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New approach for improved history matching while incorporating wettability variations in a sandstone reservoir—Field implementation
Journal of Petroleum Science and Engineering 104 (2013) 27–37
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New approach for improved history matching while incorporating wettability variations in a sandstone reservoir—Field implementation Muhammad Khurram Zahoor a,n, Mohd. Nawi Derahman b a b
Department of Petroleum and Gas Engineering, University of Engineering and Technology, G.T. Road, Lahore, Pakistan Petroleum Engineering Department, Universiti Teknologi Malaysia, Johor Bahru, Malaysia
art ic l e i nf o
a b s t r a c t
Article history: Received 2 May 2012 Accepted 19 March 2013 Available online 3 April 2013
Wettability plays an influential role in fluid distribution and flow behavior within a reservoir, due to its strong influence on capillary pressure and relative permeability. Because of its significance, it plays an important part in history matching and simulation studies. But, despite its importance and crucial role, it is usually not accorded significant attention during simulation studies. Different types of wettability may exist or co-exist within a reservoir which may alter while drilling due to invasion of mud filtrate in near wellbore area or with the passage of time due to various undergoing processes under prevailing conditions. In order to account for the effect of such wettability variations on fluid flow behavior, a set of correlations developed by us has been used in this study, to predict capillary pressure curve under any prevailing wettability conditions, when the laboratory data from the core at any known wettability condition are available. Sequentially, the methodology has been formulated, to incorporate wettability variations effects in experimental relative permeability data. The developed methodologies for including wettability alteration effect into flow affecting parameters have been implemented on Malaysian oil reservoir for improved history matching. The obtained results show better history match with the field data, thus verifying the significance of the designed approach. & 2013 Elsevier B.V. All rights reserved.
Keywords: Wettability Capillary pressure Relative permeability Simulation History matching
1. Introduction Wettability can be defined as the ability of a liquid to coat the rock surface (Shedid and Ghannam, 2005). It ranges from extremely water-wet to extremely oil-wet condition (Anderson, 1986b; Lingen et al., 1996; Tripathi and Kishore, 2007). Wettability controls the location and distribution of fluids within a reservoir (Chalbaud et al., 2009; Norman, 1990). If the rock is water-wet, the water will occupy the small pores, while the oil will occupy the larger pores and vice versa in case of oil-wet situation (Anderson, 1986b; Norman, 1990). Wettability alteration can be due to a number of reasons (Anderson, 1986b; Buckley et al., 1998; Donnez., 2007) out of which mud filtrate invasion is quite common (Ballard and Dawe, 1988; Dandekar, 2006; Yan et al., 1993). Due to such alterations, core samples cannot be considered as true representatives of the reservoir. Surfactants present in the drilling mud causes wettability alteration and can also reduce oil relative permeability in the presence of connate water (McDonald and Buller, 1992; Skalli et al., 2006; Yan et al., 1993). Core handling and conditions at which experiments have been conducted can also add to the deviation of core properties from that of the virgin reservoir (Anderson, 1986b; Zinszner and Pellerin, 2007).
Wettability affects the fluid distribution within a core and as well as in a reservoir, thus influencing the capillary pressure and relative permeability, which are the controlling parameters for fluid flow and displacement of one fluid by another (Anderson, 1987a, 1987b; Chalbaud et al., 2009; Karabakal and Bagci, 2004). As a result of it, wettability has a strong influence on history matching process during simulation studies. History matching the production data has always been an essential and challenging task requiring tremendous efforts (Muhammad and Jonathan Neil, 2010; Tillier et al., 2010; Todd Hoffman and Caers, 2005). The above mentioned parameters play a crucial role toward history matching the dynamic model and thus leading to reliable forecasts (Andres, 2010; Isha et al., 2010; Maschio et al., 2008). Engineer assigns set of parameters and also makes sensible ranges for these parameters to obtain proper or improved history match of the reservoir model (Muhammad and Jonathan Neil, 2010). To improve the history match for a Malaysian field, a developed set of correlations (Zahoor, 2011) has been used to incorporate wettability variations effect on capillary pressure curves and further methodology has been formulated for generating relative permeability data under the prevailing wettability conditions.
2. Literature review n
Corresponding author. Tel.: +92 334 4137845. E-mail address: [email protected] (M.K. Zahoor).
0920-4105/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2013.03.025
Different saturation stages of the displaced phase can be defined based on the discussion made by Anderson (1987a) and the
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
Se effective wetting phase saturation Sem effective mobile phase saturation Smphase1 saturation of mobile phase 1 Smphase1.new saturation of phase 1 under new wettability condition Snwr non-wetting phase residual saturation Srmphase1 residual saturation of phase 1 Srmphase2 residual saturation of phase 2 Sw wetting phase saturation Swr wetting phase residual saturation η function of the fraction of hydrophobic surfaces present in the system λ characteristic constant θ contact angle Δ difference
Nomenclature a c Kr Kro Krw Pc P fcw Pc.est. Pd Pdb.c PNAPL Pw SdM
pore size distribution index entry pressure relative permeability oil relative permeability water relative permeability capillary pressure capillary pressure in fractional-wet system estimated capillary pressure displacement pressure displacement pressure of base case pressure of non aqueous phase liquid (NAPL) pressure of the aqueous phase maximum saturation of the displaced phase
experiments conducted by Killins et al. (1953). Killins et al. showed the behavior of capillary pressure curve under water-wet to oil-wet conditions. Their experiments also showed that the greater the affinity (stronger adhesion) of a fluid with rock surface, the greater the displacement pressure required to initiate the displacement process, which can be explained in a simplified form, as shown in Fig. 1. Before initiating the displacement process, the phase to be displaced is continuous and when the process starts it becomes discontinuous (Anderson, 1987a). This discontinuity increases with the passage of time, demanding increase in pressure to continue the displacement process. The process stops when saturation versus capillary pressure plot becomes vertical (Anderson, 1987a). The obtained laboratory curves can be regenerated by using the Brooks and Corey (1964) model which is one of the most well known and commonly used model, because of its solid experimental validation. Mathematically P c ðSe Þ ¼
P d Se−1=λ
ð1Þ
qualitative terms rather than considering its quantitative significance. Skjaeveland et al. (2000) proposed a correlation to calculate capillary pressure for wettability states between water-wet and oil-wet conditions. They proposed that, capillary pressure can be calculated by using a sum of capillary pressure values (for oil- and water-wet system). Mathematically
Sw −Swr 1−Swr
or
Sw −Swr Se ¼ Sw ¼ 1−Swr −Snwr n
ð4Þ
14 Pc. Lab.
12 10
where effective or normalized wetting phase saturation can be given as Se ¼ Snw ¼
cw co þ ððSw −Swr Þ=ð1−Swr ÞÞaw ððSo −Sor Þ=ð1−Sor ÞÞao
P c ðSe Þ ¼
Pc(Psia)
28
8 6 4 2
ð2Þ
0 0
ð3Þ
Furthermore, correlations to generate capillary pressure curves at different wettability conditions by using the laboratory data at any other known wettability condition are present in the literature. But in these developed correlations, wettability is used in
0.1
0.2
0.3
0.4 Sw
0.5
0.6
0.7
0.8
Fig. 2. Experimental capillary pressure data.
1 Kro. Lab.
0.9
Krw. Lab.
0.8 0.7
Kr
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6 Sw
Fig. 1. Effect of increasing oil wettability on displacement pressure.
Fig. 3. Relative permeability data at 841.
0.8
1
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The limitation of above Eq. (4) is that, it always requires experimental data for calculation purposes and treats wettability as a qualitative parameter. Similarly, a set of correlations has been developed for different wettability systems by incorporating wettability in a qualitative manner and it also requires some sort
29
of experimentation studies to use such correlations (Tsakiroglou and Fleury, 1999a, 1999b). Another example of such correlations is the one developed for fractional-wet systems (Bradford and Leij, 1996). The proposed correlation can be expressed as in Eq. (5). The proposed expression
Well Oil Production Rate (Stb/day)
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0
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Well Oil Production Rate (Stb/day)
3000
2500
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1000
500
0
Well Oil Production Rate (Stb/day)
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2000
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1000
500
0
Fig. 4. History match of wells oil production rate based on laboratory data: (a) Well A1, (b)Well A4, and (c) Well A8.
30
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can be used for calculating capillary pressure under fractional wettability condition from available capillary pressure values at water-wet situation. P fcw ðSw Þ ¼ P NAPL −P w ¼ P ww c ðSw Þ−η
ð5Þ
Though the above mentioned correlations can be of great significance, but for simulation purposes correlation(s) are required in which wettability variations can be incorporated in a quantitative manner and at the same time giving liberty to generate data sets at any wettability condition without requiring repeated
0.5 Well A1 (Based on Lab. data)
0.45
Historical Data Well A1
0.4
Well water-cut
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400 Time (Days)
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800
0.5 Well A4 (Based on Lab. data)
0.45
Historical Data Well A4
0.4
Well water-cut
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
100
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400 Time (Days)
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0.5 Well A8 (Based on Lab. data)
0.45
Historical Data Well A8
0.4
Well water-cut
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
100
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400 Time (Days)
500
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800
Fig. 5. History match of water-cut profiles of different wells based on laboratory data: (a) Well A1, (b)Well A4, and (c) Well A8.
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experimentation. This will give an option that even a slight variation in this parameter can be incorporated in the capillary pressure data and later into relative permeability curves for improved history matching while conducting simulation studies.
31
3. Correlation for capillary pressure estimation by incorporating wettability in a quantitative manner At the first instance the term effective or normalized mobile phase saturation (Sem) correlation has been used (which will be
1750 Well A1 (Based on Lab. data)
Well bottom hole Pressure (Psia)
1730
Historical WBHP Well A1
1710 1690 1670 1650 1630 1610 1590 1570 1550 0
100
200
300
400 Time (Days)
500
600
700
800
1740 Well A4 (Based on Lab. data) Well bottom hole Pressure (Psia)
1730
Historical WBHP Well A4
1720 1710 1700 1690 1680 1670 1660 1650 0
100
200
300
400 Time (Days)
500
600
700
800
1730 Well A8 Based on Lab. data) Well bottom hole Pressure (Psia)
1710
Historical WBHP Well A8
1690 1670 1650 1630 1610 1590 1570 1550 0
100
200
300
400 Time (Days)
500
600
700
Fig. 6. History match of well pressure profiles: (a) Well A1, (b) Well A4, and (c) Well A8.
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P c:est: ¼ P db:c ðSem Þ−1=λ 7 k1 ðΔθÞ P db:c ðSem Þ−1=λ
ð6Þ
where k1 ¼ 1:24 10−4 and sign convention, in general, will be positive if oil wettability increases and vice versa. “Δθ” is the absolute difference of wettability in terms of contact angle. Base case wettability is calculated experimentally in terms of wettability index or number using displacement tests (Anderson, 1986a; Norman, 1990), from which it is transformed in terms of contact angle by using cosine inverse. Effective mobile phase saturation correlation (Sem) in Eq. (4) is defined based on the saturation stage. Capillary pressure curve can be divided into three regions/sections based on saturation change during displacement process, namely flow at higher displaced saturation stage, transient stage and displacement at lower saturation of the displaced phase (Zahoor, 2011; Zahoor et al., 2011). “Sem” in the developed correlation (Eq. (6)) to predict capillary pressure curve behavior during higher saturation of displaced fluid stage (H.S.S) can be incorporated as follows: −1=λ Smphase1 −Srmphase1 P c:est: ¼ P db:c 7 k1 ðΔθÞ P db:c 1−Srmphase2 −Srmphase1 −1=λ Smphase1 −Srmphase1 ð7Þ 1−Srmphase2 −Srmphase1 The boundary conditions of the above mentioned flow stage or the range of applicability of the above correlation can be given in the following manner in the most refined form, as a function of displaced phase saturation: Smphase1 ≥1−Srmphase2 −0:5625:SdM
P c:est:ðH:S:SÞ þ P c:est:ðL:S:SÞ 2
ð12Þ
4. Formulated methodology for incorporating wettability variation effect into relative permeability data After predicting the curve for capillary pressure at desired wettabililty condition, wettability variation effect can also be incorporated into laboratory relative permeability data. To explain the procedure as we know, the capillary pressure experimental data obtained at known wettability ðθ ¼ θo1 Þ can be regenerated by reducing the above Eq. (7), as follows (Δθ¼01), which is similar to the Brooks and Corey Correlation: −1=λ Smphase1 −Srmphase1 P c:est:θ1 ¼ P db:c θ1 ð13Þ 1−Srmphase2 −Srmphase1 and the capillary pressure data at any wettability condition (θ ¼ θo2 ), let us say, can be generated by using Eq. (7) as follows: Smphase1:new −Srmphase1 −1=λ P c:est:θ2 ¼ P db:c:θ þ k1 ðΔθÞ P db:c:θ 1 1 1−Srmphase2 −Srmphase1 −1=λ Smphase1:new −Srmphase1 ð14Þ 1−Srmphase2 −Srmphase1
16 Pc. Est. Pc. Lab.
14 12 10 8 6 4 2 0 0
0.1
0.2
0.3
0.4 Sw
0.5
0.6
0.7
0.8
Fig. 7. Comparative plot of capillary pressure at different wettability conditions.
1 Kro. Lab. Krw. Lab. Kro. Est. Krw. Est.
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
If
During transient flow, which exists when 1−Srmphase2 −0:6875:SdM o Smphase1 o 1−Srmphase2 −0:5625:SdM
P c:est:ðTransient:stageÞ ¼
ð8Þ
Selection of phases 1 and 2 in Eq. (7) depends on the phase for which capillary pressure needs to be generated. Capillary pressure curve during flow at lower displaced fluid saturation stage (L.S.S) can be estimated by using the following correlation: Smphase1 −Srmphase1 −1=λ P c:est: ¼ P db:c 7 k1 ðΔθÞ P db:c 1−Srmphase1 Smphase1 −Srmphase1 −1=λ ð9Þ 1−Srmphase1
Smphase1 ≤1−Srmphase2 −0:6875:SdM
capillary pressure can be estimated by averaging the capillary pressure obtained by using Eqs. (7) and (9), i.e.
Pc (Psia)
presented later), instead of effective wetting phase saturation in Eqs. (1)–(3) for standardization. It will give a broader sense and make it possible to implement the developed correlation for cases in which the wetting phase is immobile and the desaturation curve is generated by, for example, displacing oil by gas in the presence of immobile water as wetting phase (Zahoor, 2011). To generate capillary pressure data at any wettability condition, a set of correlations has been developed. This set of correlations has been developed by using a number of experimental capillary pressure data sets available at different wettability conditions. The correlations were developed by using curve matching technique (comparing the calculated values with the experimental data) and are capable of generating the data for the entire range of wettability conditions (0–1801), when the laboratory data (base case) at any wettability condition is known. The developed generalized form of correlation has been obtained by including set of parameters into the Brooks and Corey model to account for wettability variations in a quantitative manner while predicting the capillary pressure curve. In generalized form the developed correlation can be given as (Zahoor, 2011)
Kr
32
ð10Þ
0 0
0.2
0.4
0.6
0.8
1
Sw ð11Þ
Fig. 8. Comparative plot of relative permeability at different wettability conditions.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
During the displacement process in porous medium, having different wettabilities, different pressures will be required to produce the same volume of displaced phase. In other words, at the same pressure, different volumes of displaced fluid will be produced, or the displaced fluid saturation change will be different. To calculate the
33
saturation of the displaced phase “Smphase1.new” at wettability condition “θ2”, Eqs. (13) and (14) can be equated as follows. At same capillary pressure P c:est:θ1 ¼ P c:est:θ2
ð15Þ
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Well A1 (Based on Lab. data)
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Well A1 (Based on Est. data) Historical Data Well A1
0 0
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Well Oil Production Rate (Stb/day)
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1000 Well A4 (Based on Lab. data) Well A4 (Based on Est. data)
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Historical Data Well A4 0 0
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Well Oil Production Rate (Stb/day)
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Well A8 (Based on Lab. data)
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Well A8 (Based on Est. data) Historical Data Well A8 0 0
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Fig. 9. Comparative history match plots for well oil production rate based on laboratory and estimated data: (a) Well A1, (b) Well A4, and (c) Well A8.
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M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
−1=λ
¼ P db:c:θ
1
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
þk1 ðΔθÞ P db:c:θ
1
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ
Solving the above equation for “Smphase1.new”
−1=λ ð16Þ
P db:c:θ
1
Smphase1 −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ
¼ P db:c:θ
1
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ
1 þ k1 ðΔθÞ
ð17Þ
0.5 0.45
Well A1 (Based on Lab. data) Well A1 (Based on Est. data) Historical Data Well A1
0.4
Well water-cut
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400 Time (Days)
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0.5 0.45
Well A4 (Based on Lab. data)
0.4
Well A4 (Based on Est. data) Historical Data Well A4
0.35 Well water-cut
1
Smphase1 −Srmphase1 1−Srmphase2 −Srmphase1
0.3 0.25 0.2 0.15 0.1 0.05 0 0
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400 Time (Days)
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0.5 0.45
Well A8 (Based on Lab. data)
0.4
Well A8 (Based on Est. data) Historical Data Well A8
0.35 Well water-cut
P db:cθ
0.3 0.25 0.2 0.15 0.1 0.05 0 0
100
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300
400 Time (Days)
500
600
700
800
Fig. 10. Comparative history match plots for well water cut based on laboratory and estimated data: (a) Well A1, (b) Well A4, and (c) Well A8.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ ¼
ððSmphase1 −Srmphase1 Þ=ð1−Srmphase2 −Srmphase1 ÞÞ−1=λ ½1 þ k1 ðΔθÞ
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
¼ ½1 þ k1 ðΔθÞλ
Smphase1 −Srmphase1 1−Srmphase2 −Srmphase1
ð18Þ
Well bottom hole Pressure (Psia)..
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Time (Days) 1740
Well bottom hole Pressure (Psia)..
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100
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400 Time (Days)
1730 1710 1690 1670 1650 1630 1610 1590 1570 1550 0
100
200
300
ð19Þ
1750
Well bottom hole Pressure (Psia)
35
400 Time (Days)
Fig. 11. Comparative well pressure profiles based on laboratory and estimated data: (a) Well A1, (b) Well A4, and (c) Well A8.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
Similarly Smphase1:new ¼ ½1−k1 ðΔθÞλ ðSmphase1 −Srmphase1 Þ þ Srmphase1 i ¼ 1; n
ð21Þ
“n” number of “Smphase1.new” values can be calculated by using the above Eqs. (20) and (21), which are later substituted into any relative permeability model, to generate relative permeability curves at wettability “θ2”.
5. Field implementation 5.1. Field and fluid description The field is located offshore Peninsular Malaysia in the South China Sea. This oil field has dipped structure and an initial gas cap having an oil zone thickness of approximately 100 ft with an underlying water zone. It is a sandstone reservoir, having average porosity of approximately 31% and the average permeability is 757 md. Average initial reservoir pressure is 1715 psi and the initial water saturation within the reservoir is 27%. Field is produced at a constant gas–oil ratio of 0.495 scf/stb. Reservoir oil has a bubble point pressure of 1697 psi, its formation volume factor is 1.23 rb/stb and viscosity is 0.29 cp. Rock wettability is 841 at reservoir conditions and the relative permeability curves can be generated by using the following equations: krw ¼ ðSem Þ5:22
ð22Þ
kro ¼ ð1−Sem Þ2 ½1−ðSem Þ3:2
ð23Þ
The capillary pressure and relative permeability curves at this wettability condition of 841 are shown in Figs. 2 and 3, respectively. 5.2. Results and discussion Simulation study was conducted using a commercial black-oil, three phase reservoir simulator. For history matching purposes, wells are controlled based on total volume of fluid produced per day as obtained from wells and field production data. The obtained results for oil production rate, water-cut and pressure profiles for different wells are shown in Figs. 4–6, while comparing with the observed/historical data. Figs. 4–6 show proper history match particularly for well oil production and bottom hole flowing pressure profiles during the initial period of production. Later the profiles start deviating from the observed data while the well water-cut profile is not properly matched. To have a proper history match, capillary pressure curve was generated using the developed set of correlations at a wettability condition of 801 (value chosen based on past experience in the area). The estimated capillary pressure curve is shown in Fig. 7. After generating Pc data while incorporating the wettability variation, relative permeability curves under the prevailing conditions of wettability have been obtained by using the following equations: Smphase1:new −Srmphase1 5:22 krw ¼ ð24Þ 1−Srmphase2 −Srmphase1 " # Smphase1:new −Srmphase1 2 Smphase1:new −Srmphase1 3:2 1− kro ¼ 1− 1−Srmphase2 −Srmphase1 1−Srmphase2 −Srmphase1 ð25Þ
where the values for Smphase1.new have been calculated by using the formulated methodology as explained earlier and the resulting relative permeability curves are shown in Fig. 8, along with laboratory curves. History matches based on these estimated plots obtained as a result of simulation studies are shown in Figs. 9–11 while the comparative plots for the overall field history match based on both wettability conditions are shown in Figs. 12 and 13. Figs. 9–13 show improved history match based on the estimated data, beginning from a point from where the plots generated based on laboratory data started deviating. This is due to the restoration of wellbore conditions, which were altered due to the exposure of porous media to the mud filtrate. This restoration of wellbore means that the wettability condition of the near wellbore region is changed approximately to the original wellbore wettability condition, as it was before mud filtrate invasion. It can also be observed from the results that this wettability reversal is rate dependent. The profiles of the wells producing at higher rates have matched earlier with the simulation results obtained based on estimated capillary pressure and relative permeability data.
6. Conclusion Implementation of developed set of correlations to generate capillary pressure curve at any wettability condition and
1
30000 25000
0.8
20000 0.6
FOPR (Based on Lab. data) FOPR (Based on Est. data) Historical FOPR FWCT (Based on Lab. data) FWCT (Based on Est. data) Historical FWCT
15000 10000
0.4
Field water-cut
ð20Þ
Field Oil Production Rate (Stb/day)
Smphase1:new ¼ ½1 þ k1 ðΔθÞλ ðSmphase1 −Srmphase1 Þ þ Srmphase1 i ¼ 1; n
0.2
5000 0 0
100
200
300 400 Time (Days)
500
600
0 700
Fig. 12. Field history match plot of oil production rate and water cut.
1720
Historical FPR FPR (Based on Lab. data) FPR (Based on Est. data)
1710 Field Pressure Rate (Psia)
36
1700 1690 1680 1670 1660 1650 1640 0
200
400 Time (Days)
600
Fig. 13. Field history match plot of field pressure rate.
800
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correspondingly formulated methodology for generating relative permeability curves under similar prevailing conditions has resulted in an improved history match. So, proper history match can be obtained by incorporating the effect of wettability variations on capillary pressure and relative permeability curves, in a better way, acting as a base for improved and more reliable production forecasts. The obtained results also show that the wettability restoration in the zone affected by mud filtrate is rate dependent in this case, as the higher well flow rates lead to early wettability condition restoration. References Anderson, W., 1986a. Wettability literature survey—Part 2: Wettability measurement. SPE J. Pet. Technol. 38 (11), 1246–1262. Anderson, W.G., 1986b. Wettability literature survey—Part 1: Rock/oil/brine interactions and the effects of core handling on wettability. SPE J. Pet. Technol. 38 (10), 1125–1144. Anderson, W.G., 1987a. Wettability literature survey—Part 4: Effects of wettability on capillary pressure. SPE J. Pet. Technol. 39 (10), 1283–1300. Anderson, W.G., 1987b. Wettability literature survey—Part 5: The effects of wettability on relative permeability. SPE J. Pet. Technol. 39 (11), 1453–1468. Andres, R., 2010. Application of J-Functions to Prepare a Consistent Tight Gas Reservoir Simulation Model: Bossier Field, Tight Gas Completions Conference. Society of Petroleum Engineers, San Antonio, TX, USA. Ballard, T.J. and Dawe, R.A., 1988. Wettability alteration induced by oil-based drilling fluid. In: SPE Formation Damage Control Symposium. 1988 Copyright 1988, Society of Petroleum Engineers, Inc., Bakersfield, CA. Bradford, S.A., Leij, F.J., 1996. Predicting two- and three-fluid capillary pressuresaturation relationships of porous media with fractional wettability. Water Resourc. Res. 32 (2), 251–259. Brooks, R.H. and Corey, A.T., 1964. Hydraulic properties of porous media. Hydrology Papers, Paper no. 3. Buckley, J.S., Liu, Y., Monsterleet, S., 1998. Mechanisms of wetting alteration by crude oils. SPE J. 3 (1), 54–61. Chalbaud, C., et al., 2009. Interfacial tension measurements and wettability evaluation for geological CO2 storage. Adv. Water Resourc. 32 (1), 98–109. Dandekar, A.Y., 2006. Petroleum Reservoir Rock and Fluid Properties. CRC Press, Taylor & Francis Group, Florida. Donnez, P., 2007. Essentials of Reservoir Engineering. Editions Technip, Paris. Isha, S., David, S., Jessica, C.B. and Mark Alan, L., 2010.. History Match Case Study: use of Assisted History Match tools on single-well models in conjunction with a full-field history match. In: SPE Russian Oil and Gas Conference and Exhibition. Society of Petroleum Engineers, Moscow, Russia.
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Karabakal, U., Bagci, S., 2004. Determination of wettability and its effect on waterflood performance in limestone medium. Energy Fuels 18 (2), 438–449. Killins, C.R., Nielsen, R.F., Calhoun, J.C., 1953. Capillary desaturation and imbibition in porous rocks. Prod. Mon. 18 (2), 30–39. Lingen, V.P.P., Bruining, J., van Kruijsdijk, C.P.J.W., 1996. Capillary entrapment caused by small-scale wettability heterogeneities. SPE Reservoir Eng. 11 (2), 93–100. Maschio, C., Vidal, A.C., Schiozer, D.J., 2008. A framework to integrate history matching and geostatistical modeling using genetic algorithm and direct search methods. J. Pet. Sci. Eng. 63 (1–4), 34–42. McDonald, J.A., Buller, D.C., 1992. The significance of formation damage caused by the adsorption oil-based mud surfactant. J. Pet. Sci. Eng. 6 (4), 357–365. Muhammad, K. and Jonathan Neil, C., 2010.. Case studies of successfully history matched reservoir simulation models using a powerful optimization algorithm being of limited predictive value. In: Abu Dhabi International Petroleum Exhibition and Conference. Society of Petroleum Engineers, Abu Dhabi, UAE. Norman, R.M., 1990. Wettability and its effect on oil recovery. SPE J. Pet. Technol. 42 (12), 1476–1484. Shedid, S., Ghannam, T.M., 2005. Influences of droplet volume on contact angle of reservoir rocks. Energy Sources 27 (11), 1085–1097. Skalli, L., Buckley, J.S., Zhang, Y., Morrow, N.R., 2006. Surface and core wetting effects of surfactants in oil-based drilling fluids. J. Pet. Sci. Eng. 52 (1-4), 253–260. Skjaeveland, S.M., Siqveland, L.M., Kjosavik, A., Thomas, W.L.H., Virnovsky, G.A., 2000. Capillary pressure correlation for mixed-wet reservoirs. SPE Reservoir Eval. Eng. 3 (1), 60–67. Tillier, E., Enchéry, G., Le Ravalec, M., Gervais, V., 2010. Local and continuous updating of facies proportions for history-matching. J. Pet. Sci. Eng. 73 (3–4), 194–203. Todd Hoffman, B., Caers, J., 2005. Regional probability perturbations for history matching. J. Pet. Sci. Eng. 46 (1–2), 53–71. Tripathi, I. and Kishore, K.M., 2007. Flow instability associated with wettability alteration. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Anaheim, CA, USA. Tsakiroglou, C.D., Fleury, M., 1999a. Pore network analysis of resistivity index for water-wet porous media. Transp. Porous Media 35 (1), 89–128. Tsakiroglou, C.D., Fleury, M., 1999b. Resistivity index of fractional wettability porous media. J. Pet. Sci. Eng. 22 (4), 253–274. Yan, J.N., Monezes, J.L., Sharma, M.M., 1993. Wettability alteration caused by oil-based muds and mud components. SPE Drilling Completion 8 (1), 35–44. Zahoor, M.K., 2011. Predicting Effects of Wettability Variations on Displacement Behavior and their Incorporation into Inflow Performance. Ph.D Thesis, Universiti Teknologi Malaysia. Zahoor, M.K., Derahman, M.N., Yunan, M.H., 2011. WAG process design—an updated review. Brazilian J. Pet. Gas 5 (2), 109–121. Zinszner, B., Pellerin, F.M., 2007. A Geoscientists Guide to Petrophysics. Editions Technip, Paris.
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
New approach for improved history matching while incorporating wettability variations in a sandstone reservoir—Field implementation Muhammad Khurram Zahoor a,n, Mohd. Nawi Derahman b a b
Department of Petroleum and Gas Engineering, University of Engineering and Technology, G.T. Road, Lahore, Pakistan Petroleum Engineering Department, Universiti Teknologi Malaysia, Johor Bahru, Malaysia
art ic l e i nf o
a b s t r a c t
Article history: Received 2 May 2012 Accepted 19 March 2013 Available online 3 April 2013
Wettability plays an influential role in fluid distribution and flow behavior within a reservoir, due to its strong influence on capillary pressure and relative permeability. Because of its significance, it plays an important part in history matching and simulation studies. But, despite its importance and crucial role, it is usually not accorded significant attention during simulation studies. Different types of wettability may exist or co-exist within a reservoir which may alter while drilling due to invasion of mud filtrate in near wellbore area or with the passage of time due to various undergoing processes under prevailing conditions. In order to account for the effect of such wettability variations on fluid flow behavior, a set of correlations developed by us has been used in this study, to predict capillary pressure curve under any prevailing wettability conditions, when the laboratory data from the core at any known wettability condition are available. Sequentially, the methodology has been formulated, to incorporate wettability variations effects in experimental relative permeability data. The developed methodologies for including wettability alteration effect into flow affecting parameters have been implemented on Malaysian oil reservoir for improved history matching. The obtained results show better history match with the field data, thus verifying the significance of the designed approach. & 2013 Elsevier B.V. All rights reserved.
Keywords: Wettability Capillary pressure Relative permeability Simulation History matching
1. Introduction Wettability can be defined as the ability of a liquid to coat the rock surface (Shedid and Ghannam, 2005). It ranges from extremely water-wet to extremely oil-wet condition (Anderson, 1986b; Lingen et al., 1996; Tripathi and Kishore, 2007). Wettability controls the location and distribution of fluids within a reservoir (Chalbaud et al., 2009; Norman, 1990). If the rock is water-wet, the water will occupy the small pores, while the oil will occupy the larger pores and vice versa in case of oil-wet situation (Anderson, 1986b; Norman, 1990). Wettability alteration can be due to a number of reasons (Anderson, 1986b; Buckley et al., 1998; Donnez., 2007) out of which mud filtrate invasion is quite common (Ballard and Dawe, 1988; Dandekar, 2006; Yan et al., 1993). Due to such alterations, core samples cannot be considered as true representatives of the reservoir. Surfactants present in the drilling mud causes wettability alteration and can also reduce oil relative permeability in the presence of connate water (McDonald and Buller, 1992; Skalli et al., 2006; Yan et al., 1993). Core handling and conditions at which experiments have been conducted can also add to the deviation of core properties from that of the virgin reservoir (Anderson, 1986b; Zinszner and Pellerin, 2007).
Wettability affects the fluid distribution within a core and as well as in a reservoir, thus influencing the capillary pressure and relative permeability, which are the controlling parameters for fluid flow and displacement of one fluid by another (Anderson, 1987a, 1987b; Chalbaud et al., 2009; Karabakal and Bagci, 2004). As a result of it, wettability has a strong influence on history matching process during simulation studies. History matching the production data has always been an essential and challenging task requiring tremendous efforts (Muhammad and Jonathan Neil, 2010; Tillier et al., 2010; Todd Hoffman and Caers, 2005). The above mentioned parameters play a crucial role toward history matching the dynamic model and thus leading to reliable forecasts (Andres, 2010; Isha et al., 2010; Maschio et al., 2008). Engineer assigns set of parameters and also makes sensible ranges for these parameters to obtain proper or improved history match of the reservoir model (Muhammad and Jonathan Neil, 2010). To improve the history match for a Malaysian field, a developed set of correlations (Zahoor, 2011) has been used to incorporate wettability variations effect on capillary pressure curves and further methodology has been formulated for generating relative permeability data under the prevailing wettability conditions.
2. Literature review n
Corresponding author. Tel.: +92 334 4137845. E-mail address: [email protected] (M.K. Zahoor).
0920-4105/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2013.03.025
Different saturation stages of the displaced phase can be defined based on the discussion made by Anderson (1987a) and the
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
Se effective wetting phase saturation Sem effective mobile phase saturation Smphase1 saturation of mobile phase 1 Smphase1.new saturation of phase 1 under new wettability condition Snwr non-wetting phase residual saturation Srmphase1 residual saturation of phase 1 Srmphase2 residual saturation of phase 2 Sw wetting phase saturation Swr wetting phase residual saturation η function of the fraction of hydrophobic surfaces present in the system λ characteristic constant θ contact angle Δ difference
Nomenclature a c Kr Kro Krw Pc P fcw Pc.est. Pd Pdb.c PNAPL Pw SdM
pore size distribution index entry pressure relative permeability oil relative permeability water relative permeability capillary pressure capillary pressure in fractional-wet system estimated capillary pressure displacement pressure displacement pressure of base case pressure of non aqueous phase liquid (NAPL) pressure of the aqueous phase maximum saturation of the displaced phase
experiments conducted by Killins et al. (1953). Killins et al. showed the behavior of capillary pressure curve under water-wet to oil-wet conditions. Their experiments also showed that the greater the affinity (stronger adhesion) of a fluid with rock surface, the greater the displacement pressure required to initiate the displacement process, which can be explained in a simplified form, as shown in Fig. 1. Before initiating the displacement process, the phase to be displaced is continuous and when the process starts it becomes discontinuous (Anderson, 1987a). This discontinuity increases with the passage of time, demanding increase in pressure to continue the displacement process. The process stops when saturation versus capillary pressure plot becomes vertical (Anderson, 1987a). The obtained laboratory curves can be regenerated by using the Brooks and Corey (1964) model which is one of the most well known and commonly used model, because of its solid experimental validation. Mathematically P c ðSe Þ ¼
P d Se−1=λ
ð1Þ
qualitative terms rather than considering its quantitative significance. Skjaeveland et al. (2000) proposed a correlation to calculate capillary pressure for wettability states between water-wet and oil-wet conditions. They proposed that, capillary pressure can be calculated by using a sum of capillary pressure values (for oil- and water-wet system). Mathematically
Sw −Swr 1−Swr
or
Sw −Swr Se ¼ Sw ¼ 1−Swr −Snwr n
ð4Þ
14 Pc. Lab.
12 10
where effective or normalized wetting phase saturation can be given as Se ¼ Snw ¼
cw co þ ððSw −Swr Þ=ð1−Swr ÞÞaw ððSo −Sor Þ=ð1−Sor ÞÞao
P c ðSe Þ ¼
Pc(Psia)
28
8 6 4 2
ð2Þ
0 0
ð3Þ
Furthermore, correlations to generate capillary pressure curves at different wettability conditions by using the laboratory data at any other known wettability condition are present in the literature. But in these developed correlations, wettability is used in
0.1
0.2
0.3
0.4 Sw
0.5
0.6
0.7
0.8
Fig. 2. Experimental capillary pressure data.
1 Kro. Lab.
0.9
Krw. Lab.
0.8 0.7
Kr
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4
0.6 Sw
Fig. 1. Effect of increasing oil wettability on displacement pressure.
Fig. 3. Relative permeability data at 841.
0.8
1
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The limitation of above Eq. (4) is that, it always requires experimental data for calculation purposes and treats wettability as a qualitative parameter. Similarly, a set of correlations has been developed for different wettability systems by incorporating wettability in a qualitative manner and it also requires some sort
29
of experimentation studies to use such correlations (Tsakiroglou and Fleury, 1999a, 1999b). Another example of such correlations is the one developed for fractional-wet systems (Bradford and Leij, 1996). The proposed correlation can be expressed as in Eq. (5). The proposed expression
Well Oil Production Rate (Stb/day)
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0
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800
0
100
200
300
400 Time (Days)
500
600
700
800
Well Oil Production Rate (Stb/day)
3000
2500
2000
1500
1000
500
0
Well Oil Production Rate (Stb/day)
2500
2000
1500
1000
500
0
Fig. 4. History match of wells oil production rate based on laboratory data: (a) Well A1, (b)Well A4, and (c) Well A8.
30
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can be used for calculating capillary pressure under fractional wettability condition from available capillary pressure values at water-wet situation. P fcw ðSw Þ ¼ P NAPL −P w ¼ P ww c ðSw Þ−η
ð5Þ
Though the above mentioned correlations can be of great significance, but for simulation purposes correlation(s) are required in which wettability variations can be incorporated in a quantitative manner and at the same time giving liberty to generate data sets at any wettability condition without requiring repeated
0.5 Well A1 (Based on Lab. data)
0.45
Historical Data Well A1
0.4
Well water-cut
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
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400 Time (Days)
500
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0.5 Well A4 (Based on Lab. data)
0.45
Historical Data Well A4
0.4
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0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
100
200
300
400 Time (Days)
500
600
700
800
0.5 Well A8 (Based on Lab. data)
0.45
Historical Data Well A8
0.4
Well water-cut
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
100
200
300
400 Time (Days)
500
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700
800
Fig. 5. History match of water-cut profiles of different wells based on laboratory data: (a) Well A1, (b)Well A4, and (c) Well A8.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
experimentation. This will give an option that even a slight variation in this parameter can be incorporated in the capillary pressure data and later into relative permeability curves for improved history matching while conducting simulation studies.
31
3. Correlation for capillary pressure estimation by incorporating wettability in a quantitative manner At the first instance the term effective or normalized mobile phase saturation (Sem) correlation has been used (which will be
1750 Well A1 (Based on Lab. data)
Well bottom hole Pressure (Psia)
1730
Historical WBHP Well A1
1710 1690 1670 1650 1630 1610 1590 1570 1550 0
100
200
300
400 Time (Days)
500
600
700
800
1740 Well A4 (Based on Lab. data) Well bottom hole Pressure (Psia)
1730
Historical WBHP Well A4
1720 1710 1700 1690 1680 1670 1660 1650 0
100
200
300
400 Time (Days)
500
600
700
800
1730 Well A8 Based on Lab. data) Well bottom hole Pressure (Psia)
1710
Historical WBHP Well A8
1690 1670 1650 1630 1610 1590 1570 1550 0
100
200
300
400 Time (Days)
500
600
700
Fig. 6. History match of well pressure profiles: (a) Well A1, (b) Well A4, and (c) Well A8.
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P c:est: ¼ P db:c ðSem Þ−1=λ 7 k1 ðΔθÞ P db:c ðSem Þ−1=λ
ð6Þ
where k1 ¼ 1:24 10−4 and sign convention, in general, will be positive if oil wettability increases and vice versa. “Δθ” is the absolute difference of wettability in terms of contact angle. Base case wettability is calculated experimentally in terms of wettability index or number using displacement tests (Anderson, 1986a; Norman, 1990), from which it is transformed in terms of contact angle by using cosine inverse. Effective mobile phase saturation correlation (Sem) in Eq. (4) is defined based on the saturation stage. Capillary pressure curve can be divided into three regions/sections based on saturation change during displacement process, namely flow at higher displaced saturation stage, transient stage and displacement at lower saturation of the displaced phase (Zahoor, 2011; Zahoor et al., 2011). “Sem” in the developed correlation (Eq. (6)) to predict capillary pressure curve behavior during higher saturation of displaced fluid stage (H.S.S) can be incorporated as follows: −1=λ Smphase1 −Srmphase1 P c:est: ¼ P db:c 7 k1 ðΔθÞ P db:c 1−Srmphase2 −Srmphase1 −1=λ Smphase1 −Srmphase1 ð7Þ 1−Srmphase2 −Srmphase1 The boundary conditions of the above mentioned flow stage or the range of applicability of the above correlation can be given in the following manner in the most refined form, as a function of displaced phase saturation: Smphase1 ≥1−Srmphase2 −0:5625:SdM
P c:est:ðH:S:SÞ þ P c:est:ðL:S:SÞ 2
ð12Þ
4. Formulated methodology for incorporating wettability variation effect into relative permeability data After predicting the curve for capillary pressure at desired wettabililty condition, wettability variation effect can also be incorporated into laboratory relative permeability data. To explain the procedure as we know, the capillary pressure experimental data obtained at known wettability ðθ ¼ θo1 Þ can be regenerated by reducing the above Eq. (7), as follows (Δθ¼01), which is similar to the Brooks and Corey Correlation: −1=λ Smphase1 −Srmphase1 P c:est:θ1 ¼ P db:c θ1 ð13Þ 1−Srmphase2 −Srmphase1 and the capillary pressure data at any wettability condition (θ ¼ θo2 ), let us say, can be generated by using Eq. (7) as follows: Smphase1:new −Srmphase1 −1=λ P c:est:θ2 ¼ P db:c:θ þ k1 ðΔθÞ P db:c:θ 1 1 1−Srmphase2 −Srmphase1 −1=λ Smphase1:new −Srmphase1 ð14Þ 1−Srmphase2 −Srmphase1
16 Pc. Est. Pc. Lab.
14 12 10 8 6 4 2 0 0
0.1
0.2
0.3
0.4 Sw
0.5
0.6
0.7
0.8
Fig. 7. Comparative plot of capillary pressure at different wettability conditions.
1 Kro. Lab. Krw. Lab. Kro. Est. Krw. Est.
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
If
During transient flow, which exists when 1−Srmphase2 −0:6875:SdM o Smphase1 o 1−Srmphase2 −0:5625:SdM
P c:est:ðTransient:stageÞ ¼
ð8Þ
Selection of phases 1 and 2 in Eq. (7) depends on the phase for which capillary pressure needs to be generated. Capillary pressure curve during flow at lower displaced fluid saturation stage (L.S.S) can be estimated by using the following correlation: Smphase1 −Srmphase1 −1=λ P c:est: ¼ P db:c 7 k1 ðΔθÞ P db:c 1−Srmphase1 Smphase1 −Srmphase1 −1=λ ð9Þ 1−Srmphase1
Smphase1 ≤1−Srmphase2 −0:6875:SdM
capillary pressure can be estimated by averaging the capillary pressure obtained by using Eqs. (7) and (9), i.e.
Pc (Psia)
presented later), instead of effective wetting phase saturation in Eqs. (1)–(3) for standardization. It will give a broader sense and make it possible to implement the developed correlation for cases in which the wetting phase is immobile and the desaturation curve is generated by, for example, displacing oil by gas in the presence of immobile water as wetting phase (Zahoor, 2011). To generate capillary pressure data at any wettability condition, a set of correlations has been developed. This set of correlations has been developed by using a number of experimental capillary pressure data sets available at different wettability conditions. The correlations were developed by using curve matching technique (comparing the calculated values with the experimental data) and are capable of generating the data for the entire range of wettability conditions (0–1801), when the laboratory data (base case) at any wettability condition is known. The developed generalized form of correlation has been obtained by including set of parameters into the Brooks and Corey model to account for wettability variations in a quantitative manner while predicting the capillary pressure curve. In generalized form the developed correlation can be given as (Zahoor, 2011)
Kr
32
ð10Þ
0 0
0.2
0.4
0.6
0.8
1
Sw ð11Þ
Fig. 8. Comparative plot of relative permeability at different wettability conditions.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
During the displacement process in porous medium, having different wettabilities, different pressures will be required to produce the same volume of displaced phase. In other words, at the same pressure, different volumes of displaced fluid will be produced, or the displaced fluid saturation change will be different. To calculate the
33
saturation of the displaced phase “Smphase1.new” at wettability condition “θ2”, Eqs. (13) and (14) can be equated as follows. At same capillary pressure P c:est:θ1 ¼ P c:est:θ2
ð15Þ
Well Oil Production Rate (Stb/day)
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Well A1 (Based on Lab. data)
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Well A1 (Based on Est. data) Historical Data Well A1
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400 Time (Days)
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Well Oil Production Rate (Stb/day)
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Historical Data Well A4 0 0
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Well Oil Production Rate (Stb/day)
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Well A8 (Based on Lab. data)
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Well A8 (Based on Est. data) Historical Data Well A8 0 0
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Fig. 9. Comparative history match plots for well oil production rate based on laboratory and estimated data: (a) Well A1, (b) Well A4, and (c) Well A8.
34
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
−1=λ
¼ P db:c:θ
1
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
þk1 ðΔθÞ P db:c:θ
1
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ
Solving the above equation for “Smphase1.new”
−1=λ ð16Þ
P db:c:θ
1
Smphase1 −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ
¼ P db:c:θ
1
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ
1 þ k1 ðΔθÞ
ð17Þ
0.5 0.45
Well A1 (Based on Lab. data) Well A1 (Based on Est. data) Historical Data Well A1
0.4
Well water-cut
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Well A4 (Based on Lab. data)
0.4
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1
Smphase1 −Srmphase1 1−Srmphase2 −Srmphase1
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Well A8 (Based on Lab. data)
0.4
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P db:cθ
0.3 0.25 0.2 0.15 0.1 0.05 0 0
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400 Time (Days)
500
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800
Fig. 10. Comparative history match plots for well water cut based on laboratory and estimated data: (a) Well A1, (b) Well A4, and (c) Well A8.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
−1=λ ¼
ððSmphase1 −Srmphase1 Þ=ð1−Srmphase2 −Srmphase1 ÞÞ−1=λ ½1 þ k1 ðΔθÞ
Smphase1:new −Srmphase1 1−Srmphase2 −Srmphase1
¼ ½1 þ k1 ðΔθÞλ
Smphase1 −Srmphase1 1−Srmphase2 −Srmphase1
ð18Þ
Well bottom hole Pressure (Psia)..
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Time (Days) 1740
Well bottom hole Pressure (Psia)..
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400 Time (Days)
1730 1710 1690 1670 1650 1630 1610 1590 1570 1550 0
100
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300
ð19Þ
1750
Well bottom hole Pressure (Psia)
35
400 Time (Days)
Fig. 11. Comparative well pressure profiles based on laboratory and estimated data: (a) Well A1, (b) Well A4, and (c) Well A8.
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
Similarly Smphase1:new ¼ ½1−k1 ðΔθÞλ ðSmphase1 −Srmphase1 Þ þ Srmphase1 i ¼ 1; n
ð21Þ
“n” number of “Smphase1.new” values can be calculated by using the above Eqs. (20) and (21), which are later substituted into any relative permeability model, to generate relative permeability curves at wettability “θ2”.
5. Field implementation 5.1. Field and fluid description The field is located offshore Peninsular Malaysia in the South China Sea. This oil field has dipped structure and an initial gas cap having an oil zone thickness of approximately 100 ft with an underlying water zone. It is a sandstone reservoir, having average porosity of approximately 31% and the average permeability is 757 md. Average initial reservoir pressure is 1715 psi and the initial water saturation within the reservoir is 27%. Field is produced at a constant gas–oil ratio of 0.495 scf/stb. Reservoir oil has a bubble point pressure of 1697 psi, its formation volume factor is 1.23 rb/stb and viscosity is 0.29 cp. Rock wettability is 841 at reservoir conditions and the relative permeability curves can be generated by using the following equations: krw ¼ ðSem Þ5:22
ð22Þ
kro ¼ ð1−Sem Þ2 ½1−ðSem Þ3:2
ð23Þ
The capillary pressure and relative permeability curves at this wettability condition of 841 are shown in Figs. 2 and 3, respectively. 5.2. Results and discussion Simulation study was conducted using a commercial black-oil, three phase reservoir simulator. For history matching purposes, wells are controlled based on total volume of fluid produced per day as obtained from wells and field production data. The obtained results for oil production rate, water-cut and pressure profiles for different wells are shown in Figs. 4–6, while comparing with the observed/historical data. Figs. 4–6 show proper history match particularly for well oil production and bottom hole flowing pressure profiles during the initial period of production. Later the profiles start deviating from the observed data while the well water-cut profile is not properly matched. To have a proper history match, capillary pressure curve was generated using the developed set of correlations at a wettability condition of 801 (value chosen based on past experience in the area). The estimated capillary pressure curve is shown in Fig. 7. After generating Pc data while incorporating the wettability variation, relative permeability curves under the prevailing conditions of wettability have been obtained by using the following equations: Smphase1:new −Srmphase1 5:22 krw ¼ ð24Þ 1−Srmphase2 −Srmphase1 " # Smphase1:new −Srmphase1 2 Smphase1:new −Srmphase1 3:2 1− kro ¼ 1− 1−Srmphase2 −Srmphase1 1−Srmphase2 −Srmphase1 ð25Þ
where the values for Smphase1.new have been calculated by using the formulated methodology as explained earlier and the resulting relative permeability curves are shown in Fig. 8, along with laboratory curves. History matches based on these estimated plots obtained as a result of simulation studies are shown in Figs. 9–11 while the comparative plots for the overall field history match based on both wettability conditions are shown in Figs. 12 and 13. Figs. 9–13 show improved history match based on the estimated data, beginning from a point from where the plots generated based on laboratory data started deviating. This is due to the restoration of wellbore conditions, which were altered due to the exposure of porous media to the mud filtrate. This restoration of wellbore means that the wettability condition of the near wellbore region is changed approximately to the original wellbore wettability condition, as it was before mud filtrate invasion. It can also be observed from the results that this wettability reversal is rate dependent. The profiles of the wells producing at higher rates have matched earlier with the simulation results obtained based on estimated capillary pressure and relative permeability data.
6. Conclusion Implementation of developed set of correlations to generate capillary pressure curve at any wettability condition and
1
30000 25000
0.8
20000 0.6
FOPR (Based on Lab. data) FOPR (Based on Est. data) Historical FOPR FWCT (Based on Lab. data) FWCT (Based on Est. data) Historical FWCT
15000 10000
0.4
Field water-cut
ð20Þ
Field Oil Production Rate (Stb/day)
Smphase1:new ¼ ½1 þ k1 ðΔθÞλ ðSmphase1 −Srmphase1 Þ þ Srmphase1 i ¼ 1; n
0.2
5000 0 0
100
200
300 400 Time (Days)
500
600
0 700
Fig. 12. Field history match plot of oil production rate and water cut.
1720
Historical FPR FPR (Based on Lab. data) FPR (Based on Est. data)
1710 Field Pressure Rate (Psia)
36
1700 1690 1680 1670 1660 1650 1640 0
200
400 Time (Days)
600
Fig. 13. Field history match plot of field pressure rate.
800
M.K. Zahoor, M.N. Derahman / Journal of Petroleum Science and Engineering 104 (2013) 27–37
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