Coalbed methane reservoir dynamic prediction model by ... · Coalbed methane reservoir dynamic prediction model by combination of material balance equation and crossflow-diffusion

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Energy Sources, Part A: Recovery, Utilization, andEnvironmental Effects

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Coalbed methane reservoir dynamic predictionmodel by combination of material balanceequation and crossflow-diffusion

Fengpeng Lai, Zhiping Li, Yining Wang & Yingkun Fu

To cite this article: Fengpeng Lai, Zhiping Li, Yining Wang & Yingkun Fu (2016) Coalbedmethane reservoir dynamic prediction model by combination of material balance equationand crossflow-diffusion, Energy Sources, Part A: Recovery, Utilization, and EnvironmentalEffects, 38:2, 257-263, DOI: 10.1080/15567036.2013.763389

To link to this article: http://dx.doi.org/10.1080/15567036.2013.763389

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Coalbed methane reservoir dynamic prediction model bycombination of material balance equation and crossflow-diffusionFengpeng Laia, Zhiping Lia, Yining Wangb, and Yingkun Fua

aSchool of Energy Resources, China University of Geosciences, Beijing, China; bCollege of Petroleum Engineering,China University of Petroleum, Beijing, China

ABSTRACTThe proposed dynamic prediction model is based on the occurrence andmigration law of coalbed methane and on the pseudo-steady state diffusionmodel. The model integrated multi-phase flow theory and material balanceequation. It is established by considering not only desorption and diffusionbut also crossflow. Moreover, the calculation results show the rationality ofthe method used. The research analyzes the effects of crossflow factor anddiffusion coefficient on development effectiveness. The effect of crossflowfactor is obvious in the late production stage but not in the early stage. Thediffusion coefficient affects gas production during the exploitation stage.

KEYWORDSCrossflow; diffusion; dualmedia; dynamic prediction;material balance equation

1. Introduction

The porosity and permeability of coal matrix blocks are very small; thus, the Darcy flow of coalbedmethane (CBM) is very weak in pores. The main transport mechanism of CBM in pores and coalmatrix blocks is through diffusion, and Darcy flow can be ignored (Liang and Sun, 2006). Thediffusion of CBM is essentially driven from a desorption zone to a permeable fracture by concentra-tion gradient, and it is one of the important regulatory characteristics of CBM. Following Fick’s law,the diffusion of a coal matrix in the micro-pore can be spread by non-steady-state and pseudo-steady-state modeling (Yang and Wang, 1986).

A coalbed is combined with matrix and fracture systems, and the crossflow between these twosystems is obvious because of the different pressure distributions in pores. The current study did notconsider this factor (Guo et al., 2004; Tong and Zhang, 2008; Shi et al., 2011). The proposed dynamicprediction model is based on the occurrence and migration law of CBM and on the pseudo-steady statediffusion model. Considering that the diffusion is from the matrix to the fracture, the method integratedmulti-phase flow theory and material balance equation. A dynamic prediction model is established byconsidering not only the desorption and diffusion but also the crossflow. The calculation results are inline with the production changes law of CBM, reflecting the rationality of the method used. The effectsof crossflow factor and diffusion coefficient on development effectiveness were also analyzed.

2. Gas transport equations in the matrix system

Non-steady-state modeling can more accurately reflect the temporal and spatial variation of methaneconcentration and diffusion, but the calculations are very difficult and slow. Pseudo-steady statemodeling is a simplified diffusion. In the late exploitation stage, the results of these two models aresignificantly similar (Kolesar et al., 1990a, 1990b), but the calculation in pseudo-steady-state model-ing are more convenient compared with those in non-steady-state modeling.

CONTACT Dr. Fengpeng Lai laifengpeng@cugb.edu.cn School of Energy Resources, China University of Geosciences,Beijing, No.29, Xueyuan Road, Haidian District, Beijing 100083, China.© 2016 Taylor & Francis

ENERGY SOURCES, PART A: RECOVERY, UTILIZATION, AND ENVIRONMENTAL EFFECTS2016, VOL. 38, NO. 2, 257–263http://dx.doi.org/10.1080/15567036.2013.763389

The crossflow between matrix and fracture systems can be regarded as the point source term andcan be introduced into a desorption-diffusion model under the pseudo-steady state. The averageconcentration of methane in the matrix is affected both by the desorption of adsorbed gas and thecrossflow of free gas. Hence, increasing the crossflow term in the desorption-diffusion equation isnecessary to reflect the changes in the average concentration. The established new crossflow-diffusion equation in pseudo-steady state can more accurately describe the desorption transportprocess in the matrix system. Shi et al. (2011) established some equations to described gas concen-trations in the surface of the matrix element and gas average concentration in the matrix unit. Thecrossflow-diffusion equation at time n is calculated by:

qmfg� �nþ1 ¼ �FG

τ

VLpnfgpL þ pnfg

þ ϕmMpfgρsczRT

� �n

� Vnm

" #þ α

0 FGβμnmg

pnmg

� �2� pnfg� �2

� ; (1)

where qmfg is the diffusion from the matrix to the fracture, m3/(m3.d); FG is the geometry-relatedfactor, Dimensionless; τ is the desorption time, d; VL is the Langmuir volume, m3/m3; Pfg is the currentfracture pressure, MPa; PL is the Langmuir pressure, MPa; ϕm is the porosity in matrix, fraction; M isthe molecular weight of methane; ρsc is the gas density under standard conditions, t/m3; z is the gascompressibility factor, Dimensionless; R is the gas molar constant, J.mol–1.K–1; T is the temperature, K;Vm is the average gas concentration in matrix unit, m3/m3; α0 is the unit conversion factor, 8.64 × 1010;β is the crossflow factor, KPa–1; μmg is the gas viscosity in matrix, mPa.s; and Pmg is the current matrix

pressure, MPa.

3. Material balance equation of the fracture system in CBM reservoir

3.1. Assumptions

(1) For CBM reservoir, the fluid properties in the matrix and fracture systems are uniform.(2) Free gas and water are present in the fracture system of an initial CBM reservoir, and only

gas is present in the matrix system. Adsorbed gas and dissolved gas are ignored.(3) A coal reservoir is a closed system, a water and coal reservoir are slightly compressible, and

free gas is real gas.(4) Pseudo-steady diffusion follows Fick’s law; adsorption and desorption of CBM can be

described by the Langmuir isothermal equation.(5) CBM production from a coal reservoir undergoes three stages, including seepage migration,

desorption, and diffusion, all of which are in an isothermal environment.

3.2. Establishment of the material balance equation

The amount of free gas in the coal fracture system and remains in the coal fracture system underaverage reservoir pressure can be calculated by volumetric method. The gas successively desorbsfrom the coal matrix and enters the fracture system by diffusion and crossflow during the exploita-tion of CBM. These gases, moving from the matrix system to the fracture system, can be treated asthe source term in the continuity equation.

The cumulative gas production (Gp) at any time t is calculated by Eq. (2):

Gp ¼ 0:01Ahϕfi 1� Swið ÞPiZscTsc

PscZiTþ 0:01Ahqmfgt � 0:01Ahϕf 1� Sw

� � PZscTsc

PscZT; (2)

258 F. LAI ET AL.

where A is the CBM supply area, km2; h is the seam thickness, m; ϕfi is the original fracture porosity,fraction; Swi is the original water saturation in initial fracture, fraction; Pi is the reservoir initialpressure, MPa; Zsc is the gas deviation factor under standard conditions, Dimensionless; Tsc is thetemperature under standard conditions, K; Psc is the pressure under standard conditions, MPa; Zi isthe gas deviation factor under Pi, Dimensionless; ϕf is the fracture porosity under current pressure,

fraction; Sw is the average water saturation in fracture under current pressure P, fraction; and Z is thegas deviation factor under P, Dimensionless.

The matrix will swell when gas is adsorbed to the coal inner surface and will shrink when gas isdesorbed from the surface. This phenomenon is called self-regulation. In drainage, the effectivestress increases with decreasing pressure. At the same time, as CBM is desorbed from the surface ofthe matrix, the matrix shrinks, and the fracture size increases (Fu et al., 2005; Zhang and Wang,2008; Zhou et al., 2009). The changes in fracture are caused by the difference between these twoeffects.

Reservoir pressure can be calculated through Eq. (3) and then average water saturation underreservoir pressure is obtained. Gas saturation can also be obtained from water saturation. Theeffective permeability of the liquid and gas phases can be calculated at any saturation.

Gp þWpBw

Bg� 0:01Ah qmfgt þ ϕfi 1� Swið ÞPiZscTsc

PscZiT� ϕfBg

� ¼ 0:01AhϕfiSwi 1þ Cw Pi � Pð Þ½ �

Bg; (3)

where Cw is the formation water compressibility, MPa–1; Wp is the ground volume of cumulativewater production, m3; and Bw is the volume factor of formation water in CBM reservoir,Dimensionless.

4. Productivity equations

The common expression of real gas pseudo-pressure m Pfg� �

was proposed by Al-Hussainy andRamey (1996), and the calculation of radial flow deliverability of real gas can be described as in Eq.(4) (Li and Li, 2000):

qg ¼774:6KKrgh½mðPfgÞ �mðPwf Þ�

T ln rerw� 3

4

� � ; (4)

where K is the reservoir absolute permeability, mD; Krg is the gas relative permeability,Dimensionless; m Pfg

� �is the real gas pseudo-pressure under fracture average pressure, MPa2/

(mPa.s); m Pwf� �

is the real gas pseudo-pressure under bottom-hole pressure, MPa2/(mPa.s); re isthe supply radius, m; and rw is the radius diameter, m.

Water production (qw) is calculated by:

qw ¼0:54287KKrwh pfg � pw

� �Bwμfw ln re

rw� 1

2

� �;

(5)

where Krw is the water relative permeability, Dimensionless; pw is the water phase pressure, MPa; andμfw is the water viscosity in fracture, mPa.s.

ENERGY SOURCES, PART A: RECOVERY, UTILIZATION, AND ENVIRONMENTAL EFFECTS 259

5. Calculation steps

(1) The diffusion from the matrix to the fracture is zero at the initial time, and cumulative gasand water productions are known. For the material balance equations, the Newton–Raphsoniterative method was used to calculate reservoir fracture average pressure (expressed by Pn

fg).(2) Average water saturation (Sw) at time t was calculated using Pn

fg .(3) Effective permeabilities of liquid and gas phases were calculated at time t using relative

permeability curves.(4) Gas and water productions were obtained using productivity equations.(5) qmfg

� �nþ1at time (n + 1) was calculated using pnmg and Pn

fg at time n, and gas concentration

Vnþ1m and matrix pressure pnþ1

mg were then calculated.(6) By combining some results at previous time intervals, including gas production, water

production, cumulative gas production, cumulative water production, and diffusion frommatrix to fracture, the reservoir fracture average pressure at the next time interval (Pnþ1

fg )was calculated using the Newton–Raphson iterative method.

(7) Dynamic production of the CBM well was predicted by repeating steps (2) to (6).

Figure 1 shows the calculation flow, and it is consistent with the calculation step.

6. Results and discussions

Table 1 gives the basic data of a CBM reservoir in Western China, and Figure 2 shows theincremental and cumulative gas productions data. CBM production has three stages. The firststage

Figure 1. Calculation flow chart.

260 F. LAI ET AL.

is drainage and pressure lowering. In this stage, gas permeability increases. Desorption and gasproduction also gradually increase when the reservoir pressure is below the critical desorptionpressure. The second stage is the stable production stage, where more gas is produced than inthe first stage. This stage is the major production stage. An increasing amount of CBM isdesorbed and diffused into the fracture system. The third stage is the decline in gas produc-tion. This stage indicates that gas production is into failure, and measures to increaseproduction should be considered.

Water production gradually decreases during the exploitation of CBM. Table 2 shows that thedecline in water production is faster than before. Gas relative permeability increases until the timelimit (1,000 days) is reached, consequently increasing gas production. After 1,000 days, althoughwater production is still falling, but at a smaller decline, there is a small amount of water.

The crossflow factor is 1 × 10–6 KPa–1 at the previous dynamic prediction, and crossflow factorsare assumed to be 1 × 10–4 and 1 × 10–2 KPa–1. The effect of crossflow factor on gas production isanalyzed. Figure 3 indicates that the effect of crossflow factor on gas production is not obviousbecause the average concentration of gas is determined by desorption and crossflow. The average

Table 1. Parameters of coalbed methane (CBM) reservoir in western China.

Parameter Value Parameter Value

Absorbed gas geologic reserve (m3) 7.078 × 107 Absolute permeability (mD) 2.5Free gas geologic reserve (m3) 2.35 × 105 CBM viscosity (cp) 0.002Average pressure (MPa) 3.36 Water viscosity (cp) 0.6Gas reservoir area (km2) 2 Bottom hole flowing pressure (MPa) 0.7Effective thickness (m) 6.33 Supply radius (m) 800Langmuir pressure (MPa) 3.125 Borehole radius (m) 0.1Original fracture porosity 0.021 Initial diffusion 0Fracture pore compressibility (MPa–1) 0.001088 Maximum volumetric strain 0.0006Formation water compressibility (MPa–1) 0.000435 Diffusion coefficient (m2/d) 0.03Original water cut 0.95 Crossflow factor (KPa–1) 1 × 10–6

Critical pressure (MPa) 4.64 Geometric factor 6Critical temperature (K) 190.7 Shape factor (m–2) 1Reservoir temperature (K) 318.15

0

500

1000

1500

2000

2500

3000

3500

0 1000 2000 3000 4000 5000 6000

Time (d)

Gas

pro

duct

ion

(m8 /

d)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Cum

ulat

ive

gas

prod

uctio

n (1

08 m3 )

Gas productionCumulative gas production

Figure 2. Relative curve between production and production time.

ENERGY SOURCES, PART A: RECOVERY, UTILIZATION, AND ENVIRONMENTAL EFFECTS 261

concentration of gas in the matrix is greater than the surface concentration caused by a highdesorption rate. Diffusion plays a major role in gas production.

The diffusion coefficient is 0.03 m2/d at the previous prediction, and the diffusion coefficients areassumed to be 1.0 and 2.0 m2/d. The effect of diffusion coefficient on gas production is analyzed.Figure 4 shows that the diffusion coefficient affects gas production during exploitation of CBM.Higher diffusion coefficient results in higher gas production.

7. Conclusion

In the dual CBM pore-fracture system, the proposed model integrated multi-phase flow theory andmaterial balance equation, and it considered not only desorption and diffusion but also crossflow.

The effect of development effectiveness is related to the crossflow factor and diffusion coefficient.The effect of crossflow factor on gas production is not obvious in the first stage of exploitation ofCBM, but it is obvious in the late period. The diffusion coefficient affects gas production during thewhole exploitation. Higher diffusion coefficient results in higher gas production.

Funding

This research is supported by the Fundamental Research Funds of the Central Universities, the National Special Fund(No. 2011ZX05009-006), and the National Basic Research Program of China (No. 2009CB219600).

0

500

1000

1500

2000

2500

3000

3500

0 1000 2000 3000 4000 5000 6000

Time (d)

Gas

pro

duct

ion

(m3 /d

)

Crossflow factor is 0.000001/KPa

Crossflow factor is 0.0001/KPa

Crossflow factor is 0.01/KPa

Figure 3. Relative curve between gas production and time under different crossflow factor.

Table 2. Average pressure and water production at different times.

Time, dAverage Pressure,

MPaWater Production,

m3/d Time, dAverage Pressure,

MPaWater Production,

m3/d

300 3.2656 10.5257 2,700 2.6216 1.8757600 3.1835 8.2919 3,000 2.5483 1.5515900 3.1008 6.5809 3,300 2.4774 1.28981,200 3.0180 5.2589 3,900 2.3432 0.90451,500 2.9359 4.2295 4,500 2.2194 0.64621,800 2.8547 3.4223 5,100 2.1057 0.46972,100 2.7751 2.7853 5,700 2.0015 0.34722,400 2.6973 2.2796 6,000 1.9528 0.3002

262 F. LAI ET AL.

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0

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3500

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0 1000 2000 3000 4000 5000 6000

Time (d)

Gas

pro

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ion

(m3 /d

)Diffusion coefficient is 0.03 m^2/d

Diffusion coefficient is 1.0 m^2/d

Diffusion coefficient is 2.0 m^2/d

Figure 4. Relative curve between gas production and time under different diffusion coefficient.

ENERGY SOURCES, PART A: RECOVERY, UTILIZATION, AND ENVIRONMENTAL EFFECTS 263