Pre-drilling assessments of average porosity and permeability in the geothermal reservoirs of the Danish area

Many workers have discussed the correlation between porosity and permeability measurements, and despite somewhat imperfect correlations, they succeeded in establishing either exponential relationships (e.g. Tiab and Donaldson 2004) or trends defined by power functions (e.g. Doyen 1988; Mavko and Nur 1997).

A number of methods for determining permeability from porosity are described in literature, including empirical approaches and various modelling techniques. Already in 1927, Kozeny proposed to predict permeability from porosity, geometry of the pores and the specific surface of the solids in contact with the fluid. The Kozeny equation was later modified by Carman (1937, 1956) to become the Kozeny–Carman equation, which estimates the permeability of a porous media based on a grain size distribution. Block (1991) outlined a predictive approach based on the correlation between porosity and permeability in which the predictive applicability is constrained by the limits defined by a calibration dataset that includes various petrographic variables or parameters. Block (1991) even provided an equation in which the permeability is expressed as a function of grain size, sorting and rigid grain content. Evans et al. (1997) discussed a permeability prediction methodology that is based on a combination of empirical and geological modelling approaches. Originally the Kozeny equation was derived for clay-free sand with high porosity, and for that reason Walderhaug et al. (2012) suggested a modified Kozeny equation for predicting permeability in quartz-rich sandstones, even with high clay content. This modified Kozeny equation includes a parameter reflecting the type of pore system, and usually this parameter is to be considered a constant for a specific sandstone unit. With respect to modelling, Vadapalli et al. (2014) present fractal and Monte Carlo simulation approaches for permeability prediction.

Our porosity modelling is based on data from the Bunter Sandstone and Gassum formations. The permeability modelling is demonstrated by data from the Gassum formation.

Quantification of reservoir permeability, which is the single most critical factor for geothermal fluid extraction, is complicated as very few in situ measurements are available and moreover, permeability logs are not available from wells in the Danish onshore area.

Determination of reservoir permeability requires good-quality well test data, i.e. tests based on sufficiently long test intervals and flow/build-up periods of long duration. However, such high-quality test data only exist for a few Danish onshore wells (e.g. at Stenlille and Tønder; Fig. 1), and well test data are generally not available. So as an alternative, it is suggested to assess the permeability from an analysis of porosity log data supplemented by information from porosity–permeability relationships based on core analysis data. Core analysis data commonly include gas permeabilities measured at laboratory conditions, but conversion into corresponding reservoir fluid permeabilities (at field scale) is not a straightforward task. This conversion normally involves transformation of the core scale gas permeability into liquid permeability followed by upscaling from laboratory scale to field scale. The correction from gas to liquid permeability is normally carried out as a Klinkenberg correction of laboratory measurements (Klinkenberg 1941). The Klinkenberg correction is quite important for low-permeability sandstones, where a factor of 0.5 may apply, but for high permeabilities (>100 mD) the Klinkenberg correction is less pronounced and a factor in the order of 0.8–0.9 may apply (Tanikawa and Shimamoto 2009; Duan and Yang 2014). The use of core permeability values for upscaling to a full reservoir volume assumes that the sampling of the core samples is representative, which is not always the case. The arithmetic averaging used herein is based on the concept of horizontal, along layer flow, and occurring in a layered formation, which has been shown to be closely comparable to that from more advanced upscaling schemes (Kazemi et al. 2012).

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The Danish onshore core analysis database is considerably larger than the number of well tests, and the purpose of analysing core permeability data is to provide an estimate of the average permeability that characterizes a particular layer, but also to address the uncertainty of a predicted permeability estimate. A single permeability value interpreted from well test data is, however, not directly comparable to an average permeability based on core data, but the two permeability assessments are comparable at a relative scale, on the assumption that the thickness of test interval corresponds to the length of the cored interval.

The Upper Triassic–Lower Jurassic Gassum formation is present in most of the Norwegian-Danish Basin (Fig. 2), except above large salt structures and on the Ringkøbing–Fyn High (Nielsen 2003). It is also present in the northern North German Basin, but here its patchy and shallow occurrence makes it less suitable for geothermal exploration. In general, the formation thickness varies from c. 30 m to more than 300 m (Michelsen et al. 2003). Nielsen (2003) described the geological development of the Gassum formation in detail: the formation consists of shallow marine, fluvial and estuarine sandstones interbedded with marine and lagoonal mudstones, and also siltstones and minor coal beds are present. During the Late Triassic to Early Jurassic, the sedimentation was influenced by repeated sea-level fluctuations and rivers transported eroded material from the Fennoscandian Shield into the Norwegian-Danish Basin. Several sandstone layers were deposited as regressive shoreface sands, but also as fluvial sands deposited in river channels or estuaries. The Gassum formation was deposited in a humid climate, and the sandstone layers consist mainly of fine to medium-grained sand. The mudstones were primarily deposited in lagoons, lakes and offshore shelf areas. The reservoir quality of the sandstones is generally good to excellent due to the presence of rather continuous sandstone beds with low clay content (Weibel et al. submitted).

In general, the permeability reduction with burial depth is more pronounced for fine-grained sandstones than for coarse grained, but also the content of detrital clay, sorting and other elements related to variations in the depositional environments and source area proximity affect permeability (Weibel et al. submitted; Olivarius et al. 2015a, b). The presence of diagenetic cements may result in substantial permeability reduction, as the cement reduces the size of the pore throats. A synopsis of the key factors affecting porosity and permeability compiled from Olivarius et al. (2015a) and Weibel et al. (submitted) is presented in Table 1, covering both the Bunter Sandstone and Gassum formations. The tabulation summarizes the results of two diagenesis studies based on analyses of thin sections, scanning electron microscope images, cores and cuttings samples.

Well-log data of variable quality have been acquired in hydrocarbon and geothermal exploration wells drilled in the Danish onshore area during the last c. 60 years. The porosity variation is determined from the interpretation of well-log data, and log interpretation results form the basis for calculating net sand properties. Both the total and the effective porosity of the reservoir units are considered. The total porosity (PHIT) is the total volume of inter-granular porosity plus clay bound water, whereas the effective porosity (PHIE) does not include the water bound by the clay particles. The core porosity data are directly comparable to the log-derived total porosity data. The term ‘net sand’ or ‘potential reservoir layer’ is herein defined as sandstone having minimum 15 % porosity and a clay or shale content of less than 30 %. These cut-offs are adopted from hydrocarbon exploration practice and applied herein to ensure that only potential reservoir sands with sufficient storage capacity and permeability are considered. Data representing non-reservoir sandstone and mudstone intervals are thus removed prior to further data analysis.

Only a limited number of well test data are available for permeability assessments in the Danish area, since most of the data for this study originates from dry hydrocarbon exploration wells. These wells were not tested, but usually logged and sometimes also cored. Accordingly, the present study focuses on alternative ways of predicting permeability, namely a combined analysis of petrophysical well-log data and routine permeability measurements on core plugs, with the objective to provide an average gas permeability of a particular reservoir unit. For that reason a direct link between the gas permeability measured in the laboratory and the actual liquid permeability of the reservoir in the subsurface is needed for precise pre-drilling estimates of the expected performance of a given geothermal reservoir.

All available core analysis data from the Danish onshore area have been considered in order to construct a robust database of porosity and permeability data. In general, measurements were performed according to the API RP-40 standard (American Petroleum Institute), i.e. He-porosity was measured at unconfined conditions and gas permeability was measured at a confining pressure of c. 2.8 MPa (400 psi), and at a mean nitrogen gas pressure of c. 0.15 MPa (c. 1.5 bar absolute). It is not always possible to obtain information about the test conditions, as the large number of analyses were performed over several decades and by various companies. These companies were, however, expected to follow the existing standards and it is thus assumed that measurements were conducted at conditions corresponding or comparable to the API RP-40 standard. Both Klinkenberg corrected and uncorrected gas permeabilities are included in the database, but primarily uncorrected permeabilities form part of the porosity–permeability plots presented herein.

Gas permeabilities measured in the laboratory do not equal reservoir permeabilities, whereas well test data are considered to provide reservoir permeability estimates on the condition that the height (thickness) of the test interval is well-known. Strictly speaking well test data only supply a transmissivity measure (i.e. reservoir height multiplied by reservoir permeability). Permeability interpreted from well test data is commonly up to 2 times higher than the corresponding core permeability (e.g. Wolfgramm et al. 2008); for example presence of fractures in the reservoir rock usually results in permeability enhancement. The presence of fractures or overall inhomogeneities cannot be validated for geothermal reservoir rocks onshore Denmark, but our interpretations of well test data from 5 wells indicate a liquid permeability that is about 1.5 times higher than the core permeability (Fig. 3). A prerequisite is that both core and well test permeabilities exist for the same interval.

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First step in the 5-step procedure is to establish a porosity-depth model. The basic principle behind the porosity model is an assumed porosity-depth relationship, which previously has been documented for other basins (e.g. Gluyas and Cade 1997).

The porosity-depth model (step 1)

Seismic data provide information about depth and thickness of a geothermal prospect, but usually information about the average porosity must be modelled using porosity data from nearby wells. Porosities have primarily been interpreted from well-log data, and these data form the basis of establishing a correspondence between porosity and depth. The interpreted porosity log for each well is therefore averaged in order to determine the average porosity for specific reservoir intervals. A reservoir interval presumes a minimum porosity (herein >15 %) and a maximum shale content (herein <30 %) as described above. Therefore cut-offs were applied prior to calculating average porosities, i.e. the calculated porosities are average net porosities. The effect of applying cut-offs is that reservoir layers within each formation are identified and then assigned a porosity value on a well-to-well basis. One porosity-depth trend is observed for the Bunter Sandstone formation and another for the Gassum formation (Figs. 4, 5). The depth scale has been modified due to differential uplift across the Norwegian-Danish Basin (Japsen and Bidstrup 1999; Japsen et al. 2007), and hence the standard depth scale is replaced by ‘estimated maximum burial depth’.

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In step 2–4 of the 5-step procedure a regional porosity–permeability model is established (step 2), refined (step 3) and adapted to be applicable to local conditions (step 4). Step 5 suggests a method to narrow the permeability uncertainty range.

The initial permeability model (step 2)

Initially an empirical porosity–permeability relationship is established for each geological formation on the basis of available conventional core analysis data from all relevant wells using power-law-based curve fitting in line with the Kozeny law (Kozeny 1927). The resulting function used for calculating permeabilities is termed the initial permeability model and is illustrated in Fig. 6 and expressed in Eq. 1;

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$$k_{\text{ini}} = { 3}. 4 7\cdot 10^{ - 4} \cdot \varphi^{ 4. 38} ,$$

(1)

where k_ini is the initial permeability (in mD) and \(\varphi\) is the porosity (in %). The log-derived porosity was used as input data, and in this way a synthetic permeability log is assigned to all study wells, including the uncored wells. The permeability was calculated both from the effective and the total porosity, but the permeability curve calculated from the effective porosity is considered the most reliable permeability estimate, since it takes into account that the actual reservoir permeability is reduced in shaly and clayey intervals. However, the total porosity approximates the effective porosity in clean or almost shale-free reservoir intervals, and the synthetic permeability curves calculated from the total and effective porosity, respectively, are not significantly different in reservoirs with (very) low clay content.

The local permeability model (step 4)

The general permeability model represents the entire Danish onshore area and constitutes a template for constructing more refined local permeability models. Reservoir intervals of a specific formation in a local region may be characterized by deviating permeability distributions compared to the general trend (step 3). If local core permeability data are available, such data should therefore be used to calibrate the general permeability model to the local area. The use of the general permeability model in constructing local models is implemented with Eq. 3:

$$k_{\text{L}} = C \cdot k_{\text{G}},$$

(3)

where k_L is the local permeability (in mD) and C is a constant controlled by the local permeability variations, and it may be determined from permeability measurements on core plugs from one or two local wells.

Core porosity and permeability data from selected wells drilled throughout the Danish onshore area have been averaged for each well and then compared to the general trend line (Fig. 7). The figure focuses on deviations from the general model, and it appears from the figure that permeability averages deviate by a factor of up to four as indicated by the upper and lower bounds, even though some data points fit the trend line. However, this spread is caused by the use of regional data representing the entire Danish onshore area, which masks local and often more limited permeability variations. The uncertainty connected to a local dataset is less than indicated by the regional data set due to e.g. facies uniformity, and local permeability models are therefore needed for delivering locally adjusted permeability predictions.

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Permeability uncertainty range (step5)

Even a local, calibrated porosity–permeability model is associated with an uncertainty range owing to the variations in lithology and diagenetic alterations within a limited local region (geological province). Such a restricted area may only be represented by one or two wells with wireline logs and no cores, and therefore it is difficult to determine a suitable uncertainty range related to the permeability estimate representative of any local area on the basis of such a limited database. In order to get an impression of the local variability, a selected area with good well coverage has been studied in detail. 20 wells have been drilled within a restricted area at the Stenlille gas storage facility located in the eastern part of the Norwegian-Danish Basin (Fig. 1), and a large database comprising both core analysis data and well-logs exists (these data are all available from the GEUS archives). Ten of these wells are cored in the upper part of the Gassum formation, and evaluation of 3D seismic data, correlation of well-logs and interpretation of cores indicate that relatively uniform geological conditions prevailed in the area during deposition. When all unprocessed core porosity and core permeability data representing the upper part of the Gassum formation are plotted in one cross-plot, the data points show an expected substantial scattering (Fig. 8). Hence the averaged core porosity and core permeability data are used to establish a local Stenlille model for this part of the Gassum formation and most importantly, also to assess the uncertainty range of the local porosity–permeability relationship (Fig. 9). The figure shows that the scatter is notably reduced when applying the averaging technique, but also that the distribution is displaced towards lower average permeabilities compared to the general model. This displacement could be due to differences in grain size as discussed later. The key message from the figure is, however, that the high and low bounds of the uncertainty band are limited by factors of 2 and 0.5 of the model mid-line.

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The result of the porosity modelling is the establishment of two regional porosity-depth trends as presented in Figs. 4 and 5.

Applying the averaging technique for predicting permeability on the basis of core permeability data and well-log interpretations adds to the applicability of porosity–permeability relations. A “best practice” method is suggested for predicting the average permeability of a potential geothermal reservoir. The average permeability of a geothermal prospect is modelled (predicted) using the closest local permeability model, i.e. a permeability model valid for a nearby geological province. This permeability value along with estimated net sand thickness are considered the key factors, when assessing the geothermal potential of a particular prospect. From our experience, the geothermal resource is of economic value if the reservoir transmissivity is greater than 10 Darcy-metre. This assessment is in line with indicative values published in Seibt and Kellner (2003). The net sand thickness may be found from seismic isochores combined with extrapolation of net-to-gross ratios. Acquisition, interpretation and depth conversion of seismic data are thus pre-requisites for addressing the geothermal potential of a prospect. The determination of net sand thickness in a particular well is exemplified in Fig. 10.

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Porosity–permeability plots based on conventional core analysis data are commonly scattered as shown and discussed above, but the use of the averaging technique presented herein reduces the scatter and narrows the width of the uncertainty band associated with a predicted average permeability value. It is thus recognized that the uncertainty of the average value is significantly smaller than the uncertainty of a single measurement.

The use of the methodology is exemplified by the derivation of a local model as presented in Fig. 11. The model is illustrated by mid-line, high and low bounds, defining an uncertainty envelope, and it is assumed that the model is a representative of the Gassum formation at the northern basin margin. Averaged core analysis data from wells located in this area (Børglum-1 and Flyvbjerg-1; Fig. 1) point to permeabilities higher than indicated by the general model. A local model based on Eq. 3 was therefore established with the scope of predicting the average permeability of the Gassum formation in the Børglum–Flyvbjerg areas, expressed by Eq. 4:

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$$k_{\text{L}} = { 1}. 7\cdot k_{\text{G}},$$

(4)

where k_L is the local permeability (in mD) and the constant (C = 1.7) is determined from analysis of local core analysis data.

The conventional core analysis data from the Børglum-1 and Flyvbjerg-1 wells are also plotted after applying a porosity cut-off (Fig. 11). The mid-line honours calculated averages of the core analysis data from the two wells. Data points plotting below the uncertainty envelope were, however, excluded prior to calculating averages, because these data points represent sandstones with large amounts of fine-grained material (shale, mudstone and siltstone) not contributing to reservoir performance. The width of the uncertainty band is transferred from our Stenlille study, i.e. factors of 2 and 0.5 define the high and low bounds. The uncertainty band does not address the spread in the actual core analysis data, but it points out an uncertainty range that is related to the average permeability for a typical reservoir, presuming that the reservoir sandstones are almost shale-free. Furthermore, the example demonstrates that it is difficult to narrow the uncertainty range on the basis of a limited set of conventional core analysis data.

The formation brines of the potential reservoir sandstones in the Danish onshore area are characterized by hydrostatic pressure, so sandstone porosity is not affected by overpressure. Knowledge of the pressure distribution across the Danish onshore area comes from analysis of well test data. Our study indicates, however, that the sandstone porosity more likely is related to geological factors such as burial depth, lithofacies, clay content and diagenesis. In addition, the porosity-depth trend is affected by differential uplift and erosion; thus present-day depths were corrected for uplift using exhumation data presented in Japsen and Bidstrup (1999) and Japsen et al. (2007). For simple modelling purposes in undrilled areas, the porosity may be considered to be controlled by depth alone, provided that ‘present-day depths’ are replaced by ‘estimated maximum burial depths’.

One porosity-depth trend is observed for the Bunter Sandstone formation and another for the Gassum formation (Figs. 4, 5), despite the fact that local deviations from the general trends are observed. These deviations are presumably related to geological factors as outlined above. An empirical porosity-depth relationship by Gluyas and Cade (1997) is also plotted to compare the current porosity-depth trends with a suggested normal compaction curve. The Gluyas and Cade curve presumes no uplift and reflects mechanical compaction in uncemented sandstones from hydrocarbon fields with high porosity and no overpressure. These general porosity-depth trends (cf. Figs. 4, 5) are suggested for predicting the average porosity of potential reservoir sandstones in undrilled areas. A complete match between depth and porosity has not been fully achieved although maximum burial depths are applied, since the porosity also depends on factors other than depth, e.g. gain size (Table 1). However, for modelling purposes the use of a porosity-depth relation is considered satisfactory for porosity prediction and assessment. Consequently, modelled porosities are associated with uncertainty, even if the reservoir depth is well-known prior to drilling. In specific areas where the porosity distribution is well-known from local well and core data (e.g. at Tønder; Fig. 1), a local porosity-depth trend should be used.

Reservoir performance, including flow rates, is closely related to porosity and permeability, and information about permeability is essential for determining reservoir transmissivity unless well test data are available. Correct prediction of sandstone permeability is a challenge for potential geothermal reservoirs, since fluid flow depends on a number of factors. Compaction processes, precipitation of cement and presence of detrital clay lead to smaller pore throat sizes, thus lowering the resulting reservoir permeability.

Usually there is a relatively clear correlation between core porosity and core permeability data, but this correlation is somewhat ambiguous due to scattered data resulting from differences in the depositional environment, clay content and diagenetic development. The variability in the core analysis data means that a perfect correlation between core porosity and core permeability data cannot be obtained. Despite these uncertainties, this computed porosity–permeability relation forms the basis of calculating a log-based permeability curve for each well using the log-derived porosity as input data.

The observed variation in permeability for a fixed porosity value (Fig. 7) is most likely related to variations in the original depositional environment when the sandstone layers were formed, expressing itself in differences in diagenetic development, lithology, grain size distribution and clay content (Table 1). These variations emphasize the need of incorporating geological data, preferably both sedimentological and palaeogeographical data, and also the need for establishing local porosity–permeability models representative of local geological provinces characterized by relatively uniform geological development. Such local permeability models are herein suggested for predicting the average permeability of geothermal prospects. In fact, access to local permeability data from wells located close to the prospect in question is essential for using our method for predicting permeability.

The methods presented herein are developed on the basis of data from Danish geothermal reservoirs. The methodology for predicting average porosity and permeability is, however, considered applicable to similar sandstone reservoirs in other settings, but it has to be proved with additional data.

The width of the uncertainty band related to average permeability is addressed by analysing core analysis data from the local Stenlille field with several cored and logged wells located closely together. A petrophysical evaluation of the Gassum formation in the Stenlille-1 well is illustrated in Fig. 10. The character of the Gassum formation at Stenlille is considered representative of geothermal reservoirs found in the formation onshore Denmark. The Gassum formation consists of a number of sandstone layers interbedded with clay-rich intervals. Hence the Gassum formation is assigned a lithology column along with porosity and permeability curves as shown in Fig. 10. The porosity is interpreted from well-log data, whereas the permeability is calculated from the porosity–permeability relationship shown in Fig. 8.

A local Stenlille permeability model was initially established on the basis of averaged core porosity and permeability data from 10 Stenlille wells cored in the upper part of the Gassum formation, corresponding to the main gas storage reservoir (Fig. 9). In addition, the core analysis data are relevant for calibrating the log-based porosity and permeability interpretations (Fig. 10). Less core material exists from the lower part of the Gassum formation and the geological conditions that prevailed during deposition of the lower part differ from those of the upper part (Nielsen et al. 1989; Nielsen 2003). With respect to permeability, the core analyses show that the overall Stenlille permeability is approximately a factor 2 less than the permeability estimated from the general, basin-scale model (Fig. 9). The lower average permeability values observed at Stenlille are presumably caused by dominance of finer-grained sandstone compared to the sandstones defining the general permeability model. The presence of fine-grained reservoir sandstones at Stenlille is evidenced by sedimentological core log descriptions of the Stenlille-1 core (Nielsen et al. 1989). The permeabilities of these sandstones (Fig. 8) are generally lower than commonly seen in Gassum formation sandstones deposited elsewhere, and the lower permeability level at Stenlille may therefore be linked to grain size.

The averaged core analysis data from the Stenlille wells suggest that the uncertainty range can be expressed by multipliers of 2 and 0.5 (Fig. 9). This procedure involves multiplying and dividing the local Stenlille permeability trend by a factor of 2 in order to determine an upper and lower bound that delineate the data points, i.e. an envelope ranging from the lowest to the highest average permeability values. As the core permeability data do not follow a normal distribution, the uncertainty cannot be described by conventional standard deviation and instead, we suggest applying this uncertainty envelope. We assume the Stenlille dataset to be relevant for assessing the uncertainty range for a typical Gassum geothermal reservoir at any location in Denmark.