A method for evaluation of water flooding performance in fractured reservoirs Shaohua Gu a,b,n,1 , Yuetian Liu a , Zhangxin Chen b , Cuiyu Ma a a MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China b Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4 article info Article history: Received 19 December 2013 Accepted 3 June 2014 Available online 17 June 2014 Keywords: Water flooding Fractured reservoir Dual-porosity Imbibition Relative permeability abstract A mathematical model is developed for evaluation of water flooding performance in a highly fractured reservoir. The model transforms a dual-porosity medium into an equivalent single porosity medium by using a pseudo relative permeability method to normalize the relative permeability. This approach allows both fractures and matrix to have permeability, porosity, endpoint saturation, and endpoint relative permeability by themselves. Imbibition is also taken into account by modifying Chen's equation. Some effects, including imbibition and recovery rates are investigated. The investigation shows that imbibition can determine the potential of a fractured reservoir and a low recovery rate can improve the water flooding situation in terms of retarding water breakthrough and controlling the rise of water cut. A new chart composed by water cut vs. recovery curves is protracted to estimate the ultimate water- flooding recovery rate. The water flooding performance of two reservoirs is evaluated. Compared with numerical simulation method, the error of these two cases are not more than 2%, which proved that this method is reliable. Both lab test data and field data are applied to a further discussion of the characteristics of water flooding performance in fractured reservoirs. On comparison with the classical method, such as Tong's method and the X-plot method, the reason why the new method is more suitable to fractured reservoirs is addressed by a theoretical analysis. An appropriate application of this method can help the reservoir engineer to optimize the reservoir management with low costs and high efficiency. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Experiences from oil recovery around the globe have shown distinct water flooding performance in fractured reservoirs than in conventional reservoirs. In most cases, the recovery usually begins with a high production rate in an early stage and then declines dramatically once water breaks through due to a rapid rise in water cut, especially in some high yield wells. Moreover, the geological complexity is also a barrier for accurate estimation of the water flooding performance and the potential of a fractured reservoir. Furthermore, as everyone knows, it is significant to perform reservoir management and investment decision. For interpretation of water flooding performance in fractured reservoirs, many research papers have been published. Currently used methods can be classified as two categories: reservoir simulation and a reservoir performance analysis. The reservoir simulation methods consist of numerical simulation and physical simulation. Models of dual-porosity (Barenblatt et al., 1960) and shape factors (Warren and Root, 1963; Kazemi et al., 1976) are widely used in numerical simulation of the fractured reservoirs. But one of the main problems is that these models are over- simplified to meet the demand of computing. Another problem is that history matching is a subjective process. That is, various results may be obtained on the basis of the same data. Because of more tunable parameters in a dual-porosity model, more probable choices may be made by reservoir engineers. Some new technol- ogies, such as a discrete fracture network (DFN) model and unstructured grids (Hoteit and Firoozabadi, 2008a, 2008b; Huang et al., 2011), can characterize a fracture network more accurately. However, technical limitation on information collection of in-situ fractures and enormous amount of computing are impediments to their application. Actually, the physical simulation (Yuetian et al., 2013) provides an objective way to present the water flooding performance in fractured reservoirs, but high costs and low efficiency are bottleneck problems. Compared with the reservoir simulation methods, the reservoir performance analysis methods are easy, fast and cheap tools, which are composed of analytical models, empirical models and semi- empirical models. But these types of methods need more field data Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/petrol Journal of Petroleum Science and Engineering http://dx.doi.org/10.1016/j.petrol.2014.06.002 0920-4105/& 2014 Elsevier B.V. All rights reserved. n Corresponding author. Tel.: þ86 10 89732260. E-mail address: [email protected](S. Gu). 1 Visiting scholar of University of Calgary. Journal of Petroleum Science and Engineering 120 (2014) 130–140
Transcript
A method for evaluation of water flooding performance infractured reservoirs
Shaohua Gu a,b,n,1, Yuetian Liu a, Zhangxin Chen b, Cuiyu Ma a
a MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, Chinab Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4
a r t i c l e i n f o
Article history:Received 19 December 2013Accepted 3 June 2014Available online 17 June 2014
A mathematical model is developed for evaluation of water flooding performance in a highly fracturedreservoir. The model transforms a dual-porosity medium into an equivalent single porosity medium byusing a pseudo relative permeability method to normalize the relative permeability. This approachallows both fractures and matrix to have permeability, porosity, endpoint saturation, and endpointrelative permeability by themselves. Imbibition is also taken into account by modifying Chen's equation.Some effects, including imbibition and recovery rates are investigated. The investigation shows thatimbibition can determine the potential of a fractured reservoir and a low recovery rate can improve thewater flooding situation in terms of retarding water breakthrough and controlling the rise of water cut.A new chart composed by water cut vs. recovery curves is protracted to estimate the ultimate water-flooding recovery rate. The water flooding performance of two reservoirs is evaluated. Compared withnumerical simulation method, the error of these two cases are not more than 2%, which proved that thismethod is reliable. Both lab test data and field data are applied to a further discussion of thecharacteristics of water flooding performance in fractured reservoirs. On comparison with the classicalmethod, such as Tong's method and the X-plot method, the reason why the newmethod is more suitableto fractured reservoirs is addressed by a theoretical analysis. An appropriate application of this methodcan help the reservoir engineer to optimize the reservoir management with low costs and highefficiency.
& 2014 Elsevier B.V. All rights reserved.
1. Introduction
Experiences from oil recovery around the globe have showndistinct water flooding performance in fractured reservoirs than inconventional reservoirs. In most cases, the recovery usually beginswith a high production rate in an early stage and then declinesdramatically once water breaks through due to a rapid rise inwater cut, especially in some high yield wells. Moreover, thegeological complexity is also a barrier for accurate estimation ofthe water flooding performance and the potential of a fracturedreservoir. Furthermore, as everyone knows, it is significant toperform reservoir management and investment decision.
For interpretation of water flooding performance in fracturedreservoirs, many research papers have been published. Currentlyused methods can be classified as two categories: reservoirsimulation and a reservoir performance analysis. The reservoirsimulation methods consist of numerical simulation and physical
simulation. Models of dual-porosity (Barenblatt et al., 1960) andshape factors (Warren and Root, 1963; Kazemi et al., 1976) arewidely used in numerical simulation of the fractured reservoirs.But one of the main problems is that these models are over-simplified to meet the demand of computing. Another problem isthat history matching is a subjective process. That is, variousresults may be obtained on the basis of the same data. Because ofmore tunable parameters in a dual-porosity model, more probablechoices may be made by reservoir engineers. Some new technol-ogies, such as a discrete fracture network (DFN) model andunstructured grids (Hoteit and Firoozabadi, 2008a, 2008b; Huanget al., 2011), can characterize a fracture network more accurately.However, technical limitation on information collection of in-situfractures and enormous amount of computing are impediments totheir application. Actually, the physical simulation (Yuetian et al.,2013) provides an objective way to present the water floodingperformance in fractured reservoirs, but high costs and lowefficiency are bottleneck problems.
Compared with the reservoir simulation methods, the reservoirperformance analysis methods are easy, fast and cheap tools, whichare composed of analytical models, empirical models and semi-empirical models. But these types of methods need more field data
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/petrol
Journal of Petroleum Science and Engineering
http://dx.doi.org/10.1016/j.petrol.2014.06.0020920-4105/& 2014 Elsevier B.V. All rights reserved.
n Corresponding author. Tel.: þ86 10 89732260.E-mail address: [email protected] (S. Gu).1 Visiting scholar of University of Calgary.
Journal of Petroleum Science and Engineering 120 (2014) 130–140
and recovery experience to develop, and the predicting results alsoneed more checks with field production. The theory of Buckley andLeverett (1942) and the Welge (1952) equation were first proposedto explain the phenomena of two-phase flow in reservoirs. Accord-ing to experiments of Efros (1958), a relationship between oil cut,oil viscosity and outflow end water saturation in a process of water–oil displacement was obtained. Timmerman (1971) found a rela-tionship between cumulative oil production and an oil–water ratio(i.e., (1� fw)/fw) by field data, which was from a water floodingreservoir in Illinois. Tong (1978, 1988) studied statistical data frommore than 20 water flooding reservoirs around the globe and drewa chart for engineers to evaluate the water flooding performance.Chen (1985) deduced some water displacement curve(WDC) meth-ods by using the theory of Buckley–Leverett, the Welge equationand the relation found by Efros, and the results were consistentwith Tong's survey. As more advances in the technology of reservoirwater flooding evaluation are made, more types of reservoirs havebeen put into consideration by researchers. El-khatib (2001, 2012)applied the Buckley–Leverett displacement theory to study waterflooding in non-communicating stratified reservoirs and in inclinedcommunicating stratified reservoirs. Yang (2009) proposed a newdiagnostic analysis method for water flooding performance inconventional reservoirs.
In fact, many lessons and much experience have already beenlearned from hundreds of fractured reservoirs (Allan and Sun,2003; Sun and Sloan, 2003) during past many years (Dang et al.,2011). Many researchers have published many mathematicalmodels to interpret multi-phase flow in fractured medium, suchas the De Swaan (1978) model, the Kazemi analytical model (1992)and the Civan (1998) model. However, the existing problems ofevaluating water flooding performance in fractured reservoirshave not been figured out properly. One of the critical problemsis how to deal with oil–water flow in a dual-porosity medium.Another issue is how to detect the influence of imbibition on thein-situ flow and the performance of oil wells. This paper aims tosolve the above mentioned problems. First, a model is proposed
for water–oil flow in a matrix-fracture medium by using themethod of pseudo relative permeability curves. Then Chen's model(1982) is modified for calculation of the water breakthrough timeand water saturation at the breakthrough time. A chart is com-posed for water-flooding evaluation by estimation of the ultimaterecovery factor. Then the water flooding performance in twofractured reservoirs is evaluated. Compared with the classicalmethod, such as Tong's chart and X-plot method (1978), someanalyses are conducted and influential factors are discussed.
2. Mathematical model
2.1. Assumptions and definitions
A well group consists of one injector and one producer in ahighly fractured reservoir, and the Kazemi modeling concept(1976) is used, as shown in Fig. 1. The additional assumptionsare given as follows: the flow is linear, isothermal, and incom-pressible, and it obeys Darcy's law; in a dual-porosity model,fracture and matrix have its own irreducible water saturation,permeability, porosity and relativity permeability; the water–oildisplacement in this case is non-piston-like; finally, the reservoir iswater-wet and the imbibition effect is taken into account.
2.2. Pseudo relative permeability
Hearn (1971) used the pseudo relative permeability method tosimulate a stratified reservoir by water flooding, which means thatthe reservoir is divided into many layers. Babadagli and Ershaghi(1993) introduced this method into the dual porosity concept andproposed the effective fracture relative permeability (EFRP)method to reduce the model to a single porosity fracture networkmodel. In the stratified reservoir, each layer has its own thickness,porosity, initial water saturation, and residual oil saturation.Similarly, in a fractured reservoir, either fractures or matrix has
Nomenclature
A coefficient, dimensionlessB coefficient, dimensionlessb fracture aperture [L], mfw water cut, dimensionlessf/ wf the derivative of water cut of fracture, dimensionlessh formation thickness [L], mkf, kff conventional/intrinsic fracture permeability [L]2, μmkm matrix permeability [L]2, μmkT total permeability [L]2, μmkrof, krom, kroT oil relative permeability in fracture/matrix/total,
dimensionlesskrwf, krwm, krwT water relative permeability in fracture/matrix/
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140 131
its own properties, so they can be regarded as two different“layers”. The pseudo relative permeability method is introducedto transform a dual-porosity medium into an equivalent singleporosity medium, as displayed in Fig. 2. This process can simplifythe calculation by reducing the number of equations and para-meters. Actually, the end points of saturations of both matrix andfractures are not the same value. Therefore, the movable satura-tion ranges (from Sor to 1�Swi) of the two media, are totally
different from each other. In the process of calculation of pseudorelative permeability, normalization is a necessary procedure foreliminating the effect of the end points. The normalization processaims to transform various original saturation ranges to the normal-ized range from zero to one, which enables the end points ofmatrix and fractures to be the same value, as demonstrated inFig. 2(a) and (b). The equation is given as follows:
Snw ¼ Sw�Swi
1�Swi�Sorð1Þ
By the normalization process, all saturations are transformed tothe normalized range, and then the process for pseudo relativepermeability begins. The relative permeability can be tested andcalculated by the Welge–JBN method (Johnson et al., 1959) and thesaturation used in calculation is the water saturation at theoutflow end Swe; therefore, the water relative permeability canbe written as krw(Swe). The pseudo relative permeability of water inthe normalized range is (see derivation in Appendix A)
krwT ðSnweT Þ ¼ðkf f =kmÞU ðϕf =ϕmÞUkrwf ðSnwef ÞþkrwmðSnwemÞ
ðkf f =kmÞU ðϕf =ϕmÞþ1ð2Þ
Similarly, the pseudo relative permeability of oil in the normal-ized range is
kroT ðSnweT Þ ¼ðkf f =kmÞUðϕf =ϕmÞUkrof ðSnwef ÞþkromðSnwemÞ
ðkf f =kmÞU ðϕf =ϕmÞþ1ð3Þ
The fracture relative permeability curves seem X-shaped, asdisplayed in Fig. 2(c). They can be written as follows:
krwf ðSnwef Þ ¼ Snwef ð4Þ
krof ðSnwef Þ ¼ 1�Snwef ð5ÞFig. 1. Model of water flooding in water-wet fractured media and imbibition process.
Fig. 2. Model of water flooding in water-wet fractured medium: (a) matrix relative permeability curves in original range; (b) matrix relative permeability curves innormalized range; (c) fracture relative permeability curves in normalized range and (d) total relative permeability curve in normalized range.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140132
The matrix relative permeability curves in the normalizedrange (Fig. 2(b)) are transformed from relative permeability curvein the original range (Fig. 2(a)). Here the oil/water relativepermeability ratio can be presented in a polynomial fitting form,and why the polynomial form becomes the choice will beexplained in Section 4
where the coefficients P19–P27 can be determined by fitting. Thetotal relative permeability can be referred to as Fig. 2(d). The totalwater saturation is(see derivation in Appendix A)
SwT ¼ðϕf =ϕmÞUSwf þSwm
ðϕf =ϕmÞþ1ð9Þ
Similarly, the total oil saturation SoT, the total residual oilsaturation SorT and the total initial water SwiT saturation are asfollows, respectively:
SoT ¼ðϕf =ϕmÞUSof þSom
ðϕf =ϕmÞþ1ð10Þ
SorT ¼ðϕf =ϕmÞUSorf þSorm
ðϕf =ϕmÞþ1ð11Þ
SwiT ¼ðϕf =ϕmÞUSwif þSwim
ðϕf =ϕmÞþ1: ð12Þ
2.3. Flow in fracture medium with imbibition
Another key problem in this case is how to detect the effect ofexchange between fracture and matrix. Production from thematrix blocks can be associated with various physical mechanismsincluding oil expansion, capillary imbibition, gravity imbibition,diffusion and viscous displacement. In water flooding reservoir, oilexpansion is not significant role. Diffusion is not an obviousphenomenon. When fracture permeability is far higher thanmatrix permeability, viscous displacement is negligible as well.And the main mechanism in production from matrix to fracture isimbibition. Aronofsky et al. (1958) proposed an empirical model ofimbibition correlated with the oil recovery factor and ultimaterecovery factor, which is
Rm ¼ R0mð1�e�λtÞ ð13Þ
where λ is an imbibition index, which determines of the conver-gence rate to the ultimate recovery factor. In fact, it illustrates themagnitude of imbibition, and the unit is [1/s]. Although λ is anempirical parameter, it includes physical meaning. Kazemi et al.(1992) use an equation to characterize this parameter. And λcan also be obtained by spontaneous imbibition test or historymatching. By using the Duhamel principle, Chen and Liu (1982)deduced a new dynamic imbibition model by using the Aronofskymodel with respect to dynamic water saturation in the fracturesystem. Terez and Firoozabadi (1999) used the same model in theirresearch to interpret the experimental result. However, Chen'sequation has an error leading to an obvious calculation mistake,which will be discussed in Section 4. Thus the model needscorrection, and Chen's model can be modified as (see derivationin Appendix B)
ð14ÞSuppose that there is a horizontal, linear, water-wet, naturally
fractured oil-bearing formation of length L, as Fig. 3. The initialwater saturation distributions of the matrix and fracture are Swm(x,0)¼Swim and Swf(x, 0)¼0, respectively. Water has been injectedinto the inlet end (x¼0) since t¼0. The dimensionless parametersare introduced, such as x¼ x=L,t ¼ λt, WðtÞ ¼WðtÞ=lλ andqimb ¼ qimb=λ. The equations of dimensionless flow in the fracturedporous medium can be written as follows:
WðtÞf 0wf ðSwf Þ∂Swf
∂tþϕf
∂Swf
∂tþð1�SwimÞϕmR
0m Swf �
R t0 Swf ðx; τÞe�ðt�τÞdτ
h i¼ 0
Swf ðx;0Þ ¼ Swif ðxÞSwf ð0; tÞ ¼ 1
8>>><>>>:ϕm
dSwm
dtþqimb ¼ 0
qimb ¼ ð1�SwimÞϕmR0m
R t0 Swf ðx; τÞe�ðt�τÞdτ�Swf
h iSwmðx;0Þ ¼ SwimðxÞ
8>>><>>>:
ð15Þwhere the derivative of water cut f 0wf ðSwf Þ can be written as
f 0wf ðSwf Þ ¼μo=μw
½ððμo=μwÞ�1ÞSwf þ1�2 ð16Þ
A program is crafted to solve the two-phase flow equation fornumerical analysis. The data applied in numerical calculation canbe referred in Table 1. Some dynamic parameters can be deter-mined through this computing, including the fracture watersaturation Swf and the matrix water saturation Swm at differenttimes, as shown in Figs. 4 and 5.
According to the computing results, the average water satura-tion in fractures SAwf , the average water saturation in matrix SAwm ,the water saturation at the outflow end of fractures Swef, and theFig. 3. Imbibition process during water flooding.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140 133
water saturation at the outflow end of matrix Swem can beobtained. Here the average means that the values of saturationdistributed in the range of dimensionless length from 0 to 1 areaveraged. Once the above parameters are determined, the averagetotal water saturation in the normalized range can be obtainedaccording to Eqs. (1) and (9)–(12), which is
Similarly, the total water saturation at the outflow end in thenormalized range SnweT is
SnweT ¼ððϕf =ϕmÞUSwef þSwem=ðϕf =ϕmÞþ1Þ�SwiT
1�SwiT �SorTð18Þ
Then the curve of SnweT vs. SnAwT is plotted in Fig. 6. Fig. 6 showsthat SnweT is zero before water breaks through, and the total watersaturation at water breakthrough time in the normalized range isSnwBT , as displayed in Fig. 6. Since the water breaks through, SnweTand SnAwT have a linear relationship. Compared with the calculationdata, the Welge equation data shows a non-linear relationshipsince water breaks through. According to the calculation result ofthe SnweT vs. SnAwT curve shown in Fig. 6, an approximate equationcan be established as follows:
SnweT ¼ 0; SnAwT rSnwBT
SnweT ¼ SnAwT � SnwBT1� SnwBT
; SnwBT rSnAwT r1
8<: : ð19Þ
2.4. Fractional oil recovery and water cut
The total recovery factor in the matrix-fracture system is
RT ¼ϕf ð1�Swif ÞRf þϕmð1�SwimÞRm
ϕf ð1�Swif Þþϕmð1�SwimÞð20Þ
From Fig. 4, we can clearly see that the recovery factor infractures Rf rises to 1 when at time t¼0.05. It means that Rf rises to
Table 1Parameters of dimensionless model and correspondingvalues.
Fig. 4. Water saturation in fractures at different time. The dash line is Chen's (1982)calculation, and the solid line is our calculation result.
Fig. 5. Water saturation in matrix at different time: (a) the solid line is ourcalculation result and (b) the dash line is Chen's (1982) calculation result.
Fig. 6. Water saturation at out flow end in normalized range vs. average watersaturation in total by numerical calculation and Welge equation.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140134
1 in a short time. In this case, it can be seen as Rf¼1. Therefore thetotal ultimate recovery factor is
R0T ¼
ϕf ð1�Swif Þþϕmð1�SwimÞR0m
ϕf ð1�Swif Þþϕmð1�SwimÞð21Þ
According to the Buckley–Leverett theory and Eqs. (11) and(12), the water cut is
Then, applying the numerical solution to Eqs. (15) and (16), acurve of fw vs. SnweT can be obtained. Here the curve can be namedthe water flooding characteristic curve, since this curve cancharacterize the water flooding performance of a reservoir. FromEq. (19), the relationship of SnweT vs. SnAwT is known, and then therelationship of fw vs. SnAwT can also be obtained. However, we needto investigate the relationship of fw vs. RT. The value of oil recoveryfactor RT in total can be acquired by the following equation:
RT ¼SAwT �SwiT
1�SwiTð23Þ
In addition, the ultimate recovery factor in total is
R0T ¼
1�SwiT �SorT1�SwiT
ð24Þ
According to Eqs. (17), (23) and (24), we can know that
RT
R0T¼ SAwT �SwiT
1�SwiT �SorT¼ SnAwT ð25Þ
On the basis of Eqs. (19), (22) and (25), the equation of fw vs. RTcan be established
Finally, according to Eq. (26), the relationship between fw andRT with different RT' is obtained.
3. Computational procedure and application
3.1. Evaluation chart plotting process
The computational process has three steps
(1) The determination of a water flooding characteristic curve isnecessary. Here we provide a series of methods. The first one isto normalize the existing relative permeability curves of a coresample, and then the parameters P19–P27 can be determinedby fitting. However, this method needs sufficient data.Research shows that the number of relative permeabilitycurves should be more than twenty, which will be addressedin Section 4. Otherwise, the result by insufficient data may notbe reliable. If the relative permeability curves are deficient, theproduction data from other reservoirs of the same type can
still be used to generate the water flooding characteristic curveby regression. Another feasible method is to select one classicwater flooding characteristic curve which can represent thewater flooding performance in such type of reservoirs. Oncethe water flooding characteristic curve is confirmed, Eq. (22)can be used for fitting to determine the coefficients P19–P27.
(2) By substituting different R0 values into Eq. (21), we have thecorresponding Rm' values. Then substituting the correspondingRm' value into Eq. (15), the corresponding value SnwBT can beobtained.
(3) Substituting the parameters P19–P27, R0 and the correspondingvalue SnwBT into Eq. (26), a series of curves of fw vs. RT with R0
can be obtained, and the evaluation chart is plotted in Fig. 9.
3.2. The application of field case evaluation
There are two fractured basement reservoirs in an early stage ofwater-flooding, shen625 and Biantai, located in Damintun Basin,northeastern China. Three other mature water flood reservoirs ofthe same type are nearby, named Jingbei, Jinganbu and Dong-shengbu, as shown in Fig. 7, and the properties of these reservoirsare as Table 2. Qitai (2000) summarized four frequently-usedwater flooding characteristic curves, including the Sazpnov curve,Cipachev curve, Maksimov curve and Nazalov curve, as demon-strated in Fig. 8. The data of the three mature waterfloodreservoirs can be plotted in the same Figure. From the Fig. 8, weobserve that the Maksimov curve is the most approaching curve tothe curves of field data. So it can be selected as the representativecurve of this type of reservoirs. Then the parameters can bedetermined by using Eq. (22) for fitting, which are as follows:P11 3.702939, P12 �20.2722, P13 39.83089, P14 �20.7128, P15�29.1592, P16 24.3778, P17 34.81122, P18 �49.2859, P1916.70984.
In the second step, different values of the ultimate recoveryfactor RT are selected for calculation, which are 0.05, 0.1, 0.15, 0.2,
RTR0TrSnwBT ; f wðRT Þ ¼ 0
SnwBT rRTR0Tr1; f wðRT Þ ¼ 1
1þ μoμw
U 1
p19þp20U RT=R0Tð Þ�SnwBT
1�SnwBT
� �þp21U RT=R
0Tð Þ�SnwBT
1�SnwBT
� �2þp22U RT=R
0Tð Þ� SnwBT
1� SnwBT
� �þp23U RT=R
0Tð Þ�SnwBT
1�SnwBT
� �4�
þp24U RT=R0Tð Þ�SnwBT
1� SnwBT
� �5þp25U RT=R
0Tð Þ� SnwBT
1� SnwBT
� �6þp26U RT =R
0Tð Þ� SnwBT
1� SnwBT
� �7þp27U RT=R
0Tð Þ�SnwBT
1�SnwBT
� �8�
8>>>>>>>>><>>>>>>>>>:
ð26Þ
Fig. 7. The location of the five fractured reservoirs in Damintun Basin, north-eastern China.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140 135
0.25, 0.3, 0.35, 0.4, 0.45 and 0.5. The range ability of recovery rateW(t) of the three reservoirs is from 0.018 PV/year to 0.022 PV/year,where the average value 0.02 PV/year is selected. Then thecorresponding value SnwBT of the different ultimate recovery factorcan be determined. After the third step, the evaluation chart canbe obtained in Fig. 9.
The Shen625 oil reservoir is in a middle stage of recovery, butits water cut rises dramatically in recent time, and the fw vs. Rcurve approaches the R0 ¼0.2 curve. It indicates that field-development strategies need to be changed badly. Some measure-ments, including water-shutoff, reducing the choke size, newperforation, and even adjustment in well pattern, need to be takenfor water-cut control. Otherwise, if the current field-developmentstrategies are not changed, the ultimate recovery factor willmerely be around 20%, which is not a desired result. The situationof the Biantai oil reservoir is similar to the case of Shen625, whichcan achieve the ultimate recovery factor around 23%, as demon-strated in Fig. 9. As a contrast, we also use Petrel and Eclipse forreservoir geomodelling and simulation; the dash lines in Fig. 9 arethe numerical predicting outcomes: 19.7% for shen625, 23.12% forBiantai. The outcomes of numerical simulation do not appearmuch different from those of our method. However, the cost is
totally different, because reservoir simulation is a labor-intensiveand time-consuming work.
4. Discussions
4.1. Imbibition model
According to Eq. (14) and Fig. 9, it can be known that therecovery rate of a matrix-fracture reservoir depends on theultimate recovery factor of matrix. That is, the higher the matrixrecovery factor is, the stronger the imbibition is, and the moreslowly the water cut rises. In Chen's calculation, the parameter Rm'(ultimate recovery factor of matrix) is set to 0.1, but the variationrange of water saturation in matrix is from Swm¼0.2 to Swm¼0.88at different time, as shown in Fig. 5(b). As a result, the ultimaterecovery factor of matrix Rm' can reach 0.85, far beyond theprecondition Rm' ¼0.1, which means that model is not self-consistent. It can be seen that Chen and Liu (1982) miscalculatedthe imbibition rate, which leads to over-flow of oil from matrixand the numerical results mismatch the pre-condition. However,our computing result presents that the variation range of watersaturation in matrix is from 0.2 to 0.28, so Rm' cannot exceed0.1. Hence a correct calculation of the imbibition rate is verysignificant.
4.2. Effect of injection rate and fracture distribution
From the result of calculation using different recovery rates, itcan be found from Fig. 10 that the curves of different injectionrates show obvious distinction in total water saturation at break-through time in the normalized range SnwBT , as Fig. 10 shows.
For an analysis of the post water breakthrough stage, therelative permeability curves of two fractured core samples withthe same properties but different fracture distribution are tested,as Fig. 11 shows. Then the fw vs. Sw plot are calculated by therelative permeability curve on the basis of Eq. (22), as displayed inFig. 12. Sample (a) has joint fractures which connect the two endsof core samples, while sample (b) has two disconnected fractures.In the test, the differential pressure between the inflow end andthe outflow end is 0.1 MPa. Because of the different permeabilityby fracture connectivity, the flow rates of two samples aredifferent. The flow rate of sample (a) is 0.04 PV/min, and that ofsample (b) is 0.13 PV/min.
From Fig. 12, the water cut of sample (b) rises dramatically atfirst and then the curve approaches the Sazpnov curve, while thefw vs. Sw curve approaches the Maksimov curve and those of theother three fractured reservoirs located in Damintun Basin, asFig. 8 shows. From the above observation, we draw the conclusion
Fig. 9. Evaluation of two fractured reservoirs by our chart: (a) the solid curve isfield data and (b) the dash curve is numerical simulation result.
Fig. 10. Water saturation at breakthrough time with different recovery rates.
Fig. 8. The water flooding characteristic curves and field data.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140136
that the higher the recovery rate is, the faster the water cut risesince water break through. To check this conclusion, field datafrom another three fractured reservoirs are plotted, which areRenqiu reservoir (Qitai, 2000), Yanling reservoir and Casablancareservoir(Allan and Sun, 2003; Sun and Sloan, 2003), respectively.The annual oil recovery rate of Yanling is 8% per year, that ofRenqiu is 1.6%, and that of Casablanca is 1.2%. As shown in Fig. 12,the trend of field data also proves the conclusion.
From the above analysis, it can be known that the fracturedistribution plays a significant role in water performance as well.However, the fracture distribution is uncertain everywhere. Thenwe calculate the average curve calculated by 20 relative perme-ability curves of the three mature waterflood reservoirs. As Fig. 12
shows, the average curve is close to the Maksimov curve, whichcan represent the type of the reservoirs. That is why we recom-mend the number of relative permeability curves should bemore than twenty, because only one or two sample cores cannotrepresent the features of the whole reservoir, such as the differentpresentations of sample (a) and sample (b) in the relativepermeability test.
4.3. Comparison with Tong's method
Most of the classical methods, such as Tong's method (1988),are based on the Buckley–Leverett theory, together with theassumption of single porosity and no capillary pressure. The
Fig. 11. Image and properties of core samples.
Fig. 12. The water flooding characteristic curves, field data and lab test data.
Fig. 13. Comparison of evaluation method: (a) the solid line is our chart (b) and thedash line is Tong's chart.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140 137
empirical formula of Tong's method is as follows:
lgf w
1� f w¼ 7:5� ðR�R0Þþ1:69 ð27Þ
Our method is based on a dual-porosity model and takes theimbibition into consideration. The effect of imbibition is mainlyreflected in the following aspects: in most matrix-fracture reser-voirs without an aquifer, connate water in fractures cannot be acontinuous phase. If water can be a continuous phase, water flowsmore easily so the capillary pressure can suck the water intomatrix until no free flowing water is left in the fractures. There-fore, when reservoir recovery begins, oil in the fracture medium isproduced first, and water cut is zero during that time. Tong'schart (Fig. 13) shows that the connate water is an important factorto the ultimately recovery rate. From his plot of statistics from 24single porosity reservoirs, the initial water cut of these reservoirsis symptomatic of connate water saturation. The higher the initialwater cut is, the higher the connate water saturation is. Inaddition, the higher initial water cut also leads to a lower recoveryrate. So it is an apparent distinction of water performance betweenmatrix-fracture reservoirs and single porosity reservoirs. It isalso the reason why Tong's method is not suitable for water-wetfractured reservoirs.
4.4. Comparison with X-plot approach
Ershaghi and Omorigie (1978) and Ershaghi and Abdassah(1984) developed the X-plot waterflood-analysis method on thebasis of a semi-log linear relative permeability ratio for inter-mediate saturation values
krwðSweÞkroðSweÞ
¼ Ae�BSwe ð28Þ
It is also the main assumption that the plot of log (krw/kro) vs. Swis a straight line, as shown in Fig. 14. The real data of log (krw/kro)vs. Sw
n of matrix and fractures both have a straight line section.Actually the log (krw/kro) vs. Sw lines of matrix and fractures in asemi-log plot are not straight once water saturation in thenormalized range is close to 0, which are corresponding to theearly stage. That is why this method cannot work until the watercut reaches 50%. Compared with the matrix, the real data of log(krw/kro) vs. Sw
n of the fractures has a shorter straight line. If thestraight line assumption is used to predict the ultimate recoveryfactor of the fractures, the deviation of the fracture curve (krwf/krof)is larger, as marked in Fig. 14. That is, the more the fractures are inthe reservoir, the shorter the straight line is for the curve of log(krw/kro) vs. Sw
n of the total matrix-fracture system. It illustratesthat the methods on the basis of the assumption of Eq. (28) are nolonger suitable to a highly fractured reservoir, due to the mismatchof real data by the exponential form. Because of good fitting, thepolynomial form becomes the choice, as Fig. 14 shows.
5. Conclusions
(1) To develop the new method for evaluation of the water floodingperformance in fractured reservoirs, some unique features ofwater-wet matrix-fracture reservoirs must be taken into con-sideration, such as the imbibition process and dual-porosity.These features will lead to an obvious distinction in the fracturedreservoir water flooding performance. In addition, the recoveryrate also has some effects on water flooding performance.
(2) To study a matrix-fracture reservoir, a dual-porosity model is acommon method. But numerous operations and parametersmake the model hard to be used in simulation directly. Thepseudo relative permeability and saturation average can be asolution to this problem.
(3) In application of the modified imbibition model, the imbibi-tion flow rate can be related to the ultimate recovery factor. Itprovides a way to evaluate water flooding performance andestimate the potential of a reservoir by using the ultimaterecovery factor. A different ultimate recovery factor yields adifferent water cut curve in a matrix-fracture reservoir. Thusthe data of water cut with the recovery rate can be used forjudging how much the ultimate recovery factor can finally be.
(4) From the comparison with numerical simulation, our methodis a faster and easier tool which can provide reliable results.Compared with the classical methods, such as Tong's methodand the X-plot method, our method takes more uniquefeatures of water-wet fractured reservoirs. So it is moresuitable to the fractured reservoirs.
Acknowledgments
The authors are grateful for financial support from National Sci-ence and TechnologyMajor Project (Grant No. 2011ZX05009-004-001)
Table 2The main properties of the five fractured reservoirs in Damintun Basin
According to Eq. (B.1), the dimensionless cumulative oil pro-duction at time t is
QoðtÞ ¼ ð1�SwimÞϕmR0mð1�e�λtÞ ðB:2Þ
With respect to the effect of variation of saturation in fractures,the imbibition rate is
qimbðtÞ ¼dQoðtÞdt
¼ ð1�SwimÞϕmR0mλe
�λt ðB:3Þ
where the matrix blocks can be divided into many cells, such ascell A and cell B, as displayed in Fig. 3.
Once the injected water enters into the fracture-matrix med-ium, the imbibition process begins. During the water floodingprocess, each cell has its own imbibition flow once the water
contacts it. Hence the imbibition flow rate in fractures at time t1 is
After a period of time Δt, the newly injected water touches cellA. Meanwhile, the water that contacted cell A now flows to thefracture area near beside cell B. Cell B obeys the same rule as cell A,but water becomes less and less because some water has beenimbibed into cell A. Hence the imbibition flow rate in the fracturesat time t2 is
Then the water injection continues until the water breaksthrough, and the time at water breakthrough is tB. We assumethat the time from 0 to tB is divided into n sections. The flow rateat the water breakthrough time tB is
ðB:7Þwhere t0¼0, Δti¼tiþ1�ti, and 0rθr1. If n-1, Δti-0 and ti isreplaced by the characteristic time τ, then our modification modelis
qimbðx; y; z; tBÞ
¼ ð1�SwimÞϕmR0mλ Swf ðx; y; z; tBÞ�λ
Z tB
0Swf ðx; y; z; τÞe�λðtB � τÞdτ
� �ðB:8Þ
Appendix C. Supplementary information
Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.petrol.2014.06.002.
References
Allan, J., Sun, S.Q., 2003. Controls on recovery factor in fractured reservoirs: lessonslearned from 100 fractured fields. In: Proceedings of the SPE Annual TechnicalConference and Exhibition. Society of Petroleum Engineers. Denver, Colorado.
Aronofsky, J.S., Masse, L., Natanson, S.G., 1958. A Model for the Mechanism of OilRecovery from the Porous Matrix due to Water Invasion in Fractured Reservoirs.
Babadagli, T., Ershaghi, I., 1993. Improved modeling of oil/water flow in naturallyfractured reservoirs using effective fracture relative permeabilities. In: Proceed-ings of the SPE Western Regional Meeting. Anchorage, Alaska.
Barenblatt, G.I., Zheltov, I.P., Kochina, I.N., 1960. Basic concepts in the theory ofseepage of homogeneous liquids in fissured rocks [strata]. J. Appl. Math. Mech.24 (5), 1286–1303.
Buckley, S.E., Leverett, M.C., 1942. Mechanism of fluid displacement in sands. Trans.AIME 146 (1), 107–116.
Chen, Y., 1985. Derivation of relationships of water drive curves. Acta Pet. Sin. 6 (2),69–78.
Chen, Z., Liu, C., 1982. Theory of fluid displacement in a medium with doubleporosity. Energy Sources 6 (3), 193–214.
Civan, F., 1998. A theoretically derived transfer function for oil recovery fromfractured reservoirs by waterflooding. SPE Reserv. Eval. Eng. 1 (2), 141–147.
Dang, C.T.Q., Chen, Z., Nguyen, N.T.B., Bae, W., Phung, T.H., 2011. Lessons learnedand experiences gained in developing the waterflooding concept of a fracturedbasement-granite reservoir: a 20-year case study. J. Can. Pet. Technol. 50 (9),10–23.
De Swaan, A., 1978. Theory of waterflooding in fractured reservoirs. Soc. Pet. Eng. J.18 (2), 117–122.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140 139
Efros, D.A., 1958. Movement of water–oil mixture in the well system II problems ofunderground fluid dynamics and development in oil field. All-Russ. Sci. Res.Inst. Exp. Phys. 7 (c), 3–32.
El-Khatib, N.A.F., 2001. The Application of Buckley–Leverett Displacement toWaterflooding in Non-communicating Stratified Reservoirs, SPE Middle EastOil Show. Copyright 2001. Society of Petroleum Engineers Inc, Bahrain.
El-khatib, N.A.F., 2012. Waterflooding performance in inclined communicatingstratified reservoirs. Soc. Pet. Eng. J. 17 (1), 31–42.
Ershaghi, I., Abdassah, D., 1984. A prediction technique for immiscible processesusing field performance data (includes associated papers 13392, 13793, 15146and 19506 ). J. Pet. Technol. 36 (4), 664–670.
Ershaghi, I., Omorigie, O., 1978. A method for extrapolation of cut vs. recoverycurves. J. Pet. Technol. 30 (2), 203–204.
Hearn, C.L., 1971. Simulation of stratified waterflooding by pseudo relative perme-ability curves. J. Pet. Technol. 23 (7), 805–813.
Hoteit, H., Firoozabadi, A., 2008a. An efficient numerical model for incompressibletwo-phase flow in fractured media. Adv. Water Resour. 31 (6), 891–905.
Hoteit, H., Firoozabadi, A., 2008b. Numerical modeling of two-phase flow inheterogeneous permeable media with different capillarity pressures. Adv.Water Resour. 31 (1), 56–73.
Huang, Z., Yao, J., Wang, Y., Tao, K., 2011. Numerical study on two-phase flowthrough fractured porous media. Sci. China Technol. Sci. 54 (9), 2412–2420.
Qitai, Y., 2000. A generalized water displacement curve of forecasting oilfielddevelopment indexes. In: Proceedings of the International Oil and Gas Con-ference and Exhibition in China. Society of Petroleum Engineers Inc., Beijing,China.
Sun, S.Q., Sloan, R., 2003. Quantification of uncertainty in recovery efficiencypredictions: lessons learned from 250 mature carbonate fields. In: Proceedingsof the SPE Annual Technical Conference and Exhibition. Society of PetroleumEngineers, Denver, Colorado.
Terez, I.E., Firoozabadi, A., 1999. Water injection in water-wet fractured porousmedia: experiments and a new model with modified Buckley–Leverett theory.Soc. Pet. Eng. J. 4 (2), 134–141.
Timmerman, E.H., 1971. Predict performance of water floods graphically. Pet. Eng.43 (12), 77–80.
Tong, X., 1978. Discussion on statistical data from reservoir by natural waterflooding and water injection. Pet. Explor. Dev. 4 (6), 38–67.
Tong, X., 1988. Analyses of Oil Well Production Performance and ReservoirBehavior. Publishing House of Documentation for Science and Technology,Beijing, China.
Van Golf-Racht, T.D., 1982. Fundamentals of Fractured Reservoir Engineering.Elsevier Science, Amsterdam, Netherlands.
Warren, J.E., Root, P.J., 1963. The behavior of naturally fractured reservoirs. Soc. Pet.Eng. J. 3 (3), 245–255.
Welge, H.J., 1952. A simplified method for computing oil recovery by gas or waterdrive. J. Pet. Technol. 4 (4), 91–98.
Yang, Z., 2009. A new diagnostic analysis method for waterflood performance. SPEReserv. Eval. Eng. 12 (2), 341–351.
Yuetian, L., Zupeng, D., Kun, A., Yong, Z., Jun, W., 2013. Manufacturing method oflarge-scale fractured porous media for experimental reservoir simulation. Soc.Pet. Eng. J. 18 (06), 1081–1091.
S. Gu et al. / Journal of Petroleum Science and Engineering 120 (2014) 130–140140
