Petrophysics/Petrofísica/Petrofisica
Inter-Rock SW 2018: A Non-Resistivity Dependent Water Saturation Equation based on Capillary Pressure, Rock Quality and Reservoir Fluids
Inter-Rock SW 2018: una ecuación de saturación de agua no dependiente de resistividad basada en la presión capilar, la calidad de la roca y los fluidos de los depósitos
Inter-Rock SW 2018: uma equação de saturação de água não dependente da resistividade baseada na pressão capilar, qualidade da rocha e fluidos de depósito
Alfonso Quaglia
Ing°Geó°, Inter-Rock USA, LLC. Correo-e: quagliaa@inter-rock-ca.com
Gustavo Espinoza
Ing°Petr°, Inter-Rock, C. A. Correo-e: espinozag@inter-rock-ca.com
Rafael Panesso
Ing°Geó°, Esp. Inter-Rock Panamerican-Colombia. Correo-e: panessor@inter-rock-ca.com
Juan Carlos Porras
Ing°Geó°, MSc. Inter-Rock Panamerican. Correo-e: porrasjc@inter-rock-ca.com
Recibido: 30-9-19; Aprobado: 28-10-19
Abstract
Traditionally, the most common methods for estimating Sw are derived from resistivity, generating uncertainty in its calculation. This article proposes the construction of a water saturation model independent of resistivity and derived from Capillary Pressure (Pc), which uses the permeability-porosity relationship as a rock quality index and the fluid properties as parameters to adjust trends through the proposed equation in this innovative Model. In addition to other capillary pressure based methods in the past, the innovation of this procedure is that it does not consider the reservoir as an average, homogeneous and isotropic unit; on the contrary, it estimates Sw considering pore geometry variations above free water level (FWL) and can be applied to most reservoirs adjusting more accurately the Hydrocarbon Column Height and Water Saturation trends to any fluids system. Through mathematical models, several trends were identified, relating water saturation as a function of capillary pressure versus Rock Quality; generating the following equation:
Sw (Pc(n)) = f (K/Phi(n)) (1)
Resumen
Tradicionalmente, los métodos más comunes para estimar Sw se derivan de la resistividad, generando incertidumbre en su cálculo. Este artículo propone la construcción de un modelo de saturación de agua independiente de la resistividad y derivado de la presión capilar (Pc), que utiliza la relación permeabilidad-porosidad como índice de calidad de la roca y las propiedades del fluido como parámetros para ajustar las tendencias a través de la ecuación propuesta en este innovador modelo. Además de otros métodos basados en la presión capilar en el pasado, la innovación de este procedimiento es que no considera el reservorio como una unidad promedio, homogénea e isotrópica; por el contrario, estima que Sw considera las variaciones de la geometría de los poros por encima del nivel de agua libre (FWL) y se puede aplicar a la mayoría de los depósitos ajustando con mayor precisión las tendencias de altura de la columna de hidrocarburos y saturación de agua a cualquier sistema de fluidos. A través de modelos matemáticos, se identificaron varias tendencias, que relacionan la saturación del agua en función de la presión capilar versus la calidad de la roca; generando la siguiente ecuación:
Sw (Pc(n)) = f (K/Phi(n)) (1)
Resumo
Tradicionalmente, os métodos mais comuns para estimar Sw são derivados da resistividade, criando incerteza em seu cálculo. Este artigo propõe a construção de um modelo de saturação de água independente da resistividade e derivado da pressão capilar (Pc), que utiliza a razão permeabilidade-porosidade como índice de qualidade da rocha e propriedades fluidas como parâmetros ajustar as tendências através da equação proposta neste modelo inovador. Além de outros métodos baseados na pressão capilar no passado, a inovação deste procedimento é que ele não considera o reservatório como uma unidade média, homogênea e isotrópica; pelo contrário, considera que a Sw considera variações na geometria dos poros acima do nível livre da água (FWL) e pode ser aplicada à maioria dos tanques, ajustando mais precisamente as tendências de altura da coluna de hidrocarbonetos e a saturação de água para qualquer sistema de fluidos. Através de modelos matemáticos, foram identificadas várias tendências, que relacionam a saturação de água com base na pressão capilar versus a qualidade da rocha; gerando a seguinte equação:
Sw (Pc(n)) = f (K/Phi(n)) (1)
Palabras clave/Keywords/Palavras-chave:
Capillary pressure, estimación Sw, estimativa do Sw, geometria dos poros, geometría de los poros, modelo de saturación de agua, modelo de saturação de água, petrofísica, petrophysics, pore geometry, presión capilar, pressão capilar, resistividad, resistividade, resistivity, Sw estimation, water saturation model.
Citar así/Cite like this/Citação assim: Quaglia et al. (2019) o (Quaglia et al., 2019)
Referenciar así/Reference like this/Referência como esta:
Quaglia, A., Espinoza, G, Panesso, R., Porras, J. C. (2019, diciembre). Inter-Rock SW 2018: A Non-Resistivity Dependent Water Saturation Equation based on Capillary Pressure, Rock Quality and Reservoir Fluids. Geominas 47(80). 129-136
Introduction
Inter-Rock developed a model to calculate Sw as a function of Height Above Free Water Level (H) and Rock Type, which has been used internally to address challenging and important projects. This method was successfully applied to a great number of reservoirs worldwide and has evolved through the years to the concept presented in this article. It can be applied to any hydrocarbon system with any pore type variation; considering all the range of flow units from the FWL and throughout the reservoir's hydrocarbon column up to oil producing zones at irreducible water saturation conditions.
Background
Several authors have developed models to calculate Sw from capillary pressure data, Leverett, Thomeer, among others. Probably, the most used of these models is the J Function of Leverett.
Leverett (1941) developed a dimensionless function that relates the properties of rock and fluids to what he called J Function. As Acknowledged, the distribution of pore size is related to capillary pressure, so J Function can be expressed in terms of this pressure. J Function considers the changes in porosity, permeability, and wettability of the reservoir.
The J function can be derived from the Carman-Kozeny equation and expressed as follows:
where:
(2)
The capillary pressure data of a specific formation can be expressed in a curve of J Function versus Water Saturation, represented on, either a cartesian plot (Figure 1) or a log-log plot (Figure 2), getting a relationship to determine Sw from the value of J, where:
a = intercept of the trend line for Sw = 100%
b = slope of the trend line
Figure 1. J Function vs. Sw
Figure 2. J Function vs. Sw, (log-log)
Thomeer (1960) developed a semi-empirical mathematical model called Thomeer's Hyperbola Model, in which he used 279 capillary pressure profiles from core samples and approximated capillary pressure profiles from experimental data in a log-log plot to a hyperbolic function which formula is expressed as follows:
Sb/Sb¥ = eÙ(-Fg/log)Pc/Pd)) (3)
where:
Pc = capillary pressure (psia),
Pd = displacement pressure, graphically equivalent to threshold pressure (psia),
Fg= pore geometric factor, which defines the shape of the capillary pressure curve (dimensionless)
Sb = bulk mercury saturation, ration of mercury volume and bulk volume, (fraction),
Sb¥= bulk mercury saturation at infinite pressure assumed equal to porosity (fraction)
Low values of Fg indicate well sorted pore throats while high values indicate poorly sorted pore throats, where capillary pressure is associated to the pore throat distribution. In that sense, Fg can be associated to the equivalent of the pore throat radius. Therefore, low values of Fg are related to high permeability and high Fg are related to low permeability. After that, Thomeer generated a Sw equation through a fitted constant (BVnw) which represents the bulk volume of the non-wetting phase fluid at infinite PC pressure (Figure 3), which is shown as follows:
Sw = 1.0 - (BVnw/Phi) e^(-G/Log10(Pc/Pd)) (4)
where:
Pc = the input, corrected capillary pressure, psi (input curve is converted to psi).
BVnw = Fitted constant, represents the bulk volume of the non-wetting phase fluid at infinite PC pressure.
Phi = Input porosity
G = Fitted constant, defines the curve shape.
Pd = Fitted constant, defines the displacement pressure (PC pressure needed to start reducing saturations).
Figure 3. Schematic of Thomeer's Model Parameters (After Wu, 2004)
In 2007, Inter-Rock developed and applied a model to calculate Sw as a function of height and rock type, which was called “Relative Minimum Water Saturation” (Swminrel). (Figure 4). It was based on mercury capillary pressure tests relating (1-Shg) saturation and rock quality indicators (Pore Throat Radius). Those relationships were determined at different heights: a) close to water contact, b) at the top of the structure, and c) at intermediate heights between top of structure and oil-water contact. These relationships indicated the variation of the minimum Sw by Rock Type, as a function of Pore Throat Radius at constant heights. Since then, Inter Rock has been applying this model as a sort of QC for other Sw capillary-pressure-based models, which can be expressed as follows:
Sw = a*(PTR)^b (5)
where:
a & b = (exponential equation coefficients)
PTS= (Pore Throat Size in microns from Winland/Pittman equations)
Figure 4. Swminrel vs. Pore throat Size. (Inter-Rock)
Figure 5. (1-Shg) vs. Rock Type Index. (K/Phi)
It is widely documented in the literature that capillary pressure curves illustrate the variation of water saturation along the hydrocarbon column, and that this variation is basically associated with two factors:
1. Rock Type or Petrofacies (Pore Geometry).
2. Height above the Free Water Level, or what is the same, the thickness of the hydrocarbon column.
Methodology
Initially, the method related theoretical wetting phase saturations to PTS (Rock Type Index) for the same value of Pc for approximately 300 core samples from varied lithologies including unconsolidated to consolidated sandstones, carbonates, tight sandstones, and Silty-shaly-type reservoirs. The method consists of plotting the saturation profile of the theoretical wetting phase resulting from the capillary pressure test (1 - SHg) versus the corresponding value of the Rock Type index obtained from permeability-porosity relationship (K/Phi). Each saturation value corresponds to an injection pressure value (Figure 5), where saturation values of theoretical wetting phase decrease as the injection pressure increases. This exercise was carried out for each of the samples analyzed which were previously classified as equivalent lithologies and subsequently plotted to generate trend lines through all the saturation values corresponding to the same injection pressure, being able to obtain equations from regression models for each of the iso-pressure curves (Figure 6). These trends were extrapolated for all injection iso-pressure curves originating the potential type general equation that allows calculating Sw every single step above free water level (FWL) relating Height, Rock Type Index (K/Phi) and Fluid properties.
Each trend shown in (Figure 6) is represented by the potential type equation as follows:
Figure 6. Trend lines through all saturation values @ Iso-Pc vs K/Phi
Figure 7. Relationship between the values of A and B versus the recorded pressure of the mercury injection capillary pressure tests.
where:
1-SHg = Theoretical wetting phase saturation from Hg injection PC.
K / PHI = Permeability-Porosity Ratio (Rock Type Index)
An = Height Coefficient
Bn = Height Exponent
The terms An and Bn control the position and shape/concavity of the curves and are a direct function of the injection pressure applied to the sample. The mathematical models that best represent the behavior of the coefficient A and the exponent B were developed through regressions derived from the curves shown in (Figure 6 & 7).
The following equation describe the behavior of A and B values as a function of the injection pressure:
where:
A = Height Coefficient
B = Concavity Trend Exponent
Pciny = Injection pressure (Air-Mercury system)
Each of the pressures resulting in the capillary pressure test by mercury injection (Pciny) has a capillary reservoir pressure equivalent (Pcres), which is dependent on the phases of the fluids that saturate the porous media and the terms associated with the wettability of the rock (interfacial tension and contact angle). The equation that allows transforming the injection pressure and the reservoir pressure according to the Fluid system is the following:
(6)
(7)
(8)
(9)
where:
Pcres = Reservoir pressure for the corresponding system
Pciny = Injection pressure in the Air-Mercury system
(s*cosq)res = Interfacial tension times contact angle for corresponding reservoir system
(s*cosq)iny = Interfacial tension times contact angle for air-Mercury laboratory system
It is important to note that for an injection pressure value at laboratory conditions, the height above free water level will be a function of the fluid system and the difference of fluid densities in porous media.
The estimated height (H) though, would be given by the following expression:
(10)
where:
H = Height above the free water level for a given system
Pcres = Reservoir capillary pressure for the corresponding reservoir system
g Fluid gradient
ρw Water density (wetting phase). (1g/cc)
ρh hydrocarbon Density (non-wetting phase)
Using diverse values for fluid densities, therefore generating specific average coefficients for oil and gas specific gravities. Table #1.
Table #1. Specific average coefficients for oil and gas specific gravities.
Considering the height variation that results from the difference of the densities (water-hydrocarbon), and taking injection pressure as reference, it is possible to determine the right height values which correspond to the "A" and "B" coefficients mentioned above. The following equations describe the behavior of A and B values as a function of the height on average cases:
General Type equations:
Y= a(X)-b
Average Oil case:
A = 424 * (Height)^-0.3 (11)
B = 0.179 * (Height)^-0.035 (12)
Average Gas case:
A = 328 * (Height)^-0.3 (13)
B = 0.173 * (Height)^-0.035 (14)
Figure 8 clearly shows the effect generated by the difference in fluid densities to be considered in the determination of the height above the free water level (FWL). Several height values were determined for a variety of hydrocarbon densities vs air-mercury capillary pressure values. After that, mathematical models were developed allowing the conversion of these coefficients A and B, to a function of height above free water level (H); and the generation of a Water Saturation equation, not only as a function of Height (H), but also as a function of rock quality (K/PHI). The proposed equation can be interactively adjusted with new coefficients which are dependent on fluid density, interfacial tension, and wettability. As a result, it is possible to obtain accurate water saturation values from the following equation:
(15)
Figure 8. Fluid Density Effect on Height above FWL.
where:
Sw = Minimum water saturation at a given height above FWL
K / PHI = Permeability-Porosity Ratio (Rock Type Index)
A = Height Coefficient
B = Height Exponent
In any case, if it comes to time for trend adjustments, A coefficient would work as a Shifting Factor by varying (-b) exponent from lower values for better pore throat sizes, to higher values for Tighter rocks.
On the other hand, B would similarly work but this time as Curvature Trend adjustment coefficient.
Additionally, as shown in Figure 9, in a pilot well, as part of the validation for the Minimum Water Saturation Model proposed, two tested intervals (A & B), at a given height above a FWL previously determined, show good calibration among several Sw models applied (Track 6): a) blue (Cyan shaded) higher values Sw curve calculated from logs, b) brown thicker curve represents Sw from J function, which have intermediate values between curves a and c) blue lower Sw values curve, which represents the proposed water saturation model and yields more certainty since it is giving a bit of difference with the other two curves, justifying the 6% of produced water, having reported 94% Oil (P). Tested intervals, mainly mega to macro porous rock types, represent the best quality reservoir rocks. This example helps to understand the reservoir fluid distribution conditions at the moment of testing, which was originally thought to be at irreducible conditions because of its high structural position. As a final comment, among all considerations made, it was observed that “lithology” was not a critical factor to the established relationships between variables involved in this work.
Track 1: GR, BS, DCAL
Track 2: Resistivities
Track 3: Borehole Shape
Track 4: Den/Neut
Track 5: Effective porosity
Track 6: Water Saturation (a-Sw from Logs, b-SwJ from Leverett, c-Swminrel proposed model)
Track 7: Calculated Permeability
Track 8: Lithology & Fluids volumetrics (Shale, Sand, Oil, Water)
Track 9: Flow Units track (Rock Types)
Figure 9. Minimum Sw validation at a given Height above FWL.
Conclusions
1. The proposed new water saturation model (Inter-Rock Sw 2018), does not consider the reservoir as an average, homogeneous and isotropic unit, but it takes in account that Sw is affected by pore throat size variations in rock types every step above the free water level (FWL) and can be applied to most types of existing reservoir qualities.
2. The minimum relative Sw model can be applied to any hydrocarbon system, oil-water or gas-water.
3. The calibration of the model is based on a relatively simple development of mathematical models that allowed to generate adjustment coefficients, both in terms of height of the hydrocarbon column (A coefficient) and the concavity trend effect of the curves that related the Sw with rock quality index (B coefficient).
4. The equation is adjusted interactively either with proposed coefficients A and B or with those generated in particular cases, taking into account that these depend on Height (H), which implicitly depends on the buoyancy pressure, the density of the fluids in the reservoir, the interfacial tension and the wettability.
5. Even though the equation was modeled using K/Phi ratio as Rock Quality Index, it is possible that other rock quality index may be used.
Nomenclatures/Key words
Swmin: Minimum water saturation
Pc: Capillary Pressure
Non-Resistivity: Calculation method independent of Resistivity
Rock Quality: Basically, Dominant Pore Throat Size
K/PHI: Dimensionless Rock Quality Index based on K/Phi ratio
PTR: Pore Throat Radius
PTS: Pore Throat Size
FWL: Free Water Level
H: Height
HC: hydrocarbon Column
J (Sw): J Function
Pd: displacement pressure (equivalent to threshold pressure)
Fg: pore geometric factor
Sb: bulk mercury saturation
Sb∞: bulk mercury saturation at infinite pressure (aprox. porosity fraction)
Bvnw: Constant, bulk volume of the non-wetting phase at infinite PC pressure.
G: Constant, defines the curve shape.
Swirr: Irreducible water saturation
s: Interfacial Tension
q:Contact Angle
Pcres = Reservoir pressure for the corresponding system
Pciny = Injection pressure in the Air-Mercury system
g Fluid gradient
ρw Water density (wetting phase)
ρh Hydrocarbon Density (non-wetting phase)
f: Porosity
K: Permeability
A: Height Coefficient
B: Concavity Exponent
Acknowledgment
We would like to sincerely thank the Inter Rock board of directors for allowing us to develop this work and to publish this method that gave rise to this new, practical, and precise equation of minimum water saturation calculation, independent of resistivity and based on capillary pressure, fluid types, and reservoir rock quality. With this method, we seek to add an alternate option for water saturation calculation above free water level in reservoirs where the electric logs response has some type of inconvenience, doubt, limitation, or uncertainty.
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The author(s) declare(s) that she/he/they has/have no conflict of interest related to hers/his/their publication(s), furthermore, the research reported in the article was carried out following ethical standards, likewise, the data used in the studies can be requested from the author(s), in the same way, all authors have contributed equally to this work, finally, we have read and understood the Declaration of Ethics and Malpractices.