Water bot

Introduction

Water PVT and viscosity can be computed as a function of pressure, temperature, and salinity. The amount of gas in solution in water can also be included as a fourth variable. This section explains how correlations can be used to estimate the water density and viscosity which is relevant for reservoir simulation and pipe-flow calculations. The summary is based on Chapter 9 of Whitson and Brulé\({^3}\)

Water Salinity

The salinity of water represents the amount of "total dissolved solids" (TDS) in the water, usually consisting mainly of NaCl. The salinity can range from 0 (fresh water) to \(\approx\)300,000 ppm (upper limit is temperature dependent). Seawater has a salinity of about 35,000 ppm. Salinity can be expressed in different quantities: weight fraction, \(w_s\); mole fraction, \(x_s\); molality, \(c_{sw}\); weight concentration in parts per million, \(C_{sw}\); and volume concentration in parts per million, \(C_{sv}\).

The limiting concentration for NaCl brine is [3]

$$\begin{equation} C_{sw}=262180+72T + 1.06T^2 \end{equation}$$ with temperature \(T\) in Celsius.

Note

The input for water salinity in whitson⁺ is weight concentration \(C_{sw}\) (ppm = mg TDS/kg salt water = \(10^6m_s/(m_s+m_w^0)\)), which is converted internally to a weight fraction \(w_s=10^{-6}C_{sw}\) and molality \(c_{sw}=17.1C_{sw}/(10^6-C_{sw})\) used in computations.

Water Density

The mass-density of water at a specified pressure, temperature, and salinity is calculated using a water formation volume factor (FVF) through the relationship $$\begin{equation} \rho_w=\frac{\rho_{\bar{w}}^0}{B_w} \end{equation}$$ where the water FVF is calculated in a two-step procedure

1. At standard pressure, temperature, and salinity, compute the water FVF, \(B_{w}^0(T,w_s)\).
2. At pressure, temperature, and salinity, compute the water FVF, \(B_{w}(p,T,w_s)\), by correcting \(B_{w}^0(T,w_s)\) with a correction factor \(C=f(p,T,w_s)\).

Surface-Water Density

The water density at standard conditions \(\rho_{\bar{w}}^0\) is computed by the correlation $$\begin{equation} \rho_{\bar{w}}^0 = (1.0009 -0.7114w_s+0.26066w_s^2)^{-1} \end{equation}$$

Water FVF

The water FVF at standard pressure, temperature, and salinity \((B_w^0)\) is calculated by the following correlations $$\begin{equation} B_w^0 = \frac{\rho_{w}^0}{\rho_{\bar{w}}^0} = \frac{A_0+A_1w_s+A_2w_s^2}{\rho_{\bar{w}}^0} \end{equation}$$ where $$\begin{align} A_0&=5.91635 -0.01035794T + 0.9270048\cdot 10^{-5}T^2 -1127.522T^{-1}+100674.1T^{-2}\\ A_1&=-2.5166+0.0111766T-0.170552\cdot 10^{-4}T^2\\ A_2&=2.84851-0.0154305T-0.223982\cdot 10^{-4}T^2 \end{align}$$ with temperature, \(T\), in K.

The water FVF at pressure, temperature, and salinity is calculated by the following correlations $$\begin{equation} B_w=B_w^0\bigg(\frac{A_3+A_4p}{A_3+A_4p_{sc}}\bigg)^{-1/A_4} \end{equation}$$ where $$\begin{align} A_3 &= 10^6\big[0.314 +0.58w_s+1.9\cdot 10^{-4} T-1.45\cdot 10^{-6} T^2\big]\\ A_4 &= 8 + 50w_s -0.125w_sT \end{align}$$ with pressure, \(p\), in psia, temperature, \(T\) in °F.

Water Compressibility

The water compressibility can be derived from the water FVF by the definition of compressibility $$\begin{equation} c_w = -\frac{1}{B_w}\frac{dB_w}{dp}\bigg|_{x_s, T}=(A_3 + A_4p)^{-1} \end{equation}$$

Water Viscosity

Viscosity

The water viscosity is calculated by the following correlation $$\begin{equation} \mu_w=(1+A_5p)\hat{\mu}_w \end{equation}$$ where $$\begin{equation} \log\frac{\hat{\mu}_w}{\mu_w^0}=A_6+A_7\log\frac{\mu_w^0}{\mu_{w20}^0} \end{equation}$$ with $$\begin{align} A_5 &= 10^{-3}\big[0.8+0.01(T-90)\exp(-0.25c_{sw})\big]\\ A_6 &= \sum_{i=1}^3a_{1i}c_{sw}^i\\ A_7 &= \sum_{i=1}^3a_{2i}c_{sw}^i \end{align}$$ and $$\begin{equation} \log\frac{\mu_w^0}{\mu_{w20}^0}=\sum_{i=1}^4 a_{3i}\frac{(20-T)^i}{96+T}, \quad \mu_{w20}^0=1.002 cp \end{equation}$$

The coefficients \(a_{1i}\), \(a_{2i}\), and \(a_{3i}\) are: $$\begin{align} a_{11}=3.324\cdot 10^{-2},\quad a_{12}&=3.624\cdot 10^{-3},\quad a_{13}=-1.879\cdot 10^{-4}\\ a_{21}=-3.96\cdot 10^{-2},\quad a_{22}&=1.020\cdot 10^{-2},\quad a_{23}=-7.020\cdot 10^{-4}\\ a_{31}=1.2378,\quad a_{32}=-1.303\cdot10^{-3}&,\quad a_{33}=3.060\cdot 10^{-6},\quad a_{34}=2.550\cdot 10^{-8} \end{align}$$ where viscosity, \(\mu\), is in cp, temperature, \(T\), is in °F, and pressure, \(p\), is in MPa.

Viscosibility

The resevoir simulators Eclipse 100 and OPM Flow compute the water viscosity as a function of pressure through the use of a so-called "viscosibility" defined as: $$\begin{equation} C_{\mu}=\frac{1}{\mu_w}\frac{d\mu_w}{dp}\bigg|_{T,w_s} \end{equation}$$ Assuming that the viscosibility is constant, then the viscosity can be expressed as an exponential function of pressure by $$\begin{equation} \mu_w=\mu_w(p_{0})\exp(C_{\mu}(p-p_{0})) \end{equation}$$ where \(p_0\) is the reference pressure at which a corresponding water viscosity \(\mu_w(p_{0})\) is provided together with the viscosibility.

For the water-viscosity expression above, the water viscosibility is expressed by $$\begin{equation} C_{\mu}=\frac{A_5}{1+A_5p} \end{equation}$$

Computing Water Properties Using an EOS Model

Introduction

Water properties like FVF, gas-water ratio (GWR), density and compressibility are determined using an EOS model.

Required Input

The required input data for generating water properties are:

• the composition of the initial in-situ reservoir fluid (\(z_{\mathrm{bo}i}\))
• the reservoir temperature (\(T_{res}\))
• brine salinity(\(c_{sw}\))
• an Equation of State (EOS) model tuned to the relevant PVT data (not including water)

Procedure

EOS Models Tuning

Two EOS models are developed, one for the aqueous phase and another one for the non-aqueous phase. Soreide- Whitson\({^1}\) and Yan-Huang-Stenby\({^2}\) provide the equations to calculate binary interaction parameters (BIP's) for both phases (aqueous and non-aqueous) for the Peng-Robinson EOS model. These equations are provided in the tables below, and are used as initial values in the EOS regression.

Aqueous Phase BIP's for PR EOS Model used for water/hydrocarbon system.

Component	Equation
\(Hydrocarbon\)	\(k_\mathrm{ij,aq} = (1+a_\mathrm{0}c_\mathrm{sw})A_\mathrm{0}+(1+a_\mathrm{1}c_\mathrm{sw})A_\mathrm{1}T_\mathrm{ri}+(1+a_\mathrm{2}c_\mathrm{sw})A_\mathrm{2}T_\mathrm{ri}^2,\) \(a_\mathrm{0}=0.01747,a_\mathrm{1}=0.033516,a_\mathrm{2}=0.011478\) \(A_{0}=1.112-1.7369\omega_i^{-0.1},A_{1}=1.1001+0.83\omega_i,A_{2}=-0.15742-1.0988\omega_i\)
\(N_\mathrm{2}\)	\(k_{ij,aq}=-1.70235(1+0.025587c_{sw}^{0.75})+0.44338(1+0.08126c_{sw}^{0.75})T_{ri}\)
\(CO_\mathrm{2}\)	\(k_{ij,aq}=0.30823655+0.11820367c_{sw}-9.5381166\times10^{-4}c_{sw}^2\)             \(-126.42095/T-6.2924435\times10^{-4}c_{sw}T+9.2946667\times10^{-7}c_{sw}T^2\)
\(H_\mathrm{2}S\)	\(k_{ij,aq}=-0.20441+0.23426T_{ri}\)

Non-aqueous Phase BIP's for PR EOS Model used for water/hydrocarbon system.

Component	Equation/Value
\(C_1\)	\(k_{ij,non-aq} = 0.4850\)
\(C_2\)	\(k_{ij,non-aq} = 0.4920\)
\(C_3\)	\(k_{ij,non-aq} = 0.5525\)
\(C_4\)	\(k_{ij,non-aq} = 0.5091\)
\(C_{5+}\)	\(k_{ij,non-aq} = 0.5000\)
\(N_2\)	\(k_{ij,non-aq} = 0.4778\)
\(CO_2\)	\(k_{ij,non-aq} = 0.1896\)
\(H_2S\)	\(k_{ij,non-aq}=-0.19031+0.05965T_{ri}\)

The modified \(\alpha\)-term in the EOS constant \(a\) for the brine component is calculated as a function of water reduced temperature and brine salinity using the following equation:

$$\begin{equation} \alpha_{H_2O}^{0.5} = 1+0.453[1-T_{rH_2O}(1-0.0103c_{sw}^{1.1})+0.0034(T_{rH_2O}^{-3}-1)] \label{eq:mod-alapha} \end{equation}$$

Brine Density. Brine density at each pressure is calculated using Rowe-Chou correlation. Brine salinity (\(c_{sw}\)) and reservoir temperature (\(T_{res}\)) are used as inputs to this correlation. Then the aqueous EOS model is tuned to match the brine densities by regressing on the critical properties of water.

Brine Viscosity. Brine viscosity at each pressure is calculated using the Kestin correlation. Brine salinity (\(c_{sw}\)) and reservoir temperature (\(T_{res}\)) are used as inputs to this correlation. The third (P3) and forth (P4) LBC correlation parameters are used to match the aqueous EOS model viscosities to those predicted by the Kestin correlation.

Given that the water critical properties of the aqueous EOS model are modified in the steps above (to match the brine density and brine viscosity). This modified aqueous EOS model is tuned to match the saturation pressure determined from the original aqueous EOS model (with BIPs from the tables presented above).

Calculating Water Properties

After the EOS model is tuned to the brine density and viscosity data, the water properties are calculated as follows:

1. Flash a mixture of water (50 mol-%) and \(C_{4-}\) of initial in-situ reservoir fluid (\(z_{boi}\)) (50 mol-%) at reservoir temperature (\(T_{res}\)) and desired pressure using the modified aqueous EOS model.
2. Equilibrium liquid and gas from (1) are brought to 60\(^oF\) and 14.7 psia. Then \(B_w, R_{sw}, \rho_w, \mu_w \) are calculated for aqueous phase.
3. Steps 1 and 2 are repeated using the non-aqueous EOS model to determine \(R_{vw}\) and \(B_{gw}\).
4. Saturated \(c_w\) is calculated using the equation below:

$$\begin{equation} c_w=\frac{(B_{gw}-B_wR_{vw})R_{sw}^{'}}{B_w(1-R_{sw}R_{vw})}-\frac{B_w^{'}}{B_w} \label{eq:walsh} \end{equation}$$       where \(R_{sw}^{'}\) and \(B_w^{'}\) are derivative of \(R_{sw}\) and \(B_w\) for a decreasing pressure change in the saturated state.
5. Under-saturated \(c_w\) is calculated from the same equation, where \(R_{sw}^{'}\) = 0 and \(B_w^{'}\) is calculated for an increasing pressure change in the under-saturated state: $$\begin{equation} c_w=-\frac{B_w^{'}}{B_w} \label{eq:sat-cw} \end{equation}$$

References

[1] Søreide, I. and Whitson, C.H.: "Peng-Robinson Predictions for Hydrocarbons, CO2, N2 and H2S With Pure Water and NaCl-Brines," Fluid Phase Equilibria (1992)

[2] Wei Yan, Shengli Huang, Erling H. Stenby,"Measurement and modeling of CO2 solubility in NaCl brine and CO2–saturated NaCl brine density, International Journal of Greenhouse Gas Control", Volume 5, Issue 6, 2011,Pages 1460-1477,ISSN 1750-5836

[3] Whitson, C.H. and Brulé M.: "Phase Behavior", SPE Monograph Series vol.20, ISBN: 978-1-55563-087-4