US20160040532A1 - Reservoir water volume shift model - Google Patents

Abstract

A method of estimating properties of a reservoir fluid in an underground reservoir includes: calculating the density ρ of the mixture containing water; determining component-specific volume shift parameters c_i; replacing the component specific volume shift parameter for water with a volume shift parameter c_H2O; determining a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture; determining a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; and recalculating density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume.

Description

TECHNICAL FIELD
• This invention is pertinent to the thermodynamic modeling of water using cubic equation of state (EOS) models and auxiliary volume shift density correction methods.
BACKGROUND
• Thermodynamic equations of state (EOS) models relate known state variables, such as pressure and temperature, to unknown state variables, such as volume, density, and fugacity, as well as other fluid parameters. Cubic EOS models are a subclass of the broader class of general thermodynamic EOS models that relate pressure, temperature, and volume through a cubic polynomial function. Such models are used to predict phase behavior and fluid properties in a broad range of applications, including the petrophysical, geophysical, refrigeration, aerospace, and chemical process design industries. Their broad adoption and application is due to the relative simplicity of the model form, ease of computational implementation, limited number of fluid property inputs required for operation, and overall computational robustness.
• The cubic EOS model subclass was first developed by van der Waals as an extension to the ideal gas law, where both attractive and repulsive molecular forces are included. Since the initial formulation of a cubic EOS model by van der Waals, numerous modifications to the model form have been made in an effort to increase predictive accuracy and general applicability. Of note are the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) model forms, which achieved broad adoption and use in industrial applications. Both the SRK and PR models may expressed in a universal form:
• $\begin{array}{cc}P=\frac{\mathrm{RT}}{V-b}-\frac{a\alpha }{V^2+u_1\mathrm{bV}+u_2b^2},& \left(1\right)\end{array}$
• where P is pressure, T is temperature, V is volume, R is the universal gas constant, a and b are attractive and repulsive parameters specific to each pure substance or mixture, and u₁ and u₂ are model-specific constants. The values and model forms for a, α, b, u₁, and u₂ for the SRK and PR model forms are listed in Table 1 below. Expressed in its cubic polynomial form, the SRK and PR models may be written as:
• 
  Z ³−(1+B−u ₁ B)Z ²+(A+u ₂ B ² −u ₁ B−u ₁ B ²)Z−(AB+u ₂ B ² +u ₂ B ³)=0,   (2)
• where Z is the compressibility factor, with
• $\begin{array}{cc}A=\frac{{\left(a\alpha \right)}_{\mathrm{mix}}P}{R^2T^2},\mathrm{and}B=\frac{b_{\mathrm{mix}}P}{\mathrm{RT}}.& \left(3\right)\end{array}$
• The attractive and repulsive terms for mixtures are calculated using standard mixing rules and are given by:
• 
  (aα)_mix=Σ_i ^NΣ_j ^N [x _i x _j√{square root over (a _i a _jα_iα_j)}(1−k _ij)],   (4)
• 
  b_mix=Σ_i ^Nx_ib_i,   (5)
• where x_i is the mole fraction of each component in the phase of interest (liquid or gas) and k_ij are binary interaction coefficients.

• TABLE 1
  SRK and PR cubic EOS parameters.

  Model	a	α	b	Ωa	Ωb	u1	u2

  SRK	$\frac{{\Omega }_aR^2T_c^2}{P_c}$	[1 + fω(1 − T0.5 r)]2 fω = 0.48 + 1.574ω − 0.176ω2	$\frac{{\Omega }_bRT_c^2}{P_c}$	0.42748	0.08664	1	0
  PR	$\frac{{\Omega }_aR^2T_c^2}{P_c}$	[1 + fω(1 − T0.5 r)]2 fω = 0.37464 + 1.54226ω − 0.2699ω2	$\frac{{\Omega }_bRT_c^2}{P_c}$	0.45724	0.07780	2	−1

• Given physical properties for each mixture component, a cubic EOS may be used to iteratively solve for the compressibility factor Z for each potential phase (Z_liq and Z_vap for liquid and vapor phases respectively). This cubic EOS iterative method is part of a “flash algorithm” and upon convergence of the iterative portion of the flash algorithm, the liquid-phase density ρ_liq and the vapor-phase density ρ_vap may be computed through the use of the converged Z factor values:
• $\begin{array}{cc}{\rho }_{\mathrm{liq}}=\frac{P}{\mathrm{RT}}\frac{M_{\mathrm{liq}}}{Z_{\mathrm{liq}}},& \left(6\right)\\ {\rho }_{\mathrm{vap}}=\frac{P}{\mathrm{RT}}\frac{M_{\mathrm{vap}}}{Z_{\mathrm{vap}}},& \left(7\right)\end{array}$
• where M_liq and M_vap are the average molecular weights of the liquid and vapor compositions.
SUMMARY
• According to one embodiment, an apparatus for estimating conditions of reservoir fluid in an underground reservoir that includes a sensor for measuring one or more measured parameters of that fluid, the measured parameters including at least one of: temperature and pressure of the fluid a processor is disclosed. The processor is configured to: receive data representing the one or more measured parameters; calculate the density ρ of a mixture containing water; determine component-specific volume shift parameters c_i; replace the component specific volume shift parameter for water with a volume shift parameter c_H2O; determine a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture; determine a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; and recalculate density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume.
• According to another embodiment, a method of estimating properties of a reservoir fluid in an underground reservoir is disclosed. The method includes: calculating the density ρ of the mixture containing water; determining component-specific volume shift parameters c_i; replacing the component specific volume shift parameter for water with a volume shift parameter c_H2O; determining a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture; determining a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; and recalculating density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume.
BRIEF DESCRIPTION OF THE DRAWINGS
• The subject matter, which is regarded as the invention, is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other features and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings, wherein like elements are numbered alike, in which:
• FIG. 1 shows an example drilling system according to one embodiment; and
• FIG. 2 is a flow diagram illustrating an example of a method according to one embodiment.
DETAILED DESCRIPTION
• It has been discovered that two-parameter cubic EOS models, such as the SRK and PR model forms, generally exhibit poor accuracy in their predictions of liquid-phase density values. While gas-phase density predictions from cubic EOS models are generally accurate (less than 5% error), liquid-phase density errors in excess of 20% are common. This drawback stems from the fact that only two of the three critical state variables are directly solved for through two free model parameters (the attractive coefficient a and repulsive coefficient b) and are formulated so that only two critical state variables (critical pressure and critical temperature) of any component are properly represented. Without any remaining available model parameters to adjust, the third state variable of critical compressibility factor becomes a consequential result for the given EOS model form. Note that density, molar volume, compressibility factor are all related by:
• $\begin{array}{cc}\rho =\frac{M}{v}=\mathrm{MZ}& \left(8\right)\end{array}$
• The fixed or consequential values of critical compressibility factor for the SRK and PR models are Z_c=0.333 and Z_c=0.307, respectively. However, the physical value of Z_c for any specific component or mixture may vary and not be one of these values. As the true value of Z_c for a component or mixture departs from the Z_c value predicted by the EOS model, greater liquid-phase density errors are expected. As such, the SRK and PR EOS models have a demonstrated difficultly predicting the liquid phase densities of both water (Z_c,H2O=0.2295) and heavier hydrocarbons (Z_c,C20=0.1865, Z_c,C48=0.1882) due to the dramatic difference between the actual critical compressibility values for these compounds and the fixed or consequential values generated by application of the EOS models.
• A broadly-adopted strategy for improving the liquid-phase density prediction of cubic EOS models is to apply a “volume shift” model to the baseline results initially calculated by the EOS model. Volume shift models apply a linear correction factor c to the molar volume value v calculated by the EOS,
• 
  v _corrected =v _EOS −c   (9)
• In the equation above, molar volume is related to density by v=M/ρ, and thus the corrected molar volume results in a directly corrected density value as well.
• The two most commonly used volume shift models are given by Peneloux and Rauzy (Peneloux) for the SRK EOS and by Jhaveri & Youngren (Jhaveri) for the PR EOS. The Peneloux model form relates the component critical parameters for the mixture and the component Rackett factors, Z_RA, in order to formulate the volume correction parameter c model form:
• $\begin{array}{cc}c={\Sigma }_1^Nc_iz_i,& \left(10\right)\\ c_i=0.40768\left(\frac{{\mathrm{RT}}_c}{P_c}\right)\left(0.29441-Z_{\mathrm{RA}}\right),& \left(11\right)\end{array}$
• where c_i is the volume correction per component of a mixture, and z_i is the mole fraction of a component in a mixture. Thus, for single-component mixture scenarios, c=c_i.
• In the Jhaveri model, the molecular weight of each component is correlated to the dimensionless volume correction factor S_i:
• $\begin{array}{cc}c={\Sigma }_1^Nc_iz_i,& \left(12\right)\\ c_i=S_ib_i,& \left(13\right)\\ S_i=1-\frac{d}{M^e},& \left(14\right)\\ b_i=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c},& \left(15\right)\end{array}$
• where d=2.258 and e=0.1823 for n-alkane hydrocarbons (different values of d and e are also provided for aromatic and naphthenic hydrocarbons). Neither one of these two model forms directly addresses the volume shift requirements for water or other polar molecules. This limitation is resolved by the embodiments disclosed herein.
• Numerous higher-order volume shifts have been developed and reported in the open literature, where the volume correction factor has been formulated as a function of temperature or as a function of both temperature and density. However, such models simply introduce further model form error and solution method concerns. One or more embodiments disclosed herein may reduce such errors.
• Referring to FIG. 1, an exemplary embodiment of a downhole drilling, monitoring, evaluation, exploration and/or production system 10 disposed in a wellbore 12 is shown. A borehole string 14 is disposed in the wellbore 12, which penetrates at least one earth formation 16 for performing functions such as extracting matter from the formation and/or making measurements of properties of the formation 16 and/or the wellbore 12 downhole. The borehole string 14 is made from, for example, a pipe, multiple pipe sections or flexible tubing. The system 10 and/or the borehole string 14 include any number of downhole tools 18 for various processes including drilling, hydrocarbon production, and measuring one or more physical quantities in or around a borehole. Various measurement tools 18 may be incorporated into the system 10 to affect measurement regimes such as wireline measurement applications or logging-while-drilling (LWD) applications.
• In one embodiment, a parameter measurement system is included as part of the system 10 and is configured to measure or estimate various downhole parameters of the formation 16, the borehole 14, the tool 18 and/or other downhole components. The illustrated measurement system includes an optical interrogator or measurement unit 20 connected in operable communication with at least one optical fiber sensing assembly 22. The measurement unit 20 may be located, for example, at a surface location, a subsea location and/or a surface location on a marine well platform or a marine craft. The measurement unit 20 may also be incorporated with the borehole string 12 or tool 18, or otherwise disposed downhole as desired.
• In the illustrated embodiment, an optical fiber assembly 22 is operably connected to the measurement unit 20 and is configured to be disposed downhole. The optical fiber assembly 22 includes at least one optical fiber core 24 (referred to as a “sensor core” 24) configured to take a distributed measurement of a downhole parameter (e.g., temperature, pressure, stress, strain and others). In one embodiment, the system may optionally include at least one optical fiber core 26 (referred to as a “system reference core” 26) configured to generate a reference signal. The sensor core 24 includes one or more sensing locations 28 disposed along a length of the sensor core, which are configured to reflect and/or scatter optical interrogation signals transmitted by the measurement unit 20. Examples of sensing locations 28 include fibre Bragg gratings, Fabry-Perot cavities, partially reflecting mirrors, and locations of intrinsic scattering such as Rayleigh scattering, Brillouin scattering and Raman scattering locations. If included, the system reference core 26 may be disposed in a fixed relationship to the sensor core 24 and provides a reference optical path having an effective cavity length that is stable relative to the optical path cavity length of the sensor core 24.
• In one embodiment, a length of the optical fiber assembly 22 defines a measurement region 30 along which distributed parameter measurements may be taken. For example, the measurement region 30 extends along a length of the assembly that includes sensor core sensing locations 28.
• The measurement unit 20 includes, for example, one or more electromagnetic signal sources 34 such as a tunable light source, a LED and/or a laser, and one or more signal detectors 36 (e.g., photodiodes). Signal processing electronics may also be included in the measurement unit 20, for combining reflected signals and/or processing the signals. In one embodiment, a processing unit 38 is in operable communication with the signal source 34 and the detector 36 and is configured to control the source 34, receive reflected signal data from the detector 36 and/or process reflected signal data.
• In one embodiment, the measurement system is configured as a coherent optical frequency-domain reflectometry (OFDR) system. In this embodiment, the source 34 includes a continuously tunable laser that is used to spectrally interrogate the optical fiber sensing assembly 22.
• The optical fiber assembly 22 and/or the measurement system are not limited to the embodiments described herein, and may be disposed with any suitable carrier. That is, while an optical fiber assembly 22 is shown, any type of now known or later developed manners of obtaining information relative a reservoir may be utilized to measure various information (e.g., temperature, pressure, salinity and the like) about fluids in a reservoir. Thus, in one embodiment, the measurement system may not employ any fibers at all and may communicate data electrically.
• A “carrier” as described herein means any device, device component, combination of devices, media and/or member that may be used to convey, house, support or otherwise facilitate the use of another device, device component, combination of devices, media and/or member. Exemplary non-limiting carriers include drill strings of the coiled tube type, of the jointed pipe type and any combination or portion thereof. Other carrier examples include casing pipes, wirelines, wireline sondes, slickline sondes, drop shots, downhole subs, bottom-hole assemblies, and drill strings.
• In support of the teachings herein, various analysis components may be used, including a digital and/or an analog system. Components of the system, such as the measurement unit 20, the processor 38, the processing assembly 50 and other components of the system 10, may have components such as a processor, storage media, memory, input, output, communications link, user interfaces, software programs, signal processors (digital or analog) and other such components (such as resistors, capacitors, inductors and others) to provide for operation and analyses of the apparatus and methods disclosed herein in any of several manners well appreciated in the art. It is considered that these teachings may be, but need not be, implemented in conjunction with a set of computer executable instructions stored on a computer readable medium, including memory (ROMs, RAMs), optical (CD-ROMs), or magnetic (disks, hard drives), or any other type that when executed causes a computer to implement the method of the present invention. These instructions may provide for equipment operation, control, data collection and analysis and other functions deemed relevant by a system designer, owner, user or other such personnel, in addition to the functions described in this disclosure.
• Further, various other components may be included and called upon for providing for aspects of the teachings herein. For example, a power supply (e.g., at least one of a generator, a remote supply and a battery), cooling unit, heating unit, motive force (such as a translational force, propulsional force or a rotational force), magnet, electromagnet, sensor, electrode, transmitter, receiver, transceiver, antenna, controller, optical unit, electrical unit or electromechanical unit may be included in support of the various aspects discussed herein or in support of other functions beyond this disclosure.
• One embodiment disclosed here consists of a new custom, higher-order volume shift methodology that is specific to correcting the density of water. The volume shift calculated is reported as a volume shift correction factor c_i, so that it may be used in conjunction with linear Peneloux or Jhaveri volume shift model forms.
• As discussed above, cubic equations of state (EOS) models are known to have deficiencies in calculating liquid-phase densities. This deficiency is remedied by the inclusion of a molar volume adjustment factor that is applied to the liquid and vapor phase molar volume once the flash solver has achieved a converged solution. In particular, a temperature- and pressure-dependent, water-specific volume shift model is disclosed that may improve the accuracy of water density calculations. The model calculates the dimensionless volume shift for water S_H2O as a function of reduced pressure P_r and reduced temperature T_r. For the majority of the pressure-temperature phase space, the dimensional volume shift is given by
• $\begin{array}{cc}{S}_{H2O}=\frac{a+b\exp \left(T_r\right)+c\exp \left(P_r\right)+d\exp \left(T_r\right)\exp \left(P_r\right)}{1+f\exp \left(T_r\right)+g\exp \left(P_r\right)+h\exp \left(T_r\right)\exp \left(P_r\right)},& \left(16\right)\\ P_r=\frac{P}{P_c},& \left(17\right)\\ \mathrm{and}T_r=\frac{T}{T_c},& \left(18\right)\end{array}$
• where the model parameters a-h are provided in Table 2. The critical parameters of water used are P_c=22.050 MPa and T_c=647.14 K. The dimensionless volume shift S_H2O may be converted to a dimensional volume shift c_H2O by:
• 
  c_H2O=b_H2OS_H2O,   (19)
• where
• $\begin{array}{cc}{b}_{H2O}=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c}=\{\begin{array}{cc}2.114176\times {10}^{-5}\frac{m^3}{\mathrm{mol}}& \left(\mathrm{SRK}\right)\\ 1.898464\times {10}^{-5}\frac{m^3}{\mathrm{mol}}& \left(\mathrm{PR}\right)\end{array}& \left(20\right)\end{array}$
• For relatively high temperature, low pressure conditions, the dimensionless volume shift is modified to the form:
• $\begin{array}{cc}{S}_{H2O}=S_{H2O}^{*}\exp \left[-k\left(T_r-T_r^{*}\right)\right],& \left(21\right)\\ S_{H2O}^{*}=\frac{a+b\exp \left(T_r^{*}\right)+c\exp \left(P_r\right)+d\exp \left(T_r^{*}\right)\exp \left(P_r\right)}{1+f\exp \left(T_r^{*}\right)+g\exp \left(P_r\right)+h\exp \left(T_r^{*}\right)\exp \left(P_r\right)},& \left(22\right)\\ T_r^{*}={b^{\prime }\left[\ln \left(\frac{P_r-d^{\prime }}{a^{\prime }}\right)\right]}^{-1}-c^{\prime },& \left(23\right)\\ k=\{\begin{array}{cc}1000& P_r\le 1.0\\ {10}^{m\exp \left({\mathrm{nP}}_r\right)}& P_r>1.0\end{array}.& \left(24\right)\end{array}$
• The model coefficients a′-d′ and m-n are provided in Table 2.
• This secondary model form is applicable when P_r<P_r*, where
• $\begin{array}{cc}{P}_r^{*}=a^{\prime }\exp \left(\frac{b^{\prime }}{T_r+c^{\prime }}\right)+d^{\prime }.& \left(25\right)\end{array}$

• TABLE 2
  Custom water volume shift model parameters optimized for use with SRK
  and PR EOS models.

  Model
  Parameter	SRK EOS	PR EOS
  a	1.686599 × 10−1	9.906580 × 10−2
  b	−2.414245 × 10−2	−1.499797 × 10−2
  c	5.641665 × 10−3	4.148741 × 10−3
  d	2.638684 × 10−3	1.960847 × 10−3
  f	−3.329183 × 10−1	−3.495340 × 10−1
  g	3.916869 × 10−2	4.703695 × 10−2
  h	−3.927259 × 10−3	−6.664902 × 10−3
  m	4.1508	4.1508
  n	−0.34354	−0.34354
  a′	7.356856 × 102	7.356856 × 102
  b′	−6.241360	−6.241360
  c′	−5.620670 × 10−2	−5.620670 × 10−2
  d′	−1.061800 × 10−6	−1.061800 × 10−6

• The model parameters were determined by fitting un-volume-shifted water density data results from calculations with either the SRK or PR EOS models to water density data from the IAPWS EOS model. Water density data was generated using the cubic EOS models and the IAPWS model over a temperature range from 32° F. to 1000° F. at the following isobars: 15, 200, 500, 1000, 2000, 3198, 5000, 6000, 7000, 8000, 10000, 15000, 20000, and 25000 psia (2815 total data points). The density data from the cubic EOS models and IAPWS models was converted to molar volume data by the equation:
• $\begin{array}{cc}{v}_{H2O}\left(P,T\right)=\frac{M_{H2O}}{{\rho }_{H2O}\left(P,T\right)}& \left(26\right)\end{array}$
• where M_H2O=18.015 g/mol. The difference between the calculated IAPWS and EOS model volumes yields the required total volume shift that any volume shift model must reproduce:
• 
  ĉ _H2O(P,T)=v _H2O ^EOS(P,T)−v _H2O ^IAPWS(P,T).   (27)
• The required volume shift was then converted to a dimensionless volume shift form by:
• $\begin{array}{cc}{\hat{S}}_{H2O}\left(P,T\right)=\frac{{\hat{c}}_{H2O}\left(P,T\right)}{b_{H2O}}.& \left(28\right)\end{array}$
• The collection of Ŝ_H2O(P,T) data points (one data set for each cubic EOS model form) was then used to determine the model parameter values listed in Table 2. A nonlinear regression was performed to minimize the error (relative error squared) between Ŝ^H2O(P,T) and the functional model form of S_H2O(P,T).
• In one embodiment, the volume shift method for water is implemented by correcting the original un-volume-shifted density value calculated by a cubic EOS algorithm, in this case, specifically the SRK and PR EOS algorithms as used in the method developed here. The overall volume shift algorithm and the application of the water volume shift method are shown in FIG. 2. The method of calculating the required volume shift to correct the cubic EOS prediction of water density occurs in steps 205-206 for the SRK EOS and steps 216-217 for the PR EOS.
• The disclosed water volume shift method may provide for accurately calculating water density and has been validated for a temperature range of 32-1000° F. (273.2-810.9 K) and a pressure range of 15-25000 psia (0.1034-172.4 MPa). The method is not intended for predicting the density of solid phase water or ice, and thus, may not be valid below the triple point of water (T_t,H2O=273.16 K). For the purpose of implementing this method in a two-phase flash solver that is not intended to predict solid phase formation, the required volume shift for water c_H2O at a (P,T) state where T<T_t,H2O, the value of water density should be calculated as if T=T_t,H2O.
• In more detail, at block 202 a PR or SRK flash for the mixture and the measured pressure and temperature is performed. This may involve, in one embodiment, iteratively solving and determining the EOS coefficients based on a measured temperature and pressure. This may generally include solving one or more equations (1)-(7) above and result in determining values for liquid-phase density ρ_liq and vapor-phase density ρ_vap for at least water (and possibly, other components).
• Block 203 determines whether the SRK or PR EOS model is to be used. This may be a user defined determination in one embodiment. In another embodiment, both may be done. In such a case it shall be understood that portions of the method shown in FIG. 2 may be repeated.
• Assuming that the SRK EOS model is to be used, processing continues in block 204. At block 204 the component-specific volume shift parameters c_i are estimated using the Peneloux volume shift model. In one embodiment, this may include solving:
• $c_i=0.40768\left(\frac{{\mathrm{RT}}_c}{P_c}\right)\left(0.29441-Z_{\mathrm{RA}}\right),$
• where c_i is the volume correction per component of a mixture.
• At block 205, S_H2O with a water volume shift model with SRK specific model parameters is calculated. This may include, in one embodiment, solving equations (16)-(18) above. In cases, however, where P_r<P_r*, equations (16)-(18) may be modified as described in equations (21)-(24) above, where P_r* is found by equation (25).
• At block 206 c_i for water is replaced with c_H2O per equations (19)-(20) above.
• Alternatively, assuming that the PR EOS model is to be used, processing continues in block 214. At block 214 the molecular weight of each component is correlated to the dimensionless volume correction factor S_i according to the below:
• $S_i=1-\frac{d}{M^e}$
• where d=2.258 and e=0.1823 for n-alkane hydrocarbons (different values of d and e are also provided for aromatic and naphthenic hydrocarbons).
• At block 215, for each component c_i is solved per Jhaveri where
• $c_i=S_ib_i,b_i=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c},$
• At block 216, S_H2O with a water volume shift model with PR specific model parameters is calculated. This may include, in one embodiment, solving equations (16)-(18) above. In cases, however, where P_r<P_r*, equations (16)-(18) may be modified as described in equations (21)-(24) above, where P_r* is found by equation (25).
• At block 217 c_i for water is replaced with c_H2O per equations (19)-(20) above.
• Regardless of how c_H2O is calculated, at block 208 the total volume shift to be applied to liquid and vapor phases is calculated. This may include solving:
• 
  c_liq=Σ_i ^Nx_ic_i; and
• 
  c_vap=Σ_i ^Ny_ic_i,
• where x and y are liquid and vapor compositional mole fractions, respectively. Thus, for single-component fluid mixture scenarios, c=c_i.
• At block 210, the liquid and vapor phase molar volumes are corrected according to:
• 
  v _liq ^corrected =v _liq ^EOS −c _liq; and
• 
  v _vap ^corrected =v _vap ^EOS −c _vap.
• At block 212, the corrected liquid and vapor phase volumes are utilized to update density and compressibility factors according to:
• ${\rho }_{\mathrm{liq}}^{\mathrm{corrected}}=\frac{M_{\mathrm{liq}}}{{\upsilon }_{\mathrm{liq}}^{\mathrm{corrected}}};$ ${\rho }_{\mathrm{vap}}^{\mathrm{corrected}}=\frac{M_{\mathrm{vap}}}{{\upsilon }_{\mathrm{vap}}^{\mathrm{corrected}}};$ $Z_{\mathrm{liq}}^{\mathrm{corrected}}=\frac{1}{{\upsilon }_{\mathrm{liq}}^{\mathrm{corrected}}};\mathrm{and}$ $Z_{\mathrm{vap}}^{\mathrm{corrected}}=\frac{1}{{\upsilon }_{\mathrm{vap}}^{\mathrm{corrected}}}.$
• Of course, additional processing could also be performed in one or more embodiments.
• While the invention has been described with reference to exemplary embodiments, it will be understood that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications will be appreciated to adapt a particular instrument, situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims.

Claims (15)

What is claimed is:

1. An apparatus for estimating conditions of reservoir fluid in an underground reservoir, the apparatus comprising:

a sensor for measuring one or more measured parameters of that fluid, the measured parameters including at least one of: temperature and pressure of the fluid; and

a processor, the processor configured to:

receive data representing the one or more measured parameters;

calculate the density ρ of a mixture containing water;

determine component-specific volume shift parameters c_i;

replace the component specific volume shift parameter for water with a volume shift parameter c_H2O;

determine a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture;

determine a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; and

recalculate density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume.

2. The apparatus of claim 1, wherein the density is determined based by solving:

${\rho }_{\mathrm{liq}}=\frac{P}{\mathrm{RT}}\frac{M_{\mathrm{liq}}}{Z_{\mathrm{liq}}};\mathrm{and}$ ${\rho }_{\mathrm{vap}}=\frac{P}{\mathrm{RT}}\frac{M_{\mathrm{vap}}}{Z_{\mathrm{vap}}}.$

3. The apparatus of claim 2, wherein determining the component-specific volume shift parameters c_i; includes solving:

$c_i=0.40768\left(\frac{{\mathrm{RT}}_c}{P_c}\right)\left(0.29441-Z_{\mathrm{RA}}\right).$

4. The apparatus of claim 2, wherein determining the component-specific volume shift parameters c_i; includes solving:

$c_i=S_ib_i,S_i=1-\frac{d}{M^e},b_i=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c},$

where d=2.258 and e=0.1823 for n-alkane hydrocarbons (different values of d and e are also provided for aromatic and naphthenic hydrocarbons).

5. The apparatus of claim 2, wherein replacing the component specific volume shift parameter for water with a volume shift parameter c_H2O, includes solving:

$c_{H2O}=b_{H2O}S_{H2O},S_{H2O}=\frac{a+b\exp \left(T_r\right)+c\exp \left(P_r\right)+d\exp \left(T_r\right)\exp \left(P_r\right)}{1+f\exp \left(T_r\right)+\exp \left(P_r\right)+h\exp \left(T_r\right)\exp \left(P_r\right)},b_{H2O}=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c}=\{\begin{array}{c}2.114176\times {10}^{-5}\frac{m^3}{\mathrm{mol}}\left(SRK\right)\\ 1.898464\times {10}^{-5}\frac{m^3}{\mathrm{mol}}\left(PR\right)\end{array}P_r=\frac{P}{P_c}\mathrm{and}T_r=\frac{T}{T_c}$

6. The apparatus of claim 2, wherein determining a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture; includes solving:

$c_{\mathrm{liq}}=\sum _i^Nx_ic_i,c_{\mathrm{vap}}=\sum _i^Ny_ic_i.$

7. The apparatus of claim 2, wherein determining a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; includes solving:

v _liq ^corrected =v _liq ^EOS −c _liq,

v _vap ^corrected =v _vap ^EOS −c _vap.

8. The apparatus of claim 2, wherein recalculating the density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume; includes solving:

${\rho }_{\mathrm{liq}}^{\mathrm{corrected}}=\frac{M_{\mathrm{liq}}}{{\upsilon }_{\mathrm{liq}}^{\mathrm{corrected}}},{\rho }_{\mathrm{vap}}^{\mathrm{corrected}}=\frac{M_{\mathrm{vap}}}{{\upsilon }_{\mathrm{vap}}^{\mathrm{corrected}}},Z_{\mathrm{liq}}^{\mathrm{corrected}}=\frac{1}{{\upsilon }_{\mathrm{liq}}^{\mathrm{corrected}}},Z_{\mathrm{vap}}^{\mathrm{corrected}}=\frac{1}{{\upsilon }_{\mathrm{vap}}^{\mathrm{corrected}}}.$

9. A method of estimating properties of a reservoir fluid in an underground reservoir, the apparatus comprising:

receiving, at a computing device, data representing one or more measured parameters of a mixture containing water in the underground reservoir.

calculating, with the computing device, the density ρ of the mixture containing water;

determining component-specific volume shift parameters c_i;

replacing the component specific volume shift parameter for water with a volume shift parameter c_H2O;

determining a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture;

determining a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; and

recalculating density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume.

10. The method of claim 9, wherein determining the component-specific volume shift parameters c_i; may include solving:

$c_i=0.40768\left(\frac{{\mathrm{RT}}_c}{P_c}\right)\left(0.29441-Z_{\mathrm{RA}}\right).$

11. The method of claim 9, wherein determining the component-specific volume shift parameters c_i; may include solving:

$c_i=S_ib_i,S_i=1-\frac{d}{M^e},b_i=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c},$

where d=2.258 and e=0.1823 for n-alkane hydrocarbons.

12. The method of claim 9, wherein replacing the component specific volume shift parameter for water with a volume shift parameter c_H2O, includes solving:

$c_{H2O}=b_{H2O}S_{H2O},S_{H2O}=\frac{a+b\exp \left(T_r\right)+c\exp \left(P_r\right)+d\exp \left(T_r\right)\exp \left(P_r\right)}{1+f\exp \left(T_r\right)+\exp \left(P_r\right)+h\exp \left(T_r\right)\exp \left(P_r\right)},b_{H2O}=\frac{{\Omega }_b{\mathrm{RT}}_c}{P_c}=\{\begin{array}{c}2.114176\times {10}^{-5}\frac{m^3}{\mathrm{mol}}\left(SRK\right)\\ 1.898464\times {10}^{-5}\frac{m^3}{\mathrm{mol}}\left(PR\right)\end{array}P_r=\frac{P}{P_c}\mathrm{and}T_r=\frac{T}{T_c}$

13. The method of claim 9, wherein determining a total volume shift c required from the sum of the component specific volume shifts and the mole fraction of each component in the mixture; includes solving:

$c_{\mathrm{liq}}=\sum _i^Nx_ic_i,c_{\mathrm{vap}}=\sum _i^Ny_ic_i.$

14. The method of claim 9, wherein determining a corrected molar volume of the liquid and vapor by subtracting the total volume shift c from an equation of state (EOS) calculated molar volume v; includes solving:

v _liq ^corrected =v _liq ^EOS −c _liq,

v _vap ^corrected =v _vap ^EOS −c _vap.

15. The method of claim 9, wherein recalculating the density ρ and compressibility factor Z for the liquid and vapor phases with the corrected molar volume; includes solving:

${\rho }_{\mathrm{liq}}^{\mathrm{corrected}}=\frac{M_{\mathrm{liq}}}{{\upsilon }_{\mathrm{liq}}^{\mathrm{corrected}}},{\rho }_{\mathrm{vap}}^{\mathrm{corrected}}=\frac{M_{\mathrm{vap}}}{{\upsilon }_{\mathrm{vap}}^{\mathrm{corrected}}},Z_{\mathrm{liq}}^{\mathrm{corrected}}=\frac{1}{{\upsilon }_{\mathrm{liq}}^{\mathrm{corrected}}},Z_{\mathrm{vap}}^{\mathrm{corrected}}=\frac{1}{{\upsilon }_{\mathrm{vap}}^{\mathrm{corrected}}}.$

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