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The application of petrophysics to resolve fluid flow units and reservoir quality in the Upper Cretaceous Formations: Abu Sennan oil field, Egypt
Journal of African Earth Sciences 102 (2015) 61–69
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Journal of African Earth Sciences journal homepage: www.elsevier.com/locate/jafrearsci
The application of petrophysics to resolve fluid flow units and reservoir quality in the Upper Cretaceous Formations: Abu Sennan oil field, Egypt Amir Maher Sayed Lala a,⇑, Nahla Abd El-Aziz El-sayed b a b
Geophysics Department, Fac. of Science, Ain Shams University, Egypt Egyptian Petroleum Research Institute, Egypt
a r t i c l e
i n f o
Article history: Received 18 March 2014 Received in revised form 18 October 2014 Accepted 20 October 2014 Available online 24 November 2014 Keywords: Mean hydraulic radius Pore throat radius Flow zone indicator
a b s t r a c t Petrophysical flow unit concept can be used to resolve some of the key challenges faced in the characterization of hydrocarbon reservoirs. The present study deals with petrophysical evaluation of some physical properties of the Upper Cretaceous rock samples obtained from the Abu-Roash and the Bahariya Formations at southwest of Sennan oil field in the Western Desert of Egypt. The aim of this study was achieved through carrying out some petrophysical measurements of porosity, bulk density, permeability, mean hydraulic radius (Rh), irreducible water saturation, and radius of pore throat at mercury saturation of 35% in order to determine reservoir characteristics. In this study, the relationships obtained between the measured petrophysical properties such as porosity, permeability and pore throat flow unit types were established for 53 sandstone core samples obtained from two different stratigraphic units. Flow zone indicator (FZI) has been calculated to quantify the flow character of the Abu-Roash and Bahariya reservoir rocks based on empirically derived equations of robust correlation coefficients. The correlations among porosity, permeability, bulk density, mean hydraulic radius and pore throat flow properties reflect the most important reservoir behavior characteristics. The calculated multiple regression models indicate close correlation among petrophysical properties and Rh and R35%. The obtained models are able to predict Rh and R35% by using porosity and permeability, to map reservoir performance and predict the location of stratigraphic traps. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction There are many applications for pore geometry in reservoir quality studies and characterization. In many developed reservoirs, capillary pressure tests are not available while only porosity and permeability are available as they are routinely measured. In the present work, it is important to predict, either empirically or theoretically, the needed mean hydraulic and pore throat radius at certain mercury saturation values from available petrophysical properties to resolve reservoir fluid flow units. Pittman (1992) presented empirical relationships between porosity, permeability and the pore aperture size that corresponds to the displacement pressure and the apex of mercury injection plot. The study of petrophysical properties of rocks and theoretical models based on laboratory measurements and the application of empirical statistical correlations is often applied in these studies. The mean hydraulic radius (Rh) is a measure of pore channel flow efficiency. Flow speed along the channel depends on its ⇑ Corresponding author. http://dx.doi.org/10.1016/j.jafrearsci.2014.10.018 1464-343X/Ó 2014 Elsevier Ltd. All rights reserved.
cross-sectional shape (among other factors) and the hydraulic radius is a characterization of the channel efficiency. Based on the ‘constant shear stress at the boundary’ assumption, hydraulic radius is defined by Carman (1956), and Georgi and Menger (1994) as the cross section area normal to the flow section divided by the wetted pore perimeter (P), which is written as:
Rh ¼ fluid filled volume wetted area : Rh ¼ ð2Þ ðA=PÞ ¼ ðpr 2 =2prÞ ¼ ðr=2Þ
ð1Þ
where Rh: the hydraulic radius (L); A: the cross sectional area normal to flow (L2); P: the wetted perimeter (L) and r: pore throat radius of the rock sample. Thus, wetted perimeter times the hydraulic radius is equal to the area of irregular section flow. Normal flow occurs when the water surface slope (Sws), is the same as the bottom slope (So). When normal flow is approached, the velocity Manning (1891) equation is used to compute hydraulic radius which is proportional with the velocity of the fluid flow. The greater the hydraulic radius, the greater the efficiency of the channel and the more volume it
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can carry. For reservoir characterization some derived parameters are used by Georgi and Menger (1994) as;
K ¼ ð1=C 2 Þ ðu3 =ð1 uÞ3 Þ
ð2Þ
where K: permeability (md), u: porosity (%).
C ¼ ðXÞ1=2 ðSm Þ ðTÞ
ð3Þ
where X: Kozeny–Carman cross-section shape factor, Sm: matrix surface area (the specific surface area per unit grain volume) (cm1), T: tortuosity, defined as (ratio of the true path of the diffusion path of fluid particle diffusing in the porous medium, and the straight line distance between the starting and finishing points of the particle’s diffusion). These parameters are related to the flow zone indicator (FZI) which introduced by Amaefule et al. (1993). The reservoir rock specific parameter FZI characterizes the permeability versus porosity correlation for a porous rock and is assumed to be constant within the same hydraulic unit, but varies from one unit to another in the heterogenic reservoir. Flow zone indicator is a unique and useful parameter to quantify the flow character of a reservoir and it offers a relationship between petrophysical properties at small-scale, such as core plugs and at a large-scale such as well bore level. In addition, the term FZI provides the representation of the flow zones based on the surface area and hydraulic tortuosity. The (FZI) units can facilitate the tracking of waterfront displacement and predict the water flooding results. Flow zone indicator (FZI) values are estimated using Amaefule et al. (1993) relationship:
ðFZIÞ ¼ ðRQI=ur Þ
ð4Þ
where RQI is the reservoir quality index defines as:
ðRQIÞ ¼ ð0:0314Þ ðK=uÞ1=2
ð5Þ
pore to grain volume ratio ður Þ ¼ ððuÞ=ð1 ðuÞÞÞ
ð6Þ
During the past years, many studies have been done on the pore aperture corresponding to 35th percentile of cumulative mercury saturation curve which was developed by H.D. Winland, as stated in Pittman (1992) and it is focused on predicting of R35 from other data such as porosity and permeability. The R35 of a given rock sample type reflects its depositional and digenetic fabric and influences of fluid flow and defines reservoir performance (Hartmann and Coalson, 1990; Lala and Ahmed, 2012). Consequently, estimating R35 from logs, using the Winland’s model (Kolodzie, 1980), or directly from capillary pressure core data (when available) provides the basis for a common zonation that can be used by both geologists and reservoir engineers. Four petrophysical flow units with different reservoir performances are distinguished by ranges of pore throat radii and prove to be an excellent delineator of hydraulic units (Hartmann and Beaumount, 1999; Lala and Ahmed, 2012). The irreducible water saturation (Swir) is the percentage of the pore space that the mercury could not enter at infinite pressure (Jennings, 1987). In this study, a separate technique based on range of R35 flow unit type and Rh values was utilized to create a better correlation among effective petrophysical properties in all studies rock samples. This indicates that, pore throat radius and mean
Fig. 1. Location map of the study area, Western Desert, Egypt.
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hydraulic radius parameters are the primary controlling parameters of the relationship between porosity and permeability. Flow units can be identified from the calculation of pore throat radii at the R35% pore volume, which subdivides the data samples into units having similar and predictable flow characteristics, using laboratory measured mercury saturation curves. The approach presented in this paper differs with other procedures for resolving reservoir quality parameters in that it is based on assembling data with different R35, Rh and FZI, and hence potential reservoir performance. This approach enables facies and rock types to be quantitatively represented in terms of petrophysical flow unit types with distinctive ranges of petrophysical characteristics. Flow unit distribution scaled up to create new relationships between porosity and permeability improves permeability prediction using empirically derived model of high correlation coefficients.
2. Materials and methods A suite of 53 sandstone samples of Upper Cretaceous age from the southwest of Sennan Oil Field were selected for petrophysical analysis (Fig. 1). A data set of laboratory measurements which include bulk density, porosity, permeability (K), mean hydraulic radius (Rh), irreducible water saturation (Swir), R35, pore to grain volume ratio, reservoir quality index and flow zone indicator (FZI) for 53 sandstone core samples has been derived for the Upper Cretaceous sandstones of Abu-Roash and Bahariya Formations. Prior to measuring the petrophysical parameters in the laboratory, the rock samples were cleaned of pore fluid and other contaminants using the Soxhlet cleaning technique (Soxhlet, 1879) and dried in an electric oven by heating the samples to 98 °C, readying them for laboratory measurements.
Table 1 Laboratory generated petrophysical parameters for the Upper Cretaceous sandstone core samples from the Abu Sennan Oil Field, Egypt. No.
Formation
u (%)
log K (md)
Swi (%)
log R35 (lm)
log Rh (lm)
qb (gm/cc)
RQI
ur
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya
18.70821 18.67353 11.13822 11.48387 13.9267 14.7152 21.81612 12.24573 12.14688 26.8611 24.63116 17.66209 15.06982 14.8883 15.98642 4.115741 10.46734 9.815563 14.57766 18.23008 11.78556 10.37937 20.25234 18.34108 16.66019 20.19491 21.80093 17.09651 12.78836 16.52936 23.33072 22.53071 24.28992 16.4129 24.92778 24.88826 17.04025 16.31162 11.89696 12.64232 10.6923 13.33893 24.97723 25.31794 11.87687 14.1902 16.12715 14.25198 24.07556 24.34465 23.54725
0.50332 0.63609 0.87656 0.57402 0.79563 0.81974 0.77356 0.33262 0.25984 2.43571 2.307258 0.550014 0.34662 0.710672 0.8734 1.12243 1.31424 0.82685 0.008479 0.7528 1.10331 0.98238 1.76216 1.44698 0.19686 1.161548 2.316109 1.29498 1.4237 0.992661 2.30453 2.21538 2.624252 0.608546 2.043845 2.001419 0.68837 0.71222 0.52023 0.56206 0.88368 1.05878 2.41482 2.3765 0.68247 0.80469 0.96651 0.25854 2.40132 2.265076 2.166535
19.57 20.01 22.82 19.74 20.68 21.8 21.95 21.22 18.35 10.46 8.75 17.56 17.86 20.1 18.65 26.89 23.98 18.19 18.53 21.88 20.94 24.48 13.17 14.08 17.04 14.52 10.31 22.18 20.51 17.18 12.59
1.04721 1.10791 1.87615 1.79778 1.65365 1.60906 1.76447 1.84164 1.73518 0.68825 0.068186 0.98632 1.55284 1.64016 1.451 1.96658 1.98464 1.93181 1.34872 1.30627 1.51286 1.9431 0.321184 0.67366 0.95156 0.47756 0.758912 1.92812 1.97469 0.47756 0.739572
0.86328 0.89963 1.21467 1.21467 1.04576 1.05061 1.1549 1.21467 0.9914 0.08991 0.368473 0.23657 0.75449 1.02228 1.61979 1.56864 1.39794 0.61439 0.64207 1.05552 1.46852 0.693727 0.534026 0.50724 0.426511 0.868056 1.69897 1.74473 0.378398 0.85248
2.106262 2.106285 2.380595 2.365667 2.305961 2.287918 2.183707 2.348813 2.346151 1.936525 2.005602 2.202849 2.261124 2.264177 2.248496 2.617985 2.446363 2.443111 2.286117 2.23619 2.428208 2.384445 2.109763 2.190035 2.226331 2.115789 2.080425 2.349497 2.455424 2.210581 2.031525 2.051885
14.15 11.86 12.53 21.11 21.74 19.65
0.90309 0.514548 0.514548 1.16941 1.4318 1.90309
0.074451 0.598791 0.598791 0.84771 0.95468 1.05552
23.19 19.78 7.49 10.59 20.98 20.93 17.53 23.99 10.2 11.08 9.83
1.91721 1.88107 0.802774 0.745465 1.70115 1.83416 1.67778 1.69465 0.790285 0.705864 0.620136
1.20066 1.34679 0.910091 0.857332 1.26761 1.35655 1.38722 1.34679 0.904716 0.816241 0.730782
0.040668 0.034936 0.034296 0.04785 0.033666 0.031855 0.027591 0.061182 0.0668 0.31639 0.284981 0.140739 0.120555 0.184435 0.028731 0.04251 0.021374 0.038686 0.083047 0.030913 0.02568 0.031452 0.530606 0.387889 0.061328 0.266123 0.967713 0.0171 0.017048 0.242177 0.923061 0.847681 1.307202 0.156174 0.661471 0.630437 0.034435 0.034243 0.050015 0.046237 0.034718 0.025408 1.012902 0.962644 0.041528 0.033006 0.025698 0.112011 1.015784 0.863507 0.783839
0.230137 0.229612 0.125343 0.129738 0.161801 0.172542 0.279036 0.139546 0.138264 0.367262 0.326808 0.214507 0.177438 0.174927 0.190284 0.042924 0.116911 0.108839 0.170654 0.222944 0.133601 0.115815 0.253955 0.224606 0.199907 0.253053 0.278788 0.206222 0.146636 0.198026 0.304303 0.290834 0.320828 0.196357 0.332051 0.33135 0.205404 0.194909 0.135035 0.144719 0.119724 0.153921 0.332929 0.33901 0.134776 0.165368 0.192281 0.166208 0.317099 0.321784 0.307997
The black areas indicate no measurements.
2.228102 1.996642 1.9966 2.20992 2.245038 2.410551 2.340542 2.389367 2.519631 1.996486 1.984696 2.422515 2.29725 2.250845 2.318151 2.020109 2.017301 2.030921
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The porosity of the samples was measured by the Frank Jones helium porosimeter which is also considered as a volume-measuring instrument, which is normally used for determination of the volume of grains or the volume of pores in a sample. The principle of the instrument depends on the gas expansion theory, where a definite volume of the helium gas (of known pressure) is isothermally expanded into unknown void volume. After expansion, equilibrium will occur between the void volume and gas volume allowing the pressure to be measured. By using the resultant measurements and applying Boyle’s law, the void volume can then be calculated. The dried core samples were immersed in mercury in
Fig. 2. A plot of porosity versus permeability for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 3. A plot of porosity versus bulk density for sandstones of the Bahariya and Abu-Roash Formations.
calibrated pycnometer. The volume of mercury displaced by the sample is weighed. Knowing the mercury density at the measured temperature, the bulk volume of the sample can be determined. The porosity of a rock is defined as the ratio of the rock void spaces to its bulk volume expressed in percent. The Ruska gas permeameter (cat. no. 6121) was used for measuring the permeability of the samples using nitrogen gas. The clean and dry core samples were pushed into a rubber stopper of the corresponding hole size. Delicate or fragile core samples are first imbedded with a sealing wax into metal sleeve and placed in a core holder with the tapered end down.
Fig. 4. A plot of the bulk density versus permeability for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 5. A plot of bulk density versus log Rh for sandstones of the Bahariya and AbuRoash Formations.
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Our samples were tested for the capillary pressure measurement using mercury injection method. In this method, mercury is forced into the sample at the low pressure (1.169 kPa up to 779.44 kPa). The volume of mercury injected into the sample at each increasing pressure step is recorded after the stabilized condition has been achieved. The determination of capillary pressure was performed utilizing core laboratory mercury injection panel. The obtained mercury injection capillary pressure data were converted from laboratory (air/Hg system) into reservoir conditions (oil/water system) using equations from El Sayed (1993a) and El Sayed (1993b) resulting in the derivation of pore throat size distribution, R35-parameter and irreducible water saturation (Swir).
The petrophysical properties of the studied sandstone core samples cover a wide range of exploration interests with bulk density ranging from 1940 kg/m3 to 2620 kg/m3, porosity from 4.11% to 26.86%, permeability from 0.037 md to 420.97 md, mean hydraulic radius from 0.018 lm to 8.13 lm, irreducible water saturation from 7.49% to 26.89%, pore to grain volume ratio from 0.043 to 0.37, reservoir quality index from 0.017 lm to 1.31 lm, flow zone indicator from 0.083 lm to 4.07 lm and R35 from 0.01 lm to 6.35 lm. The pore aperture radii were determined graphically from the mercury injection curves corresponding to the 35th percentiles of mercury saturation whereas the irreducible water saturation (Swir)
Fig. 6. A plot of porosity versus permeability for Rh < 0.1 lm. Fig. 8. A plot of porosity versus permeability for R35 < 0.1 lm.
Fig. 7. A plot of porosity versus permeability for Rh > 0.1 lm.
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Fig. 9. A plot for porosity versus permeability for R35 > 0.1 lm.
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was estimated graphically at maximum pressure required for mercury entrance into the core sample on the water saturation scale (Sw) (Table 1). The laboratory measured core data of porosities and permeabilities were used to calculate the pore to grain volume ratio (ur), reservoir quality index (RQI) and flow zone index (FZI) for the studied sandstone core samples representing sandstone layers at different depths. Statistical analysis system (SPSS) software of multiple regression equations were used to establish the various empirical relationships. Multiple regressions are an extension of the regression analysis, which incorporates additional independents variables in the predictive equations (Balan et al., 1995).
permeability values larger than permeability cutoff value which corresponding to R35 equal 0.1 lm will improve the porosity–permeability relationship and the predicted permeability values from measured porosity become more accurate for all studied sandstone samples. Below this cutoff value the reservoir cannot produce any hydrocarbon. The relationship between the graphically estimated (R35) and the calculated mean hydraulic radius (Rh) for the studied core samples is shown in Fig. 10 and indicate a strong correlation between the two parameters with a correlation coefficient of 96%. The
3. Results and discussion The bulk density, porosity, permeability, mean hydraulic radius, irreducible water saturation and R35 have been measured for all the core samples in order to validate the performance of the different reservoir formations. The effect of petrophysical flow unit types on the relationship between porosity and permeability for all the studied core samples and their influences on the formation evaluation was investigated. The porosity values of the studied sandstones show a direct relationship with the corresponding permeability (Fig. 2) and are characterized by a high correlation coefficient (r = 83%) for all the studied samples. This relationship can be used in permeability prediction for other samples of Upper Cretaceous rocks in the Sennan Oil Field when permeability measurements are not available. The porosity, permeability and mean hydraulic radius values are correlated with the bulk density. The porosity, permeability and mean hydraulic radius values decrease with the increase of bulk density (Figs. 3–5). The correlation between the bulk density and the rock porosity is perfect, while the correlations are reliable for both permeability and mean hydraulic radius (Rh), indicating the presence of the other factors which may control their relationships. The permeability–porosity relationships depend on the hydraulic radius (Rh) for the studied samples. The permeability–porosity relation for samples having Rh < 0.1 lm is illustrated in Fig. 6 and shows a cloud of points relationship without any orientation. This is possible because at these ranges of mean hydraulic radius, the velocity of fluid flow is very small and so the channel efficiency impedes its flow. In Fig. 7 a depiction of the porosity–permeability relationship for the samples having Rh > 0.1 lm show a reliable relationship with a high correlation coefficient (r = 87%). Improvement of this relationship occurs when the mean hydraulic radius (Rh) is Rh > 0.1 lm. This is attributed to the increase of the connectivity between pores resulting in increase of the flow velocity values and in turn causing an increase of the rock permeability. The equation of the line of correlation is shown in Fig. 7. The analysis of data shows that there is no relationship between porosity and permeability for samples with R35 < 0.1 lm of the nano flow unit type (Fig. 8). This is plausible, because at these ranges of pore throat radii with R35 smaller than 0.1 lm, permeability is too low and no fluid flow exists. The relationship between porosity and permeability for sandstone samples having the R35 > 0.1 lm of meso and macro flow unit type shows a very strong relationship with a high correlation coefficient (Fig. 9). The improvement of the porosity–permeability relationship is clear in this relation at this range of pore throat radii (R35) for this flow unit. The permeability values increase with increasing of R35 values and are directly related to fluid flow. The calculated regression equation is shown in the graph and is characterized by a robust correlation coefficient (r = 95%) (Fig. 9). Using
Fig. 10. A plot of log R35 versus log Rh for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 11. A plot of irreducible water saturation versus permeablity.
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values of the mean hydraulic radii (Rh) are closely related to the measured pore throat radii values at these different flow unit types (Fig. 10). It appears from the results of this study, that the sample data points are scattered where weak relationships exist at the nano scale of the pore throat range values, due to Manning equation and its friction factor which is determined for ordinary channel flows. It is not accurately applied to this scale of rock pore channel flow. Other reasons could be that the flow depths which have inverse relationship with the flow velocity and boundary roughness size are both closely related to the mean hydraulic radius. The inverse relationship between the irreducible water saturation (Swir) and the rock permeability (log K) is illustrated in Fig. 11
Fig. 12. A plot of the quality index versus pore to grain ratio for sandstones of the Bahariya and Abu-Roash Formations.
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showing a strong correlation between them. The increase of irreducible water saturation (Swir) can be caused by either small pore throat radius or mean hydraulic radius leading to the decrease of fluid flow velocity and permeability. For all studied sandstone samples the porosity and permeability relationship is very weak at the scale of nano flow unit type because pore throat radii are too small and impede the fluid flow. However, at the micro, meso and macro flow unit types, the improvement of the relationship between porosity and permeability is caused by increasing the pore throat size and the amount of fluid flow. By using the mean hydraulic radius Rh and the graphically estimated R35, one is able to discriminate between the reservoir and non reservoir zones. In spite of this, some reservoir zones have higher porosity greater than the porosity cutoff (8–10%). These reservoir zones have low value of Rh and/or R35 of nano or micro flow unit type and so they are considered as non reservoirs. These non reservoir zones are considered as containing high grain roughness and channel tortuosity with bad connected small or large pores channels. Hence we cannot use the porosity cutoff concept to differentiate between the reservoir and non reservoir zones instead we can apply the permeability cutoff, according to the Rh and R35, making this discrimination more realistic and acceptable. The concept of permeability cutoff based on R35, is available in literature but is based on different data sets of different lithologic types (Lala and Ahmed, 2012). The relationship between reservoir quality index (RQI) and pore to grain volume ratio (ur) for the studied samples is illustrated in Fig. 12. The hydraulic unitization of the studied core samples is based on the geological attributes of pore geometry and tortuosity. The flow zone indicator (FZI) multiple straight lines represent different hydraulic units with homogenous pore network and similar tortuosity and fluid flow capacity. FZI values allow us to represent the reservoir by subdividing it into flow units with specified flow zone indicator. There is an excellent correlations between Swir (%) obtained from capillary pressure data, log K (md) and FZI (lm) versus depth for the studied sandstone core samples which are demonstrated in Figs. 13–15. As expected, irreducible water saturation (Swir) decreased with increasing FZI and log K. The application of the flow
Fig. 13. Plots of log k, Swi and FZI versus depth (1984–2000 m) for sandstones of the Bahariya and Abu-Roash Formations.
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Fig. 14. Plots of log k, Swi and FZ1 versus depth (2000–2060 m) for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 15. Plots of log k, Swi and FZI versus depth (2072–2080 m) for sandstones of the Bahariya and Abu-Roash Formations.
unit approach in this study allowed us to model in time and space the hydraulic flow units and reservoir performance by identifying the layers that have the most important contribution in the fluid flow in both the Abu-Roash and Bahariya Formations in the Abu Sennan Oil Field.
4. Regression models Linear regression techniques have become popular for predicting permeabilities from other variables. The techniques are based on correlations between measurements that have plausible
connections to permeability without investigating the physics behind the equations. Many theoretical models, such as the Kozeny–Carman model, can be expressed in linear regression format. However, the limitation of regression models is that they are only valid within the range of variables from which the relationships are derived. To get a more general empirical equation, an analysis of relatively large and diverse data set is needed. In the present study, multiple regressions statistical analysis was done on a large data set. In general these relationships among the different investigated petrophysical parameters yielded good results in the prediction of petrophysical parameters and high coefficients of correlation. For example, using the permeability
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(K in md) as the dependent variable in the multiple regression involving radius of the pore aperture corresponding to the mercury saturation at 35% (R35 in m), mean hydraulic radius (Rh in m), porosity (u in %) and bulk density qb (kg/m3) yielded the following relationship:
log k ¼ 1:72 þ 0:022ðuÞ þ 0:95 logðRh Þ þ 0:28 logðR35 Þ 0:84ðqbÞ
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Acknowledgments The authors are indebted to the Egyptian petroleum authority for permission to publish these laboratory results. The authors are also grateful to anonymous reviewers whose constructive comments helped to improve this manuscript. References
ð7Þ
The above equation has a correlation coefficient of 95%. Predictions of log k using the measured porosity, mean hydraulic radius, pore throat radius at 35% of mercury saturation and bulk density yielded reliable and acceptable results. Contribution factors (1.72, 0.022, 0.95, 0.28 and 0.84 respectively for constant a0, u, log Rh, log R35 and qb) indicate that the most important variable in this regression is u and log R35 because small values of their multipliers.
5. Conclusions The petrophysical methods described in this paper lead us to make the following conclusions: (i) The studied reservoir parameters like Rh and R35%, have been shown to be an excellent predictor of permeability in both the Bahariya and Abu-Roash formations. (ii) Mean hydraulic radius and pore throat radii at the 35% pore volume provide the best basis for defining reservoir flow units. (iii) FZI can be used for delineating the number of layers (hydraulic unit) required for estimating both geological and petrophysical parameters in reservoir simulation process. (iv) The multiple regression models obtained in this study are characterized by robust and high correlation coefficients which allowed accurate estimation of permeability values for the studied rock samples.
Amaefule, J.O., Altubay, M.H., Tiab, D., Kersey, D.G., Keelan, D.K., 1993. Enhanced Reservoir Description Using Core and Log Data to Identify Hydraulic Flow Units and Predict Permeability in Uncored Intervals/Wells. Society of Petroleum Engineers, Paper No. 26436, pp. 205–220. Balan, B., Mohagegh, S., Ameri, S., 1995. State of Art of Permeability Determination from Well Log Data, Part 1 – A Comprehensive Study, Model Development. Society of Petroleum Engineers, Paper No. 30978, pp. 70–79. Carman, P.C., 1956. Flow of Gases through Porous Media. Academic Press Inc., and Butterworths Scientific Publications. El Sayed, A.M.A., 1993a. Relationship of porosity and permeability to mercury injection derived parameters for sandstones of the Tortel Formation, Hungary. Geophys. Trans. 38, 35–46. El Sayed, A.M.A., 1993b. Inter-correlation of capillary pressure for sandstones of the Tortel Formation, Hungary. J. Petrol. Sci. Eng. 10, 47–53. Georgi, D.T., Menger, S.K., 1994. Reservoir Quality, Porosity and Permeability Relationships, Trans. 14. Mintrop Seminar (Beilae) DGMK Und Ruhr-Universitat Bochum, pp. 163/1–163/35. Hartmann, D.J., Beaumount, E.A., 1999. Prediction reservoir system quality and performance. In: Treatise of Petroleum Geology/Handbook of Petroleum Geology: Exploring for Oil and Gas Traps, pp. 1–154 (Chapter 9). Hartmann, D.J., Coalson, E.B., 1990. Evaluation of the Marrow sandstone in the Sorrento field, Cheyenne County, Colorado. In: Sonnenberg, S.A., Shannon, L.T., Rader, K., Von Drehle, W.F., Martin, G.W. (Eds.), Morrow Sandstones of Southeast Colorado and Adjacent Areas. The Rocky Mountain Association of Geologists, pp. 91–100. Jennings, J.B., 1987. Capillary pressure techniques: application to exploration and development geology. Am. Assoc. Petrol. Geol. Bull. 71, 1196–1209. Kolodzie, S., 1980. The Analysis of Pore Throat Size and the Use of the WaxmanSmits Equation to Determine OOIP in the Spindle Field, Colorado: Society of Petroleum Engineers. 55th Annual Fall Technical Conference. Society of Petroleum Engineers, Paper No. 9382, 4-2. Lala, A.M., Ahmed, A.S., 2012. Investigation of pore geometry and flow unit concepts in reservoir characterization and prediction of performance. Aust. J. Basic Appl. Sci. 6, 269–285. Manning, R., 1891. On the flow of water in open channels and pipes. Trans. Inst. Civil Eng. Ireland 20, 161–207. Pittman, E.D., 1992. Relationship of porosity and permeability to various parameters derived from mercury injection-capillary pressure curves for sandstone. Am. Assoc. Petrol. Geol. Bull. 76, 191–198. Soxhlet, F., 1879. Die gewichtsanalytische Bestimmung des Milchfettes. Dingler’s Polytech. J. 232, 461–465 (in German).
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Journal of African Earth Sciences journal homepage: www.elsevier.com/locate/jafrearsci
The application of petrophysics to resolve fluid flow units and reservoir quality in the Upper Cretaceous Formations: Abu Sennan oil field, Egypt Amir Maher Sayed Lala a,⇑, Nahla Abd El-Aziz El-sayed b a b
Geophysics Department, Fac. of Science, Ain Shams University, Egypt Egyptian Petroleum Research Institute, Egypt
a r t i c l e
i n f o
Article history: Received 18 March 2014 Received in revised form 18 October 2014 Accepted 20 October 2014 Available online 24 November 2014 Keywords: Mean hydraulic radius Pore throat radius Flow zone indicator
a b s t r a c t Petrophysical flow unit concept can be used to resolve some of the key challenges faced in the characterization of hydrocarbon reservoirs. The present study deals with petrophysical evaluation of some physical properties of the Upper Cretaceous rock samples obtained from the Abu-Roash and the Bahariya Formations at southwest of Sennan oil field in the Western Desert of Egypt. The aim of this study was achieved through carrying out some petrophysical measurements of porosity, bulk density, permeability, mean hydraulic radius (Rh), irreducible water saturation, and radius of pore throat at mercury saturation of 35% in order to determine reservoir characteristics. In this study, the relationships obtained between the measured petrophysical properties such as porosity, permeability and pore throat flow unit types were established for 53 sandstone core samples obtained from two different stratigraphic units. Flow zone indicator (FZI) has been calculated to quantify the flow character of the Abu-Roash and Bahariya reservoir rocks based on empirically derived equations of robust correlation coefficients. The correlations among porosity, permeability, bulk density, mean hydraulic radius and pore throat flow properties reflect the most important reservoir behavior characteristics. The calculated multiple regression models indicate close correlation among petrophysical properties and Rh and R35%. The obtained models are able to predict Rh and R35% by using porosity and permeability, to map reservoir performance and predict the location of stratigraphic traps. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction There are many applications for pore geometry in reservoir quality studies and characterization. In many developed reservoirs, capillary pressure tests are not available while only porosity and permeability are available as they are routinely measured. In the present work, it is important to predict, either empirically or theoretically, the needed mean hydraulic and pore throat radius at certain mercury saturation values from available petrophysical properties to resolve reservoir fluid flow units. Pittman (1992) presented empirical relationships between porosity, permeability and the pore aperture size that corresponds to the displacement pressure and the apex of mercury injection plot. The study of petrophysical properties of rocks and theoretical models based on laboratory measurements and the application of empirical statistical correlations is often applied in these studies. The mean hydraulic radius (Rh) is a measure of pore channel flow efficiency. Flow speed along the channel depends on its ⇑ Corresponding author. http://dx.doi.org/10.1016/j.jafrearsci.2014.10.018 1464-343X/Ó 2014 Elsevier Ltd. All rights reserved.
cross-sectional shape (among other factors) and the hydraulic radius is a characterization of the channel efficiency. Based on the ‘constant shear stress at the boundary’ assumption, hydraulic radius is defined by Carman (1956), and Georgi and Menger (1994) as the cross section area normal to the flow section divided by the wetted pore perimeter (P), which is written as:
Rh ¼ fluid filled volume wetted area : Rh ¼ ð2Þ ðA=PÞ ¼ ðpr 2 =2prÞ ¼ ðr=2Þ
ð1Þ
where Rh: the hydraulic radius (L); A: the cross sectional area normal to flow (L2); P: the wetted perimeter (L) and r: pore throat radius of the rock sample. Thus, wetted perimeter times the hydraulic radius is equal to the area of irregular section flow. Normal flow occurs when the water surface slope (Sws), is the same as the bottom slope (So). When normal flow is approached, the velocity Manning (1891) equation is used to compute hydraulic radius which is proportional with the velocity of the fluid flow. The greater the hydraulic radius, the greater the efficiency of the channel and the more volume it
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can carry. For reservoir characterization some derived parameters are used by Georgi and Menger (1994) as;
K ¼ ð1=C 2 Þ ðu3 =ð1 uÞ3 Þ
ð2Þ
where K: permeability (md), u: porosity (%).
C ¼ ðXÞ1=2 ðSm Þ ðTÞ
ð3Þ
where X: Kozeny–Carman cross-section shape factor, Sm: matrix surface area (the specific surface area per unit grain volume) (cm1), T: tortuosity, defined as (ratio of the true path of the diffusion path of fluid particle diffusing in the porous medium, and the straight line distance between the starting and finishing points of the particle’s diffusion). These parameters are related to the flow zone indicator (FZI) which introduced by Amaefule et al. (1993). The reservoir rock specific parameter FZI characterizes the permeability versus porosity correlation for a porous rock and is assumed to be constant within the same hydraulic unit, but varies from one unit to another in the heterogenic reservoir. Flow zone indicator is a unique and useful parameter to quantify the flow character of a reservoir and it offers a relationship between petrophysical properties at small-scale, such as core plugs and at a large-scale such as well bore level. In addition, the term FZI provides the representation of the flow zones based on the surface area and hydraulic tortuosity. The (FZI) units can facilitate the tracking of waterfront displacement and predict the water flooding results. Flow zone indicator (FZI) values are estimated using Amaefule et al. (1993) relationship:
ðFZIÞ ¼ ðRQI=ur Þ
ð4Þ
where RQI is the reservoir quality index defines as:
ðRQIÞ ¼ ð0:0314Þ ðK=uÞ1=2
ð5Þ
pore to grain volume ratio ður Þ ¼ ððuÞ=ð1 ðuÞÞÞ
ð6Þ
During the past years, many studies have been done on the pore aperture corresponding to 35th percentile of cumulative mercury saturation curve which was developed by H.D. Winland, as stated in Pittman (1992) and it is focused on predicting of R35 from other data such as porosity and permeability. The R35 of a given rock sample type reflects its depositional and digenetic fabric and influences of fluid flow and defines reservoir performance (Hartmann and Coalson, 1990; Lala and Ahmed, 2012). Consequently, estimating R35 from logs, using the Winland’s model (Kolodzie, 1980), or directly from capillary pressure core data (when available) provides the basis for a common zonation that can be used by both geologists and reservoir engineers. Four petrophysical flow units with different reservoir performances are distinguished by ranges of pore throat radii and prove to be an excellent delineator of hydraulic units (Hartmann and Beaumount, 1999; Lala and Ahmed, 2012). The irreducible water saturation (Swir) is the percentage of the pore space that the mercury could not enter at infinite pressure (Jennings, 1987). In this study, a separate technique based on range of R35 flow unit type and Rh values was utilized to create a better correlation among effective petrophysical properties in all studies rock samples. This indicates that, pore throat radius and mean
Fig. 1. Location map of the study area, Western Desert, Egypt.
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hydraulic radius parameters are the primary controlling parameters of the relationship between porosity and permeability. Flow units can be identified from the calculation of pore throat radii at the R35% pore volume, which subdivides the data samples into units having similar and predictable flow characteristics, using laboratory measured mercury saturation curves. The approach presented in this paper differs with other procedures for resolving reservoir quality parameters in that it is based on assembling data with different R35, Rh and FZI, and hence potential reservoir performance. This approach enables facies and rock types to be quantitatively represented in terms of petrophysical flow unit types with distinctive ranges of petrophysical characteristics. Flow unit distribution scaled up to create new relationships between porosity and permeability improves permeability prediction using empirically derived model of high correlation coefficients.
2. Materials and methods A suite of 53 sandstone samples of Upper Cretaceous age from the southwest of Sennan Oil Field were selected for petrophysical analysis (Fig. 1). A data set of laboratory measurements which include bulk density, porosity, permeability (K), mean hydraulic radius (Rh), irreducible water saturation (Swir), R35, pore to grain volume ratio, reservoir quality index and flow zone indicator (FZI) for 53 sandstone core samples has been derived for the Upper Cretaceous sandstones of Abu-Roash and Bahariya Formations. Prior to measuring the petrophysical parameters in the laboratory, the rock samples were cleaned of pore fluid and other contaminants using the Soxhlet cleaning technique (Soxhlet, 1879) and dried in an electric oven by heating the samples to 98 °C, readying them for laboratory measurements.
Table 1 Laboratory generated petrophysical parameters for the Upper Cretaceous sandstone core samples from the Abu Sennan Oil Field, Egypt. No.
Formation
u (%)
log K (md)
Swi (%)
log R35 (lm)
log Rh (lm)
qb (gm/cc)
RQI
ur
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Abu-Roash Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya Bahariya
18.70821 18.67353 11.13822 11.48387 13.9267 14.7152 21.81612 12.24573 12.14688 26.8611 24.63116 17.66209 15.06982 14.8883 15.98642 4.115741 10.46734 9.815563 14.57766 18.23008 11.78556 10.37937 20.25234 18.34108 16.66019 20.19491 21.80093 17.09651 12.78836 16.52936 23.33072 22.53071 24.28992 16.4129 24.92778 24.88826 17.04025 16.31162 11.89696 12.64232 10.6923 13.33893 24.97723 25.31794 11.87687 14.1902 16.12715 14.25198 24.07556 24.34465 23.54725
0.50332 0.63609 0.87656 0.57402 0.79563 0.81974 0.77356 0.33262 0.25984 2.43571 2.307258 0.550014 0.34662 0.710672 0.8734 1.12243 1.31424 0.82685 0.008479 0.7528 1.10331 0.98238 1.76216 1.44698 0.19686 1.161548 2.316109 1.29498 1.4237 0.992661 2.30453 2.21538 2.624252 0.608546 2.043845 2.001419 0.68837 0.71222 0.52023 0.56206 0.88368 1.05878 2.41482 2.3765 0.68247 0.80469 0.96651 0.25854 2.40132 2.265076 2.166535
19.57 20.01 22.82 19.74 20.68 21.8 21.95 21.22 18.35 10.46 8.75 17.56 17.86 20.1 18.65 26.89 23.98 18.19 18.53 21.88 20.94 24.48 13.17 14.08 17.04 14.52 10.31 22.18 20.51 17.18 12.59
1.04721 1.10791 1.87615 1.79778 1.65365 1.60906 1.76447 1.84164 1.73518 0.68825 0.068186 0.98632 1.55284 1.64016 1.451 1.96658 1.98464 1.93181 1.34872 1.30627 1.51286 1.9431 0.321184 0.67366 0.95156 0.47756 0.758912 1.92812 1.97469 0.47756 0.739572
0.86328 0.89963 1.21467 1.21467 1.04576 1.05061 1.1549 1.21467 0.9914 0.08991 0.368473 0.23657 0.75449 1.02228 1.61979 1.56864 1.39794 0.61439 0.64207 1.05552 1.46852 0.693727 0.534026 0.50724 0.426511 0.868056 1.69897 1.74473 0.378398 0.85248
2.106262 2.106285 2.380595 2.365667 2.305961 2.287918 2.183707 2.348813 2.346151 1.936525 2.005602 2.202849 2.261124 2.264177 2.248496 2.617985 2.446363 2.443111 2.286117 2.23619 2.428208 2.384445 2.109763 2.190035 2.226331 2.115789 2.080425 2.349497 2.455424 2.210581 2.031525 2.051885
14.15 11.86 12.53 21.11 21.74 19.65
0.90309 0.514548 0.514548 1.16941 1.4318 1.90309
0.074451 0.598791 0.598791 0.84771 0.95468 1.05552
23.19 19.78 7.49 10.59 20.98 20.93 17.53 23.99 10.2 11.08 9.83
1.91721 1.88107 0.802774 0.745465 1.70115 1.83416 1.67778 1.69465 0.790285 0.705864 0.620136
1.20066 1.34679 0.910091 0.857332 1.26761 1.35655 1.38722 1.34679 0.904716 0.816241 0.730782
0.040668 0.034936 0.034296 0.04785 0.033666 0.031855 0.027591 0.061182 0.0668 0.31639 0.284981 0.140739 0.120555 0.184435 0.028731 0.04251 0.021374 0.038686 0.083047 0.030913 0.02568 0.031452 0.530606 0.387889 0.061328 0.266123 0.967713 0.0171 0.017048 0.242177 0.923061 0.847681 1.307202 0.156174 0.661471 0.630437 0.034435 0.034243 0.050015 0.046237 0.034718 0.025408 1.012902 0.962644 0.041528 0.033006 0.025698 0.112011 1.015784 0.863507 0.783839
0.230137 0.229612 0.125343 0.129738 0.161801 0.172542 0.279036 0.139546 0.138264 0.367262 0.326808 0.214507 0.177438 0.174927 0.190284 0.042924 0.116911 0.108839 0.170654 0.222944 0.133601 0.115815 0.253955 0.224606 0.199907 0.253053 0.278788 0.206222 0.146636 0.198026 0.304303 0.290834 0.320828 0.196357 0.332051 0.33135 0.205404 0.194909 0.135035 0.144719 0.119724 0.153921 0.332929 0.33901 0.134776 0.165368 0.192281 0.166208 0.317099 0.321784 0.307997
The black areas indicate no measurements.
2.228102 1.996642 1.9966 2.20992 2.245038 2.410551 2.340542 2.389367 2.519631 1.996486 1.984696 2.422515 2.29725 2.250845 2.318151 2.020109 2.017301 2.030921
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The porosity of the samples was measured by the Frank Jones helium porosimeter which is also considered as a volume-measuring instrument, which is normally used for determination of the volume of grains or the volume of pores in a sample. The principle of the instrument depends on the gas expansion theory, where a definite volume of the helium gas (of known pressure) is isothermally expanded into unknown void volume. After expansion, equilibrium will occur between the void volume and gas volume allowing the pressure to be measured. By using the resultant measurements and applying Boyle’s law, the void volume can then be calculated. The dried core samples were immersed in mercury in
Fig. 2. A plot of porosity versus permeability for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 3. A plot of porosity versus bulk density for sandstones of the Bahariya and Abu-Roash Formations.
calibrated pycnometer. The volume of mercury displaced by the sample is weighed. Knowing the mercury density at the measured temperature, the bulk volume of the sample can be determined. The porosity of a rock is defined as the ratio of the rock void spaces to its bulk volume expressed in percent. The Ruska gas permeameter (cat. no. 6121) was used for measuring the permeability of the samples using nitrogen gas. The clean and dry core samples were pushed into a rubber stopper of the corresponding hole size. Delicate or fragile core samples are first imbedded with a sealing wax into metal sleeve and placed in a core holder with the tapered end down.
Fig. 4. A plot of the bulk density versus permeability for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 5. A plot of bulk density versus log Rh for sandstones of the Bahariya and AbuRoash Formations.
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Our samples were tested for the capillary pressure measurement using mercury injection method. In this method, mercury is forced into the sample at the low pressure (1.169 kPa up to 779.44 kPa). The volume of mercury injected into the sample at each increasing pressure step is recorded after the stabilized condition has been achieved. The determination of capillary pressure was performed utilizing core laboratory mercury injection panel. The obtained mercury injection capillary pressure data were converted from laboratory (air/Hg system) into reservoir conditions (oil/water system) using equations from El Sayed (1993a) and El Sayed (1993b) resulting in the derivation of pore throat size distribution, R35-parameter and irreducible water saturation (Swir).
The petrophysical properties of the studied sandstone core samples cover a wide range of exploration interests with bulk density ranging from 1940 kg/m3 to 2620 kg/m3, porosity from 4.11% to 26.86%, permeability from 0.037 md to 420.97 md, mean hydraulic radius from 0.018 lm to 8.13 lm, irreducible water saturation from 7.49% to 26.89%, pore to grain volume ratio from 0.043 to 0.37, reservoir quality index from 0.017 lm to 1.31 lm, flow zone indicator from 0.083 lm to 4.07 lm and R35 from 0.01 lm to 6.35 lm. The pore aperture radii were determined graphically from the mercury injection curves corresponding to the 35th percentiles of mercury saturation whereas the irreducible water saturation (Swir)
Fig. 6. A plot of porosity versus permeability for Rh < 0.1 lm. Fig. 8. A plot of porosity versus permeability for R35 < 0.1 lm.
Fig. 7. A plot of porosity versus permeability for Rh > 0.1 lm.
65
Fig. 9. A plot for porosity versus permeability for R35 > 0.1 lm.
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was estimated graphically at maximum pressure required for mercury entrance into the core sample on the water saturation scale (Sw) (Table 1). The laboratory measured core data of porosities and permeabilities were used to calculate the pore to grain volume ratio (ur), reservoir quality index (RQI) and flow zone index (FZI) for the studied sandstone core samples representing sandstone layers at different depths. Statistical analysis system (SPSS) software of multiple regression equations were used to establish the various empirical relationships. Multiple regressions are an extension of the regression analysis, which incorporates additional independents variables in the predictive equations (Balan et al., 1995).
permeability values larger than permeability cutoff value which corresponding to R35 equal 0.1 lm will improve the porosity–permeability relationship and the predicted permeability values from measured porosity become more accurate for all studied sandstone samples. Below this cutoff value the reservoir cannot produce any hydrocarbon. The relationship between the graphically estimated (R35) and the calculated mean hydraulic radius (Rh) for the studied core samples is shown in Fig. 10 and indicate a strong correlation between the two parameters with a correlation coefficient of 96%. The
3. Results and discussion The bulk density, porosity, permeability, mean hydraulic radius, irreducible water saturation and R35 have been measured for all the core samples in order to validate the performance of the different reservoir formations. The effect of petrophysical flow unit types on the relationship between porosity and permeability for all the studied core samples and their influences on the formation evaluation was investigated. The porosity values of the studied sandstones show a direct relationship with the corresponding permeability (Fig. 2) and are characterized by a high correlation coefficient (r = 83%) for all the studied samples. This relationship can be used in permeability prediction for other samples of Upper Cretaceous rocks in the Sennan Oil Field when permeability measurements are not available. The porosity, permeability and mean hydraulic radius values are correlated with the bulk density. The porosity, permeability and mean hydraulic radius values decrease with the increase of bulk density (Figs. 3–5). The correlation between the bulk density and the rock porosity is perfect, while the correlations are reliable for both permeability and mean hydraulic radius (Rh), indicating the presence of the other factors which may control their relationships. The permeability–porosity relationships depend on the hydraulic radius (Rh) for the studied samples. The permeability–porosity relation for samples having Rh < 0.1 lm is illustrated in Fig. 6 and shows a cloud of points relationship without any orientation. This is possible because at these ranges of mean hydraulic radius, the velocity of fluid flow is very small and so the channel efficiency impedes its flow. In Fig. 7 a depiction of the porosity–permeability relationship for the samples having Rh > 0.1 lm show a reliable relationship with a high correlation coefficient (r = 87%). Improvement of this relationship occurs when the mean hydraulic radius (Rh) is Rh > 0.1 lm. This is attributed to the increase of the connectivity between pores resulting in increase of the flow velocity values and in turn causing an increase of the rock permeability. The equation of the line of correlation is shown in Fig. 7. The analysis of data shows that there is no relationship between porosity and permeability for samples with R35 < 0.1 lm of the nano flow unit type (Fig. 8). This is plausible, because at these ranges of pore throat radii with R35 smaller than 0.1 lm, permeability is too low and no fluid flow exists. The relationship between porosity and permeability for sandstone samples having the R35 > 0.1 lm of meso and macro flow unit type shows a very strong relationship with a high correlation coefficient (Fig. 9). The improvement of the porosity–permeability relationship is clear in this relation at this range of pore throat radii (R35) for this flow unit. The permeability values increase with increasing of R35 values and are directly related to fluid flow. The calculated regression equation is shown in the graph and is characterized by a robust correlation coefficient (r = 95%) (Fig. 9). Using
Fig. 10. A plot of log R35 versus log Rh for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 11. A plot of irreducible water saturation versus permeablity.
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values of the mean hydraulic radii (Rh) are closely related to the measured pore throat radii values at these different flow unit types (Fig. 10). It appears from the results of this study, that the sample data points are scattered where weak relationships exist at the nano scale of the pore throat range values, due to Manning equation and its friction factor which is determined for ordinary channel flows. It is not accurately applied to this scale of rock pore channel flow. Other reasons could be that the flow depths which have inverse relationship with the flow velocity and boundary roughness size are both closely related to the mean hydraulic radius. The inverse relationship between the irreducible water saturation (Swir) and the rock permeability (log K) is illustrated in Fig. 11
Fig. 12. A plot of the quality index versus pore to grain ratio for sandstones of the Bahariya and Abu-Roash Formations.
67
showing a strong correlation between them. The increase of irreducible water saturation (Swir) can be caused by either small pore throat radius or mean hydraulic radius leading to the decrease of fluid flow velocity and permeability. For all studied sandstone samples the porosity and permeability relationship is very weak at the scale of nano flow unit type because pore throat radii are too small and impede the fluid flow. However, at the micro, meso and macro flow unit types, the improvement of the relationship between porosity and permeability is caused by increasing the pore throat size and the amount of fluid flow. By using the mean hydraulic radius Rh and the graphically estimated R35, one is able to discriminate between the reservoir and non reservoir zones. In spite of this, some reservoir zones have higher porosity greater than the porosity cutoff (8–10%). These reservoir zones have low value of Rh and/or R35 of nano or micro flow unit type and so they are considered as non reservoirs. These non reservoir zones are considered as containing high grain roughness and channel tortuosity with bad connected small or large pores channels. Hence we cannot use the porosity cutoff concept to differentiate between the reservoir and non reservoir zones instead we can apply the permeability cutoff, according to the Rh and R35, making this discrimination more realistic and acceptable. The concept of permeability cutoff based on R35, is available in literature but is based on different data sets of different lithologic types (Lala and Ahmed, 2012). The relationship between reservoir quality index (RQI) and pore to grain volume ratio (ur) for the studied samples is illustrated in Fig. 12. The hydraulic unitization of the studied core samples is based on the geological attributes of pore geometry and tortuosity. The flow zone indicator (FZI) multiple straight lines represent different hydraulic units with homogenous pore network and similar tortuosity and fluid flow capacity. FZI values allow us to represent the reservoir by subdividing it into flow units with specified flow zone indicator. There is an excellent correlations between Swir (%) obtained from capillary pressure data, log K (md) and FZI (lm) versus depth for the studied sandstone core samples which are demonstrated in Figs. 13–15. As expected, irreducible water saturation (Swir) decreased with increasing FZI and log K. The application of the flow
Fig. 13. Plots of log k, Swi and FZI versus depth (1984–2000 m) for sandstones of the Bahariya and Abu-Roash Formations.
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Fig. 14. Plots of log k, Swi and FZ1 versus depth (2000–2060 m) for sandstones of the Bahariya and Abu-Roash Formations.
Fig. 15. Plots of log k, Swi and FZI versus depth (2072–2080 m) for sandstones of the Bahariya and Abu-Roash Formations.
unit approach in this study allowed us to model in time and space the hydraulic flow units and reservoir performance by identifying the layers that have the most important contribution in the fluid flow in both the Abu-Roash and Bahariya Formations in the Abu Sennan Oil Field.
4. Regression models Linear regression techniques have become popular for predicting permeabilities from other variables. The techniques are based on correlations between measurements that have plausible
connections to permeability without investigating the physics behind the equations. Many theoretical models, such as the Kozeny–Carman model, can be expressed in linear regression format. However, the limitation of regression models is that they are only valid within the range of variables from which the relationships are derived. To get a more general empirical equation, an analysis of relatively large and diverse data set is needed. In the present study, multiple regressions statistical analysis was done on a large data set. In general these relationships among the different investigated petrophysical parameters yielded good results in the prediction of petrophysical parameters and high coefficients of correlation. For example, using the permeability
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(K in md) as the dependent variable in the multiple regression involving radius of the pore aperture corresponding to the mercury saturation at 35% (R35 in m), mean hydraulic radius (Rh in m), porosity (u in %) and bulk density qb (kg/m3) yielded the following relationship:
log k ¼ 1:72 þ 0:022ðuÞ þ 0:95 logðRh Þ þ 0:28 logðR35 Þ 0:84ðqbÞ
69
Acknowledgments The authors are indebted to the Egyptian petroleum authority for permission to publish these laboratory results. The authors are also grateful to anonymous reviewers whose constructive comments helped to improve this manuscript. References
ð7Þ
The above equation has a correlation coefficient of 95%. Predictions of log k using the measured porosity, mean hydraulic radius, pore throat radius at 35% of mercury saturation and bulk density yielded reliable and acceptable results. Contribution factors (1.72, 0.022, 0.95, 0.28 and 0.84 respectively for constant a0, u, log Rh, log R35 and qb) indicate that the most important variable in this regression is u and log R35 because small values of their multipliers.
5. Conclusions The petrophysical methods described in this paper lead us to make the following conclusions: (i) The studied reservoir parameters like Rh and R35%, have been shown to be an excellent predictor of permeability in both the Bahariya and Abu-Roash formations. (ii) Mean hydraulic radius and pore throat radii at the 35% pore volume provide the best basis for defining reservoir flow units. (iii) FZI can be used for delineating the number of layers (hydraulic unit) required for estimating both geological and petrophysical parameters in reservoir simulation process. (iv) The multiple regression models obtained in this study are characterized by robust and high correlation coefficients which allowed accurate estimation of permeability values for the studied rock samples.
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