NeqSim Field Development - Mathematical Reference

This document provides the complete mathematical foundations for the field development framework, linking thermodynamic calculations to economic evaluation and decision support.

Thermodynamic Foundations
Production Modeling
Flow Assurance & Hydraulics
Economic Calculations
Decision Analysis
Uncertainty Quantification
Emissions & Sustainability

1. Thermodynamic Foundations

1.1 Equations of State

NeqSim supports multiple equations of state. The general cubic EoS form:

\[P = \frac{RT}{V-b} - \frac{a(T)}{(V+\delta_1 b)(V+\delta_2 b)}\]

EoS	$\delta_1$	$\delta_2$	Use Case
SRK	1	0	General hydrocarbon systems
Peng-Robinson	$1+\sqrt{2}$	$1-\sqrt{2}$	Liquid density improvement
CPA	SRK base	+ association	Water, glycols, alcohols

Critical Properties

\[a_c = \Omega_a \frac{R^2 T_c^2}{P_c}, \quad b = \Omega_b \frac{R T_c}{P_c}\]

Temperature Dependence (Soave-type)

\[\alpha(T) = \left[1 + m\left(1 - \sqrt{T_r}\right)\right]^2\] \[m = 0.48 + 1.574\omega - 0.176\omega^2\]

1.2 Mixing Rules

Classical (van der Waals): $a_{mix} = \sum_i \sum_j x_i x_j \sqrt{a_i a_j}(1-k_{ij})$ $b_{mix} = \sum_i x_i b_i$

CPA Association Term: $Z^{assoc} = -\frac{1}{2}\sum_i x_i \sum_{A_i} \left(1 - X_{A_i}\right)$

Where $X_{A_i}$ is the fraction of sites A on molecule i NOT bonded.

1.3 Flash Calculations

PT Flash Objective (Rachford-Rice): $f(\beta) = \sum_i \frac{z_i(K_i - 1)}{1 + \beta(K_i - 1)} = 0$

Where:

$\beta$ = vapor fraction
$z_i$ = overall composition
$K_i = y_i/x_i$ = equilibrium ratio

Fugacity Coefficient: $\ln \phi_i = \frac{1}{RT}\int_V^\infty \left(\frac{\partial P}{\partial n_i}\bigg|_{T,V,n_{j\neq i}} - \frac{RT}{V}\right)dV - \ln Z$

2. Production Modeling

2.1 Inflow Performance Relationship (IPR)

Vogel’s Equation (oil wells below bubble point): $\frac{q_o}{q_{o,max}} = 1 - 0.2\left(\frac{P_{wf}}{P_r}\right) - 0.8\left(\frac{P_{wf}}{P_r}\right)^2$

Darcy’s Law (single-phase liquid): $q = \frac{2\pi k h}{\mu B \ln(r_e/r_w)} (P_r - P_{wf})$

Productivity Index: $J = \frac{q}{P_r - P_{wf}} \quad \text{[Sm³/d/bar]}$

2.2 Vertical Lift Performance (VLP)

Pressure Traverse: $\frac{dP}{dL} = \frac{\rho_m g \sin\theta}{1 - \frac{\rho_m v_m v_{sg}}{P}} + \frac{f \rho_m v_m^2}{2D} + \frac{\rho_m v_m dv_m}{dL}$

Components:

Gravity: $(\rho_m g \sin\theta)$
Friction: $(f \rho_m v_m^2 / 2D)$
Acceleration: $(\rho_m v_m dv_m/dL)$

2.3 Decline Curve Analysis

Arps Hyperbolic: $q(t) = \frac{q_i}{(1 + b D_i t)^{1/b}}$

Where:

$q_i$ = initial rate
$D_i$ = initial decline rate
$b$ = hyperbolic exponent (0 ≤ b ≤ 1)

Cumulative Production: $N_p = \frac{q_i}{D_i(1-b)}\left[1 - \left(\frac{q}{q_i}\right)^{1-b}\right]$

2.4 TransientWellModel Equations

Drawdown (radial flow): $P_{wf} = P_i - \frac{q \mu B}{4\pi kh}\left[\ln\left(\frac{4kt}{\phi \mu c_t r_w^2 \gamma}\right) + 2S\right]$

Dimensionless Pressure: $P_D = \frac{2\pi kh(P_i - P_{wf})}{q \mu B}$

Dimensionless Time: $t_D = \frac{kt}{\phi \mu c_t r_w^2}$

3. Flow Assurance & Hydraulics

3.1 Beggs & Brill Correlation

No-Slip Liquid Holdup: $\lambda_L = \frac{v_{SL}}{v_m}$

Where superficial velocities: $v_{SL} = \frac{Q_L}{A}, \quad v_{SG} = \frac{Q_G}{A}, \quad v_m = v_{SL} + v_{SG}$

Flow Pattern Boundaries:

Transition	Equation
$L_1$	$316 \lambda_L^{0.302}$
$L_2$	$0.0009252 \lambda_L^{-2.4684}$
$L_3$	$0.10 \lambda_L^{-1.4516}$
$L_4$	$0.5 \lambda_L^{-6.738}$

Liquid Holdup (Segregated): $H_L(0) = \frac{0.980 \lambda_L^{0.4846}}{Fr^{0.0868}}$

Liquid Holdup (Intermittent): $H_L(0) = \frac{0.845 \lambda_L^{0.5351}}{Fr^{0.0173}}$

Liquid Holdup (Distributed): $H_L(0) = \frac{1.065 \lambda_L^{0.5824}}{Fr^{0.0609}}$

Inclination Correction: $H_L(\theta) = H_L(0) \cdot \psi(\theta)$

\[\psi = 1 + C\left[\sin(1.8\theta) - \frac{1}{3}\sin^3(1.8\theta)\right]\]

3.2 Pressure Drop Calculation

Two-Phase Friction Factor: $\Delta P_f = \frac{f_{tp} \rho_n v_m^2 L}{2D}$

Normalized Friction Factor: $f_{tp} = f_n \cdot e^S$

Where $S$ depends on: $y = \frac{\lambda_L}{H_L^2}$

Elevation Component: $\Delta P_g = \rho_m g L \sin\theta$

Mixture Density: $\rho_m = \rho_L H_L + \rho_G (1 - H_L)$

3.3 Hydrate Formation

Hammerschmidt Equation (inhibitor depression): $\Delta T = \frac{K_H \cdot w}{M(100-w)}$

Where:

$\Delta T$ = temperature depression (°C)
$K_H$ = inhibitor constant (1297 for MEG)
$w$ = weight percent inhibitor
$M$ = molecular weight of inhibitor

CSMGem-type correlation (implemented in NeqSim): $\ln\left(\frac{f_w^H}{f_w^L}\right) = \frac{\Delta\mu_w^0}{RT} + \sum_i \ln(1-\theta_i)$

3.4 Wax Appearance Temperature

Coutinho Model: $\ln(\gamma_i^s x_i^s) = \frac{\Delta H_{fus,i}}{R}\left(\frac{1}{T_m} - \frac{1}{T}\right) + \frac{\Delta C_p}{R}\left(\frac{T_m-T}{T} + \ln\frac{T}{T_m}\right)$

4. Economic Calculations

4.1 Net Present Value (NPV)

\[NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}\]

Cash Flow: $CF_t = Revenue_t - OPEX_t - CAPEX_t - Tax_t$

4.2 Norwegian Petroleum Tax

Corporate Tax (22%): $Tax_C = 0.22 \times (Revenue - OPEX - DD\&A - Interest)$

Special Petroleum Tax (56%): $Tax_S = 0.56 \times (Revenue - OPEX - Uplift - Special\ DD\&A)$

Uplift Calculation: $Uplift = 0.208 \times CAPEX_{eligible}$

Depreciation (6-year linear): $DD\&A_t = \frac{CAPEX_{t-6} + CAPEX_{t-5} + ... + CAPEX_{t-1}}{6}$

After-Tax Cash Flow: $CF_{at} = Revenue - OPEX - CAPEX - Tax_C - Tax_S$

4.3 Internal Rate of Return (IRR)

Find $r$ such that: $\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t} = 0$

4.4 Payback Period

\[Payback = t^* \text{ where } \sum_{t=0}^{t^*} CF_t \geq 0\]

4.5 Capital Efficiency Metrics

Profitability Index: $PI = \frac{NPV + CAPEX_{PV}}{CAPEX_{PV}}$

Return on Investment: $ROI = \frac{\sum CF_{positive}}{CAPEX_{total}}$

4.6 Breakeven Price

Find $P_{oil}$ such that $NPV = 0$: $\sum_{t=0}^{n} \frac{(P_{oil} \cdot Q_t - OPEX_t - CAPEX_t - Tax_t(P_{oil}))}{(1+r)^t} = 0$

5. Decision Analysis

5.1 Multi-Criteria Decision Analysis (MCDA)

Weighted Sum Model: $S_i = \sum_{j=1}^{m} w_j \cdot \tilde{s}_{ij}$

Min-Max Normalization:

For “higher is better”: $\tilde{s}_{ij} = \frac{s_{ij} - s_j^{min}}{s_j^{max} - s_j^{min}}$

For “lower is better”: $\tilde{s}_{ij} = \frac{s_j^{max} - s_{ij}}{s_j^{max} - s_j^{min}}$

Weight Normalization: $w_j^{norm} = \frac{w_j}{\sum_{k=1}^{m} w_k}$

5.2 Portfolio Optimization

Objective Function: $\max Z = \sum_{i=1}^{n} x_i \cdot NPV_i$

Budget Constraint (by year): $\sum_{i=1}^{n} x_i \cdot CAPEX_{i,t} \leq Budget_t \quad \forall t$

Binary Selection: $x_i \in \{0, 1\}$

Expected Monetary Value: $EMV_i = P_i \cdot NPV_i - (1-P_i) \cdot C_{dry}$

Where:

$P_i$ = probability of success
$C_{dry}$ = cost of failure (sunk costs)

5.3 Value of Information (VoI)

\[VoI = EMV_{with\ info} - EMV_{without\ info}\]

Perfect Information: $EVPI = \sum_s P(s) \cdot \max_a \{V(a,s)\} - \max_a \{\sum_s P(s) \cdot V(a,s)\}$

5.4 Sensitivity Analysis

Tornado Analysis (one-at-a-time): $\Delta NPV_i = NPV(x_i^{high}) - NPV(x_i^{low})$

Elasticity: $E_i = \frac{\partial NPV / NPV}{\partial x_i / x_i} = \frac{\partial \ln(NPV)}{\partial \ln(x_i)}$

6. Uncertainty Quantification

6.1 Monte Carlo Simulation

Expected Value: $E[Y] \approx \frac{1}{N} \sum_{i=1}^{N} f(X_i)$

Variance: $Var[Y] \approx \frac{1}{N-1} \sum_{i=1}^{N} (Y_i - \bar{Y})^2$

Confidence Interval: $CI_{95\%} = \bar{Y} \pm 1.96 \frac{s}{\sqrt{N}}$

6.2 Distribution Functions

Triangular: $f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)} & a \leq x \leq c \\ \frac{2(b-x)}{(b-a)(b-c)} & c < x \leq b \end{cases}$

Lognormal: $f(x) = \frac{1}{x\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)$

Beta-PERT: $E[X] = \frac{a + 4m + b}{6}, \quad \sigma^2 \approx \frac{(b-a)^2}{36}$

6.3 Percentile Calculation

P10, P50, P90: $P_{p} = X_{(k)} + d(X_{(k+1)} - X_{(k)})$

Where $k = \lfloor p(N+1) \rfloor$ and $d = p(N+1) - k$

6.4 Correlation Handling

Rank Correlation (Spearman): $\rho_s = 1 - \frac{6\sum d_i^2}{N(N^2-1)}$

Iman-Conover Method for inducing correlation in Monte Carlo samples.

7. Emissions & Sustainability

7.1 CO₂ Intensity

\[I_{CO2} = \frac{\sum_{sources} E_{source}}{Q_{oil,equiv}}\]

Units: kg CO₂/boe

7.2 Emission Sources

Fuel Gas Combustion: $E_{fuel} = Q_{fuel} \cdot \rho_{gas} \cdot \frac{M_{CO2}}{M_{CH4}} \cdot (1 + \epsilon)$

Flaring: $E_{flare} = Q_{flare} \cdot \rho_{gas} \cdot \frac{44}{16} \cdot \eta_{combustion}$

Fugitive Emissions (Tier 2): $E_{fugitive} = \sum_j N_j \cdot EF_j$

Where $EF_j$ = emission factor for equipment type $j$

7.3 Power Generation Emissions

Gas Turbine: $E_{GT} = \frac{P_{shaft}}{\eta_{th}} \cdot EF_{NG}$

Where:

$\eta_{th}$ = thermal efficiency (typically 0.30-0.40)
$EF_{NG}$ = 56.1 kg CO₂/GJ (natural gas)

Combined Cycle: $\eta_{CC} = \eta_{GT} + \eta_{ST}(1 - \eta_{GT})$

7.4 Carbon Tax Scenarios

Norwegian CO₂ Tax (2025): $Tax_{CO2} = E_{total} \times 2000 \text{ NOK/tonne}$

EU ETS Cost: $Cost_{ETS} = E_{total} \times P_{EUA}$

Implementation Notes

Numerical Methods

Flash Calculations: Newton-Raphson with line search
VLE Equilibrium: Successive substitution with acceleration
Process Simulation: Sequential modular with tear streams
Optimization: Greedy heuristics for portfolio, gradient-free for complex objectives

Convergence Criteria

Calculation	Tolerance
Flash (mole balance)	$10^{-10}$
Fugacity coefficients	$10^{-8}$
Process simulation	$10^{-6}$ (relative)
Economic NPV	$10^{-4}$ MUSD

Units Convention

Quantity	SI Unit	Field Unit
Pressure	Pa	bara
Temperature	K	°C
Volume	m³	Sm³ @ 15°C, 1.01325 bara
Mass	kg	tonnes
Energy	J	kJ, MW
Money	-	MUSD, MNOK

References

Soave, G. (1972). “Equilibrium constants from a modified Redlich-Kwong equation of state.” Chem. Eng. Sci., 27(6), 1197-1203.
Peng, D.Y. & Robinson, D.B. (1976). “A New Two-Constant Equation of State.” Ind. Eng. Chem. Fundam., 15(1), 59-64.
Beggs, H.D. & Brill, J.P. (1973). “A Study of Two-Phase Flow in Inclined Pipes.” J. Pet. Technol., 25(5), 607-617.
Vogel, J.V. (1968). “Inflow Performance Relationships for Solution-Gas Drive Wells.” J. Pet. Technol., 20(1), 83-92.
Norwegian Petroleum Directorate (2024). “Petroleum Taxation in Norway.”
SPE (2023). “Petroleum Resources Management System (PRMS).”

NeqSim Field Development - Mathematical Reference

Table of Contents

1. Thermodynamic Foundations

1.1 Equations of State

Critical Properties

Temperature Dependence (Soave-type)

1.2 Mixing Rules

1.3 Flash Calculations

2. Production Modeling

2.1 Inflow Performance Relationship (IPR)

2.2 Vertical Lift Performance (VLP)

2.3 Decline Curve Analysis

2.4 TransientWellModel Equations

3. Flow Assurance & Hydraulics

3.1 Beggs & Brill Correlation

3.2 Pressure Drop Calculation

3.3 Hydrate Formation

3.4 Wax Appearance Temperature

4. Economic Calculations

4.1 Net Present Value (NPV)

4.2 Norwegian Petroleum Tax

4.3 Internal Rate of Return (IRR)

4.4 Payback Period

4.5 Capital Efficiency Metrics

4.6 Breakeven Price

5. Decision Analysis

5.1 Multi-Criteria Decision Analysis (MCDA)

5.2 Portfolio Optimization

5.3 Value of Information (VoI)

5.4 Sensitivity Analysis

6. Uncertainty Quantification

6.1 Monte Carlo Simulation

6.2 Distribution Functions

6.3 Percentile Calculation

6.4 Correlation Handling

7. Emissions & Sustainability

7.1 CO₂ Intensity

7.2 Emission Sources

7.3 Power Generation Emissions

7.4 Carbon Tax Scenarios

Implementation Notes

Numerical Methods

Convergence Criteria

Units Convention

References