Water bot

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
$C_1$	$k_{\mathrm{ij,non-aq}} = 0.4850$
$C_2$	$k_{\mathrm{ij,non-aq}} = 0.4920$
$C_3$	$k_{\mathrm{ij,non-aq}} = 0.5525$
$C_4$	$k_{\mathrm{ij,non-aq}} = 0.5091$
$C_{5+}$	$k_{\mathrm{ij,non-aq}} = 0.5000$
$N_2$	$k_{\mathrm{ij,non-aq}} = 0.4778$
$CO_2$	$k_{\mathrm{ij,non-aq}} = 0.1896$
$H_2S$	$k_{\mathrm{ij,non-aq}} = -0.19031 + 0.05965T_{\mathrm{ri}}$

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:

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.
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.
Steps 1 and 2 are repeated using the non-aqueous EOS model to determine $R_{vw}$ and $B_{gw}$.
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