# Blowout Simulation

## Overview#

The blowout simulation is employed to estimate the potential blowout rates from reservoirs through designated wellbores to the surface.

### Printable Version#

Oliasoft Technical Documentation - Blowout

## Model Overview and Application#

### Area of Application and Limitations#

Oliasoft's blowout analysis tool offers decision support for environmental risk management in well planning and drilling activities. The tool takes into account most phases of conventional drilling. There are however some important limitations on the range of applicability which the user should be aware of. These are listed below.

• The flow simulator assumes steady-state, i.e. fluid properties do not change over time. Reservoir depletion is not taken into consideration.
• BlowFlow uses only Black Oil PVT models. Fluid compositions beyond impurities due to $CO_2$, $H_2S$ and $N_2$ are not accounted for.
• BlowFlow does not currently implement a PVT model for gas condensate. Such a reservoir may be simulated, but the PVT parameters for condensate will then use the existing Black Oil PVT models.
• For two-phase reservoirs, a mixture of oil and gas is available. Combinations of oil or gas with water are not available, nor are three- phase reservoirs.
• A combination of different fluid compositions for multi-zone reservoirs is not currently possible (i.e. the phase type is assumed the same for each reservoir zone).
• Flow paths currently handled are Topside/Subsea for Drill string/Annulus/Open hole. Blowout through the casing string or casing annulus is not handled.
• The blowout engine relies on various models, many of which are based on empirical correlations which provide estimated values. These models may in themselves impose restrictions beyond those listed here.
• Pressure losses due to acceleration is neglected

Within these restrictions the tool enables analyzing well specific blowout consequences and providing detailed results for the different flow scenarios in addition to showing the overall picture. The analysis results are given for both oil and gas.

### Model Structure#

The structure of the models constituting the blowout simulation engine is shown in Figure 2.1.

The blowout engine contains three main models:

• Blowout flow rate model, which is based on the following three sub-models
• PVT model
• Temperature model
• Black Oil PVT model
• Gas PVT model
• Inflow model
• Outflow model
• Blowout duration rate model
• Blowout volume model

The models take input from the user operating the tool. Model inputs are specified in terms of deterministic values, single probability values and probability distributions.

The output from a blowout analysis includes:

• Blowout Flow Rates. Blowout flow rates are computed for all defined scenarios, for both oil and gas. Depending on how input is represented, these rates are either deterministic or stochastic. Flow rates are presented across time for the entire duration of the blowout. As the model is steady-state, only impact of well kill mechanisms will influence flow rates over time.
• Blowout Duration. The total blowout duration is defined as the time until well kill mechanisms successfully stop the uncontrolled flow. However, as depletion is not considered then if no well kill mechanisms are defined, the blowout duration is a user-defined “cut-off” point, at which time the blowout will cease.
• Blowout Volume. The blowout volumes computed are simply the product of flow rates and blowout duration.

### Modelling Principles and Input Assessment#

The aim of developing the duration, flow rate and volume models is to establish a flexible platform for relating uncertainty statements to quantities that contribute to the uncertainty about the consequences of a given blowout. Key parameter values in the models are given by probability distributions instead of deterministic values in order to reflect uncertainty in the parameters assessments. These uncertainties contribute to uncertainty in the total blowout consequence picture. Hence, the blowout consequences are given by both expected values and the total range.

These can be used instead of expert judgements for assessment of input parameters to the analysis of the model. However, due to the uniqueness of each drilling operation and the high level of detail of the quantities for which input will be required, such data are generally scarce. The majority of input parameters will be assessed on the basis of engineering judgements.

### Probability Assessment#

Probability distributions are used as the measure of uncertainty related to reservoir characteristics and the consequences of a blowout. Probability assessment related to scenarios, kill mechanisms and well information is a main activity in the blowout analysis process. Hence, personnel involved in the analysis process or in decision making related to well planning and design must be familiar with how the probability figures shall be interpreted in this context.

Due to field-to-field and well-to-well variations in factors like water depth, lithology, pressure regimes, equipment configurations and drilling procedures the drilling of each well can be considered as a unique process. Consequently, the amount of relevant experience data suitable for supporting probability assessments related to killing a well is scarce. However, by consulting system experts including geologists, mud engineers, drilling managers and other personnel with operational experience, uncertainty can be expressed at a level of detail where system information exists. In order to establish an optimal basis for decision making and planning, the aim is to reach a maximum share of the available information in the analysis, including both the available relevant historical data and expert judgements.

### The Blowout Analysis Process#

A blowout analysis process should be performed in accordance with the following steps:

1. Assessment of input data
2. Blwoout analysis
3. Evaluation of results and decision making related to implementation of candidate measures for consequence reduction

The steps fo analysis process are described below.

Input Data Assessment

Assessment of input data is a crucial and considerable part of the blowout analysis process. Some of the input required may be found from relevant documentation related to the drilling operation, e.g. the drilling program. However, several parameters, especially the probabilities and probability distributions, are assigned on the basis of subjective considerations in combination with available experience data.

Experience from pilot studies has shown that work meetings involving personnel from various disciplines help to stimulate constructive discussions and ensure that relevant conditions are included in the considerations in a consistent manner. Since assessment of subjective probabilities may be a strenuous process, it is strongly recommended that several work meetings are held during the input assessment phase.

Blowout Analysis

After assessing the input parameters required for the blowout analysis, the overall analysis can be performed. The main results from the Blowout tool include:

• Distribution of total blowout flow rates and flow rates distributed on exit points and flow paths
• Distribution of blowout duration
• Blowout volume distribution in total and distributed on exit points and flow media

Evaluation of Results

The effect of risk-reducing measures, such as equipment modifications, change of operational routines or increased information of downhole parameters can be represented by proper adjustment of the model input. Re-analyses with adjusted input provides a basis for ranking and selection of candidate measures.

## Model Input#

### General Input Types#

The blowout simulation engine requires input on a large number of parameters. There are three main types of input parameters:

• Deterministic values, e.g. 200 m
• Single probability values, e.e. 30%
• Probability density functions, e.g. N(15,2)

The input parameters to the tool have to be assessed by the user.

### Input Categories#

The different input parameters are divided into the following categories:

• Project info. Reference information such as well and field names, comments, etc.
• Reservoir. Characteristics of the reservoir including:
• Fluid
• Reservoir zones
• Inflow models
• PVT models
• Multiphase flow models
• Platform. RT elevation and water depth
• Architecture. Input for riser, casings, planned open hole section and BOP.
• Drill String. Specification of the components which make up the drill string.
• Survey. Wellbore trajectory
• Probabilistic Scenarios. Blowout path and exit point, reservoir zone penetration depth scenario, bit location scenario and BOP opening scenario.
• Duration. Probabilistic input related to the kill mechanisms considered:
• Capping
• Relief well
• Bridging
• Coning

## Temperature Model#

The blowout engine uses a simple temperature model which essentially converts measured formation temperatures to flowing well temperatures. Being a steady-state framework, the model does not considering temperature effects over time.

The following assumptions apply for the temperature model:

• Temperature of flowing fluid is independent of radial position (ideal turbulent mixing)
• Heat transfer properties of the fluid are not considered
• Fluid composition is constant along the well flow path
• Ideal heat transfer from fluid to pipe wall (no losses), i.e. conservative calculation in the direction of giving too efficient cooling

The model use the following inputs:

• Overall heat transfer coefficient, $U$
• Formation temperatures, $T_{fm1}$,...$T_{fmn}$ (Minimum two measured points)
• Fluid mass flow rates, $m_o$,$m_g$
• Fluid heat capacities, $C_o$,$C_g$
• Flow area, $A$

Formation temperatures, and corresponding well temperatures, are assumed linear between measured points. Hence, interpolation or extrapolation is used to determine temperatures between or outside the defined intervals.

The expression for the flowing well temperature at a given node, indexed i, may be expressed as

$T_{w,i}=T_{w,i-1}-\frac{\dot{Q}}{W} \textrm{if} T_{w,i-1}-\frac{\dot{Q}}{W} \geq T_{fm}$

$T_w = T_{fm}\;\;\;\;\;\; \textrm{otherwise}$

and

$T_{w,0}=T_{fm}$

where

$\dot{Q}=UA(T_{w,i-1}-T_{fm})$

$W=C_o m_o + C_g m_g$

## Input#

The first necessary input for the inflow model is the description of the reservoir, containing fluid type, reservoir fluid properties, and temperature profile. And more details in the reservoir zone are required, such as top pressure, oil/gas gravity distribution, top depth. In the blowout inflow model, either the productivity indices are provided or other key parameters, such as permeability and the skin factor are important for the simulations. Additionally, a temperature gradient model is used as input to the flow rate simulations. Other required inputs are formation temperatures for the minimum two points.

The outflow model depends on the geometry of the flowing well. The geometry includes well trajectory, drill string, casing, and open hole, technically speaking, inclinations, depths, inner and outer diameters of each section in wellbores.

## Inflow Performance Relationship#

Five different inflow models are implemented, which are Oil Basic (oil/gas), Oil Fractured (oil/gas), Oil Explicit (oil/gas), Gas Deliverability (Gas/Condensate) and Gas Explicit (gas/condensate) to calculate flow rates under the different penetration (100%, 50% and 5m), where the scenario of 100% penetration is necessary within the calculation of inflow performance relationship (IPR) [1].

The IPR curve is the relation between the flowing bottom-hole pressure $P_{wf}$ and liquid production rate $q$. Undersaturated oil reservoirs exist as single-phase reservoirs where pressures are above the bubble point pressure. The linear IPR model is given as,

$q = J (P_{res} - P_{wf})$

where $J$ is the productivity index to describe the trends of the IPR curve and $P_{res}$ represents the reservoir pressure.

The solution gas escapes from the oil and becomes free gas when the flowing bottom-hole pressure $P_{wf}$ is below the bubble point pressure $P_b$ [1]. Therefore, Vogel established an empirical equation for two-phase reservoirs in 1968 [2], and it is still widely used in the industry.

If the reservoir pressure $P_{res} < P_b$ ,

$q = q_{max} (1 - 0.2 \frac{P_{wf}}{P_{res}} - 0.8 \frac{P_{wf}}{P_{res}}^2)$

and,

$q_{max} = \frac{JP_{res}}{1.8}$

Otherwise the following is being used,

$q = J (P_{res} - P_b) + \frac{JP_{b}}{1.8} (1 - 0.2 \frac{P_{wf}}{P_{b}} - 0.8 \frac{P_{wf}}{P_{b}}^2)$

As the productivity index $J$ is an unknown variable, it can be estimated according to different flow types.

### Oil basic#

Productivity index for vertical and deviated wells in rectangular drainage areas with constant pressure boundaries.

$J = \frac{kh}{18.7 B_o \mu_o (0.5 \log \frac{2.2 x_e y_e}{c_a r_w^2} + S ) }$

### Oil Fractured#

$J = \frac{kh}{18.7 B_o \mu_o (0.5 \log \frac{2.2 x_e y_e}{c_a r_w^2} + S ) }$

or when the production time is smaller than Permeability time,

$J = \frac{kh}{21.5 B_g \mu_g (\log \frac{kt}{\textsf{poro} \mu_g c r_w^2} - 3.1 + 0.87S ) }$

### Gas Deliverability#

Transient productivity index (i.e. productivity index of a well which has not yet seen any of the boundaries (radial flow) is used in this part. Most DST/WFT fall into this category) which can be used to determine the infinite-acting period.

$J = \frac{kh}{21.5 B_g \mu_g (\log \frac{kt}{\textsf{poro} \mu_g c r_w^2} - 3.1 + 0.87S ) }$

### Forcheimer's Model#

High velocity flow in porous media and fractures is modeled by the Forchheimer equation in gas reservoir when the reservoir pressure exceeds a cut-off value numerically.

$P_{wf}= P_{res} - a \cdot q - b \cdot q^2$

The parameters a and b are estimated based on pseudo pressure correlations:

$a = \frac{\hat{P}_a \mu_f z_f}{P_{res}} \hspace{3mm}\textit{and} \hspace{3mm} b = \frac{\hat{P}_b \mu_f z_f}{P_{res}}$

Forchheimer equation can be performed for the gas systems, where the nonlinear flow is much more significant, due to the lower gas viscosity which will give high numbers for the same velocity as in liquid systems.

## Vertical Lift Performance Relationship#

The Vertical lift performance (VLP), known as the outflow model, decribes the relationship between the bottom-hole pressure and the flow rate. Widely used multiphase flow models are implemented to describe the VLP relationship. The analyzation of different flow regimes is important in the empirical models, which are bubble flow, slug flow, transition flow, mist flow, segregated flow, intermittent, distributed flow, plug flow, and froth flow.

### Hagedorn-Brown Correlation#

The Hagedorn-Brown Correlation applies only to vertical wells. It is a combination of two correlations: Hagedorn-Brown correlation for slug flow and Griffith correlation for bubble flow. Thus, it is necessary to determine the flow pattern before we proceed to the next.

$A = 1.071 - \frac{0.2218 (v_{sl} + v_{sg})^2 }{0.3048^2 d}$

$B = \frac{v_{sg}}{v_{sg} + v_{sl}}$

If $B-A \geq 0$ , continue with the Hagedorn-Brown correlation, or else the Griffith correlation is under consideration.

Griffith Correlation

Liquid holdup

$\lambda = 1 - 0.5 + [ 1 + \frac{v_m}{0.24384} - \sqrt{([1 + \frac{v_m}{0.24384} ]^2 - 4 \frac{v_{sg}}{0.24384}} ) ]$

Hagedorn-Brown Correlation

Calculate liquid viscosity number and coefficient

$N_L = \mu_L \left[ \frac{g}{\rho_L \sigma^3_L} \right]^{1/4}$

$CN_L = \frac{0.0019 + 0.0322 N_L - 0.6642 N^2_L + 4.9551 N^3_L}{ 1 - 10.0147 N_L + 33.8696 N^2_L + 277.2817 N^3_L}$

Calculate liquid, gas velocity number, and pipe diameter number

$N_{Lv} = v_{sl} \left[ \frac{\rho_L}{g \sigma_L} \right]^{1/4}$

$N_{Gv} = v_{sg} \left[ \frac{\rho_L}{g \sigma_L} \right]^{1/4}$

$N_{d} = d \left[ \frac{g \rho_L}{\sigma_L} \right]^{1/2}$

$\phi = \frac{N_{Lv}}{N^{0.575}_{GV}} \left[ \frac{\overline{P} }{14.7} \right]^{0.10} \left[ \frac{ CN_L }{ N_d } \right]$

$\xi = \left[ \frac{ 0.0047 + 1123.32 \cdot \phi + 729489.64 \cdot \phi^2 }{1 + 1097.1566 \cdot \phi + 722153.97 \cdot \phi^2} \right]^{0.5}$

Calculate liquid holdup

$\lambda = \phi \cdot \xi$

At last, the frictional pressure gradient is

$\left[\frac{dp}{dx}\right]_f = \frac{2f_{tp} \rho_{ns} v^2_m}{d} \cdot \frac{\rho_{ns}}{\rho_s}$

### Beggs & Brill Correlation#

The Beggs & Brill model is developed for tubing strings in inclined wells and pipelines for hilly terrain. This model was developed from experiments using air and water as test fluids over a wide range of parameters. Beggs & Brill uses the no-slip friction factor to calculate frictional pressure losses.

Calculate total flux rate

$v_m = v_{sg} + v_{sl}$

Calcualte no-slip holdup

$\lambda_{ns} = \frac{v_{sl}}{v_{sg} + v_{sl}}$

Calculate the Froude number

$N_{FR} = \frac{v^2_m}{gd}$

Calculate liquid velocity number $N_{Lv}$ (14).

In the following calculation, we use the no-slip holdup and the Froude number to determine the flow patterns, such as segregated, transition, intermittent, and distributed.

Calculate the horizontal holdup

$\lambda_{o} = \frac{a\lambda^b_{ns}}{N^c_{FR}}$

Calculate the inclination correction factor coefficient

$C = (1 - \lambda_{ns})ln(d\lambda^e_{ns}N^f_{Lv}N^g_{FR})$

where the values of parameters $a,b,c,d,e,f$ and $g$ are dependent of flow patterns and flow conditions (uphill or downhill).

Calculate the liquid holdup inclination correction factor

$\phi = 1 + c[sin(1.8 \theta) - 0.333 sin^3(1.8\theta)]$

where $\theta$ is the deviation from horisontal axis.

Calculate the liquid holdup

$\lambda = \lambda_o \phi,$

and more correlations can be added for different flow patterns and conditions.

Calculate the friction factor ratio

$\frac{f_{tp}}{f_{ns}} = e^S$

where $S$ is a function of $\lambda_{ns}/\lambda^2$.

Calculate no-slip Reynolds number

$(N_{Re})_{ns} = \rho_{ns} v_m de/\mu_{ns}$

The two-phase friction factor is

$f_{tp} = f_{ns} \cdot e^S$

where $f_{ns}$ is the fanning friction factor.

At last, the frictional pressure gradient is

$\left[\frac{dp}{dx}\right]_f = \frac{2f_{tp} \rho_{ns} v^2_m}{d}$

### Okrizewski#

The Orkiszewski correlation is applicable for two-phase pressure drops in vertical wells for different flow regimes (bubble, slug, transition, annular mist). The correlation has been proven accurate in the estimation of high-velocity flow in both gas condensate wells and oil wells.

In this correlation, the superficial velocity for each phase has been estimated to find out the flow regimes. The liquid hold up corresponding to different flow regimes can be obtained, such as Griffith correlation for bubble flow, Duns & Ros correlation for transition/mist flow and Chierici et. al. (19874)'s work for slug flow. And then, the Reynolds number and friction factor yielded in each flow regime, thus the frictional pressure loss can be calculated.

### Gray Correlation#

A vertical flow correlation for gas condensate wells was developed by H. E. Gray. It is an empirical model, which employs a pressure balance equation with the gas volume fraction $\xi$ .

The pressure balance equation

$dP = \frac{g}{g_c} [\xi \rho_g + (1-\xi)\rho_l ]dh + \frac{f_t G^2}{2g_c D P_{mf} dh} + \frac{G^2}{g_c}d\left[ \frac{1}{\rho_{mi}} \right]$

The gas volume fraction

$\xi = \frac{1 - \exp \left[ -0.2314 \left[ N_v \left[ 1+\frac{205.0}{N_D} \right] \right]^B \right] }{R + 1}$

$B = 0.0814 \left[ 1 - 0.0554 \ln \left[ 1+ \frac{730R}{R+1} \right] \right]$

### Gray Modified#

Gray modified is developed based on the Gray model combined with the hydrostatic pressure loss, calculated using no slip density.

### Duns & Ros Correlation#

The Duns & Ros correlation is developed for slip velocity and friction factor for each of the three flow regimes.

Calculate liquid viscosity number $N_L$ in equation (12), liquid velocity number $N_{Lv}$ (14), gas velocity number $N_{Gv}$ using equation (15), and pipe diameter number $N_d$ (16).

Calcualte dimensionless quantities

$L_s = 50 + 36 N_{Lv}$

$L_m = 75 + 84 N^{0.75}_{Lv}$

By comparison of $N_{Gv}$ and the above two values combining $N_{Lv}$, the flow regimes can be determined. Then, slip factor $S$ is obtained corresponding to different flow regimes.

Calculate slip velocity

$v_s = \frac{S}{ (\rho_L/(\sigma_L g))^{0.25} }$

Calculate liquid holdup

$H_L= \frac{ v_s - v_{sg} - v_{sl} + \sqrt{ ( v_s - v_{sg} - v_{sl} )^2 - 4 v_s v_{sl} } }{ 2 v_s }$

Calculate Reynolds number

$N_{Re} = \frac{\rho_L v_{sl} d }{\mu_L}$

At last, the firctional pressure gradient for bubble and slug flow

$\left[\frac{dp}{dx}\right]_f = \frac{f_m \rho_L v_{sl} v_m }{2 g_c d}$

The frictional pressure gradient for mist flow

$\left[\frac{dp}{dx}\right]f = \frac{f \rho_g v^2{sg} }{2 g_c d}$

In the transition zone, a linear interpolation between the flow regime boundaries is applied to obtain the frictional pressure gradient.

## PVT Model#

The pressure-volume-temperature (PVT) handling of fluids in many fluid flow simulations describes the phase behavior of gas, oil, and water at different conditions. A mixture with known composition consists of defined number of phases, phase amounts, phase compositions, phase properties (molecular weight, density, and viscosity), and the interfacial tension between phases. In addition, it is important to define the phase behavior of mixtures at a specific pressure and acquire the derivatives of all phase properties corresponding to pressure and composition.

### Vasquez-Beggs Oil#

Vasquez-Beggs is a generally applicable correlation containing equations for solution gas oil ratio $R_s$ , oil formation volume factor (FVF) $B_o$ , and oil compressibility $c_o$. The correlation was developed based on experimental data collected from fields all over the world. The data used in the development of the correlation covers a wide range of pressures, temperatures, and oil properties.

The bubble point pressure

$p_b = \left( \frac{R_s}{C_1 \gamma_g \exp \left( C_3 \left( \frac{\gamma_o}{T + 460.0} \right) \right) } \right) ^{\frac{1}{C_2}}$

where the parameters $C_1, C_2$, and $C_3$ have two groups of values when the oil gravity $\gamma_o$ is at a critical value of 30 API.

Solution Gas Oil Ratio

$R_s = C_1 \gamma_g p ^{C_2} \exp \left( C_3 \left( \frac{\gamma_o}{T + 460.0} \right) \right)$

Oil FVF - Saturated

$B_o = 1 + A_1R_s + S_2(T - 60) \left( \frac{\gamma_{API}}{\gamma_g} \right) + A_3 R_s(T - 60) \left( \frac{\gamma_{API}}{\gamma_g} \right)$

Oil compressibility at gas saturated condition

$c_o = \frac{ \left( B_g - \frac{dB_o}{dR_s} \right) \cdot \frac{dR_s}{dp}}{B_o}$

Oil FVF - Undersaturated

$B_0 = B_{ob}e(c_o(p_b - p))$

Oil compressibility at undersaturated condition

$c_o = \frac{ -1433 + 5 R_s + 17.2 T - 1180 \gamma_g + 12.61 \gamma_o }{ 10^5 \cdot p}$

### Standing Oil#

The Standing correlation contains equations for estimating bubble point pressure, solution gas oil ratio, and oil formation volume factor for California oils.

The bubble point pressure

$p_b = 18.2 \left( \left( \frac{R_s}{\gamma_g} \right)^{0.83} \frac{ 10^{0.00091T} }{10^{\gamma_{API}} } \right) - 1.4$

Solution Gas Oil Ratio

$R_s = \left( \left( \frac{p}{18.2} + 1.4 \right) \frac{ 10^{0.00091T} }{10^{\gamma_{API}} } \right)^{\frac{1}{0.83}} \gamma_g$

Oil FVF - Saturated

$B_o = 0.972 + 1.47 10 ^{-4} \left( R_s \left( \frac{\gamma_g}{\gamma_o} \right) + 1.25 T \right) ^{1.175}$

Oil FVF - Undersaturated applies equation (46).

The oil compressibility used in this equation is obtained from the Vasquez-Beggs correlation.

### De Ghetto Oil#

The De Ghetto et al. correlation contains modified PVT correlations for estimating bubble point pressure, solution gas oil ratio, oil formation volume factor (FVF), oil compressibility for heavy and extra-heavy oils.

Heavy oil

The bubble point pressure

$p_b = \left( \frac{56.434 \cdot R_s}{ \gamma_g 10^{10.9267} \left( \frac{\gamma_{API}}{T + 459.67} \right) } \right) ^{\frac{1}{1.2057}}$

Solution Gas Oil Ratio

$R_s = \frac{\gamma_g \cdot p^{1.2057} }{56.434} \cdot 10^{10.9267} \cdot \frac{\gamma_{API}}{T + 459.67}$

Oil FVF uses equation (44) and (46).

Oil compressibility - Saturated

$c_o = \frac{-2841.8 + 2.9646 R_{sb} + 25.5439 T - 1230.5 \gamma_g + 41.91 \gamma_{API} }{p \cdot 10^5} + \frac{B_g}{5.6145 B_o} \cdot \frac{dR_s}{dp}$

Oil compressibility - Undersaturated

$c_o = \frac{-2841.8 + 2.9646 R_{sb} + 25.5439 T - 1230.5 \gamma_g + 41.91 \gamma_{API} }{p \cdot 10^5}$

Extra heavy Oil

The bubble point pressure

$p_b = \left( \frac{R_s}{\gamma_g} \right)^{\frac{1}{1.1128}} \cdot \frac{10.7025}{10^{ (0.0169 \gamma_{API} - 0.00156T) }}$

Solution Gas Oil Ratio

$R_s = \gamma_g \left( \frac{p}{10.7025} \cdot 10 ^ {(0.00169 \gamma_{API} - 0.00156 )} \right) ^{1.1128}$

Oil FVF uses equation (44) and (46).

Oil compressibility - Saturated

$c_o = \frac{-889.6 + 3.1374 R_{sb} + 20 T - 627.3 \gamma_g + 81.4476 \gamma_{API} }{p \cdot 10^5} + \frac{B_g}{5.6145 B_o} \cdot \frac{dR_s}{dp}$

Oil compressibility - Undersaturated

$c_o = \frac{-889.6 + 3.1374 R_{sb} + 20 T - 627.3 \gamma_g + 81.4476 \gamma_{API} }{p \cdot 10^5}$

### Glasø Oil#

The Glasø correlation contains equations for estimating bubble point pressure, solution gas oil ratio, and oil formation volume factor for North Sea oils.

The bubble point pressure

$\log(p_b) = 1.7669 + 1.7447 \log(\chi_1) - 0.30218 \log(\chi_1)^2$

Solution Gas Oil Ratio

$R_s = \left( \frac{\chi_2 \gamma^{0.989}_{API}}{T^{0.172}} \right)^{\frac{1}{0.816}} \gamma_g$

Oil FVF - Saturated

$\log(B_o - 1) = - 6.58511 + 2.91329 \log(y) - 0.27683 \log(y)^2$

Oil FVF undersaturated condition uses equation (46)

The oil compressibility use the equation obtained from the Vasquez-Beggs correlation.

### Oil Viscosity Models#

The oil viscosity models are presented in this section.

Vasquez-Beggs

Vasquez-Beggs correlation from 1978 was based on a large database, and is therefore applicable to a wide range of oils.

First, Beggs and Robinson developed an empirical correlation for determining the viscosity of dead oil.

$\mu_{od} = 10^x -1$

Second, the viscosity at saturated condition

$\mu_{os} = 10.715 (R_s + 100)^{-0.515} (\mu_{od}^{5.44((R_s + 150)^{-0.338}) })$

Third, the viscosity at Undersaturated condition

$\mu_{o} = \mu_{os} \cdot \left(\frac{p}{p_b}\right) ^ {C_1 p^{C_2} (C_3 + C_4)^p}$

Standing

This model, often referred to as either Beal or Standing, was developed by Beal and fitted by Standing.

$\mu_{od} = (0.32 + 1.8 \cdot 10^7/\gamma_o^{4.53} ) \left(\frac{360}{T + 200} \right)^{(10^{0.43 + 8.33/\gamma_o})}$

Viscosity - Saturated

$\mu_{os} = ( 0.20 + 0.80 \cdot 10^{-0.00081 Rs}) ) \mu_{od} ^ {0.43 + 0.57 \cdot 10^{ -0.00072 Rs}}$

Viscosity - Undersaturated

$\mu_{o} = \mu_{os} + 0.001(p - p_b) (0.024 \mu^{1.6}_{os} + 0.038 \mu^{0.56}_{os} )$

Eqbogah

The Egbogah correlation contains two methods for calculating dead oil viscosity using a modified Beggs and Robinson viscosity correlation and a correlation that uses the pour point temperature $T_{pp}$.

$\log(\mu_{od} + 1) = -1.7095 - 0.0087917 T_{pp} + 2.7523 \gamma_o + (-1.2943 + 0.0033214 T_pp + 0.9581957 \gamma_o) \log(T - T_{pp})$

Viscosity - Saturated use equation (63).

Viscosity - Undersaturated

$\mu_{o} = \mu_{ob} \left( \frac{p}{p_b} \right)^{(2.6 p^{1.187} \exp(-11.513 - 8.98 \cdot 10^{-5} p ) )}$

De Ghetto

The De Ghetto et al. correlation estimates the oil viscosity for both heavy and extra-heavy oils.

Heavy Oil

$\log(\mu_{od} + 1) = 2.06429 - 0.0179 \gamma_{API} - 0.70226 \log(T)$

Viscosity - Saturated

$\mu_{os} = 0.6311 + 1.078 x - 0.003653 x^2$

Viscosity - Undersaturated

$\mu_{o} = 0.9886 \mu_{os} + 0.002763(p - p_b) (-0.01153 \mu^{1.7933}_{os} + 0.0316 \mu^{1.5939}_{os} )$

Extra heavy Oil

$\log(\mu_{od} + 1) = 1.090296 - 0.021619 \gamma_{API} - 0.61784 \log(T)$

Viscosity - Saturated

$\mu_{os} = 2.3945 + 0.8927 x + 0.001576 x^2$

Viscosity - Undersaturated

$\mu_{o} =\mu_{os} - (1 - \frac{p}{p_b} ) \left( \frac{10^{-2.19} \mu^{1.055}_{od} p^{0.3132}_b }{10^{0.0099 \gamma_o}} \right)$

### Gas PVT Model#

The Gas PVT model contains the calculation of gas viscosity estimation, gas pseudo critical model, and gas compressibility Z-factor model with known composition. Z-factor in the PVT property always needs accurate determination in a gas condensate reservoir.

Gas Viscosity Models

The gas viscosity models employ Lee and Lee Modified.

Lee

$\mu_g = 10^{-4} \frac{(9.4 + 0.02 M) T^{1.5}}{209 + 19 M + T} \exp [X \rho_g^{2.4 - 0.2 X}]$

where $M$ is the molar weight and $X = 3.5 + 986/T + 0.01M$.

Lee modified

$K = \frac{(k_1 + k_2 M) T^k_3}{k_4 + k_5 M + T} \\ X = x_1 + x_2 / T + x_3M \\ Y = y_1 + y_2X \\ \mu_g = 10^{-4} K \exp(X \rho_g^Y)$

## References#

[1] Nilsen, T.: Retningslinjer for beregning av utblåsningsrater og -varighet til bruk ved analyse av miljørisiko, OLF/Statoil, Stavanger, Norway, 2004

[2] Andersen, L. B., Aven, T., Nilsen, T.: KickRisk - Stochastic Modelling for The Quantification of Kick and Blowout Risk in Exploration Drilling - Guidelines for assessing subjective probabilities, Report RT - 199/229. RF - Rogaland Research, Stavanger, Norway, 1998

[3] Berg, A., Fosse, F. and Nævdal, G.: Production Monitoring - Specification Document Report RF - 2001/323, RF - Rogaland Research, Stavanger/Bergen, Norway, 2002

[4] Lage, A. C. V. M.: Two-phase Flow Models and Experiments for Low-Head and Underbalanced Drilling, PhD thesis at University of Stavanger, Norway, 2000

[5] Rommetveit, R.: A numerical solution model for gas kicks in oil based drilling fluids, Dr. Scient thesis at University of Bergen, Norway, 1998

[6] Ford, E. P. and Daireaux, B.: BlowFlow - User Manual, Report IRIS - 2010/045, IRIS, Stavanger, Norway, 2010

[7] Masella, J. M., Faille, I. and Gallouet, T.: On an approximate Godunov scheme, Int. J. Computational Fluid Dynamics, 1999

[8] Gavaga, S. B.: Analyse numérique des modèles hydrodynamiques d'écoulements diphasiques instationnaires dans les réseaux de production pétrolière. Thèse, ENS Lyon, France, 1991

[9] Inglis, T. A.: Petroleum engineer and development studies, Vol. 2: Directional drilling, Graham & Trotman Limited, UK, 1987

[10] ALAdwani, F. A.: Application of mechanistic models in predicting flow behavior in deviated wells under UBD conditions, M.Sc. Thesis at Louisiana State University and Agricultural and Mechanical College, USA, 2003.

[11] Colebrook, C. F.: Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws, Journal of the Institution of Civil Engineers, London, UK. 1939.

[12] Lyons, W. C. and Plisga, G. J.: Standard handbook of Petroleum & Natural gas engineering (2nd Edition), Elsevier, UK, 2005

[13] J. V. Vogel. Inflow performance relationship for solution gas drive wells. Journal of Petroleum Technology, pages 83-93, 1968.

[14] Boyun, G., Lyons, W. C. and Ghalambor, A.: Petroleum Production Engineering - A Computer-Assisted Approach, Elsevier Science & Technology Books, February 2007.

[15] Statoil: Excel DST design_all in one V2_Rev4.xls, 08.08.2011

[16] Larsen, L.: Well Testing - Analysis of Pressure Transient Data, University of Stavanger, October 2010

[17] Larsen, L.: General Productivity Models for Wells in Homogeneous and Layered Reservoirs, SPE 71613,2001.

[18] Beggs, H. D.: Production Optimization Using NodalTM Analysis, 2nd Edition, OGCI and Petroskills Publications, Tulsa, Oklahoma, USE, May 2003

[19] Vasquez, M. and Beggs, H. D.: Correlations for Fluid Physical Property Prediction, Journal of Petroleum Technology, June 1980, pp. 968-970

[20] Moradi, B. et.al.: Bubble Point Pressure Empirical Correlations, SPE 132759, June 2010

[21] Standing, M. B. Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, Society of Petroleum Engineers of AIME, Dallas, USA, 1977

[22] Sutton, R. P. and Farshad, F.: Evaluation of Empirically Derived PVT Properties for Gulf of Mexico Crude Oil, SPE Reservoir Engineering, February 1990, pp. 79-86

[23] De Ghetto, G., Paone, F. and Villa, M.: Reliability Analysis on PVT Correlations, SPE 28904, 1994

[24] De Ghetto, G., Paone, F. and Villa, M.: Pressure-Volume-Temperature Correlations for Heavy and Extra Heavy Oil, SPE 30316, 1995.

[25] Beggs, H. D. and Robinson, J. R.: Estimating the Viscosity of Crude Oil Systems, Journal of Petroleum Technology, September 1975, pp. 1140-41

[26] Chew, J. and Connally, C. A. Jr.: A Viscosity Correlation for Gas-Saturated Crude Oils, AIME (1959) 2016, pp. 23-25.

[27] Egbogah, E. O. and Ng, J. T.: An improved temperature-viscosity correlation for crude oil systems, Journal of Petroleum Science and Engineering, 5 (1990) 197-200

[28] Lake, L. W. et. al.: Petroleum Engineering Handbook, Vol. I, Society of Petroleum Engineers, 2006

[29] Muskat, M.: Physical Principles of Oil Production, Mc-Graw-Hill Book Co., Inc., New York, 1949.]

[30] Trube, A. S.: Compressibility of Undersaturated Hydrocarbon Reservoir Fluids, SPE 899-G, Vol. 210, 1957, pp. 341-344

[31] Standing, M. B. and Kratz, D. L.: Density of Natural Gases, Trans. American Institute of Mechanical Engineers, Vol. 146, p. 140-149, 1942

[32] Dranchuk, P. M., Purvis, R. A. and Robinson, D. B.: Computer Calculation of Natural Gas Compressibility Factors Using the Standing and Katz Correlation, Institute of Petroleum Technical Series, No. IP 74-008, 1974

[33] Ahmed, T.: Reservoir Engineering Handbook, Elsevier, 2010.

[34] Shi, C.: Flow Behavior of Gas-Condensate Wells, Stanford University, March 2009.

[35] Lee, A. L., Gonzalez, M. H. and Eakin, B. E.: The Viscosity of Natural Gases, Journal of Petroleum Technology, August 1966, pp. 997-1000.

[36] Londono, F. E., Archer, R. A. and Blasingame, T. A.: Simplified Correlations for Hydrocarbon Gas Viscosity and Gas Density - Validation and Correlation of Behavior Using a Large-Scale Database, SPE 75721, 2002

[37] Hagedorn, A. R., and Brown, K. E.: Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small Diameter Vertical Conduits, Journal of Petroleum Technology, April 1965.

[38] Brown, K. E. and Beggs, H. D.: The Technology of Artificial Lift Methods, Vol. 1, Pennwell Books, 1977.

[39] Griffith, P.: Two-Phase Flow in Pipes, Special Summer Program. Massachusetts Institute of Technology, Cambridge, Mass., 1962

[40] Maurer Engineering Inc.: Well Control Model (WELCON2) - Theory and User's Manual, Houston, Texas, February 1993

[41] Beggs, H. D. and Brill, J. P.: A Study of Two-Phase Flow in Inclined Pipes, Journal of Petroleum Technology, May 1973

[42] Beggs, H. D. and Brill, J. P.: Two-Phase Flow in Pipes, 6th Edition, Tulsa University Press, January 1991

[43] Orkiszewski, J.: Predicting Two-Phase Pressure Drops in Vertical Pipe, Journal of Petroleum Technology, June 1967

[44] Brill, J. P.: Discontinuities in the Orkiszewski Correlation for Predicting Pressure Gradients in Wells, Journal of Energy Resources Technology, Vol. 111/35, March 1989

[45] Chierici, G. L., Ciucci, G. M. and Sclocchi, G.: Two-Phase Vertical Flow in Oil Wells - Prediction of Pressure Drop, Journal of Petroleum Technology, August 1974, pp. 927-938.

[46] Takács, G.: Gas Lift Manual, PennWell Books, July 2005

[47] Griffith, P. and Wallis, G. B.: Two-phase slug flow, Journal of Heat Transfer, Trans. ASME, August 1961, pp. 307-320

[48] Duns, H. Jr. and Ros, N. C. J.: Vertical Flow of Gas and Liquid Mixtures from Wells, Proc. Sixth World Petroleum Congress, Frankfurt, Section II, 22-PD 6, June 1963

[49] Gray, H. E.: Vertical Flow Correlation in Gas Wells, User Manual for API 14B, Subsurface Controlled Safety Valve Sizing Computer, 1978

[50] Scandpower. Blowout and well release frequencies - Based on SINTEF Offshore Blowout Database, 2008, 2009

[51] B. Guo, W. C. Lyons, and A. Ghalambor. Petroleum production engineering, a computer-assisted approach. Elseiver Science & Technology Books, 2007