DELIVERING A SIMPLE CONVINCING SATURATION MODEL WITH INFINITE ROCK TYPE USING ITERATED FRACTAL FUNCTION SYSTEM
Yudiyoko Ega Sugiharto
Benard Ralphie
Dzulfadly B Johare
PETRONAS, Kuala Lumpur, Malaysia
ABSTRACT
The purpose of this paper is to design a saturation height function which can overcome the measurement insufficiency and can also can be applied to the reservoir models where core measurements are not available at all for all kind of siliciclastic reservoirs.
This paper adopts a quantitative approach. The data is collected from the integration of core and log data. This approach is based on the assumption that sandstone reservoirs having similar Rock Quality Index (RQI), has similar capillary pressure behavior as regards water saturation. Universal capillary pressure curves were generated from 22 core plugs in Malaysia, using actual core data. Empirical relations between curve fitting parameters and core plug RQI values were developed. The universal capillary pressure curves were then used in other wells to calculate water saturation which has no core data. For those wells, the pseudo RQI values were calculated using the porosity and permeability derived from the log. A water saturation function was derived, as a function of RQI and HAFWL, with these pseudo RQI values and height above the Free Water Level.
has similar capillary pressure behavior as regards water saturation. Universal capillary pressure curves were generated from 22 core plugs in Malaysia, using actual core data. Empirical relations between curve fitting parameters and core plug RQI values were developed. The universal capillary pressure curves were then used in other wells to calculate water saturation which has no core data. For those wells, the pseudo RQI values were calculated using the porosity and permeability derived from the log. A water saturation function was derived, as a function of RQI and HAFWL, with these pseudo RQI values and height above the Free Water Level.
The results of this study portray that Saturation modelled through this saturation height function matched very well with resistivity derived saturation. Water saturation resulting from generalized capillary curves was contrasted in wells without core data to that computed from resistivity logs. These water saturation results were found to be more realistic than those calculated purely from resistivity logs. The results confirmed the assumption that the universal capillary pressure curves can be used without core data to predict water saturation in wells. A well that has intercepted a hydrocarbon accumulation effectively represents a massive core sample of the reservoir. Because it is now possible to calculate reliably the permeability and porosity in any well. RQI is known over the entire hydrocarbon column. At the saturation anywhere in a reservoir can be determined, and the FWL known or estimated with reasonable certainty. It is in principle viable to evaluate Sw directly based on the drainage capillary theory.
This study provides an insight into the universal saturation height function as a basic formulation in designing, developing, and appropriate strategies to model water saturation.
This study not only helps subsurface study team in modelling water saturation but it also offers new insights concerning the ability and capability of the strong direct relationship among rock quality index, height above free water level, and water saturation.
This study uses an extended concept of Cuddy, et al. (1993) to describe how water saturation varies with height above the free water level (FWL). Cuddy proposed the relationship of bulk Volume of Water vs. Height above the FWL to derive water saturation. This approach requires conversion from BVW to SW. The proposed function called k-function does not require the conversion.
Introduction
Calculation of water saturation is a crucial and integral part of any petrophysical assessment. Calculated using resistivity logs, water saturation Sw is still the preferred standard method in the oil and gas sector. In hydrocarbon-bearing reservoirs with low resistivity to formation, spuriously high water saturation was often computed.
It is very important to determine a reservoir’s saturation profile. Saturation interpreted from well logs does not always give the complete picture. Accuracy of the saturation computed from logs is influenced by thin-beds, insufficient column penetration, invasion, uncertainty in the Archie and shaly sand parameters and reservoir heterogeneity (Amabeoku et al., 2006) and (Darling, 2005).
Invariably, this has resulted in underestimating hydrocarbon resources. A number of methods were developed over the years to overcome this problem. Some are relatively simple and some more complicated, including resistivity modeling based on resistivity imaging tools or implementing advanced resistivity logging software such as the Triaxial Induction Resistivity tool. The findings were rather mixed. Another common approach is to use Central research capillary pressure pressure data. While this has a sound technical base, the main drawback is that most wells still lack adequate core data. Several researchs have been written on this topic and the methods described are based on capillary pressure data to derive resistivity-independent water saturation. In this paper it is proposed that universal capillary pressure curves produced using core data from several standard sandstone reservoirs may be used to measure water saturation from different fields, areas or regions in other specific sandstone reservoirs. This is based on the basis that clastic reservoirs that have similar “Rock Quality Index or RQI,” expressed by the square root ratio of absolute permeability and porosity, will exhibit similar capillary pressure behavior with respect to the saturation of the wetting phase, namely water saturation.
Using actual core details from 22 core plugs in Malaysia, the authors produced universal capillary pressure curves. The Cuddy process theory is applied to perform curve fitting on the capillary pressure curves determined. Empirical relationships between the curve fitting parameters and the RQI values of those core plugs are then established. Global capillary curves are produced on the basis of those empirical relationships. These uniform capillary pressure curves are then used in other wells to measure water saturation which has no core data. For these wells, the pseudo RQI values can be produced using the log dependent porosity and permeability, calculated either from accurate porosity-permeability transformations or empirical equations. A water saturation Sw curve is generated based on these pseudo RQI values and “Height over Free Water Level.” In test wells the resulting water saturation, determined from the universal capillary pressure curves, was confirmed. In other wells, the water saturation obtained from uniform capillary pressure curves was contrasted with that measured from resistivity logs, without core data. The values of water saturation, obtained from the uniform capillary pressure curves, were found to be more accurate than those measured from only resistivity logs. Results in key wells, available in some of the wells, have verified the validity of the method used and the accuracy of the measured saturation of the water. The study results show that universal capillary pressure data can be used to measure water saturation for different fields, areas or regions in sandstone reservoirs, provided that the RQI values of these reservoirs are identical to those from the wells where the core data were taken. Based on these universal capillary pressure curves a saturation height function was also developed.
Capillary Pressure in Reservoir Rocks
If a porous medium is considered as a bundle of capillary tubes of different radii, the pressure required to force the entry of non-wetting fluid to start displacing the wetting fluid. Generally the lower the permeability of the rock, the lower the largest pore size and higher the entry pressure would be for a constant wettability. The capillaries of smaller and smaller radii are invaded as the pressure of the non-wetting fluid is increased. A porous reservoir rock, represented as a bundle of capillaries of different diameter, would have a thickness above Free Water Level in which water saturation is gradually reduced to the minimum possible or connate water saturation (Figure 1). It would have a transition zone in which water saturation is gradually reduced to connate water saturation at its top. The left-part of the diagram shows a schematic bundle of capillary tubes standing vertically in an open water container, which defines Free Water Level at its top. A minimum capillary displacement pressure (Pcd) is needed to force entry of oil into the largest capillary, therefore, for a certain height above Free Water Level, water saturation would be 100%. The capillary entry pressure increases as the radius of capillary decreases. The vertical dimension in this diagram could be considered as representing capillary pressure (right-part) or height above Free Water Level.
If pore diameters in a rock are relatively uniform in radii then little additional pressure is required to desaturate them and the plot of pressure versus saturation (or height above Free Water Level versus saturation) would be almost flat (a plateau on the curve until connate or irreducible water saturation is reached). If pore diameters in a rock are very heterogeneous, then capillary pressure curve would show a more gradual transition to irreducible water saturation. A typical capillary pressure curve (schematic) is shown (Figure 2). The capillary pressure curve changes when the fluids involved are changed.
Figure 1. A porous reservoir rock represented as a bundle of capillaries of different diameter.
Saturation History and Capillary Pressure
The capillary pressure is dependent on the direction of saturation change and saturation history. The saturation history of reservoirs is often complex and must be worked out before saturation could be properly modelled based on capillary pressure. All reservoir rocks are considered to be saturated and wetted by water initially, and subsequently, hydrocarbons migrated into them later. Drainage is the process in which the wetting phase (water) is removed from the reservoir due to the injection of hydrocarbons (non-wetting phase). When the hydrocarbons (non-wetting phase) move into a rock for the very first time and expel water (wetting phase), then the process is called primary drainage. Hydrocarbons could escape out of a reservoir due to leakage (or another mechanism) then water (wetting phase) could soak back into the reservoir from the aquifer to fill in the space vacated by hydrocarbons (non-wetting phase) and this process is called imbibition. The hydrocarbons can not fully escape out of a reservoir and water can not fully saturate it because some hydrocarbons are left trapped in the reservoir (residual hydrocarbons). Therefore, the imbibition process in which water saturation is increasing at the expense of hydrocarbons, never attains 100 per cent water saturation (Figure 2).
It can be stated that capillary pressure and water saturation in a reservoir rock exhibits hysteresis. Due to hysteresis, a higher water saturation will result for a given capillary pressure if the porous media is being desaturated (drainage) than if it is being resaturated (imbibition).
Figure 2. This diagram represents a water-wet reservoir, which consists of a rock whose pore geometry is uniform.
Definition of Fluid Contacts
It is essential to identify fluid contacts before water saturation can be modelled. It is essential also to clearly define fluid contacts and distinguish between various kinds of fluid contacts. The subject of fluid contacts is very complex. So, it useful to describe it first in the simplest situation. It is assumed that an oil and cap-gas reservoir (water-wet) consists of a rock whose pore geometry is uniform hence it can be described by a single capillary pressure curve (Figure 3). The reservoir contains cap-gas and oil below which there is water saturated rock having same pore geometry as the reservoir. A cross-section of this reservoir on the right shows water leg, water-oil transition zone, oil zone, gas-oil transition zone and gas cap. On the left capillary and saturation versus height curve is shown. The diagram shows various fluid interfaces in relation to its position on its capillary pressure or saturation versus height curve. The fluid interfaces start from Free Water Level, 100 per cent water saturated zone above Free Water Level, water-oil transition zone, oil-water contact, oil zone, Free Oil Level, oil-gas transition zone and gas cap.
A capillary pressure curve can be considered as a curve plotting water saturation as a function of height above the level of zero capillary pressure (Figure 3). The drainage capillary pressures are normally considered to be appropriate for describing the initial fluid distribution in a reservoir. An entry pressure must be reached before the non-wetting phase can enter the pores (for example when oil is migrating up-structure). In a reservoir, the wetting fluid saturation (water saturation) will start to decrease above the level at which the entry pressure is reached. Hence, the level of zero capillary pressure is below the level of 100 per cent water saturation. The HAFWL difference between these two levels is a function of the largest pore throat in the reservoir. If the largest pores are relatively large, then, the difference between these levels will be relatively small. The oil-water contact can be highly variable in a reservoir whereas there may only be a single flat Free Water Level (Figure 2.17). It has same Free Water Level but its oil-water contact (IOWC or Initial Oil Water Contact) is variable and is shallower towards the left. After Dahlberg (1995).
Figure 3. A schematic cross-section of a reservoir in which reservoir quality decreases towards left (pore throats are getting smaller).
Free Water Level (FWL)
This is the equilibrium level of the oil-water or gas-water contact in an open borehole. The FWL is the datum in reservoir at which capillary pressure is zero. In water-wet reservoir it lies below 100 per cent water saturated rock and it is below the oil-water contact. In gas reservoirs it lies below 100 per cent water saturated rock and lies below gas-water contact. It is defined generally at the cross-over point of water and hydrocarbon pressure trends versus elevation that are defined by point pressures taken by formation tester tool.
Cuddy Method of Saturation Height Modeling based on Log Saturations
Cuddy et al. (1993) proposed that the product of porosity and water saturation (bulk volume water) is a function of height alone. In many reservoirs, it is obvious that porosity is related inversely to water saturation above the transition zones. As porosity decreases, the water saturation increases and vice versa. Therefore, bulk volume water (product of porosity and water saturation) is a constant above the transition zone. Cuddy et al. (1993) uses log-derived water saturations to fit the function and ignores the special core analysis tests to determine capillary pressure curves. They plot bulk volume water (BVW) versus height above Free Water Level on bi-logarithmic scales and fit the following equation.
where H is HAFWL (height above Free Water Level), a and b are constants found by regression. This technique is virtually independent of permeability. The method, however, is biased towards fitting the water saturation data in better quality sands and does not consider lithology. It is also preferable to use only such saturation data that are 1 meter or more away from a bed boundary so that it is not affected by vertical logging tool responses, i.e., “shoulder effect”. The method works well in reservoirs with short transition zones. In reservoirs where large transition zones exist, the method will not work well because bulk volume water based relationships break down in transition zones (Harrison and Jing, 2001).
METHODS
After performing quality control of available core data, ten capillary pressure curves were chosen to cover a wide range of Rock Quality Index (RQI) in different reservoir types.
- Rock Quality Index (RQI) calculation
The RQI is computed using the following equation
RQI
Where: RQI = Rock Quality Index
K = Permeability in md
PHI= Porosity in decimal fraction
- Capillary Pressure Conversion to Reservoir Conditions
After correcting the measured capillary pressure in laboratory for equipment related, stress and clay bound water (CBW) effects, it is converted to reservoir conditions and fluid systems. The capillary pressure measured in the laboratory fluid system is converted to the reservoir conditions fluid system using:
Typical values of σ and cosine θ, are listed below (Table 2.1).
Table 2.1. Default values of contact angle and interfacial tension for laboratory and reservoir fluids. The default values of interfacial tension and contact angle may not work properly in few cases.
- Curve Fitting Functions of Capillary Pressure Data
A plot of capillary pressure test points versus non-wetting phase saturation need to be fitted with a continuous curve. This curve can be fitted by interpolating between measured data points either linearly or non-linearly. It is much more useful to have the curve described mathematically so that non-wetting phase saturation or height above free water level could be calculated from capillary pressure. All corrections such as closure, stress and clay bound water (Appendix A) should be applied before the curve fitting. Usually, a curve is fitted to data in laboratory units because it is easier to transform the curve equation into reservoir condition units. At reservoir conditions, the capillary pressure can be replaced by height above free water level. There are a large number of mathematical functions that are used to fit capillary pressure data. The commonly used functions are described below.
Core Air-Brine Capillary Pressure Measurements
- Advanced Exponential Function
A more elaborated exponential function has also been proposed to fit capillary pressure data. The curve fits using this exponential function are good for all samples including both low and very high permeability samples.
The advanced exponential function is used to fit capillary pressure data for three samples.
- Averaging Capillary Pressure Data
It is essential to reduce the amount of the measured capillary pressure data, which is done by averaging. The reservoir simulators require a function for saturation height input, which is obtained by averaging the capillary pressure data.
Provided there is a good relationship between permeability and porosity, classifying the data and averaging by porosity groups works. The capillary pressure data may also be grouped by geological or sedimentological facies if they have good correlations to porosity and permeability. Commonly, RQI is the best parameter to use in classifying capillary pressure data.
A good averaging technique should preserve the characteristics of individual capillary pressure curves. It can be done by averaging the calculated results from each capillary pressure curve rather than averaging the curves themselves. The grouping of capillary pressure data into porosity classes is the most common method of averaging. However, the grouping can also be done by permeability, entry pressure, or curve shape.
A key issue in using average saturation height curves is that permeability has to be predicted in uncored sections of the wells. This is easy if there is a clear relationship between porosity and permeability. In most cases, permeability measured on core has to be related to logs, which are then used to predict it in uncored sections.
multiple functions sorted by grouping core-derived porosity–permeability relationships and other rock-type indicators are widely practiced. Solutions are somehow obtained for every hydrocarbon field. Commonly, these are curve-fitting exercises that lack technical integrity. The rationale for the number and type of functions is contentious, and so is the method and choice for rock-type delineation. Interestingly, the number of functions commonly increases for large core data sets.
- Conversion of Reservoir Capillary Pressure to Height above Free Water Level
The reservoir capillary pressure can be converted to height above Free Water Level using the following equation:
where Pcres is capillary pressure at reservoir conditions in psi, g is acceleration due to gravity in ft/sec2, and Δρ is difference of brine and oil density at reservoir conditions, ρw and ρo are reservoir densities of brine and oil in lbs/ft3. It can also be stated as below:
where GRAD_W and GRAD_HC are pressure gradients of water and hydrocarbon in psi/foot.
Saturation v Height Functions
- Determination of Water saturation vs. Height function
The SwH function describes how water saturation varies by height above the level of free water (FWL).
Water saturation (Sw) calculated from log and core data interpretation can only reflect the reservoir within a few feet surrounding the well bore. Sw can not be measured as it depends on many factors, including porosity and the height above the local FWL.
In a field reservoir model, the SwH functions are used to determine Sw away from well locations so that hydrocarbons can be assessed initially for location. The reserve error that comes from an equation that poorly describes the reservoir can be important.
Data Interpolation Method
Either individual curves or average capillary pressure curves can be interpolated. Linear interpolation may be used between points at the same pressure on adjacent curves. The capillary pressure curves are smoothed before interpolation. The weighted averaging or linear interpolation of the wetting phase saturations at any particular capillary pressure is done using the equation given below.
………………………………………………………. (1)
UNDERLY CONCEPTION
Reservoirs with similar RQI will exhibit similar capillary pressure behavior with respect to wetting phase saturation, namely water saturation. Thus, the Saturation Height function derived from core plugs taken from the Malay Basin offshore Malaysia can be used to predict water saturation in similar sandstone reservoirs i.e.
PROOF OF FUNCTIONS
The Saturation Height Function, developed using the method outlined above and based on uniform capillary pressure curves, has been tested in many wells, both in Malaysia and abroad. The results in test wells which have their own core data were confirmed. Water saturation derived from universal capillary curves was compared in wells without core data to that measured from resistivity logs. Such water saturation values were found to be more accurate than those measured solely from resistivity logs. Well test results have also verified the validity of the method used and the accuracy of the computed water saturation, where applicable. The results of these experiments support the assumption that without core data, universal capillary curves can be used to estimate water saturation at wells.
Figures 9 to 16 show the comparison in several test wells between the Sw measured from resistivity logs and the Sw derived from universal capillary pressure curves. We can see that from resistivity maps, the Sw from standard capillary pressure curves is either similar or more favorable than Sw. The red curve is Sw from resistivity logs and from universal capillary pressure curves, the blue curve Sw. These examples indicate that the function of the saturation height works well in both gas and oil-bearing reservoirs in Malaysia, Myanmar and Egypt from various fields.
IMPACT OF SATURATION HEIGT MODELLING
Accurate computation of water saturation Sw will result in better hydrocarbon volumes, leading to a more reliable assessment of the oil or gas field. That will have a big impact on a project’s economic viability. A robust Saturation Height Function will result in more accurate geo-cellular and reservoir simulation models. This will subsequently lead to better prediction of production forecast, reservoir management plan, and monetization scheme for the field. Therefore, a substantial amount of revenue, in terms of millions of dollars, can be generated by a proper and accurate estimate of water saturation using this Saturation Height Function.
EQUATIONS LIMITATION AND UNCERTAINTY
After finishing all research processes, the author finds that the equation can be an interim approach. The final saturation model should be calibrated with core information. The authors recommend the future improvement for the height above free water level below 0.75 m by making the I intercept constant close to 0. As a consequence, the HAFWL lesser than 0.75 m using from this function has uncertainty. However, the impact of saturation computation in the reservoir study in the interval where HAFWL is lesser than 0.75 m is negligible.
CONCLUSION
The results from the experiments conducted in this study confirm the assumption that sandstone reservoirs with similar RQI and wettability will exhibit the same capillary pressure behavior concerning water saturation. Hence, the capillary pressure curves created using capillary pressure data from a cored well can be used to predict water saturation in wells without core data, provided that the sandstone reservoirs have similar rock quality indexes.
This paper presents a robust new approach for the determination of capillary water saturation in clastic reservoir rocks. It offers a solution to problems that current methods cannot solve satisfactorily. Implementation of this approach, however, requires a change of mindset to recognize that the complex inter-relationship of permeability and water saturation can be solved simultaneously with conventional log evaluation. The use of log data for this purpose makes use of the largest database of well information in any hydrocarbon field. Although core data can increase confidence levels for log-derived results, the traditional wisdom that cores are fundamental to petrophysical solutions may be incorrect. Methods that attempt to force solutions by fitting curves to core data are flawed. Laboratory test results may not be sufficiently accurate; or, worse still, core data simply may not provide an adequate representative sampling of the reservoir.
In conclusion, this paper presents an integrated procedure for the determination of capillary water saturation in clastic reservoirs. The procedure described here is cost effective, as it is based largely on logs that are almost always available in hydrocarbon fields.
A novel approach is presented here that effectively eliminates the need for core capillary data. A well that has intercepted a hydrocarbon accumulation effectively represents a massive core sample of the reservoir. Because it is now possible to calculate reliably the saturation in any well.
ACKNOWLEDGEMENT
The authors wish to express their gratitude to the management of PETRONAS and PETRONAS Carigali for granting permission to present this paper.
REFERENCES
Cuddy, S., Allinson, G.A., Steele, R. 1993, A Simple, Convincing Model for Calculating Water Saturations in Souther North Sea Gas Fields : SPWLA 34th Annual Logging Symposium 1-5.
Cuddy, S. 2017, Using Fractals to Determine a Reservoir’s Hydrocarbon Distribution: SPWLA 58th Annual Logging Symposium, 1-16.
Heseldin, G. M., 1974, A Method Of Averaging Capillary Pressure Curves: SPWLA’s The Log Analyst, Volume 15, Issue 4, 3-6.
ABOUT THE MAIN AUTHORS