qu−QwA,where A is the annulus sectional area, m2; dP is the pressure difference between the two ends of the microsegment dh fluid, Pa; fw and λw are the Fanning friction coefficient and Darcy friction coefficient of water, dimensionless; ρw is the density of water, kg/m3; vw is the velocity of water, m/s; D1 and D2 are the inner diameter of casing and outer diameter of oil tube, respectively, m; μw is the viscosity of formation water, mPa·s; e is the absolute roughness of pipe wall; e=0.016mm is taken for new oil tube and casing; Rew is the Reynolds number of single-phase water flow; and Qu and Qw are the pump displacement and water production, respectively, m3/s.


#### 2.1.2. VMM for BHFP of Double-Layer Water Production Stage

When both reservoir I and reservoir II produce water only, according to the assumed conditions, the DLL is above reservoir I. The physical model is shown in Figure1.Figure 1
Flow state in well for double-layer water production.In Figure1, point A is the wellhead, points B and C, respectively, represent the roof and floor of reservoir I, and points D and E, respectively, represent the roof and floor of reservoir II. The depth of DLL is point O. Figure 1(a) is the complete model diagram. The original liquid accumulation in the well and the produced water flow to the bottom of the well are pumped out of the wellhead from the oil tube. Figure 1(b) is a local enlarged drawing of Figure 1(a), showing the principle of variable mass flow. Reservoir I is divided into m sublayers, which is assumed to be five sublayers. The amount of water flowing through each sublayer is different. For the first sublayer of reservoir I, the water flow is the flow of section OB, while the second sublayer is the flow of section OB plus the first sublayer and so on. For section CD, the flow is the sum of section OB flow and the total water flow of the five sublayers. The principle of water flow through each sublayer of reservoir II is the same.It can be seen that from the roof of reservoir I downward, the mass flow through each well section (BC, CD, DE) is increasing. At the same time, the change of flow mass will also produce acceleration pressure drop. So, there is a great difference between the DLCP and single-layer production, and the VMM including acceleration pressure drop applicable to the reservoir section is obtained through equation (3).In equation (3),
(6)dρwAvw2=vwdρwAvw+ρwAvwdvw,dρwAvw=ρwqIwdh,dvw=qIwdhA,fw=λw4,A=π4D12−D22.Then, equation (7) is obtained.
(7)−dPdh=λwρwvw22D1−D2−ρwg+2vwρwqIwA,where qIw is the volume flow of radial inflow water phase per unit length, m3/(s·m).In addition to deriving the VMM, it is also necessary to derive a special velocity equation suitable for each section according to the different water flow through each well section.The flow through section OB is the difference between the pump displacement and the total water production of the double layers, which is equal to the reduction rate of fluid column in the well. The flow of section CD is the difference between the pump displacement and the water production of reservoir II, which is formed by the original fluid accumulation in the well and the water production of reservoir I. Real-time changing velocity equations in section BC and section DE are calculated according to the change of the number of sublayers. Thus, the flow velocity equations of each section are derived and shown in equations (8)–(11):
(8)vOB=vswOB=Qu−qIw1hBC−qIw2hDEA,(9)vBCj=vswBCj=Qu−n−jqIw1hBC/n−qIw2hDEA,(10)vCD=vswCD=Qu−qIw2hDEA,(11)vDEj=vswDEj=Qu−n−jqIw2hDE/nA.For the CMM, the flow velocity of section OB is adopted for all well sections. Comparing (8) and (10), it is found that they differ in the water production of reservoir I. When the water production or the thickness of reservoir is large, the flow velocity of section CD will be significantly different from that of section OB. If the flow velocity of section OB is used uniformly, it will bring great errors. The same is true for the flow velocity of BC and DE segment.The subscript “sw” indicates the apparent velocity of water, which is only used in the gas-water two-phase flow stage as below, and is equal to the actual flow velocity in the single-phase water production stage.When both reservoirs produce water, the BHFP of reservoir II is composed of wellhead casing pressureP0, static gas column pressure Psg of section AO, and dynamic liquid column pressure Pl of section OE. Among them, the VMM of equation (7) is used to calculate the pressure drop in the reservoir section (BC and DE), and the CMM of equation (4) is used in the nonreservoir section (OB and CD). The BHFP of reservoir II and reservoir I is calculated by equation (12), and the pressure drop equation of each section is obtained by transforming equation (4) and equation (7) to equations (13)–(16).
(12)Pwf2=P0+Psg+Pl=P0+PAO+POB+PBC+PCD+0.5PDE,Pwf1=P0+PAO+POB+0.5PBC,(13)POB=ρwghOB−λOBρwvOB2hOB2D1−D2,(14)PBC=∑j=1nρwghBCn−λBCjρwvBCj22D1−D2hBCn−2vBCjρwqIw1AhBCn,(15)PCD=ρwghCD−λCDρwvCD2hCD2D1−D2,(16)PDE=∑j=1nρwghDEn−λDEjρwvDEj22D1−D2hDEn−2vDEjρwqIw2AhDEn,where n is the number of sublayers in the reservoir, and the meaning of each parameter is the same as that of the corresponding parameters in the basic model. The subscript represents the corresponding value of the specific section. qIw1 and qIw2 are the radial inflow water per unit length in reservoir I and reservoir II, m3/(s·m).In the flow velocity equations,qIw1 and qIw2 are unknown parameters, which need to be solved with two other equations jointly. The CBM production equation (17) is selected.
(17)Qwi=qIwihi=2πkihiPei−Pwfi×10−6μwlnrei/rw−3/4+Ski,where i=1,2 represents reservoir I and reservoir II, respectively, hi is the layer thickness, m; Ki is the single-phase flow permeability, 10-3 μm2; Pei is the original pressure of the reservoir, MPa; rei and rw are drainage radius and well radius, respectively, m; Ski is the skin coefficient, dimensionless; and Pwf1 and Pwf2 are the BHFP of reservoir I and II, respectively, MPa.It is found that when the DLL is measured twice, that is, the DLL depths of timet1 and t2 are H1 and H2, respectively; then, VOB=H2−H1/t2−t1, combined with equation (13), the relationship between qIw1 and qIw2 can be obtained, see equation (18). From a practical point of view, the DLL value is easy to obtain by online remote real-time monitoring equipment on site.
(18)qIw1=Qu−vOBA−qIw2hDEhBC=Qu−AH2−H1/t2−t1−qIw2hDEhBC.


#### 2.1.3. Calculation of BHFP in Double-Layer Water Production Stage

The BHFP of double layers is solved by using the process shown in Figure2.Figure 2
Flow chart of BHFP calculation for double-layer water production.Figure2 is the flow chart of BHFP calculation for double-layer water production, and the specific process is shown below.First, input the basic parameters and then ① assign a value ton and assume the BHFP Pwf2 of reservoir II, ② substitute Pwf2 into equation (17) to obtain qIw2 and combine equation (18) to obtain qIw1, ③ substitute qIw1 and qIw2 into equations (8)–(11) to obtain the flow velocity of each section and then use equations (13)–(16) to obtain the pressure drop of each section, ④ use equation to obtain the BHFP Pwf2′ of reservoir II, ⑤ compare Pwf2 and Pwf2′, if it meets the accuracy requirements, it is the BHFP of reservoir II. Otherwise, let Pwf2′ replace Pwf2 and repeat ②~⑤ until it meets the accuracy requirements. ⑥ Again, use equation (12) to obtain the BHFP of reservoir I.


### 2.2. Establishment of BHFP Model in Double-Layer Gas Production Stage

The CMM of single-layer gas production is given, and then the BHFP model of double-layer gas production is obtained. The BHFP is composed of wellhead casing pressure and dynamic gas column pressure drop.


#### 2.2.1. CMM of Single-Layer Gas Production

Referring to the “Dynamic liquid column pressure drop,” a CMM of single-layer gas production for calculating dynamic gas column pressure drop is established.(19)−AdPdh+ρggA+λg4ρgvg22πD1+D2=dρgAvg2dh.Similarly, according to equation (19), the CMM without radial inflow is obtained.
(20)−dPdh=−λgρgvg22D1−D2−ρgg.See AppendixA for the calculation equation of each parameter in equation (19), where λg is the Darcy friction coefficient between gas and tubing casing annulus wall, dimensionless; ρg is the density of down-hole gas, kg/m3; and vg is the velocity of gas, m/s.


#### 2.2.2. VMM for BHFP of Double-Layer Gas Production Stage

When the DLL is close to the bottom of the well, reservoirs I and II no longer produce water, but only gas. The physical model is shown in Figure3.Figure 3
Flow state in well when both reservoirs produce gas only.The lable of each depth point in Figure3 is the same as that in Figure 1. Figure 3(a) shows the complete model diagram. Figure 3(b) is a local enlarged drawing of Figure 3(a). The gas in double layers flows upward out of the wellhead, and the gas volume in each sublayer is represented by an arrow. The more arrows in cross-sections at different depths, the more gas flows through.It is obvious that more and more gas is produced from bottom to top in the section BE, belonging to variable mass flow. Similar to the double-layer water production situation, the VMM including acceleration pressure drop is established, and velocity equations of gas flow are derived. The results are shown in equations (21)–(25), respectively.
(21)−dPdh=−λgρgvg22D1−D2−ρgg+2vgρgqIgA,where qIg is the volume flow of gas phase radial inflow per unit length, m3/(s·m).
(22)vABj=vsgABj=BgABjQscA,(23)vBCj=vsgBCj=BgBCjn−j/nQsc1+Qsc2A,(24)vCDj=vsgCDj=BgCDjQsc2A,(25)vDEj=vsgDEj=BgDEjn−j/nQsc2A.Equations (22)–(25) are derived from equation (A.4). Gas flow through section AB comes from the total gas of double reservoirs, while the gas flow through section CD is only the gas production of reservoir II. For the reservoir sections BC and DE, the gas flow at different depths changes with the number of sublayers. However, the gas velocity of all well sections for the CMM adopts the velocity of section AB.The subscript “sg” of the flow calculation equation of each section indicates the apparent velocity of the gas, which is only used in the gas-water two-phase flow stage below, and is equal to the actual gas velocity in the single-phase gas production stage.The pressure drop equation of each section is obtained by transforming equation (20) and equation (21) to equation (28)–(31), and then equations (26) and (27) are used to calculate the BHFP of the two reservoirs.
(26)Pwf2=P0+Pdg=P0+PAB+PCD+PBC+0.5PDE,(27)Pwf1=P0+PAB+0.5PBC,(28)PABj=λABjρgABjvABj22D1−D2+ρgABjghABn,(29)PBCj=ρgBCjghBCn+λBCjρgBCjvBCj22D1−D2hBCn−2vBCjρgBCjAQsc1BgBCjn,(30)PCDj=λCDjρgCDjvCDj22D1−D2+ρgCDjghCDn,(31)PDEj=ρgDEjghDEn+λDEjρgDEjvDEj22D1−D2hDEn−2vDEjρgDEjAQsc2BgDEjn.Because the wellhead can only obtain the total gas volume and cannot directly obtain the respective gas production of each reservoir, only for the section AB, the flow velocity can directly be calculated by the iterative method, while the flow velocity calculation of section BC, CD, and DE sections needs to be combined with the gas productivity.(32)Qsci=774.6kgihiPei2−Pwfi286400TiZiμgilnrei/rw+SKi,where Kgi is the gas permeability of the reservoir.


#### 2.2.3. Calculation of BHFP Double-Layer Gas Production Stage

Using the process shown in Figure4, the BHFP of the two reservoirs is solved in two stages.Figure 4
Flow chart of BHFP calculation for double-layer gas production.In Figure4, the dotted box on the left is the iterative calculation procedure of section AB without radial inflow. In the right part, the surface gas production of reservoir I is assumed, and then the production of double layers and the pressure drop of each well section are calculated iteratively according to the process on the left, so as to obtain the BHFP of the double layers.(1) Pressure Drop Calculation for Section AB. The calculation steps are as follows:① Taking the wellhead casing pressureP0 as the starting point, assuming that the pressure drop of the first subsection hAB/n is dP, the pressure at the lower end of the first subsection is P1=P0+dP, and the temperature T1=T0+jhAB/n/M; ② calculate the average temperature T0+T1/2 and average pressure P0+P1/2; ③ the average compressibility coefficient is obtained by substituting the average temperature and average pressure into equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the pressure drop dP′ from equation (A.1)–(A.3), (A.8), (22), and (28); and ⑤ compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+dP′ and return to step ②. On the contrary, take (P0+dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the roof of reservoir I; that is, the dynamic gas column pressure drop PAB of section AB is obtained.(2) Pressure Drop Calculation for Sections BC, CD, and DE. This section mainly calculates Qsc1 and Qsc2. The specific steps are as follows:① AssumingQsc1 value, so, Qsc2=Qsc−Qsc1. Taking PAB as the starting point, assuming that the pressure drop of the first subsection hBC/n of section BC is dP, the pressure at the lower end of the first subsection is P1=P0+PAB+dP, and the temperature T1=T0+hAB+jhBC/n/M; ② calculate the average temperature T0+T1/2 and average pressure P0+PAB+P1/2; ③ the average compressibility coefficient is obtained by using equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the dP’ from equations (A.1)–(A.3), (A.8), (23), and (29); and ⑤ compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+PAB+dP′ and return to step ②. On the contrary, take (P0+PAB+dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the floor of reservoir I, the dynamic gas column pressure drop PBC is obtained. ⑥ The BHFP of reservoir I is calculated by equation (27) and substituted into (32) to obtain Qsc1′; ⑦ compare Qsc1 and Qsc1′, if it does not meet the accuracy requirements, use Qsc1′ instead of Qsc1, repeat steps ① to ⑦ until it meets the accuracy requirements, then get PBC and determine the values of Qsc1 and Qsc2; and ⑧ taking the pressure at the floor of reservoir I as the starting point, PCD and PDE are obtained by using equations (24), (30), (25), and (31), respectively, and then the BHFP of reservoir II is obtained.


### 2.3. Establishment of BHFP Model in Gas-Water Coproduction Stage

In gas-water coproduction stage, the mixed gas-liquid column has different flow patterns according to the different gas-water ratio. For low-yield CBM Wells, only bubble flow or slug flow is generated during single-layer production [40], while annular flow may be produced when the gas volume increases during DLCP.


#### 2.3.1. CMM of Gas-Water Coproduction of Single Layer

The gas-water mixture column is divided into three flow patterns, as shown in Figure5.Figure 5
Physical model of two-phase flow in mixed gas-liquid column.
(a)
Bubble flow(b)
Slug flow(c)
Annular flowFigure5 shows three flow patterns that may occur during DLCP. The downward velocity is positive, and the parameter in the figure can be seen in the following equations.(1) CMM for Bubble Flow in Single-Layer Production. Referring to the “Dynamic liquid column pressure drop”, the CMM for bubble flow in single-layer production is established.
(33)−AdPdh−λpm4ρpmvpm2πD1+D22+ρpmgA=dρpmAvpm2dh,(34)−dPdh=λpmρpmvpm22D1−D2−ρpmg.See AppendixB for the calculation equation of each parameter in equation (33), where λpm is the friction coefficient of bubble flow, dimensionless, ρpm is the density of mixed fluid under down-hole conditions, kg/m3, and vpm is the velocity of mixed fluid, m/s.(2) CMM for Slug Flow in Single-Layer Production. Literature [41] is cited here, the CMM for slug flow in single-layer production is established.
(35)−LAdPdh+LAρdmg−Ltbλltbρwvltb2+Lsλllsρsvs2πD1+D28=LdρdmAvdm2dh.Similarly, according to equation (35), the CMM without radial inflow is obtained.
(36)−dPdh=Ltbλltbρwvltb2+Lsλllsρsvs22LD1−D2−ρdmg.See AppendixC for the calculation equation of each parameter in equation (35), where L, Ls, and Ltb are the total length of slug section, liquid slug section, and Taylor bubble section, respectively, m; ρdm is the average density of the whole slug, kg/m3; λtb and λlls are the friction coefficient of Taylor bubble section and liquid slug section, respectively, dimensionless; vs and ρs are the velocity m/s and density kg/m3 of gas-liquid mixture in liquid slug section, respectively; vdm is the average velocity of the whole slug, m/s; and vltb is the falling velocity of Taylor bubble liquid film, m/s.(3) CMM for Annular Flow in Single-Layer Production. Referring to the “Dynamic liquid column pressure drop,” the CMM for annular flow in single-layer production is established[42, 43].
(37)−AgdPdh+Fgw+ρggAg=dρgAgvg2dh,(38)−AwdPdh−Fgw−λHLρwvw2πD1+D28+ρwgAw=dρwAwvw2dh,where
(39)HHL=AwA,vw=vswHHL,vg=vsg1−HHL,(40)HHL=Hltb=1−vsgvgtb,(41)λHL=1.14−2lgeD1−D2+21.25ReHL0.9−2,(42)ReHL=D1−D2ρwvswμw.Add equations (37) and (38) to get equation (43):
(43)−AdPdh−λHLρwvw2πD1+D28+ρwgAw+ρggAg=dρwAwvw2dh+dρgAgvg2dh.Make 0 on the right of equation (43) and then combine equation (39) to obtain the CMM without radial inflow.
(44)−dPdh=λHLρwvsw22D1−D2HHL2−ρwgHHL−ρgg1−HHL,where Ag and Aw are the cross-sectional areas of the flow space occupied by the gas phase and water phase in the annular flow, m2; Fgw is the mutual friction between gas and water interface, N; λHL is the friction coefficient between the annular liquid film and the pipe wall, dimensionless; HHL is the liquid holdup ratio of annular flow, dimensionless; and ReHL is the Reynolds number of annular flow liquid film, dimensionless.


#### 2.3.2. VMM for BHFP of Gas-Water Coproduction Stage

The physical model for gas-water coproduction stage is shown in Figure6.Figure 6
Flow state in the well for gas-water production at the same time.In Figure6, “O’” represents the depth of the DLL after falling from point “O”. The label of other depth points is the same as that in Figure 1. Figure 6(a) shows the complete model diagram. Figure 6(b) is a local enlarged drawing, in which only the water flow direction is drawn on the left and only the gas flow direction is drawn on the right.Similar to the situation of double-layer water or gas production, it is also necessary to establish a VMM including acceleration pressure drop, while the water and gas velocities of each section adopt the equations in Parts 2.1.2 and 2.2.2, respectively.For bubble flow, in equation (33),
(45)dρpmAvpm2dh=vpmdρpmAvpmdh+ρpmAvpmdvpmdh,(46)dρpmAvpmdh=ρwqIw−ρgqIg,(47)dvpm=ρwqIwdh−ρgqIgdhρpmA.Thus, equation (33) becomes
(48)−dPdh=λpmρpmvpm22D1−D2−ρpmg+2vpmρwqIw−ρgqIgA.For slug flow, in equation (35),
(49)dρdmAvdm2dh=vdmρwqIw−ρgqIg+ρdmAvdmdvdmdh,dvdm=ρwqIwdh−ρgqIgdhρdmA.Thus, equation (35) becomes
(50)−dPdh=2vdmρwqIw−ρgqIgA+Ltbλltbρwvltb2+Lsλllsρsvs22LD1−D2−ρdmg.For annular flow, in equation (43),
(51)dρwAwvw2dh=vwρwqIw+ρwAwvwdvwdh=2vwρwqIw,(52)dρgAgvg2dh=vgρgqIg+ρgAgvgdvgdh=2vgρgqIg.Thus, equation (43) becomes
(53)−dPdh=λHLρwvsw22D1−D2HHL2−ρwgHHL−ρgg1−HHL+2vswρwqIwAHHL+2vsgρgqIgA1−HHL.


#### 2.3.3. Calculation of BHFP of Gas-Water Coproduction Stage

First, after determining the position of the DLL, calculate the pressure drop of the dynamic gas column above the DLL. Then, the flow pattern [44, 45] of the two-phase flow below the DLL is judged, and the pressure drop is calculated iteratively downward according to the corresponding flow pattern. When the DLL is below the roof of reservoir I, there are three cases, as shown in Figure 7.
(54)Annularflow:vsg≥3.1gσρw−ρgρg20.25,(55)Slugflow:37vsw+2.57vrc<vsg<3.1gσρw−ρgρg20.25,(56)Bubbleflow:vsg≤37vsw+2.57vrc.Figure 7
Calculation process of BHFP for gas-water production at the same time.
(a)
General process(b)
Section OB calculation process(c)
Section BC, CD, and DE calculation processFigure7 (a) is the general flow chart, which is divided into four cases according to the depth of the DLL, and each case is divided into different sections and different flow patterns. The pressure drop of all dynamic gas column sections is calculated by using the relevant equation in Part 2.2.3. For gas-liquid mixed column, the calculation method for section OB and other sections below the roof of reservoir I are shown in Figures 7(b) and 7(c), respectively.(1) The DLL above the Roof of Reservoir I.
(1)
The dynamic gas column pressure dropPAO is obtained by using the equation without radial inflow in Part I of Section 2.2.3(2)
Calculate the pressure dropPOB of the gas-liquid mixed column section above the roof of reservoir I: ① the DLL is the starting point, where the pressure is equal to P0+PAO. Set the pressure drop of depth increment dh as dP, and then the pressure at the lower end of dh is P1=P0+PAO+dP and the temperature T1=T0+hAO+dh/M; ② calculate the average temperature T0+T1/2 and average pressure P0+P1/2 of dh; ③ the average compressibility coefficient is obtained by using equation (A.7), and the gas volume coefficient is obtained by substituting into equation (A.5); ④ use the calculated average value to calculate the parameters required for flow pattern judgment, including the gas density calculated by equation (A.3), vsgOBj calculated by equation (22), vswOB calculated by equation (8), and the average interfacial tension calculated by equation (B.8); and ⑤ judge the flow pattern. For annular flow pattern, use equations (40)–(42) and (44) to calculate the dP′. For slug flow pattern, use equations (C.1)–(C.18) and (36) to calculate the dP′. For bubble flow pattern, use equations (34) and (B.1)–(B.7) to calculate the dP′. ⑥ Compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+PAO+dP′ and return to step ②. On the contrary, take P0+PAO+dP′ as the pressure starting point for the next depth increment and repeat steps ②~⑥ until the depth goes down to the roof of reservoir I, and the pressure drop POB of the gas-liquid column section above the roof of reservoir I is obtained(3)
Calculate the pressure drop of the well section under the roof of reservoir I: starting from the roof of reservoir I downward, for calculating the apparent velocity of gas and water, we need to know the water production and gas production of each reservoir, respectively① Assuming that the BHFP of reservoir II ispwf2, the bottom hole temperature is obtained according to equation (A.6), and then the bottom hole average compressibility coefficient Z and gas volume coefficient BgDE are obtained by combination with pwf2. Substitute the above parameters into equation (32) to obtain Qsc2, and then qIw2 is obtained by substituting pwf2 into equation (17); ② calculate Qsc1=Qsc−Qsc2; ③ calculate qIw1 according to equation (18); ④ starting from the roof of reservoir I, the pressure here is equal to P0+PAO+POB, and the pressure drop of the first segment dh from top to bottom is assumed to be dP; so, the pressure at the lower end of dh is P1=P0+PAO+POB+dP, and the temperature is T1=T0+hAO+hOB+dh/M; ⑤ calculate the average temperature T0+T1/2 and average pressure P0+P1/2; ⑥ the average compressibility coefficient of dh is obtained by using equation (A.7), the gas volume coefficient is obtained by using equation (A.5), the gas density is obtained by using equation (A.3), and the average interfacial tension is obtained by using equation (B.8); then, calculate vsgBCj and vswBCj according to equations (23) and (9); and ⑦ judge the flow pattern. For annular flow pattern, use equations (53) and (40)–(42) to calculate the dP′. For slug flow pattern, use equations (C.1)–(C.18) and (50) to calculate the dP′. For bubbe flow pattern, use equations (48) and (B.1)–(B.7) to calculate the dP′. ⑧ Compare the difference between dP′ and dP. If it does not meet the accuracy requirements, make P1=P0+PAO+POB+dP′ and return to step ⑤. On the contrary, take P1=P0+PAO+POB+dP′ as the pressure starting point for the second segment, repeat steps ④~⑧ until the depth goes down to the floor of reservoir I, and calculate the pressure drop PBC of reservoir I. Similarly, using the method of steps ④~⑧ to obtain PCD and PDE, in step ⑥, use equations (10), (11), (24), and (25) to calculate the apparent velocity of water and gas of section CD and DE, respectively. ⑨ Calculate the pressure of reservoir II by Pwf2′=P0+PAO+POB+PBC+PCD+0.5PDE. Compare the assumed Pwf2 with the calculated Pwf2′, if the accuracy requirements are met, the calculated value represents the BHFP of reservoir II. If the accuracy requirements are not met, Pwf2 is replaced by Pwf2′ and calculate again from step ① until the accuracy requirements are met. Then, the BHFP of reservoir I is obtained.If the absolute values of one of the calculatedQsc1, Qsc2, qIw1, and qIw2 are less than or equal to zero, it indicates that the BHFP of the corresponding reservoir is greater than the desorption pressure (Qsc1, Qsc2) or the original reservoir pressure (qIw1, qIw2); so, the calculated value is equal to zero.(2) The DLL below the Roof of Reservoir I.
(1)
The dynamic gas column pressure dropPAB is obtained by using the equation without radial inflow in Part I of Section 2.2.3(2)
Use the method in Part I of Section2.2.3 to calculate the dynamic gas column pressure drop PBO and the mixed gas-liquid column POE of the section under the roof of reservoir I: ① if the DLL is within the range of section BC, the dynamic gas column velocity in section BO is calculated by equation (23), and the pressure drop is calculated iteratively by equation (29). ② If the DLL is within the range of section CD, the dynamic gas column in section BO can be divided into section BC and section CO. The velocity is calculated by equation (23) and equation (24), respectively, and the corresponding pressure drop is calculated iteratively by equations (29) and (30). ③ If the DLL is within the range of section DE, the dynamic gas column velocity in section BO is calculated by equations (23), (24), and (25), respectively, and the corresponding pressure drop is calculated iteratively by equations (29), (30), and (31)For mixed gas-liquid columnPOE, take the pressure at the DLL for the above three cases as the starting point and calculate through flow pattern judgment and iteration. Refer to Part I of Section 2.2.3 for the calculation process.


## 2.1. Establishment of BHFP Model in Double-Layer Water Production Stage

### 2.1.1. CMM of Single-Layer Water Production

When a single layer only produces water, there is a static gas column above the DLL, and the BHFP is composed of wellhead casing pressure, static gas column pressure drop, and dynamic liquid column pressure drop.(1) Static Gas Column Pressure Drop Model. Wellhead casing pressure can be obtained directly, and static gas column pressure PAO can be calculated directly by equation (1) [38].
(1)P=P0+P0Z01+hMT0M/Mg−1,(2)Mg=103RMwg,where P is the downhole pressure, MPa; P0 is the wellhead casing pressure, MPa; h is the DLL depth, m; M is the geothermal heating rate, m/K; Mg is the self-heating rate of gas, m/K, R is the general gas constant, 8.314 J/(mol•K), MW is the molar mass of the gas, g/mol; g is the gravitational acceleration, m/s2; T0 is the wellhead thermodynamic temperature, K; and Z0 is the compressibility coefficient of wellhead gas, dimensionless.(2) Dynamic Liquid Column Pressure Drop Model. In the CBM drainage stage, in order to reach the gas-water production stage as soon as possible, it is necessary to reduce the DLL to decrease the BHFP; so, the water pump displacement is required to be greater than the sum of the formation water production and the reduction of down-hole fluid accumulation. Therefore, the flow direction of the water in the annulus is downward, and the water is subject to upward friction, which is different from the conventional upward flow of gas and water.Referring to reference [39], the microsegment dh is selected, the direction is positive downward, and the basic model of single-layer water production is established through the momentum conservation equation, as shown in equation (3).
(3)−AdP+ρwgdhA−fwρwvw22πD1+D2dh=dρwAvw2.In equation (3), the three terms on the left are the pressure difference between the two ends of the microsegment, gravity, and friction, and the right side is the momentum change, which is equivalent to the acceleration pressure drop. In the general CMM, it is considered that the mass of the whole well is constant, the flow rate is constant, and the right side is taken as 0. The model is shown in equation (4)
(4)−dPdh=λwρwvw22D1−D2−ρwg,where
(5)λw=1.14−2lgeD1−D2+21.25Rew0.9−2,Rew=1000D1−D2ρwvwμw,vw=Qu−QwA,where A is the annulus sectional area, m2; dP is the pressure difference between the two ends of the microsegment dh fluid, Pa; fw and λw are the Fanning friction coefficient and Darcy friction coefficient of water, dimensionless; ρw is the density of water, kg/m3; vw is the velocity of water, m/s; D1 and D2 are the inner diameter of casing and outer diameter of oil tube, respectively, m; μw is the viscosity of formation water, mPa·s; e is the absolute roughness of pipe wall; e=0.016mm is taken for new oil tube and casing; Rew is the Reynolds number of single-phase water flow; and Qu and Qw are the pump displacement and water production, respectively, m3/s.


### 2.1.2. VMM for BHFP of Double-Layer Water Production Stage

When both reservoir I and reservoir II produce water only, according to the assumed conditions, the DLL is above reservoir I. The physical model is shown in Figure1.Figure 1
Flow state in well for double-layer water production.In Figure1, point A is the wellhead, points B and C, respectively, represent the roof and floor of reservoir I, and points D and E, respectively, represent the roof and floor of reservoir II. The depth of DLL is point O. Figure 1(a) is the complete model diagram. The original liquid accumulation in the well and the produced water flow to the bottom of the well are pumped out of the wellhead from the oil tube. Figure 1(b) is a local enlarged drawing of Figure 1(a), showing the principle of variable mass flow. Reservoir I is divided into m sublayers, which is assumed to be five sublayers. The amount of water flowing through each sublayer is different. For the first sublayer of reservoir I, the water flow is the flow of section OB, while the second sublayer is the flow of section OB plus the first sublayer and so on. For section CD, the flow is the sum of section OB flow and the total water flow of the five sublayers. The principle of water flow through each sublayer of reservoir II is the same.It can be seen that from the roof of reservoir I downward, the mass flow through each well section (BC, CD, DE) is increasing. At the same time, the change of flow mass will also produce acceleration pressure drop. So, there is a great difference between the DLCP and single-layer production, and the VMM including acceleration pressure drop applicable to the reservoir section is obtained through equation (3).In equation (3),
(6)dρwAvw2=vwdρwAvw+ρwAvwdvw,dρwAvw=ρwqIwdh,dvw=qIwdhA,fw=λw4,A=π4D12−D22.Then, equation (7) is obtained.
(7)−dPdh=λwρwvw22D1−D2−ρwg+2vwρwqIwA,where qIw is the volume flow of radial inflow water phase per unit length, m3/(s·m).In addition to deriving the VMM, it is also necessary to derive a special velocity equation suitable for each section according to the different water flow through each well section.The flow through section OB is the difference between the pump displacement and the total water production of the double layers, which is equal to the reduction rate of fluid column in the well. The flow of section CD is the difference between the pump displacement and the water production of reservoir II, which is formed by the original fluid accumulation in the well and the water production of reservoir I. Real-time changing velocity equations in section BC and section DE are calculated according to the change of the number of sublayers. Thus, the flow velocity equations of each section are derived and shown in equations (8)–(11):
(8)vOB=vswOB=Qu−qIw1hBC−qIw2hDEA,(9)vBCj=vswBCj=Qu−n−jqIw1hBC/n−qIw2hDEA,(10)vCD=vswCD=Qu−qIw2hDEA,(11)vDEj=vswDEj=Qu−n−jqIw2hDE/nA.For the CMM, the flow velocity of section OB is adopted for all well sections. Comparing (8) and (10), it is found that they differ in the water production of reservoir I. When the water production or the thickness of reservoir is large, the flow velocity of section CD will be significantly different from that of section OB. If the flow velocity of section OB is used uniformly, it will bring great errors. The same is true for the flow velocity of BC and DE segment.The subscript “sw” indicates the apparent velocity of water, which is only used in the gas-water two-phase flow stage as below, and is equal to the actual flow velocity in the single-phase water production stage.When both reservoirs produce water, the BHFP of reservoir II is composed of wellhead casing pressureP0, static gas column pressure Psg of section AO, and dynamic liquid column pressure Pl of section OE. Among them, the VMM of equation (7) is used to calculate the pressure drop in the reservoir section (BC and DE), and the CMM of equation (4) is used in the nonreservoir section (OB and CD). The BHFP of reservoir II and reservoir I is calculated by equation (12), and the pressure drop equation of each section is obtained by transforming equation (4) and equation (7) to equations (13)–(16).
(12)Pwf2=P0+Psg+Pl=P0+PAO+POB+PBC+PCD+0.5PDE,Pwf1=P0+PAO+POB+0.5PBC,(13)POB=ρwghOB−λOBρwvOB2hOB2D1−D2,(14)PBC=∑j=1nρwghBCn−λBCjρwvBCj22D1−D2hBCn−2vBCjρwqIw1AhBCn,(15)PCD=ρwghCD−λCDρwvCD2hCD2D1−D2,(16)PDE=∑j=1nρwghDEn−λDEjρwvDEj22D1−D2hDEn−2vDEjρwqIw2AhDEn,where n is the number of sublayers in the reservoir, and the meaning of each parameter is the same as that of the corresponding parameters in the basic model. The subscript represents the corresponding value of the specific section. qIw1 and qIw2 are the radial inflow water per unit length in reservoir I and reservoir II, m3/(s·m).In the flow velocity equations,qIw1 and qIw2 are unknown parameters, which need to be solved with two other equations jointly. The CBM production equation (17) is selected.
(17)Qwi=qIwihi=2πkihiPei−Pwfi×10−6μwlnrei/rw−3/4+Ski,where i=1,2 represents reservoir I and reservoir II, respectively, hi is the layer thickness, m; Ki is the single-phase flow permeability, 10-3 μm2; Pei is the original pressure of the reservoir, MPa; rei and rw are drainage radius and well radius, respectively, m; Ski is the skin coefficient, dimensionless; and Pwf1 and Pwf2 are the BHFP of reservoir I and II, respectively, MPa.It is found that when the DLL is measured twice, that is, the DLL depths of timet1 and t2 are H1 and H2, respectively; then, VOB=H2−H1/t2−t1, combined with equation (13), the relationship between qIw1 and qIw2 can be obtained, see equation (18). From a practical point of view, the DLL value is easy to obtain by online remote real-time monitoring equipment on site.
(18)qIw1=Qu−vOBA−qIw2hDEhBC=Qu−AH2−H1/t2−t1−qIw2hDEhBC.


### 2.1.3. Calculation of BHFP in Double-Layer Water Production Stage

The BHFP of double layers is solved by using the process shown in Figure2.Figure 2
Flow chart of BHFP calculation for double-layer water production.Figure2 is the flow chart of BHFP calculation for double-layer water production, and the specific process is shown below.First, input the basic parameters and then ① assign a value ton and assume the BHFP Pwf2 of reservoir II, ② substitute Pwf2 into equation (17) to obtain qIw2 and combine equation (18) to obtain qIw1, ③ substitute qIw1 and qIw2 into equations (8)–(11) to obtain the flow velocity of each section and then use equations (13)–(16) to obtain the pressure drop of each section, ④ use equation to obtain the BHFP Pwf2′ of reservoir II, ⑤ compare Pwf2 and Pwf2′, if it meets the accuracy requirements, it is the BHFP of reservoir II. Otherwise, let Pwf2′ replace Pwf2 and repeat ②~⑤ until it meets the accuracy requirements. ⑥ Again, use equation (12) to obtain the BHFP of reservoir I.


## 2.1.1. CMM of Single-Layer Water Production

When a single layer only produces water, there is a static gas column above the DLL, and the BHFP is composed of wellhead casing pressure, static gas column pressure drop, and dynamic liquid column pressure drop.(1) Static Gas Column Pressure Drop Model. Wellhead casing pressure can be obtained directly, and static gas column pressure PAO can be calculated directly by equation (1) [38].
(1)P=P0+P0Z01+hMT0M/Mg−1,(2)Mg=103RMwg,where P is the downhole pressure, MPa; P0 is the wellhead casing pressure, MPa; h is the DLL depth, m; M is the geothermal heating rate, m/K; Mg is the self-heating rate of gas, m/K, R is the general gas constant, 8.314 J/(mol•K), MW is the molar mass of the gas, g/mol; g is the gravitational acceleration, m/s2; T0 is the wellhead thermodynamic temperature, K; and Z0 is the compressibility coefficient of wellhead gas, dimensionless.(2) Dynamic Liquid Column Pressure Drop Model. In the CBM drainage stage, in order to reach the gas-water production stage as soon as possible, it is necessary to reduce the DLL to decrease the BHFP; so, the water pump displacement is required to be greater than the sum of the formation water production and the reduction of down-hole fluid accumulation. Therefore, the flow direction of the water in the annulus is downward, and the water is subject to upward friction, which is different from the conventional upward flow of gas and water.Referring to reference [39], the microsegment dh is selected, the direction is positive downward, and the basic model of single-layer water production is established through the momentum conservation equation, as shown in equation (3).
(3)−AdP+ρwgdhA−fwρwvw22πD1+D2dh=dρwAvw2.In equation (3), the three terms on the left are the pressure difference between the two ends of the microsegment, gravity, and friction, and the right side is the momentum change, which is equivalent to the acceleration pressure drop. In the general CMM, it is considered that the mass of the whole well is constant, the flow rate is constant, and the right side is taken as 0. The model is shown in equation (4)
(4)−dPdh=λwρwvw22D1−D2−ρwg,where
(5)λw=1.14−2lgeD1−D2+21.25Rew0.9−2,Rew=1000D1−D2ρwvwμw,vw=Qu−QwA,where A is the annulus sectional area, m2; dP is the pressure difference between the two ends of the microsegment dh fluid, Pa; fw and λw are the Fanning friction coefficient and Darcy friction coefficient of water, dimensionless; ρw is the density of water, kg/m3; vw is the velocity of water, m/s; D1 and D2 are the inner diameter of casing and outer diameter of oil tube, respectively, m; μw is the viscosity of formation water, mPa·s; e is the absolute roughness of pipe wall; e=0.016mm is taken for new oil tube and casing; Rew is the Reynolds number of single-phase water flow; and Qu and Qw are the pump displacement and water production, respectively, m3/s.


## 2.1.2. VMM for BHFP of Double-Layer Water Production Stage

When both reservoir I and reservoir II produce water only, according to the assumed conditions, the DLL is above reservoir I. The physical model is shown in Figure1.Figure 1
Flow state in well for double-layer water production.In Figure1, point A is the wellhead, points B and C, respectively, represent the roof and floor of reservoir I, and points D and E, respectively, represent the roof and floor of reservoir II. The depth of DLL is point O. Figure 1(a) is the complete model diagram. The original liquid accumulation in the well and the produced water flow to the bottom of the well are pumped out of the wellhead from the oil tube. Figure 1(b) is a local enlarged drawing of Figure 1(a), showing the principle of variable mass flow. Reservoir I is divided into m sublayers, which is assumed to be five sublayers. The amount of water flowing through each sublayer is different. For the first sublayer of reservoir I, the water flow is the flow of section OB, while the second sublayer is the flow of section OB plus the first sublayer and so on. For section CD, the flow is the sum of section OB flow and the total water flow of the five sublayers. The principle of water flow through each sublayer of reservoir II is the same.It can be seen that from the roof of reservoir I downward, the mass flow through each well section (BC, CD, DE) is increasing. At the same time, the change of flow mass will also produce acceleration pressure drop. So, there is a great difference between the DLCP and single-layer production, and the VMM including acceleration pressure drop applicable to the reservoir section is obtained through equation (3).In equation (3),
(6)dρwAvw2=vwdρwAvw+ρwAvwdvw,dρwAvw=ρwqIwdh,dvw=qIwdhA,fw=λw4,A=π4D12−D22.Then, equation (7) is obtained.
(7)−dPdh=λwρwvw22D1−D2−ρwg+2vwρwqIwA,where qIw is the volume flow of radial inflow water phase per unit length, m3/(s·m).In addition to deriving the VMM, it is also necessary to derive a special velocity equation suitable for each section according to the different water flow through each well section.The flow through section OB is the difference between the pump displacement and the total water production of the double layers, which is equal to the reduction rate of fluid column in the well. The flow of section CD is the difference between the pump displacement and the water production of reservoir II, which is formed by the original fluid accumulation in the well and the water production of reservoir I. Real-time changing velocity equations in section BC and section DE are calculated according to the change of the number of sublayers. Thus, the flow velocity equations of each section are derived and shown in equations (8)–(11):
(8)vOB=vswOB=Qu−qIw1hBC−qIw2hDEA,(9)vBCj=vswBCj=Qu−n−jqIw1hBC/n−qIw2hDEA,(10)vCD=vswCD=Qu−qIw2hDEA,(11)vDEj=vswDEj=Qu−n−jqIw2hDE/nA.For the CMM, the flow velocity of section OB is adopted for all well sections. Comparing (8) and (10), it is found that they differ in the water production of reservoir I. When the water production or the thickness of reservoir is large, the flow velocity of section CD will be significantly different from that of section OB. If the flow velocity of section OB is used uniformly, it will bring great errors. The same is true for the flow velocity of BC and DE segment.The subscript “sw” indicates the apparent velocity of water, which is only used in the gas-water two-phase flow stage as below, and is equal to the actual flow velocity in the single-phase water production stage.When both reservoirs produce water, the BHFP of reservoir II is composed of wellhead casing pressureP0, static gas column pressure Psg of section AO, and dynamic liquid column pressure Pl of section OE. Among them, the VMM of equation (7) is used to calculate the pressure drop in the reservoir section (BC and DE), and the CMM of equation (4) is used in the nonreservoir section (OB and CD). The BHFP of reservoir II and reservoir I is calculated by equation (12), and the pressure drop equation of each section is obtained by transforming equation (4) and equation (7) to equations (13)–(16).
(12)Pwf2=P0+Psg+Pl=P0+PAO+POB+PBC+PCD+0.5PDE,Pwf1=P0+PAO+POB+0.5PBC,(13)POB=ρwghOB−λOBρwvOB2hOB2D1−D2,(14)PBC=∑j=1nρwghBCn−λBCjρwvBCj22D1−D2hBCn−2vBCjρwqIw1AhBCn,(15)PCD=ρwghCD−λCDρwvCD2hCD2D1−D2,(16)PDE=∑j=1nρwghDEn−λDEjρwvDEj22D1−D2hDEn−2vDEjρwqIw2AhDEn,where n is the number of sublayers in the reservoir, and the meaning of each parameter is the same as that of the corresponding parameters in the basic model. The subscript represents the corresponding value of the specific section. qIw1 and qIw2 are the radial inflow water per unit length in reservoir I and reservoir II, m3/(s·m).In the flow velocity equations,qIw1 and qIw2 are unknown parameters, which need to be solved with two other equations jointly. The CBM production equation (17) is selected.
(17)Qwi=qIwihi=2πkihiPei−Pwfi×10−6μwlnrei/rw−3/4+Ski,where i=1,2 represents reservoir I and reservoir II, respectively, hi is the layer thickness, m; Ki is the single-phase flow permeability, 10-3 μm2; Pei is the original pressure of the reservoir, MPa; rei and rw are drainage radius and well radius, respectively, m; Ski is the skin coefficient, dimensionless; and Pwf1 and Pwf2 are the BHFP of reservoir I and II, respectively, MPa.It is found that when the DLL is measured twice, that is, the DLL depths of timet1 and t2 are H1 and H2, respectively; then, VOB=H2−H1/t2−t1, combined with equation (13), the relationship between qIw1 and qIw2 can be obtained, see equation (18). From a practical point of view, the DLL value is easy to obtain by online remote real-time monitoring equipment on site.
(18)qIw1=Qu−vOBA−qIw2hDEhBC=Qu−AH2−H1/t2−t1−qIw2hDEhBC.


## 2.1.3. Calculation of BHFP in Double-Layer Water Production Stage

The BHFP of double layers is solved by using the process shown in Figure2.Figure 2
Flow chart of BHFP calculation for double-layer water production.Figure2 is the flow chart of BHFP calculation for double-layer water production, and the specific process is shown below.First, input the basic parameters and then ① assign a value ton and assume the BHFP Pwf2 of reservoir II, ② substitute Pwf2 into equation (17) to obtain qIw2 and combine equation (18) to obtain qIw1, ③ substitute qIw1 and qIw2 into equations (8)–(11) to obtain the flow velocity of each section and then use equations (13)–(16) to obtain the pressure drop of each section, ④ use equation to obtain the BHFP Pwf2′ of reservoir II, ⑤ compare Pwf2 and Pwf2′, if it meets the accuracy requirements, it is the BHFP of reservoir II. Otherwise, let Pwf2′ replace Pwf2 and repeat ②~⑤ until it meets the accuracy requirements. ⑥ Again, use equation (12) to obtain the BHFP of reservoir I.


## 2.2. Establishment of BHFP Model in Double-Layer Gas Production Stage

The CMM of single-layer gas production is given, and then the BHFP model of double-layer gas production is obtained. The BHFP is composed of wellhead casing pressure and dynamic gas column pressure drop.


### 2.2.1. CMM of Single-Layer Gas Production

Referring to the “Dynamic liquid column pressure drop,” a CMM of single-layer gas production for calculating dynamic gas column pressure drop is established.(19)−AdPdh+ρggA+λg4ρgvg22πD1+D2=dρgAvg2dh.Similarly, according to equation (19), the CMM without radial inflow is obtained.
(20)−dPdh=−λgρgvg22D1−D2−ρgg.See AppendixA for the calculation equation of each parameter in equation (19), where λg is the Darcy friction coefficient between gas and tubing casing annulus wall, dimensionless; ρg is the density of down-hole gas, kg/m3; and vg is the velocity of gas, m/s.


### 2.2.2. VMM for BHFP of Double-Layer Gas Production Stage

When the DLL is close to the bottom of the well, reservoirs I and II no longer produce water, but only gas. The physical model is shown in Figure3.Figure 3
Flow state in well when both reservoirs produce gas only.The lable of each depth point in Figure3 is the same as that in Figure 1. Figure 3(a) shows the complete model diagram. Figure 3(b) is a local enlarged drawing of Figure 3(a). The gas in double layers flows upward out of the wellhead, and the gas volume in each sublayer is represented by an arrow. The more arrows in cross-sections at different depths, the more gas flows through.It is obvious that more and more gas is produced from bottom to top in the section BE, belonging to variable mass flow. Similar to the double-layer water production situation, the VMM including acceleration pressure drop is established, and velocity equations of gas flow are derived. The results are shown in equations (21)–(25), respectively.
(21)−dPdh=−λgρgvg22D1−D2−ρgg+2vgρgqIgA,where qIg is the volume flow of gas phase radial inflow per unit length, m3/(s·m).
(22)vABj=vsgABj=BgABjQscA,(23)vBCj=vsgBCj=BgBCjn−j/nQsc1+Qsc2A,(24)vCDj=vsgCDj=BgCDjQsc2A,(25)vDEj=vsgDEj=BgDEjn−j/nQsc2A.Equations (22)–(25) are derived from equation (A.4). Gas flow through section AB comes from the total gas of double reservoirs, while the gas flow through section CD is only the gas production of reservoir II. For the reservoir sections BC and DE, the gas flow at different depths changes with the number of sublayers. However, the gas velocity of all well sections for the CMM adopts the velocity of section AB.The subscript “sg” of the flow calculation equation of each section indicates the apparent velocity of the gas, which is only used in the gas-water two-phase flow stage below, and is equal to the actual gas velocity in the single-phase gas production stage.The pressure drop equation of each section is obtained by transforming equation (20) and equation (21) to equation (28)–(31), and then equations (26) and (27) are used to calculate the BHFP of the two reservoirs.
(26)Pwf2=P0+Pdg=P0+PAB+PCD+PBC+0.5PDE,(27)Pwf1=P0+PAB+0.5PBC,(28)PABj=λABjρgABjvABj22D1−D2+ρgABjghABn,(29)PBCj=ρgBCjghBCn+λBCjρgBCjvBCj22D1−D2hBCn−2vBCjρgBCjAQsc1BgBCjn,(30)PCDj=λCDjρgCDjvCDj22D1−D2+ρgCDjghCDn,(31)PDEj=ρgDEjghDEn+λDEjρgDEjvDEj22D1−D2hDEn−2vDEjρgDEjAQsc2BgDEjn.Because the wellhead can only obtain the total gas volume and cannot directly obtain the respective gas production of each reservoir, only for the section AB, the flow velocity can directly be calculated by the iterative method, while the flow velocity calculation of section BC, CD, and DE sections needs to be combined with the gas productivity.(32)Qsci=774.6kgihiPei2−Pwfi286400TiZiμgilnrei/rw+SKi,where Kgi is the gas permeability of the reservoir.


### 2.2.3. Calculation of BHFP Double-Layer Gas Production Stage

Using the process shown in Figure4, the BHFP of the two reservoirs is solved in two stages.Figure 4
Flow chart of BHFP calculation for double-layer gas production.In Figure4, the dotted box on the left is the iterative calculation procedure of section AB without radial inflow. In the right part, the surface gas production of reservoir I is assumed, and then the production of double layers and the pressure drop of each well section are calculated iteratively according to the process on the left, so as to obtain the BHFP of the double layers.(1) Pressure Drop Calculation for Section AB. The calculation steps are as follows:① Taking the wellhead casing pressureP0 as the starting point, assuming that the pressure drop of the first subsection hAB/n is dP, the pressure at the lower end of the first subsection is P1=P0+dP, and the temperature T1=T0+jhAB/n/M; ② calculate the average temperature T0+T1/2 and average pressure P0+P1/2; ③ the average compressibility coefficient is obtained by substituting the average temperature and average pressure into equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the pressure drop dP′ from equation (A.1)–(A.3), (A.8), (22), and (28); and ⑤ compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+dP′ and return to step ②. On the contrary, take (P0+dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the roof of reservoir I; that is, the dynamic gas column pressure drop PAB of section AB is obtained.(2) Pressure Drop Calculation for Sections BC, CD, and DE. This section mainly calculates Qsc1 and Qsc2. The specific steps are as follows:① AssumingQsc1 value, so, Qsc2=Qsc−Qsc1. Taking PAB as the starting point, assuming that the pressure drop of the first subsection hBC/n of section BC is dP, the pressure at the lower end of the first subsection is P1=P0+PAB+dP, and the temperature T1=T0+hAB+jhBC/n/M; ② calculate the average temperature T0+T1/2 and average pressure P0+PAB+P1/2; ③ the average compressibility coefficient is obtained by using equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the dP’ from equations (A.1)–(A.3), (A.8), (23), and (29); and ⑤ compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+PAB+dP′ and return to step ②. On the contrary, take (P0+PAB+dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the floor of reservoir I, the dynamic gas column pressure drop PBC is obtained. ⑥ The BHFP of reservoir I is calculated by equation (27) and substituted into (32) to obtain Qsc1′; ⑦ compare Qsc1 and Qsc1′, if it does not meet the accuracy requirements, use Qsc1′ instead of Qsc1, repeat steps ① to ⑦ until it meets the accuracy requirements, then get PBC and determine the values of Qsc1 and Qsc2; and ⑧ taking the pressure at the floor of reservoir I as the starting point, PCD and PDE are obtained by using equations (24), (30), (25), and (31), respectively, and then the BHFP of reservoir II is obtained.


## 2.2.1. CMM of Single-Layer Gas Production

Referring to the “Dynamic liquid column pressure drop,” a CMM of single-layer gas production for calculating dynamic gas column pressure drop is established.(19)−AdPdh+ρggA+λg4ρgvg22πD1+D2=dρgAvg2dh.Similarly, according to equation (19), the CMM without radial inflow is obtained.
(20)−dPdh=−λgρgvg22D1−D2−ρgg.See AppendixA for the calculation equation of each parameter in equation (19), where λg is the Darcy friction coefficient between gas and tubing casing annulus wall, dimensionless; ρg is the density of down-hole gas, kg/m3; and vg is the velocity of gas, m/s.


## 2.2.2. VMM for BHFP of Double-Layer Gas Production Stage

When the DLL is close to the bottom of the well, reservoirs I and II no longer produce water, but only gas. The physical model is shown in Figure3.Figure 3
Flow state in well when both reservoirs produce gas only.The lable of each depth point in Figure3 is the same as that in Figure 1. Figure 3(a) shows the complete model diagram. Figure 3(b) is a local enlarged drawing of Figure 3(a). The gas in double layers flows upward out of the wellhead, and the gas volume in each sublayer is represented by an arrow. The more arrows in cross-sections at different depths, the more gas flows through.It is obvious that more and more gas is produced from bottom to top in the section BE, belonging to variable mass flow. Similar to the double-layer water production situation, the VMM including acceleration pressure drop is established, and velocity equations of gas flow are derived. The results are shown in equations (21)–(25), respectively.
(21)−dPdh=−λgρgvg22D1−D2−ρgg+2vgρgqIgA,where qIg is the volume flow of gas phase radial inflow per unit length, m3/(s·m).
(22)vABj=vsgABj=BgABjQscA,(23)vBCj=vsgBCj=BgBCjn−j/nQsc1+Qsc2A,(24)vCDj=vsgCDj=BgCDjQsc2A,(25)vDEj=vsgDEj=BgDEjn−j/nQsc2A.Equations (22)–(25) are derived from equation (A.4). Gas flow through section AB comes from the total gas of double reservoirs, while the gas flow through section CD is only the gas production of reservoir II. For the reservoir sections BC and DE, the gas flow at different depths changes with the number of sublayers. However, the gas velocity of all well sections for the CMM adopts the velocity of section AB.The subscript “sg” of the flow calculation equation of each section indicates the apparent velocity of the gas, which is only used in the gas-water two-phase flow stage below, and is equal to the actual gas velocity in the single-phase gas production stage.The pressure drop equation of each section is obtained by transforming equation (20) and equation (21) to equation (28)–(31), and then equations (26) and (27) are used to calculate the BHFP of the two reservoirs.
(26)Pwf2=P0+Pdg=P0+PAB+PCD+PBC+0.5PDE,(27)Pwf1=P0+PAB+0.5PBC,(28)PABj=λABjρgABjvABj22D1−D2+ρgABjghABn,(29)PBCj=ρgBCjghBCn+λBCjρgBCjvBCj22D1−D2hBCn−2vBCjρgBCjAQsc1BgBCjn,(30)PCDj=λCDjρgCDjvCDj22D1−D2+ρgCDjghCDn,(31)PDEj=ρgDEjghDEn+λDEjρgDEjvDEj22D1−D2hDEn−2vDEjρgDEjAQsc2BgDEjn.Because the wellhead can only obtain the total gas volume and cannot directly obtain the respective gas production of each reservoir, only for the section AB, the flow velocity can directly be calculated by the iterative method, while the flow velocity calculation of section BC, CD, and DE sections needs to be combined with the gas productivity.(32)Qsci=774.6kgihiPei2−Pwfi286400TiZiμgilnrei/rw+SKi,where Kgi is the gas permeability of the reservoir.


## 2.2.3. Calculation of BHFP Double-Layer Gas Production Stage

Using the process shown in Figure4, the BHFP of the two reservoirs is solved in two stages.Figure 4
Flow chart of BHFP calculation for double-layer gas production.In Figure4, the dotted box on the left is the iterative calculation procedure of section AB without radial inflow. In the right part, the surface gas production of reservoir I is assumed, and then the production of double layers and the pressure drop of each well section are calculated iteratively according to the process on the left, so as to obtain the BHFP of the double layers.(1) Pressure Drop Calculation for Section AB. The calculation steps are as follows:① Taking the wellhead casing pressureP0 as the starting point, assuming that the pressure drop of the first subsection hAB/n is dP, the pressure at the lower end of the first subsection is P1=P0+dP, and the temperature T1=T0+jhAB/n/M; ② calculate the average temperature T0+T1/2 and average pressure P0+P1/2; ③ the average compressibility coefficient is obtained by substituting the average temperature and average pressure into equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the pressure drop dP′ from equation (A.1)–(A.3), (A.8), (22), and (28); and ⑤ compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+dP′ and return to step ②. On the contrary, take (P0+dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the roof of reservoir I; that is, the dynamic gas column pressure drop PAB of section AB is obtained.(2) Pressure Drop Calculation for Sections BC, CD, and DE. This section mainly calculates Qsc1 and Qsc2. The specific steps are as follows:① AssumingQsc1 value, so, Qsc2=Qsc−Qsc1. Taking PAB as the starting point, assuming that the pressure drop of the first subsection hBC/n of section BC is dP, the pressure at the lower end of the first subsection is P1=P0+PAB+dP, and the temperature T1=T0+hAB+jhBC/n/M; ② calculate the average temperature T0+T1/2 and average pressure P0+PAB+P1/2; ③ the average compressibility coefficient is obtained by using equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the dP’ from equations (A.1)–(A.3), (A.8), (23), and (29); and ⑤ compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+PAB+dP′ and return to step ②. On the contrary, take (P0+PAB+dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the floor of reservoir I, the dynamic gas column pressure drop PBC is obtained. ⑥ The BHFP of reservoir I is calculated by equation (27) and substituted into (32) to obtain Qsc1′; ⑦ compare Qsc1 and Qsc1′, if it does not meet the accuracy requirements, use Qsc1′ instead of Qsc1, repeat steps ① to ⑦ until it meets the accuracy requirements, then get PBC and determine the values of Qsc1 and Qsc2; and ⑧ taking the pressure at the floor of reservoir I as the starting point, PCD and PDE are obtained by using equations (24), (30), (25), and (31), respectively, and then the BHFP of reservoir II is obtained.


## 2.3. Establishment of BHFP Model in Gas-Water Coproduction Stage

In gas-water coproduction stage, the mixed gas-liquid column has different flow patterns according to the different gas-water ratio. For low-yield CBM Wells, only bubble flow or slug flow is generated during single-layer production [40], while annular flow may be produced when the gas volume increases during DLCP.


### 2.3.1. CMM of Gas-Water Coproduction of Single Layer

The gas-water mixture column is divided into three flow patterns, as shown in Figure5.Figure 5
Physical model of two-phase flow in mixed gas-liquid column.
(a)
Bubble flow(b)
Slug flow(c)
Annular flowFigure5 shows three flow patterns that may occur during DLCP. The downward velocity is positive, and the parameter in the figure can be seen in the following equations.(1) CMM for Bubble Flow in Single-Layer Production. Referring to the “Dynamic liquid column pressure drop”, the CMM for bubble flow in single-layer production is established.
(33)−AdPdh−λpm4ρpmvpm2πD1+D22+ρpmgA=dρpmAvpm2dh,(34)−dPdh=λpmρpmvpm22D1−D2−ρpmg.See AppendixB for the calculation equation of each parameter in equation (33), where λpm is the friction coefficient of bubble flow, dimensionless, ρpm is the density of mixed fluid under down-hole conditions, kg/m3, and vpm is the velocity of mixed fluid, m/s.(2) CMM for Slug Flow in Single-Layer Production. Literature [41] is cited here, the CMM for slug flow in single-layer production is established.
(35)−LAdPdh+LAρdmg−Ltbλltbρwvltb2+Lsλllsρsvs2πD1+D28=LdρdmAvdm2dh.Similarly, according to equation (35), the CMM without radial inflow is obtained.
(36)−dPdh=Ltbλltbρwvltb2+Lsλllsρsvs22LD1−D2−ρdmg.See AppendixC for the calculation equation of each parameter in equation (35), where L, Ls, and Ltb are the total length of slug section, liquid slug section, and Taylor bubble section, respectively, m; ρdm is the average density of the whole slug, kg/m3; λtb and λlls are the friction coefficient of Taylor bubble section and liquid slug section, respectively, dimensionless; vs and ρs are the velocity m/s and density kg/m3 of gas-liquid mixture in liquid slug section, respectively; vdm is the average velocity of the whole slug, m/s; and vltb is the falling velocity of Taylor bubble liquid film, m/s.(3) CMM for Annular Flow in Single-Layer Production. Referring to the “Dynamic liquid column pressure drop,” the CMM for annular flow in single-layer production is established[42, 43].
(37)−AgdPdh+Fgw+ρggAg=dρgAgvg2dh,(38)−AwdPdh−Fgw−λHLρwvw2πD1+D28+ρwgAw=dρwAwvw2dh,where
(39)HHL=AwA,vw=vswHHL,vg=vsg1−HHL,(40)HHL=Hltb=1−vsgvgtb,(41)λHL=1.14−2lgeD1−D2+21.25ReHL0.9−2,(42)ReHL=D1−D2ρwvswμw.Add equations (37) and (38) to get equation (43):
(43)−AdPdh−λHLρwvw2πD1+D28+ρwgAw+ρggAg=dρwAwvw2dh+dρgAgvg2dh.Make 0 on the right of equation (43) and then combine equation (39) to obtain the CMM without radial inflow.
(44)−dPdh=λHLρwvsw22D1−D2HHL2−ρwgHHL−ρgg1−HHL,where Ag and Aw are the cross-sectional areas of the flow space occupied by the gas phase and water phase in the annular flow, m2; Fgw is the mutual friction between gas and water interface, N; λHL is the friction coefficient between the annular liquid film and the pipe wall, dimensionless; HHL is the liquid holdup ratio of annular flow, dimensionless; and ReHL is the Reynolds number of annular flow liquid film, dimensionless.


### 2.3.2. VMM for BHFP of Gas-Water Coproduction Stage

The physical model for gas-water coproduction stage is shown in Figure6.Figure 6
Flow state in the well for gas-water production at the same time.In Figure6, “O’” represents the depth of the DLL after falling from point “O”. The label of other depth points is the same as that in Figure 1. Figure 6(a) shows the complete model diagram. Figure 6(b) is a local enlarged drawing, in which only the water flow direction is drawn on the left and only the gas flow direction is drawn on the right.Similar to the situation of double-layer water or gas production, it is also necessary to establish a VMM including acceleration pressure drop, while the water and gas velocities of each section adopt the equations in Parts 2.1.2 and 2.2.2, respectively.For bubble flow, in equation (33),
(45)dρpmAvpm2dh=vpmdρpmAvpmdh+ρpmAvpmdvpmdh,(46)dρpmAvpmdh=ρwqIw−ρgqIg,(47)dvpm=ρwqIwdh−ρgqIgdhρpmA.Thus, equation (33) becomes
(48)−dPdh=λpmρpmvpm22D1−D2−ρpmg+2vpmρwqIw−ρgqIgA.For slug flow, in equation (35),
(49)dρdmAvdm2dh=vdmρwqIw−ρgqIg+ρdmAvdmdvdmdh,dvdm=ρwqIwdh−ρgqIgdhρdmA.Thus, equation (35) becomes
(50)−dPdh=2vdmρwqIw−ρgqIgA+Ltbλltbρwvltb2+Lsλllsρsvs22LD1−D2−ρdmg.For annular flow, in equation (43),
(51)dρwAwvw2dh=vwρwqIw+ρwAwvwdvwdh=2vwρwqIw,(52)dρgAgvg2dh=vgρgqIg+ρgAgvgdvgdh=2vgρgqIg.Thus, equation (43) becomes
(53)−dPdh=λHLρwvsw22D1−D2HHL2−ρwgHHL−ρgg1−HHL+2vswρwqIwAHHL+2vsgρgqIgA1−HHL.


### 2.3.3. Calculation of BHFP of Gas-Water Coproduction Stage

First, after determining the position of the DLL, calculate the pressure drop of the dynamic gas column above the DLL. Then, the flow pattern [44, 45] of the two-phase flow below the DLL is judged, and the pressure drop is calculated iteratively downward according to the corresponding flow pattern. When the DLL is below the roof of reservoir I, there are three cases, as shown in Figure 7.
(54)Annularflow:vsg≥3.1gσρw−ρgρg20.25,(55)Slugflow:37vsw+2.57vrc<vsg<3.1gσρw−ρgρg20.25,(56)Bubbleflow:vsg≤37vsw+2.57vrc.Figure 7
Calculation process of BHFP for gas-water production at the same time.
(a)
General process(b)
Section OB calculation process(c)
Section BC, CD, and DE calculation processFigure7 (a) is the general flow chart, which is divided into four cases according to the depth of the DLL, and each case is divided into different sections and different flow patterns. The pressure drop of all dynamic gas column sections is calculated by using the relevant equation in Part 2.2.3. For gas-liquid mixed column, the calculation method for section OB and other sections below the roof of reservoir I are shown in Figures 7(b) and 7(c), respectively.(1) The DLL above the Roof of Reservoir I.
(1)
The dynamic gas column pressure dropPAO is obtained by using the equation without radial inflow in Part I of Section 2.2.3(2)
Calculate the pressure dropPOB of the gas-liquid mixed column section above the roof of reservoir I: ① the DLL is the starting point, where the pressure is equal to P0+PAO. Set the pressure drop of depth increment dh as dP, and then the pressure at the lower end of dh is P1=P0+PAO+dP and the temperature T1=T0+hAO+dh/M; ② calculate the average temperature T0+T1/2 and average pressure P0+P1/2 of dh; ③ the average compressibility coefficient is obtained by using equation (A.7), and the gas volume coefficient is obtained by substituting into equation (A.5); ④ use the calculated average value to calculate the parameters required for flow pattern judgment, including the gas density calculated by equation (A.3), vsgOBj calculated by equation (22), vswOB calculated by equation (8), and the average interfacial tension calculated by equation (B.8); and ⑤ judge the flow pattern. For annular flow pattern, use equations (40)–(42) and (44) to calculate the dP′. For slug flow pattern, use equations (C.1)–(C.18) and (36) to calculate the dP′. For bubble flow pattern, use equations (34) and (B.1)–(B.7) to calculate the dP′. ⑥ Compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+PAO+dP′ and return to step ②. On the contrary, take P0+PAO+dP′ as the pressure starting point for the next depth increment and repeat steps ②~⑥ until the depth goes down to the roof of reservoir I, and the pressure drop POB of the gas-liquid column section above the roof of reservoir I is obtained(3)
Calculate the pressure drop of the well section under the roof of reservoir I: starting from the roof of reservoir I downward, for calculating the apparent velocity of gas and water, we need to know the water production and gas production of each reservoir, respectively① Assuming that the BHFP of reservoir II ispwf2, the bottom hole temperature is obtained according to equation (A.6), and then the bottom hole average compressibility coefficient Z and gas volume coefficient BgDE are obtained by combination with pwf2. Substitute the above parameters into equation (32) to obtain Qsc2, and then qIw2 is obtained by substituting pwf2 into equation (17); ② calculate Qsc1=Qsc−Qsc2; ③ calculate qIw1 according to equation (18); ④ starting from the roof of reservoir I, the pressure here is equal to P0+PAO+POB, and the pressure drop of the first segment dh from top to bottom is assumed to be dP; so, the pressure at the lower end of dh is P1=P0+PAO+POB+dP, and the temperature is T1=T0+hAO+hOB+dh/M; ⑤ calculate the average temperature T0+T1/2 and average pressure P0+P1/2; ⑥ the average compressibility coefficient of dh is obtained by using equation (A.7), the gas volume coefficient is obtained by using equation (A.5), the gas density is obtained by using equation (A.3), and the average interfacial tension is obtained by using equation (B.8); then, calculate vsgBCj and vswBCj according to equations (23) and (9); and ⑦ judge the flow pattern. For annular flow pattern, use equations (53) and (40)–(42) to calculate the dP′. For slug flow pattern, use equations (C.1)–(C.18) and (50) to calculate the dP′. For bubbe flow pattern, use equations (48) and (B.1)–(B.7) to calculate the dP′. ⑧ Compare the difference between dP′ and dP. If it does not meet the accuracy requirements, make P1=P0+PAO+POB+dP′ and return to step ⑤. On the contrary, take P1=P0+PAO+POB+dP′ as the pressure starting point for the second segment, repeat steps ④~⑧ until the depth goes down to the floor of reservoir I, and calculate the pressure drop PBC of reservoir I. Similarly, using the method of steps ④~⑧ to obtain PCD and PDE, in step ⑥, use equations (10), (11), (24), and (25) to calculate the apparent velocity of water and gas of section CD and DE, respectively. ⑨ Calculate the pressure of reservoir II by Pwf2′=P0+PAO+POB+PBC+PCD+0.5PDE. Compare the assumed Pwf2 with the calculated Pwf2′, if the accuracy requirements are met, the calculated value represents the BHFP of reservoir II. If the accuracy requirements are not met, Pwf2 is replaced by Pwf2′ and calculate again from step ① until the accuracy requirements are met. Then, the BHFP of reservoir I is obtained.If the absolute values of one of the calculatedQsc1, Qsc2, qIw1, and qIw2 are less than or equal to zero, it indicates that the BHFP of the corresponding reservoir is greater than the desorption pressure (Qsc1, Qsc2) or the original reservoir pressure (qIw1, qIw2); so, the calculated value is equal to zero.(2) The DLL below the Roof of Reservoir I.
(1)
The dynamic gas column pressure dropPAB is obtained by using the equation without radial inflow in Part I of Section 2.2.3(2)
Use the method in Part I of Section2.2.3 to calculate the dynamic gas column pressure drop PBO and the mixed gas-liquid column POE of the section under the roof of reservoir I: ① if the DLL is within the range of section BC, the dynamic gas column velocity in section BO is calculated by equation (23), and the pressure drop is calculated iteratively by equation (29). ② If the DLL is within the range of section CD, the dynamic gas column in section BO can be divided into section BC and section CO. The velocity is calculated by equation (23) and equation (24), respectively, and the corresponding pressure drop is calculated iteratively by equations (29) and (30). ③ If the DLL is within the range of section DE, the dynamic gas column velocity in section BO is calculated by equations (23), (24), and (25), respectively, and the corresponding pressure drop is calculated iteratively by equations (29), (30), and (31)For mixed gas-liquid columnPOE, take the pressure at the DLL for the above three cases as the starting point and calculate through flow pattern judgment and iteration. Refer to Part I of Section 2.2.3 for the calculation process.


## 2.3.1. CMM of Gas-Water Coproduction of Single Layer

The gas-water mixture column is divided into three flow patterns, as shown in Figure5.Figure 5
Physical model of two-phase flow in mixed gas-liquid column.
(a)
Bubble flow(b)
Slug flow(c)
Annular flowFigure5 shows three flow patterns that may occur during DLCP. The downward velocity is positive, and the parameter in the figure can be seen in the following equations.(1) CMM for Bubble Flow in Single-Layer Production. Referring to the “Dynamic liquid column pressure drop”, the CMM for bubble flow in single-layer production is established.
(33)−AdPdh−λpm4ρpmvpm2πD1+D22+ρpmgA=dρpmAvpm2dh,(34)−dPdh=λpmρpmvpm22D1−D2−ρpmg.See AppendixB for the calculation equation of each parameter in equation (33), where λpm is the friction coefficient of bubble flow, dimensionless, ρpm is the density of mixed fluid under down-hole conditions, kg/m3, and vpm is the velocity of mixed fluid, m/s.(2) CMM for Slug Flow in Single-Layer Production. Literature [41] is cited here, the CMM for slug flow in single-layer production is established.
(35)−LAdPdh+LAρdmg−Ltbλltbρwvltb2+Lsλllsρsvs2πD1+D28=LdρdmAvdm2dh.Similarly, according to equation (35), the CMM without radial inflow is obtained.
(36)−dPdh=Ltbλltbρwvltb2+Lsλllsρsvs22LD1−D2−ρdmg.See AppendixC for the calculation equation of each parameter in equation (35), where L, Ls, and Ltb are the total length of slug section, liquid slug section, and Taylor bubble section, respectively, m; ρdm is the average density of the whole slug, kg/m3; λtb and λlls are the friction coefficient of Taylor bubble section and liquid slug section, respectively, dimensionless; vs and ρs are the velocity m/s and density kg/m3 of gas-liquid mixture in liquid slug section, respectively; vdm is the average velocity of the whole slug, m/s; and vltb is the falling velocity of Taylor bubble liquid film, m/s.(3) CMM for Annular Flow in Single-Layer Production. Referring to the “Dynamic liquid column pressure drop,” the CMM for annular flow in single-layer production is established[42, 43].
(37)−AgdPdh+Fgw+ρggAg=dρgAgvg2dh,(38)−AwdPdh−Fgw−λHLρwvw2πD1+D28+ρwgAw=dρwAwvw2dh,where
(39)HHL=AwA,vw=vswHHL,vg=vsg1−HHL,(40)HHL=Hltb=1−vsgvgtb,(41)λHL=1.14−2lgeD1−D2+21.25ReHL0.9−2,(42)ReHL=D1−D2ρwvswμw.Add equations (37) and (38) to get equation (43):
(43)−AdPdh−λHLρwvw2πD1+D28+ρwgAw+ρggAg=dρwAwvw2dh+dρgAgvg2dh.Make 0 on the right of equation (43) and then combine equation (39) to obtain the CMM without radial inflow.
(44)−dPdh=λHLρwvsw22D1−D2HHL2−ρwgHHL−ρgg1−HHL,where Ag and Aw are the cross-sectional areas of the flow space occupied by the gas phase and water phase in the annular flow, m2; Fgw is the mutual friction between gas and water interface, N; λHL is the friction coefficient between the annular liquid film and the pipe wall, dimensionless; HHL is the liquid holdup ratio of annular flow, dimensionless; and ReHL is the Reynolds number of annular flow liquid film, dimensionless.


## 2.3.2. VMM for BHFP of Gas-Water Coproduction Stage

The physical model for gas-water coproduction stage is shown in Figure6.Figure 6
Flow state in the well for gas-water production at the same time.In Figure6, “O’” represents the depth of the DLL after falling from point “O”. The label of other depth points is the same as that in Figure 1. Figure 6(a) shows the complete model diagram. Figure 6(b) is a local enlarged drawing, in which only the water flow direction is drawn on the left and only the gas flow direction is drawn on the right.Similar to the situation of double-layer water or gas production, it is also necessary to establish a VMM including acceleration pressure drop, while the water and gas velocities of each section adopt the equations in Parts 2.1.2 and 2.2.2, respectively.For bubble flow, in equation (33),
(45)dρpmAvpm2dh=vpmdρpmAvpmdh+ρpmAvpmdvpmdh,(46)dρpmAvpmdh=ρwqIw−ρgqIg,(47)dvpm=ρwqIwdh−ρgqIgdhρpmA.Thus, equation (33) becomes
(48)−dPdh=λpmρpmvpm22D1−D2−ρpmg+2vpmρwqIw−ρgqIgA.For slug flow, in equation (35),
(49)dρdmAvdm2dh=vdmρwqIw−ρgqIg+ρdmAvdmdvdmdh,dvdm=ρwqIwdh−ρgqIgdhρdmA.Thus, equation (35) becomes
(50)−dPdh=2vdmρwqIw−ρgqIgA+Ltbλltbρwvltb2+Lsλllsρsvs22LD1−D2−ρdmg.For annular flow, in equation (43),
(51)dρwAwvw2dh=vwρwqIw+ρwAwvwdvwdh=2vwρwqIw,(52)dρgAgvg2dh=vgρgqIg+ρgAgvgdvgdh=2vgρgqIg.Thus, equation (43) becomes
(53)−dPdh=λHLρwvsw22D1−D2HHL2−ρwgHHL−ρgg1−HHL+2vswρwqIwAHHL+2vsgρgqIgA1−HHL.


## 2.3.3. Calculation of BHFP of Gas-Water Coproduction Stage

First, after determining the position of the DLL, calculate the pressure drop of the dynamic gas column above the DLL. Then, the flow pattern [44, 45] of the two-phase flow below the DLL is judged, and the pressure drop is calculated iteratively downward according to the corresponding flow pattern. When the DLL is below the roof of reservoir I, there are three cases, as shown in Figure 7.
(54)Annularflow:vsg≥3.1gσρw−ρgρg20.25,(55)Slugflow:37vsw+2.57vrc<vsg<3.1gσρw−ρgρg20.25,(56)Bubbleflow:vsg≤37vsw+2.57vrc.Figure 7
Calculation process of BHFP for gas-water production at the same time.
(a)
General process(b)
Section OB calculation process(c)
Section BC, CD, and DE calculation processFigure7 (a) is the general flow chart, which is divided into four cases according to the depth of the DLL, and each case is divided into different sections and different flow patterns. The pressure drop of all dynamic gas column sections is calculated by using the relevant equation in Part 2.2.3. For gas-liquid mixed column, the calculation method for section OB and other sections below the roof of reservoir I are shown in Figures 7(b) and 7(c), respectively.(1) The DLL above the Roof of Reservoir I.
(1)
The dynamic gas column pressure dropPAO is obtained by using the equation without radial inflow in Part I of Section 2.2.3(2)
Calculate the pressure dropPOB of the gas-liquid mixed column section above the roof of reservoir I: ① the DLL is the starting point, where the pressure is equal to P0+PAO. Set the pressure drop of depth increment dh as dP, and then the pressure at the lower end of dh is P1=P0+PAO+dP and the temperature T1=T0+hAO+dh/M; ② calculate the average temperature T0+T1/2 and average pressure P0+P1/2 of dh; ③ the average compressibility coefficient is obtained by using equation (A.7), and the gas volume coefficient is obtained by substituting into equation (A.5); ④ use the calculated average value to calculate the parameters required for flow pattern judgment, including the gas density calculated by equation (A.3), vsgOBj calculated by equation (22), vswOB calculated by equation (8), and the average interfacial tension calculated by equation (B.8); and ⑤ judge the flow pattern. For annular flow pattern, use equations (40)–(42) and (44) to calculate the dP′. For slug flow pattern, use equations (C.1)–(C.18) and (36) to calculate the dP′. For bubble flow pattern, use equations (34) and (B.1)–(B.7) to calculate the dP′. ⑥ Compare dP′ with dP, if it does not meet the accuracy requirements, make P1=P0+PAO+dP′ and return to step ②. On the contrary, take P0+PAO+dP′ as the pressure starting point for the next depth increment and repeat steps ②~⑥ until the depth goes down to the roof of reservoir I, and the pressure drop POB of the gas-liquid column section above the roof of reservoir I is obtained(3)
Calculate the pressure drop of the well section under the roof of reservoir I: starting from the roof of reservoir I downward, for calculating the apparent velocity of gas and water, we need to know the water production and gas production of each reservoir, respectively① Assuming that the BHFP of reservoir II ispwf2, the bottom hole temperature is obtained according to equation (A.6), and then the bottom hole average compressibility coefficient Z and gas volume coefficient BgDE are obtained by combination with pwf2. Substitute the above parameters into equation (32) to obtain Qsc2, and then qIw2 is obtained by substituting pwf2 into equation (17); ② calculate Qsc1=Qsc−Qsc2; ③ calculate qIw1 according to equation (18); ④ starting from the roof of reservoir I, the pressure here is equal to P0+PAO+POB, and the pressure drop of the first segment dh from top to bottom is assumed to be dP; so, the pressure at the lower end of dh is P1=P0+PAO+POB+dP, and the temperature is T1=T0+hAO+hOB+dh/M; ⑤ calculate the average temperature T0+T1/2 and average pressure P0+P1/2; ⑥ the average compressibility coefficient of dh is obtained by using equation (A.7), the gas volume coefficient is obtained by using equation (A.5), the gas density is obtained by using equation (A.3), and the average interfacial tension is obtained by using equation (B.8); then, calculate vsgBCj and vswBCj according to equations (23) and (9); and ⑦ judge the flow pattern. For annular flow pattern, use equations (53) and (40)–(42) to calculate the dP′. For slug flow pattern, use equations (C.1)–(C.18) and (50) to calculate the dP′. For bubbe flow pattern, use equations (48) and (B.1)–(B.7) to calculate the dP′. ⑧ Compare the difference between dP′ and dP. If it does not meet the accuracy requirements, make P1=P0+PAO+POB+dP′ and return to step ⑤. On the contrary, take P1=P0+PAO+POB+dP′ as the pressure starting point for the second segment, repeat steps ④~⑧ until the depth goes down to the floor of reservoir I, and calculate the pressure drop PBC of reservoir I. Similarly, using the method of steps ④~⑧ to obtain PCD and PDE, in step ⑥, use equations (10), (11), (24), and (25) to calculate the apparent velocity of water and gas of section CD and DE, respectively. ⑨ Calculate the pressure of reservoir II by Pwf2′=P0+PAO+POB+PBC+PCD+0.5PDE. Compare the assumed Pwf2 with the calculated Pwf2′, if the accuracy requirements are met, the calculated value represents the BHFP of reservoir II. If the accuracy requirements are not met, Pwf2 is replaced by Pwf2′ and calculate again from step ① until the accuracy requirements are met. Then, the BHFP of reservoir I is obtained.If the absolute values of one of the calculatedQsc1, Qsc2, qIw1, and qIw2 are less than or equal to zero, it indicates that the BHFP of the corresponding reservoir is greater than the desorption pressure (Qsc1, Qsc2) or the original reservoir pressure (qIw1, qIw2); so, the calculated value is equal to zero.(2) The DLL below the Roof of Reservoir I.
(1)
The dynamic gas column pressure dropPAB is obtained by using the equation without radial inflow in Part I of Section 2.2.3(2)
Use the method in Part I of Section2.2.3 to calculate the dynamic gas column pressure drop PBO and the mixed gas-liquid column POE of the section under the roof of reservoir I: ① if the DLL is within the range of section BC, the dynamic gas column velocity in section BO is calculated by equation (23), and the pressure drop is calculated iteratively by equation (29). ② If the DLL is within the range of section CD, the dynamic gas column in section BO can be divided into section BC and section CO. The velocity is calculated by equation (23) and equation (24), respectively, and the corresponding pressure drop is calculated iteratively by equations (29) and (30). ③ If the DLL is within the range of section DE, the dynamic gas column velocity in section BO is calculated by equations (23), (24), and (25), respectively, and the corresponding pressure drop is calculated iteratively by equations (29), (30), and (31)For mixed gas-liquid columnPOE, take the pressure at the DLL for the above three cases as the starting point and calculate through flow pattern judgment and iteration. Refer to Part I of Section 2.2.3 for the calculation process.


## 3. Analysis of Influencing Factors of BHFP for DLCP

The CMM (current method) regards the two reservoirs as a whole, it is believed that the production of gas and water at a certain time is unchanged from the depth of the reservoir to the wellhead, and only the difference of gas volume coefficient and deviation coefficient along the well caused by temperature and pressure changes is considered. However, since the production is the fundamental determinant, when the production is unchanged, the above changes will not cause significant changes in velocity and density, and the resulting acceleration pressure drop can also be ignored.The VMM adopts the same method as CMM for the well section above the upper reservoir, but for the two reservoirs and the section between reservoirs, it is considered that both gas and water production of each well section at a certain time are different, and a more refined description is made. For the gas phase, besides being affected by pressure and temperature, the velocity will change greatly due to the different gas flows through each well section. In particular, for the gas-water two phase, it will produce obvious acceleration pressure drop, which will also change the original friction. At the same time, the difference of the gas-water ratio of the two reservoirs will also lead to the change of the density of the mixed gas liquid column. The superposition of all these factors will lead to the change of pressure.The above are the main differences between the VMM and the CMM. Based on the data of an actual well, the influence on BHFP is analyzed from three aspects: acceleration pressure drop, velocity change of each well section, and density change.The field data is from a DLCP parameter well SS-2 in Qinshui Basin, Shanxi Province. The combined production layers are 3# coal seam of Shanxi formation in Lower Permian system and 15# coal seam of Taiyuan Formation in the Upper Carboniferous system. The well began drainage and pressure reduction production on September 8, 2012 and gas production on October 1, 2013. The basic parameters of the well includeD1=0.1397m, D2=0.073m, rw=0.108m, e=0.000016m, μw=0.98mPa·s, ρw=1000kg/m3, Z0=0.9999, T0=273K, and geothermal gradient 3.3°C/100 m. The roof depths of 3# and 15# are 971.75 m and 1092.55 m, respectively, the layer thickness is 6.04 m and 7.55 m, the original formation pressure is 6.38 MPa and 6.23 MPa, and the permeability is 0.2×10−3μm2, 0.1×10−3μm2, skin factor -0.67 and -0.25, and temperature 27.5°C and 31.5°C. The measured parameters are shown in Table 1.Table 1
Measured parameters of different DLL depths in three stages.
Production stageMeasurement time for DLL and pressureMeasured DLL (m)Pump displacement (m3/d)Total gas production (m3/d)Casing pressure (MPa)Measured BHP (MPa)Water production only① 2013.08.115518.600.15.21② 2013.08.2162910.800.14.56③ 2013.08.236572600.14.31① 2013.09.2777043.300.13.52② 2013.10.0179844.200.13.21Gas and water production① 2013.10.0282142.41800.22.01② 2013.10.0584445.32100.51.92① 2013.10.2582342.31370.11.81② 2013.10.2893743.22280.51.35① 2014.10.109697.81600.551.68② 2014.10.1310129.22350.941.50Gas production only① 2015.05.121100 (bottom hole)02851.121.15② 2015.05.2303401.261.28In Table1, the second column gives the measurement time of DLL and pressure, and the serial numbers ①, ②, and ③ represent the measurement sequence of each group. The third column is the measured DLL, and the fourth to seventh columns, respectively, represent the pump displacement, total gas production, casing pressure, and BHFP of the lower reservoir. It can be seen from Table 1 that only the BHFP of the lower reservoir can be obtained through field measurement, which also confirms the conclusion that the BHFP of the lower reservoir is taken as the common BHFP of the two reservoirs. With the decrease of the DLL, the water and gas production begin to increase, the BHFP decreases, and the wellhead pressure increases.


### 3.1. Acceleration Pressure Drop

The pressure generated by the fluid in the well is composed of gravity of the fluid itself, friction, and acceleration pressure drop. The proportion of each pressure component for all DLL is calculated, as shown in Table2 and Figure 8.Table 2
Composition proportion of pressure generated by fluid in well for three production stages.
Production stageDLLPwf2P0Pwf2−P0Gravity pressure dropFriction pressure dropAcceleration pressure dropValueRatioValueRatioValueRatiomMPaMPaMPaMPa%MPa%MPa%Water production only6294.78370.14.68374.683599.99491.6×10−40.00357.5×10−50.00166574.50940.14.40944.409299.99471.8×10−40.00425.0×10−50.00117983.12770.13.02773.027399.98832.8×10−40.00927.5×10−50.0025Water-gas production8442.01060.51.51061.493898.88857.2×10−30.47939.6×10−30.63229371.38710.50.88710.743283.77860.05245.90690.091510.314510121.59870.940.65870.604091.68973.2×10−30.49190.05157.8184Gas production only1100 bottom hole1.20031.120.08030.080299.8832009.4×10−50.11681.32431.260.06430.064299.8981006.6×10−50.1019Figure 8
Proportion of friction pressure drop and acceleration pressure drop under different DLL.Figure8 shows the ratio of friction pressure drop and acceleration pressure drop to the pressure drop generated by fluid in the well. The figure includes the DLL depths of each of the three production stages.It can be seen from Table2 and Figure 8 that the acceleration pressure drop accounts for a small proportion in the double-layer water production and double-layer gas production stages, which can be ignored. However, in the gas-water coproduction stage, the acceleration pressure drop and friction force increase significantly.It is the early stage of drainage and pressure reduction for double-layer water production, and the bottom hole pressure difference is small; so, the water production and water flow velocity are small, making the acceleration pressure drop and friction force relatively small.It can be seen from Table2 that in the double-layer gas production stage, the wellhead pressure accounts for 95% of the BHFP, and the pressure generated by the gas flow in the well is very small, which is mainly caused by the low productivity of the CBM well in this example (the total gas production of two layers is less than 340 m3/d). Therefore, no matter how the velocity change and acceleration pressure drop are considered, the final calculation results will not be substantially affected.It can be seen from Table2 and Figure 8 that, compared with the other two stages, the friction and acceleration pressure drop are greatly increased during gas-water coproduction stage. When the DLL is 937 m and 1012 m, the acceleration pressure drop accounts for 10.31% and 7.82% of the pressure generated by the well fluid, respectively, which is 100-10000 times that of the other two stages. Even when the gas production is just started, the contribution of acceleration pressure drop is increased hundreds of times when the DLL is 844 m. In addition, compared with the friction and acceleration pressure drop, it can be seen that the contribution of acceleration pressure drop to BHFP is 2-15 times of friction. Therefore, even if the friction with a very small value is considered in the CMM, the acceleration pressure drop that is several times larger than the friction should be taken into account; otherwise, a large error will occur. The above results show that the acceleration pressure drop in the gas-water coproduction stage has become a nonnegligible component of the BHFP, and VMM is more accurate than CMM because it takes it into account.


### 3.2. Fluid Density in the Well

In double-layer water production stage, the density remains stable no matter how the water production of the two reservoirs changes. In double-layer gas production stage, although the change of gas production will affect the gas density to a certain extent, for the low permeability and low production reservoir, as described in Part 3.1, the pressure generated by the gas in the well is very small; so, the contribution to the BHFP can be ignored; so, here we mainly analyze the density change in the gas-water coproduction stage.During single-layer production, the gas and water production are relatively stable. From the bottom to the wellhead, the density gradually decreases. During DLCP, when the fluid from the lower reservoir flows through the upper reservoir, the mixing of the fluid produced from the upper reservoir will change the original fluid density. The greater the difference between the gas-water ratio of the upper reservoir and that of the lower reservoir, the greater the density change.Figure9 shows the density change of gas-liquid mixing column from the DLL to the bottom of the well in gas-water coproduction stage during DLCP, with a sudden change at point C. Since the DLL 1012 m is below point C, there is only the second half.Figure 9
Mixed gas-liquid column density in each well section at the gas-water coproduction stage.It can be seen from Figure9 that the density curve from the DLL to the bottom is different from the continuous change process of single-layer production but is divided into two sections. The former section is from the DLL to the floor of reservoir I, that is, point C, and the latter section is from point C to the bottom of the well. The density has a sudden change near point C; that is, it suddenly rises at the boundary from the latter section to the former section. This phenomenon can be analyzed from the produced fluids of the two reservoirs. As shown in Table 3, the water-gas ratio of reservoir I is tens to hundreds of times that of reservoir II. Therefore, in comparison, reservoir I produces more water, and reservoir II produces more gas. So, the inflow of a large amount of water from reservoir I increases the water-gas ratio of the original fluid system and causes the instantaneous increase of the density near the floor of reservoir I. The reason for the instantaneous increase is that the water flows downward. Therefore, the water volume increase is the largest near the floor of reservoir I, and the density increases instantaneously. This situation does not exist in single-layer production and is a unique feature of gas-water coproduction during DLCP. If the CMM of single-layer production is adopted, the change of gas-liquid column density is not taken into account, which will inevitably lead to large errors.Table 3
Water-gas ratio in gas-water coproduction stage.
DLL (m)Qw1 (m3/d)Qg1 (m3/d)Qw2 (m3/d)Qg2 (m3/d)Water-gas ratio of reservoir IWater-gas ratio of reservoir II84442.7310.162.49199.844.210.012593735.0215.627.86212.382.240.0371012025.482.39209.5200.0114


### 3.3. Apparent Velocity

The production of gas and water flowing through each well section is different, which changes the velocity of each section. At the same time, when the density changes greatly, the pressure on the gas phase changes, and the velocity will further change.Figure10 shows the mixed fluid velocity of each well section calculated by the two models in the gas-water coproduction stage. The sections of OB, BC, CD, and DE represent that the DLLs of 844 m and 937 m are above reservoir I. The sections of BC, CO, OD, and DE represent that the DLL of 1012 m is below the roof of reservoir I.Figure 10
Mixed fluid velocity of each well section calculated by VMM and CMM.In Figure10, the upper, middle, and lower groups of curves represent the mixed fluid velocity of each section calculated by VMM and CMM, respectively. The trend of velocity change calculated by the two models is similar, which increases first and then decreases. At the same time, the calculation results of CMM are generally higher than those of VMM, and the closer to the bottom of the well, the greater the difference.Firstly, it can be seen from Table3 that reservoir I mainly produces water and reservoir II mainly produces gas. Combined with the development mode of water flow downward and gas flow upward in CBM wells, from reservoir I downward, although the water production gradually increases, the gas production decreases limited. Therefore, the two-phase apparent velocity gradually increases. In section DE, the lower the gas production is, the smaller the water production is, but the increase of water production is limited. Therefore, the total velocity of this section will decrease rapidly. Secondly, CMM uses the total gas volume and total water volume to calculate the velocity of whole well; therefore, for the DLL 844 m and 937 m, it is completely accurate in section OB, while from section BC down, the gas flow starts to be less than the total gas volume, and the gas volume is only the gas production of reservoir II in section CD, and at the floor of section DE, the gas volume is close to zero. However, in the calculation, the total gas volume is still used in all well sections. Therefore, as shown in Figure 10, the velocity calculated by CMM will be higher than the actual value (the principle of DLL 1012 m is the same).To sum up, compared with CMM, VMM takes into account the change of the density value of the mixed liquid, the decrease of the apparent velocity, and the acceleration pressure drop caused by the flow of the product from the upper reservoir into the well liquid column. The three factors work together, and the calculated BHFP is closer to the actual situation. It solves the problem that CMM adopts invariable gas and water production and does not consider the influence of the upper layer on the lower layer, which makes the calculated BHFP slightly larger. Therefore, the VMM is applicable to the calculation of BHFP in each production stage of DLCP.


## 3.1. Acceleration Pressure Drop

The pressure generated by the fluid in the well is composed of gravity of the fluid itself, friction, and acceleration pressure drop. The proportion of each pressure component for all DLL is calculated, as shown in Table2 and Figure 8.Table 2
Composition proportion of pressure generated by fluid in well for three production stages.
Production stageDLLPwf2P0Pwf2−P0Gravity pressure dropFriction pressure dropAcceleration pressure dropValueRatioValueRatioValueRatiomMPaMPaMPaMPa%MPa%MPa%Water production only6294.78370.14.68374.683599.99491.6×10−40.00357.5×10−50.00166574.50940.14.40944.409299.99471.8×10−40.00425.0×10−50.00117983.12770.13.02773.027399.98832.8×10−40.00927.5×10−50.0025Water-gas production8442.01060.51.51061.493898.88857.2×10−30.47939.6×10−30.63229371.38710.50.88710.743283.77860.05245.90690.091510.314510121.59870.940.65870.604091.68973.2×10−30.49190.05157.8184Gas production only1100 bottom hole1.20031.120.08030.080299.8832009.4×10−50.11681.32431.260.06430.064299.8981006.6×10−50.1019Figure 8
Proportion of friction pressure drop and acceleration pressure drop under different DLL.Figure8 shows the ratio of friction pressure drop and acceleration pressure drop to the pressure drop generated by fluid in the well. The figure includes the DLL depths of each of the three production stages.It can be seen from Table2 and Figure 8 that the acceleration pressure drop accounts for a small proportion in the double-layer water production and double-layer gas production stages, which can be ignored. However, in the gas-water coproduction stage, the acceleration pressure drop and friction force increase significantly.It is the early stage of drainage and pressure reduction for double-layer water production, and the bottom hole pressure difference is small; so, the water production and water flow velocity are small, making the acceleration pressure drop and friction force relatively small.It can be seen from Table2 that in the double-layer gas production stage, the wellhead pressure accounts for 95% of the BHFP, and the pressure generated by the gas flow in the well is very small, which is mainly caused by the low productivity of the CBM well in this example (the total gas production of two layers is less than 340 m3/d). Therefore, no matter how the velocity change and acceleration pressure drop are considered, the final calculation results will not be substantially affected.It can be seen from Table2 and Figure 8 that, compared with the other two stages, the friction and acceleration pressure drop are greatly increased during gas-water coproduction stage. When the DLL is 937 m and 1012 m, the acceleration pressure drop accounts for 10.31% and 7.82% of the pressure generated by the well fluid, respectively, which is 100-10000 times that of the other two stages. Even when the gas production is just started, the contribution of acceleration pressure drop is increased hundreds of times when the DLL is 844 m. In addition, compared with the friction and acceleration pressure drop, it can be seen that the contribution of acceleration pressure drop to BHFP is 2-15 times of friction. Therefore, even if the friction with a very small value is considered in the CMM, the acceleration pressure drop that is several times larger than the friction should be taken into account; otherwise, a large error will occur. The above results show that the acceleration pressure drop in the gas-water coproduction stage has become a nonnegligible component of the BHFP, and VMM is more accurate than CMM because it takes it into account.


## 3.2. Fluid Density in the Well

In double-layer water production stage, the density remains stable no matter how the water production of the two reservoirs changes. In double-layer gas production stage, although the change of gas production will affect the gas density to a certain extent, for the low permeability and low production reservoir, as described in Part 3.1, the pressure generated by the gas in the well is very small; so, the contribution to the BHFP can be ignored; so, here we mainly analyze the density change in the gas-water coproduction stage.During single-layer production, the gas and water production are relatively stable. From the bottom to the wellhead, the density gradually decreases. During DLCP, when the fluid from the lower reservoir flows through the upper reservoir, the mixing of the fluid produced from the upper reservoir will change the original fluid density. The greater the difference between the gas-water ratio of the upper reservoir and that of the lower reservoir, the greater the density change.Figure9 shows the density change of gas-liquid mixing column from the DLL to the bottom of the well in gas-water coproduction stage during DLCP, with a sudden change at point C. Since the DLL 1012 m is below point C, there is only the second half.Figure 9
Mixed gas-liquid column density in each well section at the gas-water coproduction stage.It can be seen from Figure9 that the density curve from the DLL to the bottom is different from the continuous change process of single-layer production but is divided into two sections. The former section is from the DLL to the floor of reservoir I, that is, point C, and the latter section is from point C to the bottom of the well. The density has a sudden change near point C; that is, it suddenly rises at the boundary from the latter section to the former section. This phenomenon can be analyzed from the produced fluids of the two reservoirs. As shown in Table 3, the water-gas ratio of reservoir I is tens to hundreds of times that of reservoir II. Therefore, in comparison, reservoir I produces more water, and reservoir II produces more gas. So, the inflow of a large amount of water from reservoir I increases the water-gas ratio of the original fluid system and causes the instantaneous increase of the density near the floor of reservoir I. The reason for the instantaneous increase is that the water flows downward. Therefore, the water volume increase is the largest near the floor of reservoir I, and the density increases instantaneously. This situation does not exist in single-layer production and is a unique feature of gas-water coproduction during DLCP. If the CMM of single-layer production is adopted, the change of gas-liquid column density is not taken into account, which will inevitably lead to large errors.Table 3
Water-gas ratio in gas-water coproduction stage.
DLL (m)Qw1 (m3/d)Qg1 (m3/d)Qw2 (m3/d)Qg2 (m3/d)Water-gas ratio of reservoir IWater-gas ratio of reservoir II84442.7310.162.49199.844.210.012593735.0215.627.86212.382.240.0371012025.482.39209.5200.0114


## 3.3. Apparent Velocity

The production of gas and water flowing through each well section is different, which changes the velocity of each section. At the same time, when the density changes greatly, the pressure on the gas phase changes, and the velocity will further change.Figure10 shows the mixed fluid velocity of each well section calculated by the two models in the gas-water coproduction stage. The sections of OB, BC, CD, and DE represent that the DLLs of 844 m and 937 m are above reservoir I. The sections of BC, CO, OD, and DE represent that the DLL of 1012 m is below the roof of reservoir I.Figure 10
Mixed fluid velocity of each well section calculated by VMM and CMM.In Figure10, the upper, middle, and lower groups of curves represent the mixed fluid velocity of each section calculated by VMM and CMM, respectively. The trend of velocity change calculated by the two models is similar, which increases first and then decreases. At the same time, the calculation results of CMM are generally higher than those of VMM, and the closer to the bottom of the well, the greater the difference.Firstly, it can be seen from Table3 that reservoir I mainly produces water and reservoir II mainly produces gas. Combined with the development mode of water flow downward and gas flow upward in CBM wells, from reservoir I downward, although the water production gradually increases, the gas production decreases limited. Therefore, the two-phase apparent velocity gradually increases. In section DE, the lower the gas production is, the smaller the water production is, but the increase of water production is limited. Therefore, the total velocity of this section will decrease rapidly. Secondly, CMM uses the total gas volume and total water volume to calculate the velocity of whole well; therefore, for the DLL 844 m and 937 m, it is completely accurate in section OB, while from section BC down, the gas flow starts to be less than the total gas volume, and the gas volume is only the gas production of reservoir II in section CD, and at the floor of section DE, the gas volume is close to zero. However, in the calculation, the total gas volume is still used in all well sections. Therefore, as shown in Figure 10, the velocity calculated by CMM will be higher than the actual value (the principle of DLL 1012 m is the same).To sum up, compared with CMM, VMM takes into account the change of the density value of the mixed liquid, the decrease of the apparent velocity, and the acceleration pressure drop caused by the flow of the product from the upper reservoir into the well liquid column. The three factors work together, and the calculated BHFP is closer to the actual situation. It solves the problem that CMM adopts invariable gas and water production and does not consider the influence of the upper layer on the lower layer, which makes the calculated BHFP slightly larger. Therefore, the VMM is applicable to the calculation of BHFP in each production stage of DLCP.


## 4. Application of Models

The VMM is applied to the above field well, and the pressures of different well sections in three production stages are obtained. At the same time, the corresponding verification is carried out using the existing CMM, and the calculation results of the two models are compared.


### 4.1. Application in Double-Layer Water Production Stage

First, the field data is substituted into the VMM to obtain the BHFP of the two reservoirs. Then, the CMM is substituted to obtain the BHFP of the lower reservoir. Finally, the accuracy of the two models is evaluated by comparing the calculated results with the measured values of the lower reservoir, see Table4 and Figure 11.Table 4
Pressure calculated by the two models in double-layer water production stage (MPa).
DLL (m)MeasuredPwf2VMMCMMPAOPOBPBCPCDPDEPwf1Pwf2ErrorPwf2Error6294.560.10273.35890.05921.12460.07393.59264.78384.914.78374.916574.310.10273.08450.05921.12460.07393.31824.50944.634.50954.637983.210.10291.70270.05921.12460.07391.93653.12772.563.11682.90Figure 11
The calculation error of BHFP for water production and gas production.Figure11 shows the calculation errors of the two models at different DLL depths in the stages of two-layer water production and two-layer gas production.It can be seen from Table4 that the error of the BHFP of the lower reservoir calculated by VMM is small, and the maximum error is only 4.91%. The calculation results of the CMM are almost the same as those of VMM, which indicates that both models are applicable at this stage. This result is more intuitively seen in Figure 11. Among them, when the DLL is close to 800 m, the error of CMM starts to be greater than that of VMM. The reason is that the pressure difference at the bottom of the well becomes larger, which leads to the increase of water production (see Table 1), and then the acceleration pressure drop occurs, resulting in the difference between the results of the two models. Nevertheless, the acceleration pressure drop at this time is still not enough to bring about a relatively large impact and can still be ignored.


### 4.2. Application in Double-Layer Gas Production Stage

Similar to the application in Part 4.1, the specific results are shown in Table5 and Figure 11.Table 5
Pressure calculated by the two models in double-layer gas production stage (MPa).
DLL (m)MeasuredPwf2VMMCMMPABPBCPCDPDEPwf1Pwf2ErrorPwf2Error11001.150.03542.37×10−44.49×10−32.35×10−41.19551.20034.191.20024.191.280.03892.61×10−44.94×10−32.58×10−41.31911.32433.341.32403.34It can be seen from Table5 and Figure 11 that the calculation results of VMM and CMM are very close, and the errors are small, indicating that both models are applicable to the calculation of BHFP in the double-layer gas production stage.


### 4.3. Application in Gas-Water Coproduction Stage

Similar to the application in Part 4.1, the specific results are shown in Table6 and Figure 12.Table 6
Pressure calculated by the two models in water-gas coproduction stage (MPa).
DLL (m)MeasuredPwf2VMMCMMPAOPOBPBCPCDPDEPwf1Pwf2ErrorPwf2Error8441.920.01640.74250.0360.72170.05611.27692.01064.722.211415.189371.350.01820.19670.03250.66240.05490.7311.38712.751.499111.0410121.50PABPBCPCOPODPDEPwf1Pwf2ErrorPwf2Error0.03111.79×10−41.39×10−30.59640.05640.98121.59876.581.60737.15Figure 12
The calculation error of BHFP for gas-water coproduction stage.Figure12 shows the calculation errors of the two models at different DLL depths in the gas-water coproduction stage.It can be seen from Table6 that the errors of VMM and CMM are 2.75%~6.58% and 7.15% ~15.18%, respectively; so, the VMM is more accurate. According to the requirements of the 5% error convention on site, the error of VMM is more than 5% only when the DLL is 1012 m, but it is still less than 7.15% of CMM. In general, it can basically meet the needs of the site engineering. The error of CMM is more than 5%, especially when the DLL is close to the upper reservoir, and the error is as high as 15.18%. The result can be seen more intuitively from Figure 12. In the whole gas-water coproduction stage, the accuracy of VMM is higher than that of CMM.To sum up, in the stages of two-layer water production and two-layer gas production, the calculation results of the two models are very similar, and both are suitable for BHFP calculation. However, in the stage of gas-water coproduction, the VMM is relatively accurate, while the CMM has a large error. Therefore, the CMM is not applicable to the gas-water coproduction stage, while the VMM is applicable to the whole process of DLCP.In addition, on the basis of verifying the accuracy of the VMM, the BHFPPwf1 and Pwf2 of the two reservoirs calculated by the VMM is compared, as shown in Figure 13.Figure 13
BHFP comparison between upper and lower reservoir.Figure13 shows the BHFP of upper reservoir and lower reservoir at different DLL depths calculated by VMM.It can be seen from Figure13 that in the stages of water production and gas-water co-production, regardless of the change of the DLL depth, the BHFP Pwf1 and Pwf2 differ greatly, with a difference of 1.19 MPa, 1.19 MPa, 1.19 MPa, 0.77 MPa, 0.71 MPa and 0.63 MPa from shallow to deep, reaching 24.9%, 26.4%, 38.1%, 37.5%, 49.1% and 38.9% of the BHFP of the lower reservoir, respectively. In the gas production stage, the difference is small, which are 0.0048 MPa and 0.0052 MPa, respectively, accounting for only 0.4% and 0.39% of the BHFP of the lower reservoir. Therefore, the two reservoirs can be regarded as a whole, and the production system can be adjusted with the same BHFP in gas production stage, however, it is necessary to adjust the respective production systems according to their respective BHFP in the other two stages.


## 4.1. Application in Double-Layer Water Production Stage

First, the field data is substituted into the VMM to obtain the BHFP of the two reservoirs. Then, the CMM is substituted to obtain the BHFP of the lower reservoir. Finally, the accuracy of the two models is evaluated by comparing the calculated results with the measured values of the lower reservoir, see Table4 and Figure 11.Table 4
Pressure calculated by the two models in double-layer water production stage (MPa).
DLL (m)MeasuredPwf2VMMCMMPAOPOBPBCPCDPDEPwf1Pwf2ErrorPwf2Error6294.560.10273.35890.05921.12460.07393.59264.78384.914.78374.916574.310.10273.08450.05921.12460.07393.31824.50944.634.50954.637983.210.10291.70270.05921.12460.07391.93653.12772.563.11682.90Figure 11
The calculation error of BHFP for water production and gas production.Figure11 shows the calculation errors of the two models at different DLL depths in the stages of two-layer water production and two-layer gas production.It can be seen from Table4 that the error of the BHFP of the lower reservoir calculated by VMM is small, and the maximum error is only 4.91%. The calculation results of the CMM are almost the same as those of VMM, which indicates that both models are applicable at this stage. This result is more intuitively seen in Figure 11. Among them, when the DLL is close to 800 m, the error of CMM starts to be greater than that of VMM. The reason is that the pressure difference at the bottom of the well becomes larger, which leads to the increase of water production (see Table 1), and then the acceleration pressure drop occurs, resulting in the difference between the results of the two models. Nevertheless, the acceleration pressure drop at this time is still not enough to bring about a relatively large impact and can still be ignored.


## 4.2. Application in Double-Layer Gas Production Stage

Similar to the application in Part 4.1, the specific results are shown in Table5 and Figure 11.Table 5
Pressure calculated by the two models in double-layer gas production stage (MPa).
DLL (m)MeasuredPwf2VMMCMMPABPBCPCDPDEPwf1Pwf2ErrorPwf2Error11001.150.03542.37×10−44.49×10−32.35×10−41.19551.20034.191.20024.191.280.03892.61×10−44.94×10−32.58×10−41.31911.32433.341.32403.34It can be seen from Table5 and Figure 11 that the calculation results of VMM and CMM are very close, and the errors are small, indicating that both models are applicable to the calculation of BHFP in the double-layer gas production stage.


## 4.3. Application in Gas-Water Coproduction Stage

Similar to the application in Part 4.1, the specific results are shown in Table6 and Figure 12.Table 6
Pressure calculated by the two models in water-gas coproduction stage (MPa).
DLL (m)MeasuredPwf2VMMCMMPAOPOBPBCPCDPDEPwf1Pwf2ErrorPwf2Error8441.920.01640.74250.0360.72170.05611.27692.01064.722.211415.189371.350.01820.19670.03250.66240.05490.7311.38712.751.499111.0410121.50PABPBCPCOPODPDEPwf1Pwf2ErrorPwf2Error0.03111.79×10−41.39×10−30.59640.05640.98121.59876.581.60737.15Figure 12
The calculation error of BHFP for gas-water coproduction stage.Figure12 shows the calculation errors of the two models at different DLL depths in the gas-water coproduction stage.It can be seen from Table6 that the errors of VMM and CMM are 2.75%~6.58% and 7.15% ~15.18%, respectively; so, the VMM is more accurate. According to the requirements of the 5% error convention on site, the error of VMM is more than 5% only when the DLL is 1012 m, but it is still less than 7.15% of CMM. In general, it can basically meet the needs of the site engineering. The error of CMM is more than 5%, especially when the DLL is close to the upper reservoir, and the error is as high as 15.18%. The result can be seen more intuitively from Figure 12. In the whole gas-water coproduction stage, the accuracy of VMM is higher than that of CMM.To sum up, in the stages of two-layer water production and two-layer gas production, the calculation results of the two models are very similar, and both are suitable for BHFP calculation. However, in the stage of gas-water coproduction, the VMM is relatively accurate, while the CMM has a large error. Therefore, the CMM is not applicable to the gas-water coproduction stage, while the VMM is applicable to the whole process of DLCP.In addition, on the basis of verifying the accuracy of the VMM, the BHFPPwf1 and Pwf2 of the two reservoirs calculated by the VMM is compared, as shown in Figure 13.Figure 13
BHFP comparison between upper and lower reservoir.Figure13 shows the BHFP of upper reservoir and lower reservoir at different DLL depths calculated by VMM.It can be seen from Figure13 that in the stages of water production and gas-water co-production, regardless of the change of the DLL depth, the BHFP Pwf1 and Pwf2 differ greatly, with a difference of 1.19 MPa, 1.19 MPa, 1.19 MPa, 0.77 MPa, 0.71 MPa and 0.63 MPa from shallow to deep, reaching 24.9%, 26.4%, 38.1%, 37.5%, 49.1% and 38.9% of the BHFP of the lower reservoir, respectively. In the gas production stage, the difference is small, which are 0.0048 MPa and 0.0052 MPa, respectively, accounting for only 0.4% and 0.39% of the BHFP of the lower reservoir. Therefore, the two reservoirs can be regarded as a whole, and the production system can be adjusted with the same BHFP in gas production stage, however, it is necessary to adjust the respective production systems according to their respective BHFP in the other two stages.


## 5. Conclusions and Suggestions

(1)
In order to solve the problem that the current CMM does not meet the characteristics of DLCP, the VMM of three production stages is established on the basis of the CMM, and the velocity equations of water flow and gas flow in each section are derived. The field data verification results show that the calculation errors of VMM in the three stages are 2.56%~4.91%, 2.75%~6.58%, and 3.34% ~4.19%, respectively, and the average errors are 4.03%, 4.68% and 3.77%, respectively, which are small and can meet the needs of field engineering. However, the error of CMM is less than 5% only at the stage of double-layer water production and double-layer gas production, and the error is larger at the stage of gas-water coproduction, which is 7.15%~15.18%, with an average of 11.12%. VMM considers the acceleration pressure drop, the density change, and velocity change, all caused by the interaction between two reservoir fluids, which makes the calculation result more accurate(2)
In order to solve the problem that there may be errors in adjusting the production system of the upper reservoir by taking the BHFP of the lower reservoir, the BHFP of the two reservoirs in the three stages is calculated by using VMM. The results show that the BHFP of the two reservoirs has a large difference in the stages of water production and gas-water coproduction, the maximum difference is 49.1%, and the difference is small only in the gas production stage. Therefore, in the first two stages, the current method has a large error, while VMM can easily solve the problem(3)
When the total water production and gas production of the two reservoirs are relatively low (according to the maximum water production of 45 m3/d and the maximum gas production of 340 m3/d in this example), the acceleration pressure drop can be ignored in the double-layer water production stage and the double-layer gas production stage. Both VMM and CMM can be used to calculate the BHFP. However, in the gas-water coproduction stage, the calculation error of CMM is large and can only be calculated by VMM(4)
Although the calculation results of the VMM adopted are closer to the field reality, the influence factors such as the temperature, pressure, and productivity of the studied CBM reservoir are small; so, the advantages of this model are not particularly obvious. It needs to be further verified and optimized in ordinary DLCP gas wells, especially those with deep wells and high production



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*Source: 7506870-2023-02-08.xml*