# Production Decline Behavior Analysis of a Vertical Well with a Natural Water Influx/Waterflood

Authors: Mingqiang Wei; Keyi Ren; Yonggang Duan; Qingxuan Chen; Morteza Dejam
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The production decline type curves are considered as a robust technique to interpret the production data and obtain the flow parameters, the original gas in place, etc. However, most of the previous models have focused on the primary depletion with a closed boundary, rather than on the secondary depletion with a water influx/waterflood. Therefore, in this study, a transient flow model considering the water influx/waterflood is developed. Subsequently, the functions of the production decline type curves for a vertical well with a water influx/waterflood are derived based on the material balance equation. In other words, the theory of Blasingame production decline analysis is extended to the water influx/waterflood reservoir. Further advanced Blasingame production decline type curves for a vertical well in water influx/waterflood reservoirs are generated. Compared with Blasingame type curves without a water influx/waterflood, the behavior of the ones presented in this study is quite different at the boundary. Thereafter, the effects of the relevant parameters, including the dimensionless maximum water influx, the dimensionless beginning time of the water influx, and the dimensionless external boundary radius, are studied on type curves. Finally, Blasingame type curves for a vertical well in water influx/waterflood reservoirs are verified through a field case study. This work provides very meaningful references for reservoir engineers working on the evaluation of the water influx and the estimation of the beginning time of the water influx by matching the developed type curves with the actual field data.


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## Body

## 1. Introduction

There are many water influx gas reservoirs in the world, such as Longwangmiao gas reservoir [1], California Monterey reservoir [2], and Baiyun sag reservoir in the Pearl River Mouth Basin [3]. The water influx has a significant effect on the gas well production. Therefore, the diagnosis and evaluation of the water influx rate and the water influx time are vital in exploitation of these reservoirs.Over the past decades, the production performance behavior for a vertical well in water influx/waterflood reservoirs has been studied analytically, semianalytically, and numerically. Van Everdingen and Hurst [4], Fetkovich [5], Carter and Tracy [6], Allard and Chen [7], Leung [8], and Shen et al. [9, 10] developed mathematical models for evaluation of the aquifer properties. Subsequently, Marques and Trevisan [11] compared the simulation results for four classical water influx models [4–6, 8] and discussed the applicable conditions for these models. Based on the two-phase unsteady flow theory, Cui and Zhao [12] presented a technique for calculating production index in multilayer water-drive reservoirs. Later, Omeke et al. [13] developed a new aquifer influx model for a finite aquifer system in order to describe the pseudosteady flow regime. The proposed prediction model was fast and the obtained results showed a reasonable degree of accuracy. Thereafter, Li et al. [14] presented a method for identification and prediction of water influx in a fractured-vuggy carbonate reservoir. The four diagnostic curves for identifying aquifer influx phase were proposed.Although the previous proposed models can be used for forecasting the production in water influx reservoirs, they have not taken the effect of material balance into account. Among the transient analysis models [15–17] and production analysis methods [18–23], the Blasingame type curves technique [21–23] derived using the reservoirs’ material balance equation is a very popular and practical production decline analysis. This method, which can be used to evaluate the reservoir properties, drainage volume, etc., has been widely implemented in conventional reservoirs for several decades. However, most of the previous efforts [22, 24–28] for improving the Blasingame decline type curves analysis have been mainly focused on the primary production, rather than on the secondary depletion.Considering the effect of water influx/waterflood, Doublet and Blasingame [29] proposed the model of Fetkovich decline analysis and generated the Fetkovich decline type curves. However, this model, which is based on highly idealized assumed production conditions, cannot be used to analyze the various production scenarios (including different rates and various flowing pressures). Until now, the works that have extended the Blasingame decline type curves for analyzing water influx data are limited. Therefore, the aim of this work is to study the Blasingame production decline type curves behavior for a vertical well with a water influx/waterflood.In this study, first, the transient flow model is developed considering the water influx/waterflood with a ramp rate at the external boundary. Then, the functions of the Blasingame decline type curves for a vertical well with a water influx/waterflood are derived. In other words, the theory of Blasingame production decline analysis is generalized to the water influx/waterflood reservoir. Subsequently, Blasingame production decline type curves are generated. The four flow regimes, including early unsteady flow regime, primary depletion flow regime, second unsteady flow regime, and system pseudosteady flow regime, are recognized. In addition, the effects of relevant parameters such as the dimensionless maximum water influx, the dimensionless beginning time of the water influx, and the dimensionless external boundary radius are studied on the type curves. Finally, type curves are verified through a field case study. This study can provide very meaningful references for reservoir engineers in water influx rate evaluation as well as the beginning time of the water influx estimation by matching the type curves with the actual field data.


## 2. Mathematical Model with a Natural Water Influx/Waterflood

### 2.1. Physical Model and Its Assumptions

The physical model with a water influx/waterflood at the external boundary is shown in Figure1. The model under study assumes that a vertical well is centered in a bounded circular reservoir with a constant rate at the inner boundary and water influx at the outer boundary. The other basic assumptions of the model are as follows:(1)
The reservoir is homogenous, anisotropic, and horizontal with closed top and bottom boundaries. The characteristic parameters (such as thickness, permeability, porosity, and initial pressure) of the reservoir are constant.(2)
The fluid is considered as a single phase. The other properties of the fluid remain constant.(3)
The fluid flow follows Darcy’s law. The well produces gas at a constant rate.(4)
The influx at the outer boundary is initially zero and the water influx gradually increases from zero at the initial time to a fixed value at a certain time. The waterflood case is named as the “ramp” rate case by Doublet and Blasingame [29].(5)
The reservoir is isothermal and the effect of gas gravity is neglected.Figure 1
Physical model of the water-drive reservoir.r is the radial radius, rw is the well radius, re is the radius distance of the external boundary at the water influx, and qext(t) is the water influx at the external boundary.


### 2.2. Mathematical Model and Its Solution

Based on the above assumptions, the partial differential equation describing the fluid flow can be derived in terms of pseudopressure as follows:(1)∂2ψ∂r2+1r∂ψ∂r=μϕctk∂ψ∂t,where r is the radius distance from the wellbore (m), ψ is the reservoir pseudopressure (MPa2/MPa·s), k is the permeability (10−3 μm2), ϕ is the porosity (decimal), μ is the fluid viscosity (MPa·s), ct is the total compressibility (MPa−1), and t is the time (day).The definition of the pseudopressure is(2)ψ=2∫p0ppμzdp.Initial boundary condition:(3)ψr,0=ψi.Internal (or inner) boundary condition:(4)2πkhμBr∂ψ∂rr=rw=q.External (or outer) boundary condition:(5)2πkhμBr∂ψ∂rr=re=qextt,where ψi is the initial pseudopressure (MPa2/MPa·s), rw is the well radius (m), h is the thickness of the pay zone (m), q is the production per day (m3/d), B is the fluid formation volume factor (decimal), re is the radius distance of the external boundary at the water influx (m), and qext(t) is the water influx at the external boundary (m3/day), which is a function of the time.Doublet and Blasingame [29] proposed the “Ramp” waterflood flux function as follows:(6)qextt=qext,∞1−exp−ttstart,where tstart is the beginning time of the water influx (day) and qext,∞ is the maximum water influx, which does not exceed the well production (m3/d).To simplify the derivation of equations, first, the dimensionless definition of all the variables (listed in Table1) used in the proposed mathematical model is presented.Table 1
Dimensionless definition of all the variables used in this study.
Dimensionless pressure
ψ
D
=
k
h
/
q
μ
B
Δ
ψ
Dimensionless radius
r
D
=
r
/
r
w
Dimensionless radius at the external boundary
r
eD
=
r
e
/
r
w
Dimensionless maximum water influx
q
Dext,
∞
=
q
ext,
∞
/
q
Dimensionless time
t
D
=
k
t
/
μ
φ
c
t
r
w
2
Dimensionless waterflood flux at the external boundary
q
Dext
t
D
=
q
Dext,
∞
1
−
exp
−
t
D
/
t
DstartUsing the formula of the dimensionless waterflood flux at the external boundary in Table1, the dimensionless “Ramp” waterflood fluxes at the external boundary with different qDext,∞ and tDstart are shown in Figures 2 and 3, respectively. As qDext,∞ increases, the values of qDext become larger after tDstart (Figure 2). The values of qDext increase with increasing tDstart, but finally, the qDext curves converge into one horizontal line (Figure 3). Consequently, qDext,∞ and tDstart have significant effects on the production decline behavior of the natural water influx reservoirs or the artificial waterflood reservoirs.Figure 2
The dimensionless “Ramp” waterflood flux at the external boundary versus the dimensionless time fortstartD = 103.Figure 3
Schematic of the dimensionless “Ramp” waterflood flux at the external boundary versus the dimensionless time withqDext,∞ = 0.6.Based on the definition of all the dimensionless variables listed in Table1, equations (1) and (3)–(5) of the mathematical model can be converted and written in the following dimensionless forms:(7)1rD∂∂rDrD∂ψD∂rD=∂2ψD∂rD2+1rD∂ψD∂rD=∂ψD∂tD,(8)ψDrD,tD=0=0,(9)rD∂ψD∂rDrD=1=−1,(10)rD∂ψD∂rDrD=reD=qDexttD.To solve the equations, the mathematical model is first transformed into Laplace domain. With Laplace transformation, equations (7)–(10) can be written as(11)1rD∂∂rDrD∂ψ¯D∂rD=sψ¯D,(12)ψ¯DrD,tD=0=0,(13)rDdψ¯DdrDrD=1=−1s,(14)rDdψ¯DdrDrD=reD=q¯Dexts.Combining equations (11)–(14), the pressure response solution of equation (11) can be obtained as follows [29]:(15)ψ¯DrD=1,s=1ssK0sI1reDs+K1reDsI0sK1sI1reDs−K1reDsI1s+q¯DextssreDK0sI1s+I0sK1sK1sI1reDs−K1reDsI1s,where K0(x) is the modified Bessel function of second kind (zero order), K1(x) is the modified Bessel function of second kind (first order), I0(x) is the modified Bessel function of first kind (zero order), and I1(x) is the modified Bessel function of first kind (first order).In equation (15), the first part is the “no-flow” boundary solution and the second part is the boundary flux solution [29]. In addition, the transient pressure solution for a vertical well in a natural water influx/waterflood homogenous reservoir can be generalized to a fractured natural water influx/waterflood reservoir and different well configurations.


## 2.1. Physical Model and Its Assumptions

The physical model with a water influx/waterflood at the external boundary is shown in Figure1. The model under study assumes that a vertical well is centered in a bounded circular reservoir with a constant rate at the inner boundary and water influx at the outer boundary. The other basic assumptions of the model are as follows:(1)
The reservoir is homogenous, anisotropic, and horizontal with closed top and bottom boundaries. The characteristic parameters (such as thickness, permeability, porosity, and initial pressure) of the reservoir are constant.(2)
The fluid is considered as a single phase. The other properties of the fluid remain constant.(3)
The fluid flow follows Darcy’s law. The well produces gas at a constant rate.(4)
The influx at the outer boundary is initially zero and the water influx gradually increases from zero at the initial time to a fixed value at a certain time. The waterflood case is named as the “ramp” rate case by Doublet and Blasingame [29].(5)
The reservoir is isothermal and the effect of gas gravity is neglected.Figure 1
Physical model of the water-drive reservoir.r is the radial radius, rw is the well radius, re is the radius distance of the external boundary at the water influx, and qext(t) is the water influx at the external boundary.


## 2.2. Mathematical Model and Its Solution

Based on the above assumptions, the partial differential equation describing the fluid flow can be derived in terms of pseudopressure as follows:(1)∂2ψ∂r2+1r∂ψ∂r=μϕctk∂ψ∂t,where r is the radius distance from the wellbore (m), ψ is the reservoir pseudopressure (MPa2/MPa·s), k is the permeability (10−3 μm2), ϕ is the porosity (decimal), μ is the fluid viscosity (MPa·s), ct is the total compressibility (MPa−1), and t is the time (day).The definition of the pseudopressure is(2)ψ=2∫p0ppμzdp.Initial boundary condition:(3)ψr,0=ψi.Internal (or inner) boundary condition:(4)2πkhμBr∂ψ∂rr=rw=q.External (or outer) boundary condition:(5)2πkhμBr∂ψ∂rr=re=qextt,where ψi is the initial pseudopressure (MPa2/MPa·s), rw is the well radius (m), h is the thickness of the pay zone (m), q is the production per day (m3/d), B is the fluid formation volume factor (decimal), re is the radius distance of the external boundary at the water influx (m), and qext(t) is the water influx at the external boundary (m3/day), which is a function of the time.Doublet and Blasingame [29] proposed the “Ramp” waterflood flux function as follows:(6)qextt=qext,∞1−exp−ttstart,where tstart is the beginning time of the water influx (day) and qext,∞ is the maximum water influx, which does not exceed the well production (m3/d).To simplify the derivation of equations, first, the dimensionless definition of all the variables (listed in Table1) used in the proposed mathematical model is presented.Table 1
Dimensionless definition of all the variables used in this study.
Dimensionless pressure
ψ
D
=
k
h
/
q
μ
B
Δ
ψ
Dimensionless radius
r
D
=
r
/
r
w
Dimensionless radius at the external boundary
r
eD
=
r
e
/
r
w
Dimensionless maximum water influx
q
Dext,
∞
=
q
ext,
∞
/
q
Dimensionless time
t
D
=
k
t
/
μ
φ
c
t
r
w
2
Dimensionless waterflood flux at the external boundary
q
Dext
t
D
=
q
Dext,
∞
1
−
exp
−
t
D
/
t
DstartUsing the formula of the dimensionless waterflood flux at the external boundary in Table1, the dimensionless “Ramp” waterflood fluxes at the external boundary with different qDext,∞ and tDstart are shown in Figures 2 and 3, respectively. As qDext,∞ increases, the values of qDext become larger after tDstart (Figure 2). The values of qDext increase with increasing tDstart, but finally, the qDext curves converge into one horizontal line (Figure 3). Consequently, qDext,∞ and tDstart have significant effects on the production decline behavior of the natural water influx reservoirs or the artificial waterflood reservoirs.Figure 2
The dimensionless “Ramp” waterflood flux at the external boundary versus the dimensionless time fortstartD = 103.Figure 3
Schematic of the dimensionless “Ramp” waterflood flux at the external boundary versus the dimensionless time withqDext,∞ = 0.6.Based on the definition of all the dimensionless variables listed in Table1, equations (1) and (3)–(5) of the mathematical model can be converted and written in the following dimensionless forms:(7)1rD∂∂rDrD∂ψD∂rD=∂2ψD∂rD2+1rD∂ψD∂rD=∂ψD∂tD,(8)ψDrD,tD=0=0,(9)rD∂ψD∂rDrD=1=−1,(10)rD∂ψD∂rDrD=reD=qDexttD.To solve the equations, the mathematical model is first transformed into Laplace domain. With Laplace transformation, equations (7)–(10) can be written as(11)1rD∂∂rDrD∂ψ¯D∂rD=sψ¯D,(12)ψ¯DrD,tD=0=0,(13)rDdψ¯DdrDrD=1=−1s,(14)rDdψ¯DdrDrD=reD=q¯Dexts.Combining equations (11)–(14), the pressure response solution of equation (11) can be obtained as follows [29]:(15)ψ¯DrD=1,s=1ssK0sI1reDs+K1reDsI0sK1sI1reDs−K1reDsI1s+q¯DextssreDK0sI1s+I0sK1sK1sI1reDs−K1reDsI1s,where K0(x) is the modified Bessel function of second kind (zero order), K1(x) is the modified Bessel function of second kind (first order), I0(x) is the modified Bessel function of first kind (zero order), and I1(x) is the modified Bessel function of first kind (first order).In equation (15), the first part is the “no-flow” boundary solution and the second part is the boundary flux solution [29]. In addition, the transient pressure solution for a vertical well in a natural water influx/waterflood homogenous reservoir can be generalized to a fractured natural water influx/waterflood reservoir and different well configurations.


## 3. Production Decline Analysis Theory

The dimensionless pressure solution in the Laplace domain can be obtained using the dimensionless rate solution, as proposed by Van Everdingen and Hurst [4]:(16)qDs=1s21ψ¯Ds.The subject of the production performance behavior of the natural water influx/waterflood reservoirs is significant because the water invasion has an important effect on the well production. On the other side, the water invasion behavior and water influx rate are difficult to diagnose by the dynamic monitoring technology. The response of transient production decline type curves for conventional primary depletion reservoirs with a closed boundary were presented in the works performed by Palacio and Blasingame [21]; Doublet et al. [29] and Marhaendrajana and Blasingame [30]. This section generalizes the Blasingame transient production decline type curves to a new reservoir model with a “ramp” boundary flux for the water influx reservoirs.


### 3.1. Material Balance Equation for Natural Water Influx/Waterflood Reservoirs

The material balance equation is the theoretical basis of the Blasingame decline type curves. However, the material balance equation of traditional closed boundary reservoirs is not appropriate for natural water influx/waterflood reservoirs. The material balance equation of water influx/waterflood reservoirs can be derived as follows:(17)GpBg+WpBw=GBg−Bgi+GBgicwswi+cf1−swipi−p+We,where G is the reserve of the natural water influx/waterflood reservoir; Gp and Wp are the productions of the gas and water, respectively; We is the water influx rate; cg, cw, and cf are the compressibilities of the gas, water, and rock, respectively; swi is the irreducible water saturation; p is the reservoir pressure; Bg and Bgi are the gas volume coefficients at the pressure p and the initial reservoir pressure pi, respectively; and Bw is the water volume coefficient.To facilitate the derivation of equation (17), the volume coefficient of the water, M, is defined as the ratio between water volume and pore volume in the reservoir:(18)M=VwiGBgi/1−Swi,where Vwi is the water volume in the reservoir.Considering the effect of the water expansion and the shrinkage of the rock, the expression of the water influx can be written as(19)We=Vwicf+cwΔp=MNBgiΔp1−swicf+cw,where Δp is the pressure drop from pressure pi to pressure p.Inserting equation (19) and the real gas state equation into equation (16), one can reach to(20)pz1−cwswi+cf+Mcf+cw1−swiΔp=pizi1−QpG,where zi and z are the deviation factors at the initial reservoir pressure pi and the pressure p, respectively, and Qp is the pseudocumulative production rate, which is defined as follows:(21)Qp=Gp+WpBwBg.


### 3.2. Blasingame Production Decline Analysis Theory

To simplify equation (20), a new parameter (ca) is defined as follows:(22)ca=cwswi+cf+Mcf+cw1−swi.Taking the derivative of equation (20) with respect to t gives(23)dpdt=−piqpziGz1−swipM+swicw+M+1cf+1−swicg−∂cd/∂p,where cg is the gas compressibility and qp is the pseudoproduction rate, which is defined as follows:(24)qp=dQpdt.In equation (23), ∂cd/∂p can be derived as(25)∂cd∂p=cgM+swicw+M+1cfpi−p.In order to simplify equation (23), the total compressibility ct is defined as(26)ct=M+swicw+M+1cf+1−swicg−∂cd/∂p1−swi.Combining equations (23), (25), and (26), the flow production equation in the natural water influx/waterflood reservoir can be obtained as follows:(27)qp=−ziGctpzpidpdt.Additionally, to analyze the case of the variable rate/pressure data for a gas well, Palacio and Blasingame [21] introduced the material balance pseudotime function for a gas well and proposed the Blasingame production decline type curves analysis method.The material balance pseudotime function is(28)tca=μCtiq∫0tqμCtdt.Inserting equation (27) into equation (28), the material balance pseudotime equation can be rewritten as(29)tca=μctiq∫pipziGpzpiμdp=Gμctziqpi∫pippμzdp.To simplify equation (29), the normalized pseudopressure is defined as below:(30)pp=μgzpi∫0ppμgzdp.In addition, taking equation (30) into equation (29), the material balance pseudotime equation can be rewritten as(31)tca=Gctiqppi−pp,where ppi and pp are the normalized pseudoinitial reservoir pressure and normalized pseudoreservoir pressure, respectively.According to the flow theory in porous media, the standard pseudosteady flow equation can be written as follows:(32)q=Jpp−ppwf,where J is the gas production index.Combining equations (31) and (32) results in(33)ppi−pPwfq=matca+ba,pss,where ma = 1/(G × cti) and ba,pss = (μ × z)i/(2 × J × pi).Equation (33) in water influx reservoirs is consistent with the one in dry gas reservoirs. Therefore, the model proposed above can be extended to water influx reservoirs.Based on the above similar theory derivation, Palacio and Blasingame [20] developed the theory of Blasingame production decline type curves analysis. These Blasingame decline type curves are summarized in Table 2.Table 2
Summary of the Blasingame decline type curves.
Case
Plotting function
Dimensionless normalized production curve
q
Dd versus tDd
Dimensionless normalized cumulative production integral curve
q
Ddi versus tDd
Dimensionless production integral derivative curve
q
Ddid versus tDdThe production decline type curves are generally defined using the following dimensionless parameters [20, 30].Dimensionless normalized production function:(34)qDd=ba,pssΔpptca=qDlnreD−12.Dimensionless material balance pseudotime:(35)tDd==maba,pss2reD21lnreD−1/2tD.Dimensionless normalized cumulative production integral function:(36)qDdi=1tDd∫0tDdqDdτdτ.Dimensionless production integral derivative function:(37)qDdid=−dqDdidlntDd=−tDddqDditDd.


## 3.1. Material Balance Equation for Natural Water Influx/Waterflood Reservoirs

The material balance equation is the theoretical basis of the Blasingame decline type curves. However, the material balance equation of traditional closed boundary reservoirs is not appropriate for natural water influx/waterflood reservoirs. The material balance equation of water influx/waterflood reservoirs can be derived as follows:(17)GpBg+WpBw=GBg−Bgi+GBgicwswi+cf1−swipi−p+We,where G is the reserve of the natural water influx/waterflood reservoir; Gp and Wp are the productions of the gas and water, respectively; We is the water influx rate; cg, cw, and cf are the compressibilities of the gas, water, and rock, respectively; swi is the irreducible water saturation; p is the reservoir pressure; Bg and Bgi are the gas volume coefficients at the pressure p and the initial reservoir pressure pi, respectively; and Bw is the water volume coefficient.To facilitate the derivation of equation (17), the volume coefficient of the water, M, is defined as the ratio between water volume and pore volume in the reservoir:(18)M=VwiGBgi/1−Swi,where Vwi is the water volume in the reservoir.Considering the effect of the water expansion and the shrinkage of the rock, the expression of the water influx can be written as(19)We=Vwicf+cwΔp=MNBgiΔp1−swicf+cw,where Δp is the pressure drop from pressure pi to pressure p.Inserting equation (19) and the real gas state equation into equation (16), one can reach to(20)pz1−cwswi+cf+Mcf+cw1−swiΔp=pizi1−QpG,where zi and z are the deviation factors at the initial reservoir pressure pi and the pressure p, respectively, and Qp is the pseudocumulative production rate, which is defined as follows:(21)Qp=Gp+WpBwBg.


## 3.2. Blasingame Production Decline Analysis Theory

To simplify equation (20), a new parameter (ca) is defined as follows:(22)ca=cwswi+cf+Mcf+cw1−swi.Taking the derivative of equation (20) with respect to t gives(23)dpdt=−piqpziGz1−swipM+swicw+M+1cf+1−swicg−∂cd/∂p,where cg is the gas compressibility and qp is the pseudoproduction rate, which is defined as follows:(24)qp=dQpdt.In equation (23), ∂cd/∂p can be derived as(25)∂cd∂p=cgM+swicw+M+1cfpi−p.In order to simplify equation (23), the total compressibility ct is defined as(26)ct=M+swicw+M+1cf+1−swicg−∂cd/∂p1−swi.Combining equations (23), (25), and (26), the flow production equation in the natural water influx/waterflood reservoir can be obtained as follows:(27)qp=−ziGctpzpidpdt.Additionally, to analyze the case of the variable rate/pressure data for a gas well, Palacio and Blasingame [21] introduced the material balance pseudotime function for a gas well and proposed the Blasingame production decline type curves analysis method.The material balance pseudotime function is(28)tca=μCtiq∫0tqμCtdt.Inserting equation (27) into equation (28), the material balance pseudotime equation can be rewritten as(29)tca=μctiq∫pipziGpzpiμdp=Gμctziqpi∫pippμzdp.To simplify equation (29), the normalized pseudopressure is defined as below:(30)pp=μgzpi∫0ppμgzdp.In addition, taking equation (30) into equation (29), the material balance pseudotime equation can be rewritten as(31)tca=Gctiqppi−pp,where ppi and pp are the normalized pseudoinitial reservoir pressure and normalized pseudoreservoir pressure, respectively.According to the flow theory in porous media, the standard pseudosteady flow equation can be written as follows:(32)q=Jpp−ppwf,where J is the gas production index.Combining equations (31) and (32) results in(33)ppi−pPwfq=matca+ba,pss,where ma = 1/(G × cti) and ba,pss = (μ × z)i/(2 × J × pi).Equation (33) in water influx reservoirs is consistent with the one in dry gas reservoirs. Therefore, the model proposed above can be extended to water influx reservoirs.Based on the above similar theory derivation, Palacio and Blasingame [20] developed the theory of Blasingame production decline type curves analysis. These Blasingame decline type curves are summarized in Table 2.Table 2
Summary of the Blasingame decline type curves.
Case
Plotting function
Dimensionless normalized production curve
q
Dd versus tDd
Dimensionless normalized cumulative production integral curve
q
Ddi versus tDd
Dimensionless production integral derivative curve
q
Ddid versus tDdThe production decline type curves are generally defined using the following dimensionless parameters [20, 30].Dimensionless normalized production function:(34)qDd=ba,pssΔpptca=qDlnreD−12.Dimensionless material balance pseudotime:(35)tDd==maba,pss2reD21lnreD−1/2tD.Dimensionless normalized cumulative production integral function:(36)qDdi=1tDd∫0tDdqDdτdτ.Dimensionless production integral derivative function:(37)qDdid=−dqDdidlntDd=−tDddqDditDd.


## 4. Results and Discussion

### 4.1. Model Validation and Comparison

According to the above production decline analysis theory, the Blasingame type curves can be obtained by using one of the numerical Laplace inversion methods, including Stehfest, Zakian, Fourier series, and Schapery methods [31–33]. Here, the Stehfest method is applied [34]. For this study, if the water influx/waterflood at the external boundary, qDext,∞, is equal to zero, the proposed model can be reduced to the traditional closed boundary model.Figure4 shows the type curves for a vertical well with a traditional closed boundary model and water influx/waterflood qDext,∞ = 0.7 at the external boundary model. From Figure 4, it can be observed that the rate decline curves behavior is quite different in the boundary response regime. The slope of the dimensionless normalized production curve is −1 (angle = 45°) in the pseudosteady flow regime dominated by the closed boundary, whereas the slope of the dimensionless normalized production curve is not equal to −1 in the regimes dominated by the water influx/waterflood boundary.Figure 4
Comparison of Blasingame type curves with different boundary models.In addition, according to the behavior of the Blasingame production decline curves (Figure5), type curves can be divided into the four regimes: (Ι) early unsteady flow regime (EUFR) around the vertical well, which represents the pressure wave spreads continuously away from the wellbore; (II) the primary depletion flow regime (PDFR); (III) the second unsteady flow regime (SUFR) resulting from the external pressure support such as natural water influx or waterflood; and (IV) system pseudosteady flow regime (SPSF), in which dimensionless normalized production curve has a slope of −1 (angle = 45°).Figure 5
Blasingame production decline type curves by considering natural water influx or waterflood.


### 4.2. Sensitivity Analysis

Most of the previous studies have focused on the closed boundary model, rather than on the natural water influx or waterflood model at the external boundary. Thus, the sensitivity of relevant parameters of the water influx or waterflood is discussed in detail in this section.Figure6 shows that the dimensionless maximum water influx has a significant effect on the second unsteady flow regime and system pseudosteady flow regime. From Figure 6, it can be observed that the greater the dimensionless maximum water influx, the larger the values of qDd, qDdi, and qDdid for the type curves happen during the second unsteady flow regime and system pseudosteady flow regimes. The main reason is that the external pressure support of the natural water influx or waterflood becomes greater with an increase in the dimensionless maximum water influx.Figure 6
Effect of the dimensionless maximum water influx (qDext,∞) on type curves.Figure7 illustrates the effect of the dimensionless beginning time of the water influx (tDstart) on the second unsteady flow regime of the Blasingame production decline type curves. As tDstart increases, the regime of the second unsteady flow emerges earlier. According to the emerging time of the second unsteady flow regime, the type curves can be used to estimate the tDstart by type curve history matching.Figure 7
Effect of the dimensionless beginning time of the water influx (tDstart) on type curves.Figure8 shows the effect of the dimensionless external boundary radius (reD) on production decline type curves. It can be observed that first regime are governed by reD. As the dimensionless external boundary radius increases, the values of qDd, qDdi, and qDdid for the type curves become smaller during the early unsteady flow regime. According to the early unsteady flow regime behavior of the Blasingame decline type curves, the type curves can be used to estimate the reD by type curves history matching.Figure 8
Effect of the dimensionless external boundary radius (reD) on type curves.


## 4.1. Model Validation and Comparison

According to the above production decline analysis theory, the Blasingame type curves can be obtained by using one of the numerical Laplace inversion methods, including Stehfest, Zakian, Fourier series, and Schapery methods [31–33]. Here, the Stehfest method is applied [34]. For this study, if the water influx/waterflood at the external boundary, qDext,∞, is equal to zero, the proposed model can be reduced to the traditional closed boundary model.Figure4 shows the type curves for a vertical well with a traditional closed boundary model and water influx/waterflood qDext,∞ = 0.7 at the external boundary model. From Figure 4, it can be observed that the rate decline curves behavior is quite different in the boundary response regime. The slope of the dimensionless normalized production curve is −1 (angle = 45°) in the pseudosteady flow regime dominated by the closed boundary, whereas the slope of the dimensionless normalized production curve is not equal to −1 in the regimes dominated by the water influx/waterflood boundary.Figure 4
Comparison of Blasingame type curves with different boundary models.In addition, according to the behavior of the Blasingame production decline curves (Figure5), type curves can be divided into the four regimes: (Ι) early unsteady flow regime (EUFR) around the vertical well, which represents the pressure wave spreads continuously away from the wellbore; (II) the primary depletion flow regime (PDFR); (III) the second unsteady flow regime (SUFR) resulting from the external pressure support such as natural water influx or waterflood; and (IV) system pseudosteady flow regime (SPSF), in which dimensionless normalized production curve has a slope of −1 (angle = 45°).Figure 5
Blasingame production decline type curves by considering natural water influx or waterflood.


## 4.2. Sensitivity Analysis

Most of the previous studies have focused on the closed boundary model, rather than on the natural water influx or waterflood model at the external boundary. Thus, the sensitivity of relevant parameters of the water influx or waterflood is discussed in detail in this section.Figure6 shows that the dimensionless maximum water influx has a significant effect on the second unsteady flow regime and system pseudosteady flow regime. From Figure 6, it can be observed that the greater the dimensionless maximum water influx, the larger the values of qDd, qDdi, and qDdid for the type curves happen during the second unsteady flow regime and system pseudosteady flow regimes. The main reason is that the external pressure support of the natural water influx or waterflood becomes greater with an increase in the dimensionless maximum water influx.Figure 6
Effect of the dimensionless maximum water influx (qDext,∞) on type curves.Figure7 illustrates the effect of the dimensionless beginning time of the water influx (tDstart) on the second unsteady flow regime of the Blasingame production decline type curves. As tDstart increases, the regime of the second unsteady flow emerges earlier. According to the emerging time of the second unsteady flow regime, the type curves can be used to estimate the tDstart by type curve history matching.Figure 7
Effect of the dimensionless beginning time of the water influx (tDstart) on type curves.Figure8 shows the effect of the dimensionless external boundary radius (reD) on production decline type curves. It can be observed that first regime are governed by reD. As the dimensionless external boundary radius increases, the values of qDd, qDdi, and qDdid for the type curves become smaller during the early unsteady flow regime. According to the early unsteady flow regime behavior of the Blasingame decline type curves, the type curves can be used to estimate the reD by type curves history matching.Figure 8
Effect of the dimensionless external boundary radius (reD) on type curves.


## 5. Field Application

A field case is presented to demonstrate the application of the model proposed in this study. Well A1 is located in an offshore sand reservoir in the western South China Sea. The reservoir is a typical gas reservoir driven by the edge water. The basic parameters for the production decline analysis are given in Table3.Table 3
Basic parameters for the production decline analysis.
Parameters
Unit
Value
Well radius,rw
m
0.1
Thickness of the pay zone,h
m
18
Reservoir porosity,ϕ
Decimal
0.253
Initial reservoir pressure,pi
MPa
14.6
Reservoir temperature,T
°C
80
Gas gravity,rg
Decimal
0.57
Initial water saturation,swi
Decimal
0.25
Rock compressibility,cf
MPa−1
5.55 × 10−4
Water compressibility,cw
MPa−1
4.67 × 10−4
Gas compressibility,cg
MPa−1
7.12 × 10−4The production data of well A1 are matched and analyzed by the Blasingame production decline analysis technique with the water influx/waterflood model. Figure9 shows that the Blasingame production decline type curves match the real production data well. On the other hand, the behavior of type curves of water invasion (regime III and regime IV) can be observed clearly from Figure 9. The relevant parameters (Table 4) are obtained by matching and analyzing the Blasingame decline curves of well A1.Figure 9
The matched curves of the Blasingame production decline for well A1.Table 4
Results of the Blasingame production decline analysis for well A1.
Parameters
Unit
Value
Permeability,k
mD
10.9
Distance from the well to the circular boundary,re
m
76.32
Maximum water influx,qDext,∞
Decimal
0.86
Beginning time of the water influx,tDstart
Decimal
1.52
Volume coefficient of water,M
Decimal
0.25
Well’s original gas in place,Nv
108 m3
7.62


## 6. Conclusions

This study investigates the Blasingame production decline type curves for a vertical well in water influx/waterflood reservoirs, analyses the behavior of production decline type curves, and discusses the effects of the relevant parameters on type curves. Based on the results obtained in this work, the following conclusions can be made:(1)
The production decline type curve analysis model for a vertical well in natural water influx/waterflood reservoirs is established.(2)
The Blasingame production decline curves for a vertical well in natural water influx/waterflood reservoirs are generated and divided into the four regimes: early unsteady flow regime, primary depletion flow regime, second unsteady flow regime, and system pseudosteady flow regime.(3)
The production decline type curves are affected by the natural water influx/waterflood significantly. The dimensionless maximum water influx has a significant effect on the second unsteady flow regime and system pseudosteady flow regime, and the dimensionless water influx time has mainly affected the emerging time of the second unsteady flow regime.(4)
The validation of the field case demonstrates that the Blasingame production decline type curves for water influx/waterflood boundary proposed in this study could be applied to evaluate the water influx and the time of the water influx. In other words, the work provides very meaningful references for reservoir engineers working on the dynamic analysis of the natural water influx/waterflood reservoir by matching the developed type curves with the actual field data.



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*Source: 1683989-2019-09-02.xml*