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Timestamp: 2019-04-19 14:30:53+00:00

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Evaluation of a net pay production schedule usually requires estimates of rock permeability. Precise prediction of reservoir rock permeability from open-hole wireline logs in uncored intervals is of fundamental importance. There is no single routine open-hole wireline log instrument that measures permeability directly. As a consequence, the current state of knowledge relies upon empirical relationships that use either porosity or resistivity data. Mohaghegh et al. (1997) provided a summary of empirical models based on the correlation between permeability, porosity, resistivity and the irreducible water saturation. A limited number of researchers have published empirical models that estimate permeability using open-hole log measurements (Saner et al. 1997; Xue et al. 1997; Yao & Holditch 1993). The log measurements used for permeability estimation included gamma ray (GR), induction deep resistivity, sonic and bulk density. These models show, in general, good correlations when applied, locally, to clean sandstone rocks with interparticle porosity. The applicability of these models to highly heterogeneous shaly sandstones, however, does not usually give plausible results (Perez et al. 2005). Some of these empirical models are often diffuse when used at water saturations that are not irreducible (Coates & Dumanoir 1973; Mohaghegh et al. 1997).
Nuclear magnetic resonance (NMR) log permeability models are usually constructed based on the relationship between the transversal relaxation time (T2) distribution and the pore-size distribution (Di & Jensen 2016). For well-sorted clean clastic rocks, this relationship is fairly consistent, and the prediction of permeability usually gives reasonable results (Kenyon et al. 1988). But in heterogeneous sandstones and in tight rocks, the prediction capability of permeability models using the T2 distribution poses a great challenge, and is sensitive to different rock lithofacies (Davis et al. 2006; Di & Jensen 2016).
In the above equations, k is permeability (in mD), ϕ is porosity (fraction), Rtirr is deep resistivity from a zone at Swirr (in ohm m), Rw is formation water resistivity at the formation temperature (in ohm m) and ρh is the hydrocarbon density (g cm−3).
It is clear from equations (1)–(3) that the Coates & Dumanoir (1973) model is limited to reservoir formations at irreducible water saturations.
depositional and diagenetic controls causing minor textural and geometrical pore structure variations within the same hydraulic flow unit.
(8)The above model represents an improvement in the quantitative prediction of permeability in uncored intervals, provided that precise estimates of the FZI and effective porosity values are accessible. Empirical inferences for estimating the FZI (Guo et al. 2007) are often diffuse and are plagued by large scatter, especially in highly heterogeneous reservoirs. These models generate considerable spread and are fraught by great uncertainty when applied to other locations with different depositional characteristics to the original location pertinent to the model data (Tricoranto 2002). This is possibly due to the lack of scientific foundation for selecting appropriate log measurements when coming up with FZI empirical and neural network models. A haphazard procedure is usually followed for selecting the input log measurements instead (Guo et al. 2007).
The objective of this study is to use dimensional analysis in order to explore the link between the FZI and key open-hole wireline log measurements. The following section illustrates how dimensional analysis was used to formulate empirical models for predicting the flow zone indicator and rock permeability.
(9)In the above notation, Δt is the compressional sonic wave travel time (in µs/ft), ρb is the bulk density (in g cm–3), Pe is the photoelectric absorption (in barns/electron), Rt is the true resistivity (ohm m) and Rwa is the apparent water resistivity (ohm m).
In the above notation, ϕ is the effective porosity and m is the cementation exponent. The apparent water resistivity (Rwa) is equal to the water resistivity (Rw) only in a water zone. Porosity, or the resistivity formation factor (F), might have been used in equation (9) instead of Rwa. Nevertheless, the dimensional analysis becomes erratic unit-wise. As a rule of thumb, validation of the dimensionless groups, using the relevant experimental data, requires less effort when all independent variables of the generic equation (9) have dimensions (Zendehboudi et al. 2011).
). When applied to any physical system, the Buckingham's pi theorem states that a dimensionally homogeneous equation, involving n variables, can be reduced to an equation with (n−k) dimensionless groups, with k being the number of independent reference, or fundamental, dimensions (Buckingham 1914).
The dependent variable (the FZI) must appear as a non-repeating variable in only one of the dimensionless groups.
The repeating variables have to be dimensionally independent of each other. In addition, the dimensions of any of the repeating variable must not be reproduced by some combination of products of powers of the remaining repeating variables dimensions.
All of the fundamental dimensions used in the analysis must be represented within a particular set of the repeating variables.
is denoted as the resistivity number. The contraction of the physical relationship between the flow zone indicator and the log-derived variables into a more succinct functional form given by equations (13) and (14) illustrates the main utility of dimensional analysis. For carefully defined variables contributing to the FZI variance (equation 9), dimensional analysis is likely to lead to a general relationship between the dimensional groups that may be applied for similar systems. This particular inference can only be confirmed with log data.
Log and core data are obtained for well B from an outstanding oilfield, in the Arabian Peninsula, penetrating two distinct terrigenous sandstone oil reservoirs (R1 and R2) with excellent quality and abundant oil production (Al-Sultan 2017). The log measurements consist of GR, caliper, neutron porosity, bulk density, compressional wave travel time (sonic log), laterolog deep resistivity and photoelectric absorption (Figs 1 and 2). All logs were depth shifted, and environmentally corrected, and corrected for washout effects (Al-Sultan 2017). Lithology in both reservoirs is composed mainly of sand and shale layer sequences, with a minor presence of siderite nodules, glauconite, and fragments of marine fossils and coal beds. Both reservoirs lie in an anticlinal structure composed of relatively complex geological features such as lateral lithology changes, and the existence of normal faults in some sections near the crest. The anticline structure has an elongated nose, with the crest of the structure appearing to be broad and relatively undeformed (Al-Sultan 2017). Sandstone layers of reservoirs R1 and R2 appear to belong to a sedimentary depositional environment that may be generally interpreted to be fluvial channels on a delta plane close to a river mouth (Al-Sultan 2017). Reservoir units consist of sandstone formations with different grain sizes: fine to medium grained, coarse grained and even conglomerates. Descriptive statistics of the data used to validate the dimensional analysis groups are given in Table 2. Core data of helium porosity and air permeability have been obtained for 335 depth intervals from the same well. As shown in Figures 1 and 2, core porosity measurements match reasonably well with the porosity values obtained from neutron and density logs (PHIE ND curves in Figs 1 and 2). More details about the data description and analysis are given in Appendix A.
Well-log suite for well B in reservoir R1 (Al-Sultan 2017).
Well-log suite for well B in reservoir R2 (Al-Sultan 2017).
increases. An increase in flow resistance causes a decrease in permeability, and a decrease in the FZI, as a consequence. The power-law trend is fraught by some scatter, though, with a correlation coefficient of c. 0.88. The premise of this dimensional analysis is that the dimensionless groups obtained (equations 13 and 14) appear to account for sufficient wireline log responses that may reflect the FZI of siliclastic rocks.
for reservoirs R1 and R2 penetrated by well B.
for various DRT values for reservoirs R1 and R2 penetrated by well B.
(18)In the above notation, NXRD is the normalized resistivity, NXRHO is the normalized bulk density, NXGR is the normalized gamma-ray measurement, NXSP is the normalized spontaneous potential, NXDT is the normalized compression travel time from the sonic log and NXNPH is the normalized apparent neutron log porosity.
Figure 5 illustrates a comparison between the FZI values calculated using the Guo et al. (2007) and measured FZI values, for reservoirs R1 and R2 of well B. The FZI estimates appear to be fraught with considerable uncertainty. For relatively low FZI values (less than 2 µm), the Guo et al. (2007) model appears to overestimate the predicted FZI. On the other hand, for high FZI values (greater than 2 µm), the Guo et al. (2007) model appears to underestimate the predicted FZI. Indeed, a linear regression analysis for estimating the FZI from wireline log measurements appears to be inadequate. Figure 6 illustrates a comparison between FZI values calculated using the FZI empirical model introduced in this study (equation 17) and measured FZI values, also using the same dataset applied in generating Figure 5. A satisfactory agreement is obtained between predicted and measured FZI values.
Estimated FZI using the Guo et al. (2007) model v. measured FZI values using data for reservoirs R1 and R2 of well B.
Estimated FZI using the dimensional analysis model (equation 17) v. measured FZI values using data for reservoirs R1 and R2 of well B.
(21)Therefore, including GR response, as an independent variable, for estimating the FZI becomes redundant. Neutron log porosity is also used to calculate the total porosity (see equation A4). The total porosity is used in return to calculate Rwa (equation 10). Thus, the neutron log porosity is implicitly accounted for in estimating the FZI.
, the FZI decreases as Pe increases.
The empirical model of permeability as a function of routine log measurements, given by equation (16), has been used to estimate permeability for the various rock types of reservoirs R1 and R2 of well B. As shown in Figure 7, the estimated permeability values using this approach were in satisfactory agreement with measured core data values. In general, estimated permeability values (Fig. 7), using the general empirical model given by equation (16), were much closer to the permeability values (Fig. 8) estimated using the Coates & Dumanoir (1973). Permeability estimates obtained from the Coates & Dumanoir (1973) appear to be plagued by a considerable spread and seem to be fraught with significant uncertainty.
Estimated permeability using the dimensional analysis permeability model v. measured permeability using data for reservoirs R1 and R2 of well B.
Estimated permeability using Coates & Dumanoir (1973) model v. measured permeability using data for reservoirs R1 and R2 of well B.
The uncertainty associated with the Coates & Dumanoir (1973) permeability predictions for this field case might be caused by a violation of the irreducible water-saturation condition in reservoir R1, and by a violation of the stationarity condition. An irreducible water-saturation condition, in the reservoir, would be confirmed by a constant bulk volume water (BVW) profile as a function of depth (Asquith & Krygowski 2004). It is the product of porosity by water saturation at any logged depth. Figure 9 indicates a variable BVW for all depth intervals above the water zone identified by the lowest resistivity (Rt) profile shown in Figure 1. Indeed, the resistivity profile of the middle permeable zone indicates an oil column above a water aquifer. The gradual increase in resistivity above the water aquifer indicates a capillary transition zone in reservoir R1. This fact is also confirmed by the production data of this reservoir. As indicated by the resistivity (Rt) profile of Figure 2, none of the permeable zones, of reservoir R2, containing hydrocarbon is in communication with a water aquifer. Figure 10 indicates that the BVW is relatively constant with depth for all the hydrocarbon zones of reservoir R2. Therefore, reservoir R2 does not have any capillary transition zone. Indeed, all of the permeable zones of reservoir R2 are at an irreducible water-saturation condition. This is also confirmed by the production data of this reservoir. As shown in Figure 7, the satisfactory prediction capability of permeability using the dimensional analysis model (equation 16) does not appear to be affected by the presence of a capillary transition zone in reservoir R1.
A display of bulk volume water (BVW) for cored depths of reservoir R1 in well B.
A display of bulk volume water (BVW) for cored depths of reservoir R2 in well B.
is the error in the wellbore radius.
(24)The average relative error obtained for estimating permeability, by applying the dimensional analysis model (equation 16) on reservoir R1 and R2 data, is c. 36% with a standard deviation of about 32% (Fig. 7). This relative error implies that the predicted permeability values are within less than an order of magnitude variation of the measured permeability values. As indicated from equation (24), 36% error in permeability also leads to an average relative error in the production schedule of about 36%. On the other hand, estimates of permeability using the Coates & Dumanoir (1973) model indicate, on average, about a three orders of magnitude variation from measured values (Fig. 8). Likewise, this error in permeability leads to estimates of the production schedule within a three orders of magnitude variation from the actual production schedule. The application of the dimensional analysis model for estimating permeability (equation 16) appears to improve the evaluation of the net pay production schedule considerably for the field case investigated.
Rock quality index (RQI) v. normalized porosity index (ϕz) for reservoirs R1 and R2 penetrated by well B. The straight lines are iso-DRT lines.
Rock quality index (RQI) v. normalized porosity index (ϕz) for various DRT values for reservoirs R1 and R2 penetrated by well B.
undermines the use of DRT for making a distinction between various rock types.
Figure 13 compares the rock types predicted from discrete rock type values (equation 7) with rock types predicted based on the classification functions scores (equation 28). The exact matching of rock types occurs for 288 samples out of 330 samples tested (Fig. 13), corresponding to a success rate of c. 87% (Table 7). The observations of rock type 1 falsely identified, are all identified as rock type 2. The falsely identified rock type 2 is almost equally identified as either rock type 1 or type 3. Similarly, nine out of 11 falsely identified rock type 3 are identified as rock type 2. Indeed, as indicated in Table 3, rock types 1, 2 and 3 are characterized by a low range of permeability and by a great overlap in the porosity range. This fact mitigates the effects of false identification of rock type on the estimation of permeability. Nevertheless, it seems that the DRT approach for rock type delineation is more robust for rock types in the high-permeability range than for rock types in the low-permeability range. The reasonable matching frequency between the discriminant functions rock type output and the rock types obtained using the DRT analysis asserts the sturdiness of the rock type identification using the DRT alone (equation 7). Indeed, the delineation of rock types using the DRT analysis yields reasonable correlations for permeability as a function of the effective porosity (Fig. 14). This fact confirms that the resulting rock types mimic the geological controls of rock texture and diagenetic processes affecting the structure of the hydraulic flow units.
Percentages of correctly identified rock types using the discriminant analysis on data of reservoirs R1 and R2 of well B.
Measured permeability v. effective porosity for various DRT values for reservoirs R1 and R2 penetrated by well B.
(29)The application of the GHE approach identified six global hydraulic flow units (GHE-1–GHE-6). Originally, eight hydraulic flow units were identified using the DRT analysis. A distinct trend in texture contrast appears to be associated with the global hydraulic elements. GHE-3 and GHE-4 are associated with fine and poorly sorted sands with clay- and silt-rich shales, respectively. These two global hydraulic elements have the worst reservoir rock quality (Tables 3 and 8). On the other hand, the coarse-grained well-sorted sands and the clean conglomerates are associated with GHE-8, which has the best reservoir rock quality (Tables 3 and 8).
groups that are highly correlated. Moreover, the prediction of permeability using the GHE petrotyping approach (Fig. 16) appears to yield comparable results to permeability prediction using the DRT approach (Fig. 8). The average relative error in estimated permeability using the DRT approach is about 36% with a standard deviation of 32%. The average relative error in permeability using the GHE petrotyping approach is about 35% with a standard deviation of 41%. Both petrotyping approaches yield, on average, satisfactory permeability estimates within less than an order of magnitude variation from the measured permeability.
Dimensionless FZI v. dimensionless resistivity for various GHE values for reservoir R1 and R2 penetrated by well B.
Estimated permeability using the GHE petrotyping approach v. measured permeability for data for reservoirs R1 and R2 of well B.
is obtained when the data are partitioned into unique hydraulic flow units. The dimensionless groups have been validated with sandstone reservoir data from a prominent onshore oilfield from the Middle East. A non-linear empirical model for predicting the rock FZI using open-hole log measurements has emerged as a consequence. Dimensional analysis reveals that the FZI is an intricate non-linear function of open-hole log responses. Therefore, the use of linear regression analysis models for estimating the rock FZI as a function of open-hole log measurements may be inadequate. The dimensional analysis approach introduced does not appear to require reservoir layers to be at an irreducible water-saturation condition.
The field case presented shows that it is possible to generate reliable estimates of FZI and rock permeability from wireline open-hole log measurements when rocks are properly classified into distinct hydraulic flow units. The systematic classification of hydraulic flow units using a standard reference approach, such as the discrete rock type (DRT) or the global hydraulic elements (GHE), appears to identify clear rock-quality progression trends with increasing DRT and GHE values.
The data (Figs 1 and 2) consist of gamma ray (GR), caliper, neutron porosity, bulk density, compressional wave sonic interval transit time, laterolog deep resistivity and photoelectric absorption. The data belong to two sandstone reservoirs, R1 and R2, from well B of a prominent sandstone oilfield in the Middle East.
The well is cored at 335 depth intervals. Conventional core analysis was performed on all collected samples. The permeability of the cored intervals is measured using an air permeameter at a pressure of 1000 psi. The coupled interaction between the deposition and the diagenesis processes has caused the porosity and permeability to be highly variable. Permeability values for this well vary over roughly six orders of magnitude from 0.04 mD in low-quality zones to 10 000 mD in high-quality zones. As shown in Figure A1, permeability appears to have a bimodal distribution with a positively skewed left mode and a negatively skewed right mode. Porosity varies from 5.7 to c. 35.5%, and appears to have a negatively skewed distribution (Fig. A2). As shown in Figure A3, the flow zone indicator (FZI) values calculated from the core data appear to have a bimodal distribution, with the central mode pseudo-normal. As shown in Figure A4, a plot of permeability v. porosity displays a general trend of increasing permeability with increasing effective porosity, with a sizeable amount of data scatter. As shown in Figure A4, for a fixed effective porosity of 15 or 20%, permeability varies over approximately three orders of magnitude.
Frequency histogram of measured permeability for reservoirs R1 and R2 penetrated by well B.
Frequency histogram of the effective porosity for reservoirs R1 and R2 penetrated by well B.
Frequency histogram for the calculated flow zone indicator (FZI) from cored data for reservoirs R1 and R2 penetrated by well B.
Measured permeability v. calculated effective porosity for reservoir R1 and R2 penetrated by well B.
(A4)where ϕd is the apparent density log porosity and ϕn is the apparent neutron log porosity. As indicated by Thomas & Steiber (1975), the approximation of the effective porosity using equation (A3) is appropriate for a pore-lining dispersed shale. Otherwise, an evaluation of the clay distribution is fundamental for an accurate estimation of the effective porosity. For instance, equation (A3) does not adequately approximate the effective porosity for a structural or laminated shale.
(A5)In the above notation, Swirr is the irreducible water saturation obtained from capillary pressure measurements.
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