Equations: Reservoir simulation

The equations of the “Reservoir simulation” lessons with their conditions and units, and what each symbol means in this discipline.

Depletion of a closed “tank” at constant bottomhole pressure

c_t V_p\,\frac{\dd p}{\dd t} = -J\,(p - p_{wf})\quad\Rightarrow\quad p(t) = p_{wf} + (p_i - p_{wf})\,\e^{-t/\tau},\qquad \tau = \frac{c_t V_p}{J}

metric oilfield units: MPa, m³, d (any consistent system will do)

Holds when: The pressure is the same everywhere in the volume: a pressure change caused by the well spreads instantly (this is what makes it a “tank”). The flow is single-phase (one fluid only): the pressure is above the bubble point pressure and no gas comes out of the oil. $c_t$ and $J$ are constant, and there is no inflow from beyond the boundary. The rate at reservoir conditions (volumes measured at the pressure and temperature of the reservoir) is proportional to the drawdown $p - p_{wf}$ , the pressure difference between the tank and the well.

What a reservoir simulation model is and why it is needed →

Explicit time step for the “tank”

p^{n+1} = p^{n} - \frac{\Delta t}{\tau}\,(p^{n} - p_{wf})

any consistent units ( $\Delta t$ and $\tau$ in the same units of time)

Holds when: The derivative is replaced by a difference. The rate over the whole step is taken from the pressure at the start of the step (an explicit scheme). The formula becomes the exact solution only in the limit $\Delta t \to 0$ (a step that shrinks to nothing).

What a reservoir simulation model is and why it is needed →

Fluid balance in a cell over a time step

\frac{m_i^{n+1} - m_i^{n}}{\Delta t} = \sum_{j} F_{ji} - W_i

any consistent units

Holds when: Within a cell the pressure, saturations and properties are uniform. We write the balance separately for each component: oil, gas (free gas and gas dissolved in the oil) and water. We calculate the flows and withdrawals from the pressures and saturations of the new step (an implicit scheme) or of the start of the step (an explicit one).

What a reservoir simulation model is and why it is needed →

Flow between neighbouring cells (single-phase, by Darcy's law)

q_{ij} = \frac{T_{ij}}{\mu}\,(p_j - p_i),\qquad T_{ij} = \frac{A}{\dfrac{\Delta x_i}{2k_i} + \dfrac{\Delta x_j}{2k_j}}

Holds when: The flow is single-phase and horizontal. In an inclined reservoir we add the weight of the fluid column to the pressure difference. The cells are rectangular and their centres lie on a line perpendicular to the common face. With several phases, we multiply the rate of each phase by its relative permeability, usually taken in the cell the flow comes from.

What a reservoir simulation model is and why it is needed →

Implicit time step for the “tank”

p^{n+1} = p^{n} - \frac{\Delta t}{\tau}\,(p^{n+1} - p_{wf})\quad\Rightarrow\quad p^{n+1} = \frac{p^{n} + (\Delta t/\tau)\,p_{wf}}{1 + \Delta t/\tau}

any consistent units

Holds when: We calculate the rate over the step from the pressure at the end of the step (an implicit scheme). The model is linear. In a real simulator the equations are nonlinear, and at each step we solve them by iterations.

What a reservoir simulation model is and why it is needed →

Amounts of oil, water and gas in a cell of a black oil model

N_o = \frac{V_p S_o}{B_o}, \qquad N_w = \frac{V_p S_w}{B_w}, \qquad G = V_p\left(\frac{S_g}{B_g} + \frac{R_s\,S_o}{B_o}\right)

metric; standard conditions are the ones that the property tables refer to

Holds when: The composition of the “oil” and the “gas” does not change. Their properties depend on pressure alone, and the reservoir temperature is constant. Gas dissolves in oil, but oil does not dissolve in gas. The extended model adds the content of liquid hydrocarbons in the gas. We neglect the gas dissolved in water. If necessary, we add the same kind of term for water. We get the mass of a component by multiplying by its density at standard conditions.

Types of reservoir simulation model →

Mass of a component in a cell of a compositional model

m_i = V_p \sum_{j} S_j\,C_{ij}

SI; instead of mass we often calculate the number of moles — then the concentrations are molar

Holds when: The phases in a cell are in equilibrium: the composition of each follows from the overall composition, the pressure and the temperature. We sum over all the phases in which the component is present. We usually neglect hydrocarbon components in the water.

Types of reservoir simulation model →

Dependence of oil viscosity on temperature (the Andrade equation)

\mu(T) = A\,\e^{B/T}

viscosity in the units of the constant $A$ ; temperature in kelvin only

Holds when: It is an empirical relationship (fitted to measurements): we choose the constants to match the measured viscosity of this oil at two temperatures at least. The oil is degassed (it has lost its gas), or its solution gas–oil ratio does not change. Pressure affects viscosity much less than temperature does. Over a wide range of temperatures, the straight line on a plot of $\ln\mu$ against $1/T$ bends. So for heavy oils we use more accurate relationships, and we enter a table of laboratory values into the model.

Types of reservoir simulation model →

Flow between the matrix and the fractures (single-phase)

q_{mf} = \sigma\,V_b\,\frac{k_m}{\mu}\,(p_m - p_f), \qquad \sigma = 4\left(\frac{1}{L_x^2} + \frac{1}{L_y^2} + \frac{1}{L_z^2}\right)

Holds when: All the matrix blocks in a cell are identical, and a single average value describes the pressure inside a block. The expression given for $\sigma$ is one of the common ones. Other derivations give other numerical factors, and in history matching we often adjust it. With several phases, we multiply the rate of each phase by its relative permeability and add capillary and gravity forces. These forces decide the exchange in displacement.

Types of reservoir simulation model →

Pore volume of a cell

V_p = \Delta x\,\Delta y\,\Delta z\cdot\mathrm{NTG}\cdot\phi

any consistent units (m and m³)

Holds when: The cell is rectangular. For corner-point geometry, the simulator calculates the cell volume from the coordinates of the eight corners. The porosity refers to the conditions at which the reference pressure of the pore compressibility (see below) is set.

Building a simulation model: input data and results →

Density of oil in the reservoir from the model data

\rho_o = \frac{\rho_{o,\text{sc}} + R_s\,\rho_{g,\text{sc}}}{B_o}

kg/m³; the density of water in the reservoir is $\rho_{w,\text{sc}}/B_w$

Holds when: Mass is conserved: 1 m³ of stock-tank oil together with the gas dissolved in it occupies $B_o$ m³ in the reservoir (reservoir oil density). If the oil is described by a table without a solution gas–oil ratio, the simulator calculates $\rho_o = \rho_{o,\text{sc}}/B_o$ , without the mass of the dissolved gas.

Building a simulation model: input data and results →

Pore volume at the pressure in the cell

V_p(p) = V_p(p_{\text{ref}})\left(1 + x + \frac{x^2}{2}\right),\qquad x = c_f\,(p - p_{\text{ref}})

any consistent units: $c_f$ and the pressures in the same units

Holds when: $1 + x + x^2/2$ are the first terms of the series for $e^{x}$ . For $|x| = 0.05$ the difference from the exponential is about 0.002%. The compressibility is constant. If the change of permeability with pressure matters, we give it in separate tables.

Building a simulation model: input data and results →

Phase pressures and capillary pressure in equilibrium

p_o(z) = p_{\text{ref}} + \rho_o g\,(z - z_{\text{ref}}),\qquad p_c(z) = (\rho_w - \rho_o)\,g\,(z_{\text{FWL}} - z)

metric: MPa, m, g/cm³ (the pressure in the deck is in bars)

Holds when: The fluids are at rest, and both phases are continuous over the whole height. Depths are vertical, positive downwards. The densities here are constant. The simulator adds up (integrates) the pressure with depth using a density that depends on pressure. For a reservoir tens of metres thick the difference is negligible. Above the FWL $p_c > 0$ . If $p_c$ is less than the entry pressure of the rock, the cell is fully water-saturated.

Building a simulation model: input data and results →

Misfit function of history matching

M = \frac{1}{N}\sum_{k=1}^{N} w_k\left(\frac{d_k^{\text{meas}} - d_k^{\text{calc}}}{\sigma_k}\right)^{2}

dimensionless: each discrepancy is divided by the error in the same units

Holds when: The measurement errors are independent and random. We must estimate the errors $\sigma_k$ honestly, because what counts as a good match depends on them. The engineer chooses the weights, and the result depends on them. The misfit function puts judgement into a formula, but it does not replace judgement.

History matching →

Misfit function with a prior term

S(\mathbf{m}) = \sum_{k}\left(\frac{d_k^{\text{meas}} - d_k^{\text{calc}}(\mathbf{m})}{\sigma_k}\right)^{2} + \sum_{j}\left(\frac{m_j - m_j^{\text{prior}}}{s_j}\right)^{2}

dimensionless

Holds when: The errors of the measurements and of the prior estimates are independent and normally distributed (they follow the usual bell-shaped curve). Then the minimum of $S$ corresponds to the most probable set of parameters. We obtained the prior estimates independently of the history. Otherwise we would count the same information twice.

History matching →

Incremental production and acceleration from an intervention

\Delta Q = Q_{\text{with}} - Q_{\text{without}},\qquad Q_{\text{acc}} = Q_{\text{well}} - \Delta Q

any identical units of oil volume at standard conditions

Holds when: Both runs use the same model and the same operating regimes of the other wells, apart from the changes that the intervention itself causes. The period of comparison is the same for both runs and long enough for the acceleration to cancel out: over a short period, part of the acceleration looks like an increment.

Reservoir simulation in practice: case studies →

Oil rate with a limit on gas-handling capacity

q_o = \min\left(q_{\text{pot}},\ \frac{Q_g}{R}\right)

gas and oil at standard conditions

Holds when: All the produced gas passes through the limited capacity; gas used for the field's own needs and gas sold are accounted for in $Q_g$ . The forecast gives the gas–oil ratio. When withdrawal is restricted, the gas approaches the wells more slowly, so in a model run with a gas limit on the group the gas–oil ratio rises later.

Reservoir simulation in practice: case studies →

Logarithmically spaced radial grid

r_i = r_w\,\alpha^{i}, \qquad \alpha = \left(\frac{r_e}{r_w}\right)^{1/N}, \qquad i = 0, 1, \dots, N

any identical units of length

Holds when: A vertical well at the centre of the model, with axial symmetry. The calculation point of a ring is often placed at its “logarithmic midpoint”, at the radius $\sqrt{r_{i-1} r_i}$ . The rules differ slightly between simulators.

Grids in reservoir simulation models →

Outlet concentration of a chain of N well-mixed cells

c_N(W_{iD}) = 1 - \e^{-N W_{iD}} \sum_{k=0}^{N-1} \frac{(N W_{iD})^k}{k!}

dimensionless

Holds when: One-dimensional flow, cells of equal size, constant porosity and Darcy velocity. A tracer or a miscible agent of the same viscosity is transported, so the flow does not depend on the concentration. Upstream weighting, with a time step much smaller than the time to cross a cell. An implicit scheme with a large step smears the front even more. There is no physical dispersion: the exact solution is a step at $W_{iD} = 1$ .

Grids in reservoir simulation models →

Extrapolation of the result to an infinitely fine grid

R_\infty \approx R_{2N} + \frac{R_{2N} - R_N}{2^{p} - 1}, \qquad p = \log_2\frac{R_N - R_{N/2}}{R_{2N} - R_N}

in the units of the result

Holds when: The grid is refined by a factor of two each time, with nothing else changed. The result changes monotonically (in one direction only), and the error already falls as $h^{p}$ , where $h$ is the cell size. We need at least three grids to check this. For $p = 1$ this is a linear extrapolation in $1/N$ : $R_\infty \approx 2R_{2N} - R_N$ .

Grids in reservoir simulation models →

Numerical diffusion coefficient of the upstream scheme

D_{num} = \frac{v\,\Delta x}{2}\,(1 \pm C), \qquad C = \frac{v\,\Delta t}{\Delta x}

SI (any consistent system)

Holds when: Linear transport of a tracer on a uniform one-dimensional grid. The minus sign is for an explicit scheme in time (stable only for $C \le 1$ ), and the plus sign for an implicit one (stable for any step). In the limit of a very small step, $D_{num} = v\Delta x/2$ . The mixing-cells formula corresponds to this limit.

Grids in reservoir simulation models →

Flow of a phase through the common face of two cells

F_{\alpha} = 0.0853\,T_{ij}\,\lambda_{\alpha\uparrow}\left(\Phi_{\alpha,j} - \Phi_{\alpha,i}\right),\qquad \lambda_\alpha = \frac{k_{r\alpha}}{\mu_\alpha B_\alpha},\qquad \Phi_\alpha = p_\alpha - \gamma_\alpha D

metric oilfield units: $k$ in mD (as part of $T_{ij}$ ), lengths in m, $\mu$ in mPa·s, pressures in MPa

Holds when: Darcy's law applies between the cell centres: the flow through a face depends only on the potential difference of these two cells (the two-point formula; its limits are in the last section). The properties inside each cell are uniform. The water pressure in a cell is $p_w = p_o - p_c(S_w)$ , so the capillary pressure enters the potential difference of the water.

Interblock flow and transmissibility →

Transmissibility of the connection between two cells

T_{ij} = \left(\frac{1}{t_i} + \frac{1}{t_j}\right)^{-1} = \frac{A}{\dfrac{\Delta x_i}{2k_i} + \dfrac{\Delta x_j}{2k_j}},\qquad \bar k = \frac{\Delta x_i + \Delta x_j}{\Delta x_i/k_i + \Delta x_j/k_j}

any consistent units; in this lesson $k$ is in mD, $A$ in m² and lengths in m, so $T$ is in mD·m

Holds when: The same as for the half-cells. If the cross-sections of the halves differ (a one-dimensional model of a formation of variable thickness or width), each of $t_i$ and $t_j$ gets its own cross-section. For equal cells $\bar k$ is the simple harmonic mean $2k_ik_j/(k_i + k_j)$ .

Interblock flow and transmissibility →

Pressure at the common face

p_f = \frac{t_i\,p_i + t_j\,p_j}{t_i + t_j}

any consistent units

Holds when: Single-phase flow. The pressure changes linearly inside each half.

Interblock flow and transmissibility →

Vertical and horizontal transmissibility of equal cells

\frac{T_z}{T_x} = \frac{k_v}{k_h}\left(\frac{\Delta x}{\Delta z}\right)^2

dimensionless

Holds when: Equal, uniform neighbouring cells with $\Delta x = \Delta y$ . The net-to-gross ratio is one.

Interblock flow and transmissibility →

Transmissibility between the rings of a radial grid

t_i = \frac{\theta\,h\,k_i}{\ln(r_f/r_i)},\qquad t_{i+1} = \frac{\theta\,h\,k_{i+1}}{\ln(r_{i+1}/r_f)},\qquad T = \left(\frac{1}{t_i} + \frac{1}{t_{i+1}}\right)^{-1}

$k$ in mD and lengths in m, so $T$ is in mD·m (natural logarithm)

Holds when: Inside each half-ring the flow is steady radial flow in uniform rock. If the rings have the same permeability, $T = \theta h k/\ln(r_{i+1}/r_i)$ , and the position of the boundary $r_f$ does not matter.

Interblock flow and transmissibility →

Transmissibility of half of a skewed cell

t_i = k_i\,\frac{\mathbf{A}\cdot\mathbf{c}_i}{\mathbf{c}_i\cdot\mathbf{c}_i}

$k$ in mD and lengths in m, so $t$ is in mD·m

Holds when: The two-point formula: the flow follows from the pressures of two cells only. The net-to-gross ratio and the transmissibility multipliers enter as extra factors.

Interblock flow and transmissibility →

Fault transmissibility multiplier for cells of different sizes

F = \frac{\dfrac{d_i}{k_i} + \dfrac{d_j}{k_j}}{\dfrac{d_i}{k_i} + \dfrac{d_j}{k_j} + \dfrac{l_f}{k_f}}

dimensionless; all lengths in one unit and all permeabilities in one unit

Holds when: The fault zone is thin compared with the cells. Single-phase flow across the fault. For equal cells the formula turns into the multiplier of the lesson on partially communicating faults.

Interblock flow and transmissibility →

Mobility of a phase at the face (upstream weighting)

\lambda_{\alpha\uparrow} = \begin{cases} \lambda_\alpha(S_j,\ p_j), & \Phi_{\alpha,j} \ge \Phi_{\alpha,i}\\ \lambda_\alpha(S_i,\ p_i), & \Phi_{\alpha,j} < \Phi_{\alpha,i}\end{cases}

as in the flow formula

Holds when: The direction is found for each phase separately, from the difference of its own potentials. With gravity and capillary pressure, the phases can flow in opposite directions through one face. The viscosity and the formation volume factor change little from cell to cell. We take them together with the relative permeability from the upstream cell, or as averages of the two cells. This hardly affects the result. The specific weight of the phase in the potential difference is taken as the average of the two cells, if the phase is present in both.

Interblock flow and transmissibility →

Permeability tensor in the grid axes for rotated principal axes

k_{xx} = k_1\cos^2\theta + k_2\sin^2\theta,\qquad k_{yy} = k_1\sin^2\theta + k_2\cos^2\theta,\qquad k_{xy} = (k_1 - k_2)\sin\theta\cos\theta

Holds when: A two-dimensional problem. Darcy's law $\mathbf{u} = -(\mathbf{K}/\mu)\nabla p$ with a symmetric tensor.

Interblock flow and transmissibility →

Well model: rate, cell pressure and bottomhole pressure

q_s = \frac{0.0853\,\mathrm{WI}\,(p_0 - p_{wf})}{\mu B},\qquad \mathrm{WI} = \frac{2\pi kh}{\ln(r_0/r_w) + S}

metric oilfield units; natural logarithm

Holds when: A vertical well at the centre of the cell opens its whole thickness. Single-phase inflow. Near the well the flow is close to steady-state radial flow (at distances of the order of the cell size). $r_0$ is calculated with a formula that suits the grid (see below). Neighbouring wells and the boundaries of the model are several cells away.

Well model in a grid block →

Equivalent radius of a cell in an isotropic formation (Peaceman's formula)

r_0 = 0.14\,\sqrt{\Delta x^2 + \Delta y^2}\qquad (\Delta x = \Delta y:\ \ r_0 = 0.198\,\Delta x)

any units of length (the same for all)

Holds when: Isotropic formation (permeability the same in all directions). Vertical well at the centre of the cell. Five-point scheme on a Cartesian grid. Other wells and the boundaries of the model are several cells away. Neighbouring cells have the same size. The ratio of the sides of the cell hardly affects the accuracy: on a uniform grid the formula is also right for strongly elongated cells (it differs from the exact solution by about 0.3%).

Well model in a grid block →

Equivalent radius and well index in an anisotropic formation

r_0 = 0.28\,\frac{\left[(k_y/k_x)^{1/2}\,\Delta x^2 + (k_x/k_y)^{1/2}\,\Delta y^2\right]^{1/2}}{(k_y/k_x)^{1/4} + (k_x/k_y)^{1/4}},\qquad \mathrm{WI} = \frac{2\pi\sqrt{k_xk_y}\,h}{\ln(r_0/r_w) + S}

any consistent units (the permeabilities enter as ratios and through $\sqrt{k_xk_y}$ )

Holds when: A vertical well at the centre of the cell. The grid axes match the principal directions of permeability. As in the isotropic case: a uniform grid, with other wells and model boundaries several cells away. The ratio of the sides of the cell hardly affects the accuracy.

Well model in a grid block →

Distribution of the rate between layers

q_l = J_l\,(p_l - p_{wf}),\qquad p_{wf} = \frac{\sum_l J_l\,p_l - q}{\sum_l J_l},\qquad J_l = \frac{0.0853\,\mathrm{WI}_l}{\mu B}

metric oilfield units

Holds when: Single-phase inflow. The pressure in the wellbore is the same for all layers (a thin interval; otherwise the weight of the fluid column in the wellbore is added to the $p_{wf}$ of each layer). If the cell pressure of a layer is lower than the bottomhole pressure, $q_l$ is negative: fluid leaves the wellbore into this layer (wellbore crossflow).

Well model in a grid block →

Mass balance of a cell over a time step

V_i\left[(\rho\phi)_i^{n+1} - (\rho\phi)_i^{n}\right] = \Delta t\left[(\rho q)_{i-1/2} - (\rho q)_{i+1/2} - (\rho q_w)_i\right]

SI (any consistent system)

Holds when: The flows in the square brackets are averages over the step. Read this way, the equality is exact. The flow is along one axis. In two and three dimensions, the flows through the other faces are added. The density at a face is taken from the values in the neighbouring cells.

Pressure equation for single-phase flow →

The pressure equation for single-phase flow

\phi\rho\,c_t\,\frac{\partial p}{\partial t} = \nabla\cdot\left[\frac{\rho}{\mu}\,\mathbf{K}\left(\nabla p - \rho g\,\nabla D\right)\right] - \rho\,\tilde q,\qquad c_t = \frac{1}{\rho\phi}\,\frac{\dd(\rho\phi)}{\dd p}

Holds when: One mobile fluid. Darcy's law (no inertial losses). Isothermal flow (the temperature does not change). The density, the viscosity, the porosity and the permeability may depend on the pressure. The equation does not simplify anything yet.

Pressure equation for single-phase flow →

The linear pressure equation for a heterogeneous formation

\phi\mu c_t\,\frac{\partial p}{\partial t} = \frac{\partial}{\partial x}\!\left(k_x\frac{\partial p}{\partial x}\right) + \frac{\partial}{\partial y}\!\left(k_y\frac{\partial p}{\partial y}\right) + \frac{\partial}{\partial z}\!\left(k_z\frac{\partial p}{\partial z}\right) - \mu\,\tilde q

Holds when: A slightly compressible fluid: $c\,\Delta p \ll 1$ . The change of density in the flow is not taken into account. The viscosity and the total compressibility do not depend on pressure. The porosity changes little. The coordinate axes point along the principal directions of permeability.

Pressure equation for single-phase flow →

Equation of a cell (single-phase, slightly compressible fluid)

S_i\,\frac{p_i^{n+1} - p_i^{n}}{\Delta t} = \sum_j a_{ij}\left(p_j - p_i\right) - q_i,\qquad S_i = V_{p,i}\,c_t,\qquad a_{ij} = \frac{0.0853\,T_{ij}}{\mu}

metric oilfield units: $k$ in mD, lengths in m, $\mu$ in mPa·s, $c_t$ in 1/MPa, $p$ in MPa, $t$ in d, volumes at reservoir conditions

Holds when: A slightly compressible fluid, with constant viscosity and compressibility (a linear equation). The sum runs over all cells with which cell $i$ has a common face or a non-neighbour connection. The pressures on the right side are taken at the start of the step (explicit scheme) or at the end (implicit scheme). If the cells lie at different depths, we put $(p_j - p_i) - \rho g\,(D_j - D_i)$ in place of $p_j - p_i$ , with $\rho g$ in MPa/m.

Pressure equation for single-phase flow →

Explicit scheme: the new pressure of a cell

p_i^{n+1} = p_i^{n} + \frac{\Delta t}{S_i}\left[\sum_j a_{ij}\left(p_j^{n} - p_i^{n}\right) - q_i\right]

metric oilfield units, as in the equation of a cell

Holds when: The flows and the withdrawal over the whole step are calculated from the pressures at the start of the step.

Pressure equation for single-phase flow →

The largest step of the explicit scheme

\Delta t \le \min_i \tau_i,\qquad \tau_i = \frac{S_i}{\sum_j a_{ij}}

metric oilfield units (any consistent units)

Holds when: A linear equation. The condition is sufficient: if it holds, the scheme is stable. Under it the new pressure of the cell (leaving the withdrawal aside) is a weighted average of the pressures of the cell itself and its neighbours at the start of the step, with non-negative weights. So the calculation cannot “jump over” the neighbours. For a long uniform grid the condition is exact: a larger step is unstable. For a short or non-uniform grid the allowed step is a little larger (see the advanced section).

Pressure equation for single-phase flow →

Implicit scheme: the equation of a cell in the system

-a_{i,i-1}\,p_{i-1}^{n+1} + \left(\frac{S_i}{\Delta t} + \sum_j a_{ij}\right)p_i^{n+1} - a_{i,i+1}\,p_{i+1}^{n+1} = \frac{S_i}{\Delta t}\,p_i^{n} - q_i

metric oilfield units

Holds when: A one-dimensional grid. In two and three dimensions a row has five or seven unknowns. A linear equation. In a nonlinear one the coefficients depend on the unknown pressures, and the system is solved by iterations.

Pressure equation for single-phase flow →

The largest step of the explicit scheme for a uniform grid

\Delta t_{\max} = \frac{1}{2}\left(\frac{\eta_x}{\Delta x^2} + \frac{\eta_y}{\Delta y^2} + \frac{\eta_z}{\Delta z^2}\right)^{-1},\qquad \eta_x = \frac{0.0853\,k_x}{\phi\mu c_t}

metric oilfield units: $k$ in mD, $\mu$ in mPa·s, $c_t$ in 1/MPa, dimensions in m, $\Delta t$ in d

Holds when: A uniform grid inside the formation, and a linear equation. In one dimension only one term remains, and $\Delta t_{\max} = \Delta x^2/(2\eta)$ .

Pressure equation for single-phase flow →

Phase balance in a cell over a time step

V_i\left[\left(\frac{\phi S_\alpha}{B_\alpha}\right)_i^{n+1} - \left(\frac{\phi S_\alpha}{B_\alpha}\right)_i^{n}\right] = \Delta t\left[\left(\frac{A\,u_\alpha}{B_\alpha}\right)_{i-1/2} - \left(\frac{A\,u_\alpha}{B_\alpha}\right)_{i+1/2} - q_{\alpha,i}\right]

any consistent units; phase volumes at standard conditions

Holds when: The phases do not exchange mass (the gas is dissolved and does not come out of solution), and the density at standard conditions is constant. The rates in the square brackets are averages over the step. Read this way, the equality is exact. The flow is along one axis. In two and three dimensions, the faces along the other axes are added.

Two-phase flow equations →

Continuity equation for a phase

\frac{\partial}{\partial t}\left(\frac{\phi S_\alpha}{B_\alpha}\right) + \nabla\cdot\left(\frac{\mathbf{u}_\alpha}{B_\alpha}\right) + \tilde q_\alpha = 0,\qquad \alpha = o,\ w

any consistent units

Holds when: This is the limit as $\Delta x \to 0$ and $\Delta t \to 0$ , and the properties vary smoothly. The phases do not exchange mass.

Two-phase flow equations →

Darcy's law for a phase when both phases flow together

\mathbf{u}_\alpha = -\lambda_\alpha\left(\nabla p_\alpha - \rho_\alpha g\,\nabla D\right),\qquad \lambda_\alpha = \frac{k\,k_{r\alpha}(S_w)}{\mu_\alpha}

SI; in metric oilfield units, multiply $\lambda_\alpha$ by 0.0853 ( $k$ in mD, $\mu$ in mPa·s, $p$ in MPa, lengths in m, $u$ in m/d)

Holds when: The flow is laminar (smooth, without turbulence), and the phases do not mix. Relative permeabilities depend only on the saturation and on the direction in which it changes. There are separate curves for imbibition (water pushes oil out of a water-wet rock, that is, a rock that water clings to more than oil) and for drainage (oil pushes water out). The weight of the phase acts along the $D$ axis, so in horizontal flow the term with $\nabla D$ vanishes.

Two-phase flow equations →

Closure relations

S_o + S_w = 1,\qquad p_o - p_w = p_c(S_w),\qquad k_{r\alpha} = k_{r\alpha}(S_w),\qquad B_\alpha,\ \mu_\alpha,\ \phi = f(p)

any consistent units

Holds when: The pores contain only oil and water. The $p_c$ and $k_r$ curves are chosen for the process that takes place in the reservoir: imbibition when a water-wet reservoir is flooded.

Two-phase flow equations →

Working equations of two-phase flow (one dimension)

\begin{aligned} &\frac{\partial}{\partial t}\left(\frac{\phi S_o}{B_o}\right) = \frac{\partial}{\partial x}\left[\frac{\lambda_o}{B_o}\left(\frac{\partial p_o}{\partial x} - \rho_o g\,\frac{\partial D}{\partial x}\right)\right] - \tilde q_o\\ &\frac{\partial}{\partial t}\left(\frac{\phi S_w}{B_w}\right) = \frac{\partial}{\partial x}\left[\frac{\lambda_w}{B_w}\left(\frac{\partial p_o}{\partial x} - \frac{\partial p_c}{\partial x} - \rho_w g\,\frac{\partial D}{\partial x}\right)\right] - \tilde q_w \end{aligned}

SI (any consistent system)

Holds when: The flow is one-dimensional, along $x$ . The fluid properties are functions of pressure. The relative permeabilities and $p_c$ are functions of the water saturation.

Two-phase flow equations →

Total compressibility in a two-phase cell

c_t = S_o c_o + S_w c_w + c_f,\qquad c_\alpha = -\frac{1}{B_\alpha}\frac{\dd B_\alpha}{\dd p},\qquad c_f = \frac{1}{\phi}\frac{\dd\phi}{\dd p}

1/MPa (any consistent units)

Holds when: We evaluate the properties of both phases at the oil pressure, because capillary pressure (tens of kilopascals) is small compared with the changes in reservoir pressure.

Two-phase flow equations →

Pressure equation for two-phase flow

\phi c_t\,\frac{\partial p_o}{\partial t} = \nabla\cdot\left[\lambda_t\,\nabla p_o - \lambda_w\,\nabla p_c - \left(\lambda_o\rho_o + \lambda_w\rho_w\right)g\,\nabla D\right] - \tilde q_t,\qquad \lambda_t = \lambda_o + \lambda_w

SI (any consistent system)

Holds when: The two phases do not mix, and the gas is dissolved. The formation volume factors vary little in space, so terms of the order of $c_\alpha\,\mathbf{u}_\alpha\cdot\nabla p$ are dropped. The fluid properties are evaluated at the oil pressure.

Two-phase flow equations →

Water flow in terms of the total velocity

\mathbf{u}_w = f_w\,\mathbf{u}_t + \lambda_o f_w\left[\nabla p_c + (\rho_w - \rho_o)\,g\,\nabla D\right],\qquad f_w = \frac{\lambda_w}{\lambda_t}

SI (any consistent system)

Holds when: The same as for Darcy's law for the phases. The formula is exact: it only rearranges the terms of Darcy's law.

Two-phase flow equations →

Saturation equation

\phi\,\frac{\partial S_w}{\partial t} + \nabla\cdot\left(f_w\,\mathbf{u}_t\right) + \nabla\cdot\left[\lambda_o f_w\,(\rho_w - \rho_o)\,g\,\nabla D\right] = \nabla\cdot\left(\lambda_o f_w\left|\frac{\dd p_c}{\dd S_w}\right|\nabla S_w\right) - \tilde q_w^{\text{res}}

SI (any consistent system)

Holds when: The rock and fluids are incompressible ( $\phi$ and $B_\alpha$ are constant). The rock is water-wet ( $\dd p_c/\dd S_w < 0$ ), and the total velocity is known from the pressure equation.

Two-phase flow equations →

Cell equation for a phase

V_{p,i}\,\frac{(S_\alpha/B_\alpha)_i^{n+1} - (S_\alpha/B_\alpha)_i^{n}}{\Delta t} = \sum_j 0.0853\,T_{ij}\left(\frac{k_{r\alpha}}{\mu_\alpha B_\alpha}\right)_{\!\uparrow}\left(\Phi_{\alpha,j} - \Phi_{\alpha,i}\right) - q_{\alpha,i},\qquad \Phi_\alpha = p_\alpha - \gamma_\alpha D

metric oilfield units: $k$ (inside $T_{ij}$ ) in mD, $\mu$ in mPa·s, $p$ in MPa, $D$ in m, $\Delta t$ in days; 0.0853 is the unit-conversion constant of Darcy's law

Holds when: The pore volume is constant over the step (for compressible rock it is found from the pressure). The mobility at a face is taken upstream (the reason is given below). The solution scheme decides at which time level (the start or the end of the step) the pressures and mobilities in the right-hand side are taken (IMPES and fully implicit schemes).

Two-phase flow equations →

The cell equation as a finite-difference scheme

S\,\frac{p_i^{n+1} - p_i^{n}}{\Delta t} = a\left(p_{i+1} - 2p_i + p_{i-1}\right)\quad\Longleftrightarrow\quad \frac{p_i^{n+1} - p_i^{n}}{\Delta t} = \eta\,\frac{p_{i+1} - 2p_i + p_{i-1}}{\Delta x^2}

metric oilfield units: $k$ in mD, $\mu$ in mPa·s, $c_t$ in 1/MPa, $p$ in MPa, $t$ in d; 0.0853 is the constant of Darcy's law in these units (Rock permeability)

Holds when: Equal cells of a uniform formation, the linear pressure equation, an inner cell without a well. The moment whose pressures give the right-hand side has not been chosen yet. The scheme depends on it (see below).

Finite differences: from equations to computation →

The Taylor series for neighbouring nodes

p_{i\pm1} = p_i \pm \Delta x\,p'_i + \frac{\Delta x^2}{2}\,p''_i \pm \frac{\Delta x^3}{6}\,p'''_i + \frac{\Delta x^4}{24}\,p^{(4)}_i \pm \ldots

any consistent units

Holds when: The function is smooth: it has continuous derivatives of the needed order between the nodes. The grid step is constant (a non-uniform grid is in the advanced section).

Finite differences: from equations to computation →

Difference formulas and their error

\begin{aligned} &\frac{p_{i+1} - p_i}{\Delta x} = p'_i + \frac{\Delta x}{2}\,p''_i + O(\Delta x^2) &&\text{(forward)}\\ &\frac{p_i - p_{i-1}}{\Delta x} = p'_i - \frac{\Delta x}{2}\,p''_i + O(\Delta x^2) &&\text{(backward)}\\ &\frac{p_{i+1} - p_{i-1}}{2\Delta x} = p'_i + \frac{\Delta x^2}{6}\,p'''_i + O(\Delta x^4) &&\text{(central)}\\ &\frac{p_{i+1} - 2p_i + p_{i-1}}{\Delta x^2} = p''_i + \frac{\Delta x^2}{12}\,p^{(4)}_i + O(\Delta x^4) &&\text{(second difference)} \end{aligned}

any consistent units

Holds when: A smooth function and a constant step. The values at the nodes are exact (no noise or round-off).

Finite differences: from equations to computation →

The weighted scheme for the pressure equation

\frac{p_i^{n+1} - p_i^{n}}{\Delta t} = \eta\left[\sigma\,\frac{p_{i+1}^{n+1} - 2p_i^{n+1} + p_{i-1}^{n+1}}{\Delta x^2} + (1 - \sigma)\,\frac{p_{i+1}^{n} - 2p_i^{n} + p_{i-1}^{n}}{\Delta x^2}\right]

metric oilfield units (any consistent units)

Holds when: A linear equation, a uniform formation, a uniform grid. For a heterogeneous formation the same is written for the cell equations with the coefficients $a_{ij}$ and $S_i$ . For $\sigma > 0$ each equation has three unknowns at the end of the step, and at each step we solve a tridiagonal system, as in the implicit scheme.

Finite differences: from equations to computation →

Truncation error of the weighted scheme

\psi_i^{n} = \left(\tfrac12 - \sigma\right)\Delta t\,\frac{\partial^2 p}{\partial t^2} - \frac{\eta\,\Delta x^2}{12}\,\frac{\partial^4 p}{\partial x^4} + O\!\left(\Delta t^2 + \Delta x^4\right)

any consistent units

Holds when: A smooth solution (far from sudden changes in how the wells are run, and from the wells themselves). Constant coefficients, a uniform grid.

Finite differences: from equations to computation →

Accumulation of error in a stable scheme

\begin{aligned} &e^{n+1} = \mathbf{A}\,e^{n} + \Delta t\,\mathbf{B}\,\psi^{n},\qquad \mathbf{B} = \left(\mathbf{I} - \sigma\Delta t\,\mathbf{L}\right)^{-1},\qquad \mathbf{A} = \mathbf{B}\left(\mathbf{I} + (1 - \sigma)\Delta t\,\mathbf{L}\right)\\ &\left\|e^{n}\right\| \le \left\|e^{0}\right\| + t^{n}\max_k\left\|\psi^{k}\right\| \end{aligned}

any consistent units

Holds when: A linear problem. The estimate on the right holds if $\|\mathbf{A}\| \le 1$ (the scheme is stable) and $\|\mathbf{B}\| \le 1$ , which is the case for the weighted scheme for the pressure equation. For an unstable scheme $\|\mathbf{A}\| > 1$ , and the contribution of each step is multiplied by $\|\mathbf{A}\|$ at all the following steps.

Finite differences: from equations to computation →

Amplification factor of the weighted scheme

G(\xi) = \frac{1 - 4(1 - \sigma)\,r\sin^2(\xi/2)}{1 + 4\sigma\,r\sin^2(\xi/2)},\qquad r = \frac{\eta\,\Delta t}{\Delta x^2}

dimensionless

Holds when: A linear equation with constant coefficients and a uniform grid. Boundaries are not taken into account (an infinite or periodic grid). For a heterogeneous formation we check the condition locally, with the local $\eta$ and $\Delta x$ . For a nonlinear equation we check it with the coefficients “frozen” at the step. The condition is then necessary but not always sufficient.

Finite differences: from equations to computation →

Stability conditions of the weighted scheme

\sigma = 0:\ \ r \le \tfrac12;\qquad 0 < \sigma < \tfrac12:\ \ r \le \frac{1}{2(1 - 2\sigma)};\qquad \sigma \ge \tfrac12:\ \ \text{for any } r

dimensionless

Holds when: They follow from $G \ge -1$ for the shortest, checkerboard harmonic ( $\xi = \pi$ ). The condition $G \le 1$ always holds. In two and three dimensions the explicit scheme needs $r_x + r_y + r_z \le \tfrac12$ , the same condition as for a uniform grid in the lesson on the pressure equation.

Finite differences: from equations to computation →

The Crank–Nicolson scheme for the cell equation

S_i\,\frac{p_i^{n+1} - p_i^{n}}{\Delta t} = \frac12\sum_j a_{ij}\left(p_j^{n+1} - p_i^{n+1}\right) + \frac12\sum_j a_{ij}\left(p_j^{n} - p_i^{n}\right) - q_i

metric oilfield units, as in the lesson on the pressure equation

Holds when: A linear equation. The withdrawal $q_i$ is constant during the step. The system for the pressures at the end of the step is tridiagonal (in one dimension), with the diagonal $S_i/\Delta t + \tfrac12\sum_j a_{ij}$ .

Finite differences: from equations to computation →

Error model for a convergence study

e = R - R_{\text{exact}} \approx C_x\,\Delta x^{p} + C_t\,\Delta t^{q}

in units of the result

Holds when: The steps are small enough for the leading terms of the error to dominate (the asymptotic range). The solution is smooth. Near wells, at displacement fronts and at sharp boundaries of properties, the observed order is lower than the theoretical one. The errors of the solution of the equations at each step (the tolerances of the iterations) are much smaller than the error of the scheme.

Finite differences: from equations to computation →

IMPES: the pressure equation of a cell with the mobilities at the start of the step

V_{p,i}\,c_{t,i}^{n}\,\frac{p_i^{n+1} - p_i^{n}}{\Delta t} = \sum_j a_{ij}^{n}\left(p_j^{n+1} - p_i^{n+1}\right) - q_{t,i},\qquad a_{ij}^{n} = 0.0853\,T_{ij}\,\lambda_{t,ij}^{n}

metric oilfield units: $T$ in mD·m, $\mu$ in mPa·s, $p$ in MPa, $t$ in d, volumes at reservoir conditions; 0.0853 is the constant of Darcy's law

Holds when: The mobilities and compressibilities are taken at the values at the start of the step, so the equation is linear in the pressures at the end of the step. The phases are slightly compressible, and the formation volume factors change little over a step. Capillary pressure (the pressure difference between the phases that comes from surface forces in the pores) and gravity are taken into account on the right-hand side, at the values at the start of the step.

IMPES and fully implicit schemes →

IMPES: explicit recalculation of the water saturation

V_{p,i}\,\frac{(S_w/B_w)_i^{n+1} - (S_w/B_w)_i^{n}}{\Delta t} = \sum_j 0.0853\,T_{ij}\left(\frac{k_{rw}}{\mu_w B_w}\right)_{\!\uparrow}^{\!n}\left(p_j^{n+1} - p_i^{n+1}\right) - q_{w,i}

metric oilfield units, as in the pressure equation; volumes of water at standard conditions

Holds when: Without capillary pressure and gravity. With them, we put the difference of the water potentials $\Phi_w = p_o - p_c - \gamma_w D$ in place of the pressure difference. The pore volume is constant during the step. The oil saturation is found from $S_o = 1 - S_w$ . With compressible phases, the linearisation of the pressure equation leaves a small error in the oil balance.

IMPES and fully implicit schemes →

The Courant condition for the explicit recalculation of saturation

C_i = \frac{\Delta t\,Q_i\,f_w'}{V_{p,i}} \le 1\quad\Longrightarrow\quad \Delta t \le \min_i \frac{V_{p,i}}{Q_i\,f_w'}

any consistent units: $Q$ in m³/d, $V_p$ in m³, $\Delta t$ in d

Holds when: A sufficient condition for upstream weighting in incompressible flow: if it holds, the calculation is stable. Capillary pressure and gravity add their own limits (see the advanced section). In a one-dimensional strip $Q_i = q_t$ ; in two and three dimensions it is the sum of the incoming flows. With a noticeably larger step the calculation loses monotonicity (the saturations no longer change steadily in one direction): they oscillate from cell to cell and go beyond the physical limits.

IMPES and fully implicit schemes →

The slope of the fractional flow curve for the Corey model

f_w' = \frac{\dd f_w}{\dd S_w} = \frac{f_w\,(1 - f_w)}{1 - S_{wi} - S_{or}}\left(\frac{n_w}{S_n} + \frac{n_o}{1 - S_n}\right)

dimensionless

Holds when: The Corey model (lesson on relative permeability). The viscosities are constant. At the ends of the mobile interval, where only one phase flows, the slope is zero.

IMPES and fully implicit schemes →

Residual of the balance of a phase in a cell (fully implicit method)

R_{\alpha,i} = V_{p,i}\,\frac{(S_\alpha/B_\alpha)_i^{n+1} - (S_\alpha/B_\alpha)_i^{n}}{\Delta t} - \sum_j 0.0853\,T_{ij}\left(\frac{k_{r\alpha}}{\mu_\alpha B_\alpha}\right)_{\!\uparrow}^{\!n+1}\left(\Phi_{\alpha,j} - \Phi_{\alpha,i}\right)^{n+1} + q_{\alpha,i}^{n+1} = 0

metric oilfield units; volumes of the phases at standard conditions

Holds when: The same cell equation as in the two-phase lesson, with all the quantities of the right-hand side taken at the end of the step. The unknowns of a cell are $p_o$ and $S_w$ , and the other quantities are expressed through the closure relations. In the black oil model with gas, a cell has three equations and three unknowns.

IMPES and fully implicit schemes →

A Newton iteration for a system of equations

\mathbf{J}\!\left(\mathbf{x}^{(\nu)}\right)\delta\mathbf{x} = -\mathbf{R}\!\left(\mathbf{x}^{(\nu)}\right),\qquad \mathbf{x}^{(\nu+1)} = \mathbf{x}^{(\nu)} + \delta\mathbf{x},\qquad J_{kl} = \frac{\partial R_k}{\partial x_l}

in units of the residuals and the unknowns

Holds when: The residuals are smooth functions of the unknowns. With upstream weighting the derivatives change abruptly when a flow changes direction. At each iteration we solve a linear system. The matrix $\mathbf{J}$ is not inverted.

IMPES and fully implicit schemes →

Convergence criteria of the iterations within a step

\max_i\,\frac{\Delta t\,B_\alpha\left|R_{\alpha,i}\right|}{V_{p,i}} \le \varepsilon_R,\qquad \frac{\Delta t\,B_\alpha\left|\sum_i R_{\alpha,i}\right|}{\sum_i V_{p,i}} \le \varepsilon_{MB}

dimensionless

Holds when: They are checked for each phase. A condition is often added that the corrections of the last iteration, in pressure and in saturation, are small. Typical orders of the tolerances: $\varepsilon_R$ from $10^{-3}$ to $10^{-2}$ , $\varepsilon_{MB}$ from $10^{-7}$ to $10^{-6}$ . The specific values depend on the simulator and the task.

IMPES and fully implicit schemes →

Choosing the next step from the target changes

\Delta t^{n+1} = \Delta t^{n}\cdot\min\!\left(\frac{\Delta S_{\text{target}}}{\max_i\left|\Delta S_i\right|},\ \frac{\Delta p_{\text{target}}}{\max_i\left|\Delta p_i\right|}\right),\qquad \Delta t^{n+1} \le f_{\text{growth}}\,\Delta t^{n}

$\Delta t$ in d, $p$ in MPa

Holds when: The changes per step are roughly proportional to the step: a flow without abrupt events. The step is also limited by the largest and smallest values, by report dates and by well events: after a start, a shut-in or a change in the way a well is run we begin again with a small step.

IMPES and fully implicit schemes →

A row of the pressure system: the cell equation at an implicit step

\left(\frac{S_i}{\Delta t} + \sum_j a_{ij}\right)\delta_i - \sum_j a_{ij}\,\delta_j = -q_i + \sum_j a_{ij}\left(p_j^{n} - p_i^{n}\right)

metric oilfield units

Holds when: A linear pressure equation: the properties of the fluid and the rock are constant during the step (a nonlinear case is solved by iterations, and each one has such a system). The sums run over all connections of the cell: neighbours across faces and non-neighbour connections. A well with a given bottomhole pressure adds its own coefficient to the diagonal.

Solving systems of linear equations in simulators →

A uniform shift of pressure is an eigenvector of the matrix

\mathbf{A} = \frac{S}{\Delta t}\,\mathbf{I} + \mathbf{C},\qquad \mathbf{C}\,\mathbf{1} = 0,\qquad \lambda_{\min}(\mathbf{A}) = \frac{S}{\Delta t},\qquad \kappa = 1 + \frac{\Delta t}{S}\,\lambda_{\max}(\mathbf{C})

metric oilfield units; $\kappa$ is dimensionless

Holds when: The cells have the same capacity. The outer boundaries are closed, and the wells work at given rates. A well with a given bottomhole pressure, or a constant-pressure boundary, adds to the diagonal and increases $\lambda_{\min}$ .

Solving systems of linear equations in simulators →

A step of the forward pass of Gaussian elimination

m_{ki} = \frac{A_{ki}}{A_{ii}},\qquad A_{kl} \leftarrow A_{kl} - m_{ki}\,A_{il},\qquad b_k \leftarrow b_k - m_{ki}\,b_i\qquad (k > i)

in the units of the system

Holds when: The pivot element $A_{ii}$ (the diagonal number that we divide by) is not zero. With diagonal dominance this is guaranteed, and no row swaps are needed. Rows in which $A_{ki} = 0$ do not change.

Solving systems of linear equations in simulators →

Work and memory of banded elimination

W \approx n\,b^2,\qquad M \approx n\,b;\qquad b = N_x\ \ (\text{2D}),\qquad b = N_xN_y\ \ (\text{3D})

dimensionless counts

Holds when: The leading term of the estimate for $b \ll n$ . Elimination without row swaps. For the Cholesky decomposition of a symmetric matrix the work is about half as large. For a dense matrix the work is $n^3/3$ .

Solving systems of linear equations in simulators →

The Jacobi, Gauss–Seidel and successive over-relaxation methods

\begin{aligned} &\text{Jacobi:} && x_i^{(\nu+1)} = \frac{1}{A_{ii}}\Bigl(b_i - \sum_{l \ne i} A_{il}\,x_l^{(\nu)}\Bigr)\\ &\text{Gauss–Seidel:} && x_i^{(\nu+1)} = \frac{1}{A_{ii}}\Bigl(b_i - \sum_{l < i} A_{il}\,x_l^{(\nu+1)} - \sum_{l > i} A_{il}\,x_l^{(\nu)}\Bigr)\\ &\text{SOR:} && x_i^{(\nu+1)} = x_i^{(\nu)} + \omega\,\bigl(\hat x_i - x_i^{(\nu)}\bigr)\end{aligned}

in the units of the system

Holds when: $A_{ii} \ne 0$ ; Gauss–Seidel and SOR go through the equations in the order of their numbers and use the new values at once; convergence is guaranteed with strict diagonal dominance (Jacobi and Gauss–Seidel), and for a symmetric positive definite matrix for Gauss–Seidel and for SOR with $0 < \omega < 2$ .

Solving systems of linear equations in simulators →

The speed of convergence of simple iterations

\nu \approx \frac{\ln(1/\varepsilon)}{-\ln\rho(\mathbf{G})},\qquad \rho_{GS} = \rho_J^2,\qquad \omega_{opt} = \frac{2}{1 + \sqrt{1 - \rho_J^2}},\qquad \rho_{SOR} = \omega_{opt} - 1

dimensionless

Holds when: an estimate for a large number of iterations; at the first iterations the speed depends on the initial error; the relations for Gauss–Seidel and SOR hold for “consistently ordered” matrices, a technical condition on the order of the unknowns. The five-point matrices of Cartesian grids with natural numbering satisfy it.

Solving systems of linear equations in simulators →

The convergence of the conjugate gradient method

\lVert\mathbf{e}^{(\nu)}\rVert_{\mathbf{A}} \le 2\left(\frac{\sqrt\kappa - 1}{\sqrt\kappa + 1}\right)^{\nu}\lVert\mathbf{e}^{(0)}\rVert_{\mathbf{A}},\qquad \nu \lesssim \frac{\sqrt\kappa}{2}\,\ln\frac{2}{\varepsilon}

dimensionless

Holds when: a symmetric positive definite matrix; exact arithmetic; an upper bound: in practice convergence is often faster, especially if the eigenvalues are gathered in groups.

Solving systems of linear equations in simulators →

The preconditioned system

\mathbf{M}^{-1}\mathbf{A}\,\mathbf{x} = \mathbf{M}^{-1}\mathbf{b},\qquad \mathbf{M} \approx \mathbf{A},\qquad \kappa\left(\mathbf{M}^{-1}\mathbf{A}\right) \ll \kappa(\mathbf{A})

in the units of the system

Holds when: the matrix $\mathbf{M}^{-1}$ is not built: at every iteration we solve the system $\mathbf{M}\mathbf{z} = \mathbf{r}$ ; for the conjugate gradient method $\mathbf{M}$ must be symmetric positive definite.

Solving systems of linear equations in simulators →

The stopping criterion and the link between the residual and the error

\frac{\lVert\mathbf{r}^{(\nu)}\rVert}{\lVert\mathbf{b}\rVert} \le \varepsilon,\qquad \mathbf{r}^{(\nu)} = \mathbf{b} - \mathbf{A}\,\mathbf{x}^{(\nu)},\qquad \frac{\lVert\mathbf{x} - \mathbf{x}^{(\nu)}\rVert}{\lVert\mathbf{x}\rVert} \le \kappa\,\frac{\lVert\mathbf{r}^{(\nu)}\rVert}{\lVert\mathbf{b}\rVert}

dimensionless ratios

Holds when: we can calculate the residual but not the error, because the exact solution is unknown; the inequality for the error is an upper bound.

Solving systems of linear equations in simulators →

The error of the iterations and the iteration matrix

\mathbf{e}^{(\nu+1)} = \mathbf{G}\,\mathbf{e}^{(\nu)},\qquad \mathbf{G}_J = \mathbf{D}^{-1}(\mathbf{L} + \mathbf{U}),\qquad \mathbf{G}_{GS} = (\mathbf{D} - \mathbf{L})^{-1}\,\mathbf{U}

dimensionless

Holds when: a linear system with a constant matrix; a diagonal with no zeros.

Solving systems of linear equations in simulators →

The conjugate gradient algorithm

\begin{aligned} &\mathbf{r}^{(0)} = \mathbf{b} - \mathbf{A}\mathbf{x}^{(0)},\quad \mathbf{p}^{(0)} = \mathbf{r}^{(0)};\\ &\alpha_\nu = \frac{\mathbf{r}^{(\nu)\mathsf T}\mathbf{r}^{(\nu)}}{\mathbf{p}^{(\nu)\mathsf T}\mathbf{A}\,\mathbf{p}^{(\nu)}},\quad \mathbf{x}^{(\nu+1)} = \mathbf{x}^{(\nu)} + \alpha_\nu\,\mathbf{p}^{(\nu)},\quad \mathbf{r}^{(\nu+1)} = \mathbf{r}^{(\nu)} - \alpha_\nu\,\mathbf{A}\,\mathbf{p}^{(\nu)};\\ &\beta_\nu = \frac{\mathbf{r}^{(\nu+1)\mathsf T}\mathbf{r}^{(\nu+1)}}{\mathbf{r}^{(\nu)\mathsf T}\mathbf{r}^{(\nu)}},\quad \mathbf{p}^{(\nu+1)} = \mathbf{r}^{(\nu+1)} + \beta_\nu\,\mathbf{p}^{(\nu)}\end{aligned}

in the units of the system

Holds when: a symmetric positive definite matrix; with a preconditioner, in the directions and the dot products we use $\mathbf{z} = \mathbf{M}^{-1}\mathbf{r}$ in place of $\mathbf{r}$ .

Solving systems of linear equations in simulators →

The modified equation and the numerical diffusion of the upstream scheme

\frac{\partial S}{\partial t} + v\,\frac{\partial f_w}{\partial x} = \frac{\partial}{\partial x}\!\left(D_{num}\,\frac{\partial S}{\partial x}\right),\qquad D_{num}(S) = \frac{v f_w'(S)\,\Delta x}{2}\,\bigl(1 \mp C(S)\bigr),\qquad C(S) = \frac{v f_w'(S)\,\Delta t}{\Delta x}

any consistent units: $v$ in m/d, $\Delta x$ in m, $\Delta t$ in d, $D_{num}$ in m²/d

Holds when: The upstream scheme on a uniform grid. The main error terms are included. The other terms are of the order of $\Delta x^2$ and $\Delta t^2$ . The solution is smooth. At a jump of saturation the analysis gives only a qualitative picture, and the width of the smeared front is found by calculation (next section).

Numerical dispersion →

Saturation at the face in a TVD scheme with a limiter

S_{i+1/2} = S_i + \tfrac12\,\Psi(r_i)\,(S_i - S_{i-1}),\qquad r_i = \frac{S_{i+1} - S_i}{S_i - S_{i-1}},\qquad \Psi_{\text{vL}}(r) = \frac{r + |r|}{1 + |r|}

dimensionless

Holds when: The flow is in the $+x$ direction. With $\Psi = 0$ we get the upstream scheme, and with $\Psi = 1$ a linear second-order reconstruction (a straight-line profile inside each cell). The limiter lies in the TVD region, $0 \le \Psi(r) \le \min(2r,\ 2)$ , and at extremes (peaks and dips of the profile, $r \le 0$ ) $\Psi = 0$ . The time step has two stages (Heun's scheme: a trial step and then a correction), and the Courant number is at most 0.5.

Numerical dispersion →

Equivalent permeability of a block: Darcy's law for a coarse cell

Q = \frac{k^{*}A\,\Delta p}{\mu\,L}\qquad\Longrightarrow\qquad k^{*} = \frac{Q\,\mu\,L}{A\,\Delta p}

SI (any consistent system)

Holds when: The flow is single-phase and steady, and the fluid is incompressible. Darcy's law holds in the fine cells and for the block as a whole. The rate and the pressure difference belong to one block under definite boundary conditions. $k^{*}$ depends on them (see the section on boundary conditions). If a pressure difference along one axis causes flow along another axis too, one number is not enough. We need the tensor $\mathbf{K}^{*}$ .

Permeability upscaling for single-phase flow →

Porosity and saturation of a coarse cell

\bar\phi = \frac{\sum_i V_i\,\phi_i}{\sum_i V_i},\qquad \bar S_w = \frac{\sum_i V_i\,\phi_i\,S_{w,i}}{\sum_i V_i\,\phi_i}

any equal units of volume; porosity and saturation as fractions

Holds when: The sums run over all the fine cells inside the coarse one. With a net-to-gross ratio $N_i$ : $\bar N = \sum V_iN_i/\sum V_i$ , the porosity of the reservoir rock is $\bar\phi = \sum V_iN_i\phi_i/\sum V_iN_i$ , and the weight in the saturation is $V_iN_i\phi_i$ . In this way the pore volume of the reservoir rock, $\bar N\bar\phi\sum V_i$ , is preserved. Oil and gas saturations are averaged in the same way. If the formation volume factor differs from cell to cell, we add it to the weight to get the volumes in place at standard conditions.

Permeability upscaling for single-phase flow →

Wiener bounds

\bar k_{\text{H}} = \left(\sum_i \frac{v_i}{k_i}\right)^{-1} \;\le\; k^{*} \;\le\; \bar k_{\text{A}} = \sum_i v_i\,k_i

permeabilities in any equal units

Holds when: The flow is single-phase and steady. The fine cells are isotropic (for anisotropic cells, we take the permeability of each direction separately). The bounds hold for any arrangement of the cells and for all three kinds of boundary conditions of the lesson, if $k^{*}$ is calculated from the flow in the block itself, with no buffer zone.

Permeability upscaling for single-phase flow →

Cardwell–Parsons bounds for flow along the

x

axis

\begin{aligned} \underline{k}_x &= \frac{1}{n_y}\sum_{j=1}^{n_y}\frac{n_x}{\sum_{i=1}^{n_x} 1/k_{ij}},\\ \overline{k}_x &= n_x\left[\sum_{i=1}^{n_x}\left(\frac{1}{n_y}\sum_{j=1}^{n_y} k_{ij}\right)^{-1}\right]^{-1},\\ \underline{k}_x &\le k^{*}_x \le \overline{k}_x \end{aligned}

permeabilities in any equal units; cells of equal size (otherwise the means get weights of lengths and areas)

Holds when: The flow is single-phase and steady. The block has sealed side faces, or it is periodic. For linear boundary conditions the upper bound is not guaranteed (see the advanced section). In a three-dimensional block $j$ runs over all the cells of the $y$ – $z$ cross-section. For flow along $y$ , rows and columns swap roles.

Permeability upscaling for single-phase flow →

Cell equations of a block in steady flow

\sum_{m} T_{lm}\,(p_m - p_l) + \sum_{b} T_{b}\,(p_b - p_l) = 0,\qquad T_{lm} = \frac{A_f}{\frac{\Delta x_l}{2k_l} + \frac{\Delta x_m}{2k_m}},\qquad T_b = \frac{2k_lA_f}{\Delta x_l}

metric oilfield: $k$ in mD, lengths in m, pressures in MPa. The viscosity and the constant 0.0853 are the same in all terms and cancel

Holds when: The flow is steady and the fluid is incompressible: there is no accumulation. There are no wells inside the block. The scheme has five points (seven in three dimensions): the cells are linked only through common faces, and the permeability tensor of the fine cells is diagonal (one value for each axis, with no cross terms). The equations are linear: the pressures can be written as fractions of the applied pressure difference, and $k^{*}$ does not depend on the size of the difference.

Permeability upscaling for single-phase flow →

Darcy's law for a block with a permeability tensor

\langle\mathbf{u}\rangle = -\frac{1}{\mu}\,\mathbf{K}^{*}\langle\nabla p\rangle,\qquad \mathbf{K}^{*} = \begin{pmatrix} k^{*}_{xx} & k^{*}_{xy}\\ k^{*}_{yx} & k^{*}_{yy}\end{pmatrix},\qquad k^{*}_{xx} = \frac{\mu\langle u_x\rangle}{G},\quad k^{*}_{yx} = \frac{\mu\langle u_y\rangle}{G}

SI (any consistent system)

Holds when: The averages are over the volume of the block. With periodic and linear conditions the average gradient equals the given one, and the average velocity is found from the rates through the faces (see the advanced section). The second calculation, with the gradient along $y$ , gives $k^{*}_{yy}$ and $k^{*}_{xy}$ . With these conditions a correctly calculated tensor is symmetric, $k^{*}_{xy} = k^{*}_{yx}$ , and $k^{*}_{xx}k^{*}_{yy} > (k^{*}_{xy})^2$ . With sealed side faces the average gradient acquires a transverse component that the calculation does not set. So from such calculations we take only the diagonal values.

Permeability upscaling for single-phase flow →

Transmissibility between coarse cells from the detailed model

T^{*}_{I,II} = \frac{\mu\,Q}{0.0853\,\left(\bar p_{I} - \bar p_{II}\right)}

metric oilfield: $Q$ in m³/d, $p$ in MPa, $\mu$ in mPa·s, $T^{*}$ in mD·m

Holds when: The flow is single-phase and steady. The problem is solved for the region between the centres of two cells (a local calculation) or for the whole model with wells (a global one). Like the equivalent permeability, the result depends on the boundary conditions of the local calculation. In the one-dimensional case the formula reduces to the sum of the resistances of the fine cells between the centres of the coarse ones.

Permeability upscaling for single-phase flow →

Well index of a coarse cell

\mathrm{WI}^{*} = \frac{\mu\,q}{0.0853\,\left(\bar p_0 - p_{wf}\right)}

metric oilfield, as in the lesson Well model in a grid block

Holds when: The flow to the well is single-phase and steady. The boundary conditions of the local calculation are, for example, the pressure of radial inflow on the outer boundary of the region. The index and the transmissibilities around the cell with the well are calculated together, from one calculation. If the cell pressure is taken as the average over its volume, the index of a uniform cell corresponds to a radius of about $0.35\,\Delta x$ , not Peaceman's $0.2\,\Delta x$ . The transmissibilities to the neighbours must be consistent with this. The skin factor of the wellbore is added separately, if it is not in the fine model.

Permeability upscaling for single-phase flow →

Average Darcy velocity from the rates through the faces of the block

\langle u_x\rangle = \frac{1}{V}\sum_{f} x_f\,q_f,\qquad \langle u_y\rangle = \frac{1}{V}\sum_{f} y_f\,q_f

SI (any consistent system)

Holds when: The flow is steady, and there are no sources inside the block. The sum is over all the faces of fine cells on the boundary of the block.

Permeability upscaling for single-phase flow →

Pseudo relative permeability and pseudo saturation of a coarse cell

\hat k_{rf,x} = \frac{\hat k_{f,x}}{k^{*}_x},\qquad \hat S_w = \frac{\sum_i V_i\,\phi_i\,S_{w,i}}{\sum_i V_i\,\phi_i}

permeabilities in equal units; saturation as a fraction

Holds when: $k^{*}$ and $\hat k_f$ are calculated for one direction and under the same boundary conditions. The directions $y$ and $z$ have their own values. The sums run over the fine cells inside the coarse one. How to get $\hat k_f$ depends on the method. Steady-state methods solve single-phase problems, and dynamic methods take the rates and pressures from a displacement calculation.

Two-phase upscaling: pseudo-functions →

Pseudo relative permeabilities of a layered cell in the capillary equilibrium limit

p_{c,j}(S_{w,j}) = P_c,\qquad \hat k_{rf}^{\parallel} = \frac{\sum_j k_jh_j\,k_{rf}(S_{w,j})}{\sum_j k_jh_j},\qquad \hat k_{rf}^{\perp} = \frac{\left(\sum_j h_j/\bigl(k_jk_{rf}(S_{w,j})\bigr)\right)^{-1}}{\left(\sum_j h_j/k_j\right)^{-1}}

permeabilities in mD, thicknesses in m; pseudo relative permeabilities are dimensionless

Holds when: The layers are continuous over the whole cell, and each one is uniform. Along the layers they work in parallel, and across them in series (averaging over layers). The flow is slow: capillary forces have time to equalise the pressure between the layers (the criterion is below). The relative permeability and capillary pressure curves are those of the process being modelled (for waterflooding of a water-wet rock, imbibition).

Two-phase upscaling: pseudo-functions →

Ratio of capillary and viscous forces in a cell

N_{cv} = \frac{t_x}{t_c} = \frac{k_v\,k_{ro}'\,\Delta p_c\,L}{\mu_o\,u\,H^2}

Holds when: This is an estimate of the order of magnitude, like the ratio of gravity to viscous forces in the lesson on layered reservoirs. The pressure difference $\Delta p_c/H$ over the thickness of a layer takes the place of the gravity gradient. $N_{cv} \gg 1$ is the capillary equilibrium limit. $N_{cv} \ll 1$ means that capillary crossflows (flow of water between layers, driven by capillary forces) can be neglected. In between, a calculation is needed.

Two-phase upscaling: pseudo-functions →

Pseudo relative permeabilities and pseudo capillary pressure by the Kyte–Berry method

\begin{aligned} \bar p_{f,I} &= \frac{\sum_j k_jk_{rf,j}h_j\bigl(p_{f,j} - \rho_fg(D_j - \bar D_I)\bigr)}{\sum_j k_jk_{rf,j}h_j},\qquad \Delta\bar\Phi_f = \bar p_{f,I} - \bar p_{f,II} - \rho_fg\,(\bar D_I - \bar D_{II}),\\ \hat k_{rf} &= \frac{\mu_f\,q_f\,\Delta x}{0.0853\,k^{*}A\,\Delta\bar\Phi_f},\qquad \hat p_c = \bar p_{o,I} - \bar p_{w,I} \end{aligned}

metric oilfield: $k$ in mD, $q$ in m³/d, $p$ in MPa, $\mu$ in mPa·s, lengths in m

Holds when: The reservoir and the flow are the same as in the fine-grid calculation from which the pressures and rates were taken. The pseudo relative permeabilities are assigned to the saturation of the upstream block. If a phase is immobile in the whole column (all $k_{rf,j} = 0$ ), the weight has no meaning. In the examples of the lesson, the pressure of such a phase is averaged with the weight $kh$ . $0.0853$ is the constant of Darcy's law (Darcy's law). In a horizontal reservoir the gravity terms vanish.

Two-phase upscaling: pseudo-functions →

Block pressure with pore-volume weighted averaging

\bar p_{f,I} = \frac{\sum_{i\in I} V_i\phi_i\,\bigl(p_{f,i} - \rho_fg(D_i - \bar D_I)\bigr)}{\sum_{i\in I} V_i\phi_i}

metric oilfield, as above

Holds when: The rest is the same potential difference and Darcy's law as in the Kyte–Berry method. For the last block before a well, we use the difference with the pressure at its face and the distance $\Delta x/2$ .

Two-phase upscaling: pseudo-functions →

Pseudo functions of a cross-section at capillary–gravity equilibrium

\hat S_w = \frac{\int_0^H\phi\,S_w(z)\,\dd z}{\int_0^H\phi\,\dd z},\qquad \hat k_{rf} = \frac{\int_0^H k\,k_{rf}\bigl(S_w(z)\bigr)\dd z}{\int_0^H k\,\dd z},\qquad p_c\bigl(S_w(z)\bigr) = p_c(0) + \Delta\rho\,g\,z

Holds when: Vertical equilibrium: the phase pressures equalise over the thickness faster than the front advances along the reservoir. The flow is along the reservoir: the pseudo relative permeabilities are the conductivities of the layers $k\,k_{rf}$ , added in parallel. Going through $p_c(0)$ from the largest to the smallest, we get a table $(\hat S_w, \hat k_{rw}, \hat k_{ro})$ . The capillary pressure is that of imbibition or of drainage, according to the process.

Two-phase upscaling: pseudo-functions →

The condition of scale separation

\lambda \;\ll\; \Delta \;\ll\; L

any equal units of length

Holds when: The condition is checked separately for each direction: the correlation lengths in the horizontal and the vertical directions differ by tens or hundreds of times. “Much less” means, in practice, a difference of a few times. When $\Delta$ is comparable to $\lambda$ , the equivalent permeability of the cell depends on the boundary conditions of its calculation (the figure below).

Special topics in upscaling →

Directional permeability of a tensor

k(\varphi) = \mathbf{n}^{\mathsf{T}}\mathbf{K}\,\mathbf{n} = k_{xx}\cos^2\varphi + k_{yy}\sin^2\varphi + 2k_{xy}\sin\varphi\cos\varphi

permeabilities in any equal units

Holds when: The flow is in the $x$ – $y$ plane, and the tensor is symmetric. The largest and smallest values, the principal permeabilities $k_1$ and $k_2$ , are reached along the principal axes, turned by the angle $\alpha$ : $\tan2\alpha = 2k_{xy}/(k_{xx} - k_{yy})$ . A scheme that leaves out $k_{xy}$ sees $k_{xx}\cos^2\varphi + k_{yy}\sin^2\varphi$ . Along a principal axis it is wrong by $-(1 - k_2/k_1)\sin^2 2\alpha/2$ .

Special topics in upscaling →

Darcy velocity near an injection well

u(r) = \frac{q}{2\pi r h}

m, d, m³ (any consistent units)

Holds when: Plane radial flow in a unit of constant thickness. The distribution of the inflow among the layers is not taken into account. Near other wells the picture of the flow is more complex. For an estimate of the order of magnitude, the formula is applied up to the middle of the distance between wells.

Special topics in upscaling →

First arrival of a tracer in a layered cross-section

W_{iD,1} = \frac{\Delta C_1}{\Delta F_1} = \frac{\phi_1h_1\big/\sum_j\phi_jh_j}{k_1h_1\big/\sum_j k_jh_j}

dimensionless; $k$ and $h$ in any equal units

Holds when: The layers are continuous between the rows of wells, the flow in each is linear, and the pressure difference is the same. In single-phase flow there is then no crossflow between the layers. The tracer moves with the water without lag and smearing. The rate of each layer is proportional to $k_jh_j$ , and its pore volume to $\phi_jh_j$ .

Special topics in upscaling →

Saturation and relative permeability of a bundle of capillaries in drainage

\begin{gathered}S_o = \frac{\sum_{r_i \ge r_c} N_i r_i^2}{\sum_i N_i r_i^2},\\ k_{ro} = \frac{\sum_{r_i \ge r_c} N_i r_i^4}{\sum_i N_i r_i^4},\qquad k_{rw} = 1 - k_{ro}\end{gathered}

radii in equal units; in $r_c$ consistent units ( $p_c$ in Pa, $\sigma$ in N/m, $r_c$ in m)

Holds when: The tubes have the same length and tortuosity, and each is completely filled with one phase (water films are neglected). The volume of a tube is proportional to $r^2$ , and its Poiseuille conductance to $r^4$ . Oil enters a tube as soon as $p_c$ exceeds its entry pressure. The tubes are not connected to each other.

Pore-scale flow modelling →

Conductance of a throat and the balance of rates in a pore

\begin{gathered}q_{ij} = g_{ij}\,(p_i - p_j),\qquad g_{ij} = \frac{\pi r_{ij}^4}{8\,\mu\,\ell_{ij}},\\ \sum_j g_{ij}\,(p_j - p_i) = 0\end{gathered}

SI (any consistent system)

Holds when: The flow is laminar Poiseuille flow in a cylindrical throat. For angular cross-sections a shape correction is introduced. The resistance of the pores themselves is neglected: they are much wider than the throats. The sum in the balance is over all the throats of the pore. On the inlet and outlet faces of the sample the pressures are set.

Pore-scale flow modelling →

Thresholds of imbibition: piston and snap-off

\begin{gathered}p_c^{\text{throat}} = \frac{2\sigma\cos\theta_a}{r},\qquad p_c^{\text{pore}} = \frac{2\sigma\cos\theta_a}{R},\\ p_c^{\text{snap-off}} = \frac{\sigma(\cos\theta_a - \sin\theta_a)}{r}\end{gathered}

SI: $p_c$ in Pa with $\sigma$ in N/m and radii in m

Holds when: Capillary equilibrium: the flow is so slow that the positions of the menisci are set only by the capillary pressure (a small capillary number: the viscous forces are weak compared with the capillary ones). The first two formulas are for a round cross-section. The snap-off formula is for a throat of square cross-section with water in the corners. At $\theta_a \ge 45°$ snap-off is impossible in it. A pore is filled from one side (no cooperative filling).

Pore-scale flow modelling →

The JBN method: relative permeability at the outlet face from an unsteady-state experiment

\begin{aligned} f_{o2} &= \frac{\dd N_{pD}}{\dd W_i},\qquad S_{w2} = \bar S_w - W_i\,f_{o2},\qquad I_r = \frac{q/\Delta p}{(q/\Delta p)_0},\\ \frac{k_{ro}(S_{w2})}{k_{ro}'} &= \frac{f_{o2}}{\dd\bigl(1/(W_iI_r)\bigr)/\dd(1/W_i)},\qquad k_{rw}(S_{w2}) = k_{ro}(S_{w2})\,\frac{1 - f_{o2}}{f_{o2}}\,\frac{\mu_w}{\mu_o} \end{aligned}

dimensionless; viscosities in equal units

Holds when: The assumptions of Buckley–Leverett hold: a uniform sample, incompressible phases, negligible capillary pressure and end effect, and stable displacement. The data are taken after water breakthrough. The derivatives are taken from smoothed dependences. To get the relative permeabilities as fractions of the absolute permeability, the result is multiplied by $k_{ro}'$ .

Petrophysical input for simulation: relative permeability and capillary pressure →

Normalised relative permeabilities and Corey exponents of a sample

S_n = \frac{S_w - S_{wi}}{1 - S_{wi} - S_{or}},\qquad k_{rw}^{*} = \frac{k_{rw}}{k_{rw}'},\quad k_{ro}^{*} = \frac{k_{ro}}{k_{ro}'},\qquad n_w = \frac{\sum \ln k_{rw}^{*}\,\ln S_n}{\sum (\ln S_n)^2},\quad n_o = \frac{\sum \ln k_{ro}^{*}\,\ln(1 - S_n)}{\sum \bigl(\ln(1 - S_n)\bigr)^2}

dimensionless; the sums are over the inner points of the sample

Holds when: In logarithms the Corey model is a straight line through the origin: $\ln k_r^{*} = n\ln S_n$ . The least-squares method for such a line gives the formulas above. The points at the very ends (small $S_n$ or $1 - S_n$ ) are distorted more strongly by the error of the end points, so we check them separately. If the Corey model does not describe the shape of the curves, we average the normalised relative permeabilities at equal $S_n$ , or we take a more flexible fit.

Petrophysical input for simulation: relative permeability and capillary pressure →

The Leverett J-function of a rock type and the capillary pressure of a cell

J = \frac{31.4\,p_c\sqrt{k/\phi}}{\sigma\cos\theta},\qquad S_w^{*} = \frac{S_w - S_{wirr}}{1 - S_{wirr}},\qquad J = J_d\,(S_w^{*})^{-b},\qquad p_c(S_w) = \frac{J\,\sigma\cos\theta}{31.4\sqrt{k/\phi}}

metric oilfield; $J$ is dimensionless

Holds when: $31.4 = 10^6\sqrt{0.9869\cdot10^{-15}}/10^{-3}$ is the constant of the Leverett function for MPa, mD and mN/m (the lesson on capillary pressure). The power form is the same as in the example of the lesson on capillary pressure. Its parameters are chosen from the points by least squares in logarithms. The drainage curve is for the initial distribution of fluids. For a waterflood calculation we need an imbibition curve of its own.

Petrophysical input for simulation: relative permeability and capillary pressure →

Two-point scaling of a table by the end points of a cell

S_t = S_{wi,t} + (S_w - S_{wi})\,\frac{1 - S_{wi,t} - S_{or,t}}{1 - S_{wi} - S_{or}},\qquad k_{rw}(S_w) = k_{rw,t}(S_t)\,\frac{k_{rw}'}{k_{rw,t}'}

dimensionless

Holds when: Horizontal scaling (along the saturation axis) is a linear mapping of the interval $[S_{wi}, 1 - S_{or}]$ of the cell onto the interval of the table. Three-point scaling adds an intermediate point (for example $1 - S_{or}$ between the critical and the largest water saturation) and maps the two parts separately. The end points of a cell are set by maps, or by links with the properties of the cell. The capillary pressure is scaled consistently, through the Leverett function. The shape of the curve (the exponents) is common to the rock type.

Petrophysical input for simulation: relative permeability and capillary pressure →

The Land relation: the trapped saturation of the non-wetting phase

S_{or}^{*} = \frac{S_{oi}^{*}}{1 + C\,S_{oi}^{*}},\qquad S^{*} = \frac{S}{1 - S_{wi}},\qquad C = \frac{1}{S_{or,\max}^{*}} - \frac{1}{S_{oi,\max}^{*}}

fractions

Holds when: This is an empirical relation. $C$ is chosen from imbibition experiments at different initial saturations, or from one pair of end points. For gas the same holds, with $S_{gi}^{*}$ and $S_{gr}^{*}$ : trapped gas when water invades a gas cap, in water-alternating-gas injection, and in underground gas storage.

Petrophysical input for simulation: relative permeability and capillary pressure →

Stone's second model for the oil relative permeability (normalised form)

k_{ro} = k_{rocw}\left[\left(\frac{k_{row}}{k_{rocw}} + k_{rw}\right)\left(\frac{k_{rog}}{k_{rocw}} + k_{rg}\right) - (k_{rw} + k_{rg})\right]

fractions

Holds when: A water-wet rock: water is in the small pores, gas in the large ones, and oil between them. A negative result is set to zero. This is how the model sets the residual oil of three-phase displacement. $k_{rocw}$ must be the same in both two-phase tables. Otherwise, at $S_g \to 0$ or $S_w \to S_{wi}$ the model does not return to the two-phase curves.

Petrophysical input for simulation: relative permeability and capillary pressure →

Capillary pressure at the inner face and the saturation there (the Hassler–Brunner method)

p_{cL} = \tfrac12\,\Delta\rho\,\omega^2\left(r_2^2 - r_1^2\right),\qquad S_L = \frac{\dd\,(p_{cL}\,\bar S)}{\dd p_{cL}} = \bar S + p_{cL}\,\frac{\dd\bar S}{\dd p_{cL}}

Holds when: Equilibrium is reached at each rotation speed. At the outer face $p_c = 0$ . The field of centrifugal force along the sample is taken as uniform. This is acceptable for short samples ( $r_1/r_2$ not less than about 0.7). Otherwise more accurate solutions are used. The derivative is taken from the smoothed dependence $\bar S(p_{cL})$ built from the speed steps.

Petrophysical input for simulation: relative permeability and capillary pressure →

Symbols

Symbol	Meaning and context
$\Delta t$	time step, dWhat a reservoir simulation model is and why it is needed
$\Delta Q$	incremental production of the sector from the intervention, m³Reservoir simulation in practice: case studies
$\bar k$	the permeability that gives the same transmissibility over the distance between the centres $(\Delta x_i + \Delta x_j)/2$ : $T_{ij} = \bar k A/[(\Delta x_i + \Delta x_j)/2]$ , mDInterblock flow and transmissibility
$\tilde q$	withdrawal by wells per unit volume of the formation (negative for injection), 1/sPressure equation for single-phase flow withdrawal per unit volume of the formation at reservoir conditions, 1/sPressure equation for single-phase flow
$\Delta x$	grid step, mFinite differences: from equations to computation
$\Delta p$	pressure difference between the inlet and outlet faces, PaPermeability upscaling for single-phase flow
$\Delta x,\ A$	distance between the block centres and the area of their common face, m, m²Two-phase upscaling: pseudo-functions
$\Delta x,\ \Delta t$	cell size and time step, m, sGrids in reservoir simulation models grid step and time step, m, dFinite differences: from equations to computation
$\Delta x,\ \Delta t,\ A$	cell size, time step, face area, m, d, m²Finite differences: from equations to computation
$\Delta x,\ \Delta y$	sizes of the cell seen from above, mWell model in a grid block cell dimensions along the $x$ and $y$ axes, mWell model in a grid block
$\Delta x,\ \Delta y,\ \Delta z$	cell dimensions; $\Delta z$ is the gross thickness, including non-reservoir rock, mBuilding a simulation model: input data and results dimensions of the cell, mPressure equation for single-phase flow
$\Delta x,\ \Delta z$	cell size in plan ( $\Delta x = \Delta y$ ) and in thickness, mInterblock flow and transmissibility
$\bar S,\ S_L$	average saturation of the sample (from the volume produced) and the saturation at the inner face, —Petrophysical input for simulation: relative permeability and capillary pressure
$\Delta t,\ \tau$	time step and characteristic depletion time, dWhat a reservoir simulation model is and why it is needed
$(\cdot)_{\uparrow}$	value in the upstream cell of the given phase: whichever of $i$ and $j$ has the higher potential of that phase, —Two-phase flow equations
$(\cdot)_\uparrow^{n+1}$	mobility of the phase at the end of the step from the upstream cell, 1/(mPa·s)IMPES and fully implicit schemes
$\left(k_{rw}/(\mu_w B_w)\right)_\uparrow^{n}$	water mobility at the start of the step from the upstream cell, 1/(mPa·s)IMPES and fully implicit schemes
$\langle\nabla p\rangle$	pressure gradient averaged over the volume of the block, Pa/mPermeability upscaling for single-phase flow
$\langle\mathbf{u}\rangle$	velocity of flow (Darcy velocity) averaged over the volume of the block, m/sPermeability upscaling for single-phase flow
$\lVert\cdot\rVert$	norm (size) of a vector, for example the square root of the sum of the squares, —Solving systems of linear equations in simulators
$\\|\cdot\\|,\ t^{n}$	norm of a vector (a measure of its size, for example the root-mean-square) and the time from the start of the calculation, —, dFinite differences: from equations to computation
$\lVert\mathbf{e}\rVert_{\mathbf{A}}$	energy norm $\sqrt{\mathbf{e}^{\mathsf T}\mathbf{A}\mathbf{e}}$ , —Solving systems of linear equations in simulators
$\partial^2 p/\partial t^2,\ \partial^4 p/\partial x^4$	derivatives of the exact solution in the middle of the step, MPa/d², MPa/m⁴Finite differences: from equations to computation
$\mp$	the sign “−” is for the explicit scheme, “+” for the implicit one, —Numerical dispersion
$\bar p_0$	pressure of the coarse cell in the same calculation, defined in the same way as the pressures of the neighbouring cells when the transmissibilities to them were calculated, MPaPermeability upscaling for single-phase flow
$\Delta C_1,\ \Delta F_1$	shares of the storage capacity and of the flow capacity of this layer: the first point of the F–Φ curve, —Special topics in upscaling
$A$	area of the common face of the cells, m²What a reservoir simulation model is and why it is needed a constant of the given oil, mPa·sTypes of reservoir simulation model area of the block face perpendicular to the flow, m²Permeability upscaling for single-phase flow
$\mathbf{A}$	area vector of the common face: it points perpendicular to the face, and its length equals the area of the face, m²Interblock flow and transmissibility operator of the step: what the scheme turns the error at the start of the step into, —Finite differences: from equations to computation matrix of the pressure system, m³/(d·MPa)Solving systems of linear equations in simulators
$A_f$	area of the common face, m²Permeability upscaling for single-phase flow
$a_{i,i\pm1}$	coefficients of the links with the neighbours (an end cell has no neighbour on one side, and the coefficient there is zero), m³/(d·MPa)Pressure equation for single-phase flow
$a_{ij}$	coefficient of the link between cells $i$ and $j$ : the flow between them for a pressure difference of 1 MPa, m³/(d·MPa)Pressure equation for single-phase flow link coefficient $0.0853\,T_{ij}/\mu$ , m³/(d·MPa)Solving systems of linear equations in simulators
$a_{ij}^{n}$	link coefficient of cells $i$ and $j$ with the mobilities at the start of the step, m³/(d·MPa)IMPES and fully implicit schemes
$A_{il},\ b_i$	elements of the matrix and of the right-hand side, in the units of the systemSolving systems of linear equations in simulators
$A_{kl},\ b_k$	elements of the matrix and of the right-hand side that change in the elimination, in the units of the systemSolving systems of linear equations in simulators
$A,\ \Delta x,\ k$	area of the common face, sizes and permeabilities of the cells, m², m, mDInterblock flow and transmissibility
$a = 0.0853\,kA/(\mu\Delta x)$	link coefficient of two equal neighbouring cells, m³/(d·MPa)Finite differences: from equations to computation
$b$	half-width of the band: the largest difference between the numbers of linked unknowns, —Solving systems of linear equations in simulators
$B$	a constant of the given oil showing how steeply the viscosity falls with temperature, KTypes of reservoir simulation model
$B_o$	oil formation volume factor at the pressure in the cell, m³/m³Building a simulation model: input data and results
$B_o,\ B_w,\ B_g$	formation volume factors of oil, water and gas at the pressure in the cell (each is the volume of the fluid in the reservoir divided by its volume at the surface; surface volumes are at 0.101325 MPa and 20 °C), m³/m³Types of reservoir simulation model
$B_\alpha$	formation volume factor of the phase: it converts the residual into reservoir volumes, m³/m³IMPES and fully implicit schemes
$B_\alpha,\ \mu_\alpha,\ \phi$	tables of the PVT properties of the phases (how their volume and viscosity depend on pressure) and the pore compressibility, m³/m³, mPa·s, —Two-phase flow equations
$\Delta p_c$	typical capillary pressure difference between the layers, PaTwo-phase upscaling: pseudo-functions
$C$	Courant number: the fraction of a cell that the front crosses in a step, —Grids in reservoir simulation models Land coefficient: the larger it is, the less is trapped, —Petrophysical input for simulation: relative permeability and capillary pressure
$\mathbf{C}$	matrix of links: $\sum_j a_{ij}$ on the diagonal, $-a_{ij}$ off it, m³/(d·MPa)Solving systems of linear equations in simulators
$c_f$	pore compressibility (how much the pore volume changes per unit change of pressure; change in pore volume), 1/bar (1/MPa)Building a simulation model: input data and results pore compressibility, 1/MPaTwo-phase flow equations
$\mathbf{c}_i$	vector from the centre of cell $i$ to the centre of the face, mInterblock flow and transmissibility
$C_i$	Courant number of the cell: the fraction of the cell that the saturation passes in a step, —IMPES and fully implicit schemes
$C_{ij}$	mass concentration of component $i$ in phase $j$ : the density of the phase multiplied by the mass fraction of the component in it, kg/m³Types of reservoir simulation model
$c_N$	fraction of the injected agent in the flow at the outlet of the last cell, —Grids in reservoir simulation models
$c_o,\ c_w$	compressibilities of oil and water (above the bubble-point pressure), 1/MPaTwo-phase flow equations
$c_t$	total compressibility of oil, connate water and rock together (how much their volume changes when the pressure changes), MPa⁻¹What a reservoir simulation model is and why it is needed total compressibility: compressibility of the fluid $c$ plus compressibility of the pores $c_f$ , 1/PaPressure equation for single-phase flow total compressibility, 1/MPaPressure equation for single-phase flow
$c_{t,i}^{n}$	total compressibility $S_oc_o + S_wc_w + c_f$ at the start of the step, 1/MPaIMPES and fully implicit schemes
$C_x,\ C_t$	coefficients that depend on the solution (its derivatives) and on the scheme, but not on the steps, —Finite differences: from equations to computation
$C(S)$	Courant number for the saturation $S$ : the fraction of a cell that it travels in one step (Courant–Friedrichs–Lewy condition), —Numerical dispersion
$d_i,\ d_j$	distances from the cell centres to the plane of the fault, mInterblock flow and transmissibility
$D_j,\ \bar D_I$	depth of a fine cell and of the centre of the coarse block (measured downwards), mTwo-phase upscaling: pseudo-functions
$d_k^{\text{calc}}$	calculated value of the same quantity at the same moment, units of the measurementHistory matching
$d_k^{\text{calc}}(\mathbf{m})$	calculated value of the $k$ -th measurement at the parameters $\mathbf{m}$ , units of the measurementHistory matching
$d_k^{\text{meas}}$	measured (observed) value of the $k$ -th measurement: pressure, rate, water cut, gas–oil ratio, units of the measurementHistory matching
$D_{num}$	numerical diffusion coefficient, m²/sGrids in reservoir simulation models
$D_{num}(S)$	numerical diffusion coefficient for the water saturation $S$ (it works like a real diffusion coefficient), m²/dNumerical dispersion
$\mathbf{D},\ \mathbf{L},\ \mathbf{U}$	the diagonal, and the lower and upper triangular parts of the matrix (with the opposite sign), in the units of the matrixSolving systems of linear equations in simulators
$\mathbf{e}^{(\nu)}$	error of the approximation after $\nu$ iterations, in the units of the unknownsSolving systems of linear equations in simulators error of the approximation $\mathbf{x} - \mathbf{x}^{(\nu)}$ , in the units of the unknownsSolving systems of linear equations in simulators
$e^{n},\ \psi^{n}$	vectors of the solution error and of the truncation error over all nodes at step $n$ , MPa, MPa/dFinite differences: from equations to computation
$F$	multiplier of the transmissibility of the connection without the fault, from 0 to 1, —Interblock flow and transmissibility
$f_{\text{growth}}$	largest allowed growth of the step, usually 2–3, —IMPES and fully implicit schemes
$F_{ji}$	flow from the neighbouring cell $j$ into cell $i$ (negative if the fluid leaves), kg/d or standard m³/dWhat a reservoir simulation model is and why it is needed
$f_{o2}$	oil fraction in the flow at the outlet of the sample, $1 - f_{we}$ ; the index 2 means the outlet face, as is usual in this method, —Petrophysical input for simulation: relative permeability and capillary pressure
$f_w$	fractional flow of water: its share of the total flow if gravity and capillary pressure are ignored, —Two-phase flow equations fractional flow of water $\lambda_w/\lambda_t$ (without gravity and capillary pressure), —IMPES and fully implicit schemes
$f_w\,\mathbf{u}_t$	transport of water by the total flow (viscous term), m/sTwo-phase flow equations
$f_w'$	the largest slope of the fractional flow curve between the saturation of the cell and the saturations of its upstream neighbours; to be safe for any saturations, the largest slope on the whole curve, —IMPES and fully implicit schemes
$f_w'(S)$	slope of the fractional flow curve; $vf_w'$ is the speed at which the saturation $S$ moves (frontal advance equation), —Numerical dispersion
$F_{\alpha}$	flow of phase $\alpha$ from cell $j$ into cell $i$ ; negative if the phase flows from $i$ to $j$ , m³/d (standard conditions)Interblock flow and transmissibility
$\bar p_{f,I}$	pressure of phase $f$ in block I: the average over the central column of fine cells $j$ , corrected to the depth of the block centre $\bar D_I$ , MPaTwo-phase upscaling: pseudo-functions
$\hat k_{f,x}$	equivalent phase permeability of the coarse cell along $x$ : the permeability at which the cell passes the same rate of the phase as the fine cells, mDTwo-phase upscaling: pseudo-functions
$g$	hydrostatic constant: $0.00981$ MPa/m for each 1 g/cm³ of density, MPa·cm³/(g·m)Building a simulation model: input data and results
$G$	magnitude of the average gradient in the calculation where the pressure falls along $x$ , Pa/mPermeability upscaling for single-phase flow
$g_{ij}$	hydraulic conductance of the throat, m³/(Pa·s)Pore-scale flow modelling
$\mathbf{G}_J,\ \mathbf{G}_{GS}$	iteration matrices of the Jacobi and Gauss–Seidel methods, —Solving systems of linear equations in simulators
$g,\ D$	acceleration of gravity and the depth, measured downwards, m/s², mPressure equation for single-phase flow acceleration due to gravity and depth (downwards), m/s², mTwo-phase flow equations
$G(\xi)$	how many times the amplitude of the harmonic changes over a step; $G < 0$ means that the harmonic changes sign at every step, —Finite differences: from equations to computation
$h$	thickness of the layer of rings, mInterblock flow and transmissibility thickness of the reservoir that takes the water, mSpecial topics in upscaling
$H$	thickness of the reservoir, mTwo-phase upscaling: pseudo-functions
$i$	number of the unknown being eliminated (the pivot row: the row that we subtract from the rows below it), —Solving systems of linear equations in simulators
$\hat x_i$	the value that the Gauss–Seidel formula would give, in the units of the unknownsSolving systems of linear equations in simulators
$i \in I$	all the fine cells of block I, —Two-phase upscaling: pseudo-functions
$\mathbf{I}$	identity matrix, —Finite differences: from equations to computation
$I_r$	relative injectivity. At a constant rate $I_r = \Delta p_0/\Delta p$ , where $\Delta p_0$ is the pressure drop at the start of the experiment (only oil flows, at $S_{wi}$ ), —Petrophysical input for simulation: relative permeability and capillary pressure
$\bar p_I,\ \bar p_{II}$	pressures of the coarse cells in the same calculation: the pressures at their centres, or the averages by pore volume of their fine cells, MPaPermeability upscaling for single-phase flow
$\Delta S_i,\ \Delta p_i$	changes of the saturation and pressure of the cell over the previous step, —, MPaIMPES and fully implicit schemes
$\Delta x_i,\ \Delta x_j$	sizes of the cells in the direction of flow, mWhat a reservoir simulation model is and why it is needed
$\mathbf{I},\ \mathbf{1}$	identity matrix and the vector of ones, an equal change of pressure in all cells, —Solving systems of linear equations in simulators
$J$	productivity index of the well (the rate per unit of drawdown), in reservoir volumes of liquid, m³/(d·MPa)What a reservoir simulation model is and why it is needed
$\mathbf{J}$	Jacobian matrix of size $2N \times 2N$ , in units of residuals per unit of the unknownsIMPES and fully implicit schemes
$J_d,\ b$	parameters of the rock type: the $J$ of the entry pressure and the exponent, —Petrophysical input for simulation: relative permeability and capillary pressure
$J_l$	productivity of the layer referred to the pressure of its cell, m³/(d·MPa)Well model in a grid block
$\mathbf{K}$	permeability tensor; if the model axes point along the principal directions, it is diagonal: $k_x$ , $k_y$ , $k_z$ , m²Pressure equation for single-phase flow
$k^{*}$	equivalent permeability of the block along the flow direction, m²Permeability upscaling for single-phase flow equivalent permeability of the block in any direction, mDPermeability upscaling for single-phase flow single-phase equivalent permeability between the block centres, mDTwo-phase upscaling: pseudo-functions
$\mathbf{K}^{*}$	tensor of the equivalent permeability of the block, m²Permeability upscaling for single-phase flow
$k^{*}_x$	single-phase equivalent permeability of the same cell along $x$ , mDTwo-phase upscaling: pseudo-functions
$k^{*}_{yx}$	flow along $y$ caused by a gradient along $x$ , written as a permeability, m²Permeability upscaling for single-phase flow
$k_1,\ k_2$	principal permeabilities (along and across the dominant direction), mDInterblock flow and transmissibility
$k_i$	permeability of the cell in the direction of the connection, mDInterblock flow and transmissibility permeability of the fine cell, mDPermeability upscaling for single-phase flow
$k_i,\ k_{i+1}$	permeability of the rings, mDInterblock flow and transmissibility
$k_i,\ k_j$	permeability of the cells in that direction, m²What a reservoir simulation model is and why it is needed permeabilities of the cells across the fault, mDInterblock flow and transmissibility
$k_{ij}$	permeability of a fine cell: $i$ is the number along the flow (the column), $j$ is the number of the line of cells along the flow (the row), mDPermeability upscaling for single-phase flow
$k_j,\ h_j$	permeability and thickness of the layer, mD, mTwo-phase upscaling: pseudo-functions
$k_j,\ k_{rf,j},\ h_j$	permeability, relative permeability of the phase and thickness of a fine cell of the column: the averaging weights, mD, —, mTwo-phase upscaling: pseudo-functions
$k_l,\ k_m$	permeability of the cells in the direction perpendicular to the face, mDPermeability upscaling for single-phase flow
$k_m$	matrix permeability, m²Types of reservoir simulation model
$k_{r\alpha},\ \mu_\alpha,\ B_\alpha$	relative permeability (how well the phase flows when other phases are present), viscosity and formation volume factor (the volume the phase takes at reservoir conditions per unit volume at the surface) of the phase, —, mPa·s, m³/m³Interblock flow and transmissibility
$k_{r\alpha}(S_w)$	relative permeability curves (for example, from the Corey model, a common simple formula for them), —Two-phase flow equations
$k_{ro}'$	oil relative permeability at $S_{wi}$ : the method gives the relative permeabilities as fractions of this value, —Petrophysical input for simulation: relative permeability and capillary pressure
$k_{rocw}$	oil relative permeability at irreducible water and no gas: the common end point of both tables, —Petrophysical input for simulation: relative permeability and capillary pressure
$k_{rog}(S_g),\ k_{rg}(S_g)$	oil and gas relative permeabilities from the gas–oil table (at irreducible water), at the gas saturation of the cell, —Petrophysical input for simulation: relative permeability and capillary pressure
$k_{row}(S_w),\ k_{rw}(S_w)$	oil and water relative permeabilities from the water–oil table, at the water saturation of the cell, —Petrophysical input for simulation: relative permeability and capillary pressure
$k_{rw}'/k_{rw,t}'$	vertical scaling of the end-point value; for oil in the same way with $k_{ro}'$ , —Petrophysical input for simulation: relative permeability and capillary pressure
$k_{rw}^{},\ k_{ro}^{}$	relative permeabilities as fractions of the end-point values of the sample, —Petrophysical input for simulation: relative permeability and capillary pressure
$k_{rw,t}(S_t)$	water relative permeability from the table (linear interpolation between rows), —Petrophysical input for simulation: relative permeability and capillary pressure
$k_v$	vertical permeability of the unit of layers (a group of neighbouring layers), m²Two-phase upscaling: pseudo-functions
$k_v,\ k_h$	vertical and horizontal permeability, mDInterblock flow and transmissibility
$\underline{k}_x,\ \overline{k}_x$	lower and upper bounds of the equivalent permeability along $x$ , mDPermeability upscaling for single-phase flow
$k_x,\ k_y$	principal permeabilities in the horizontal plane (the cell sides run along them), mDWell model in a grid block
$k_x,\ k_y,\ k_z$	principal permeabilities; they may change from point to point, m²Pressure equation for single-phase flow
$k_{xx},\ k_{yy},\ k_{xy}$	components of the tensor in the grid axes, mDInterblock flow and transmissibility components of the equivalent permeability tensor in the axes of the grid, mDSpecial topics in upscaling
$k,\ h$	permeability of the cell and the open thickness of the formation in it, mD, mWell model in a grid block
$k,\ \phi$	permeability and porosity of the sample or of the cell, mD, fractionPetrophysical input for simulation: relative permeability and capillary pressure
$k,\ \phi,\ \mu,\ c_t$	permeability, porosity, viscosity, total compressibility, mD, —, mPa·s, 1/MPaPressure equation for single-phase flow
$k,\ k_{r\alpha}$	absolute permeability and relative permeability of the phase, m², —Two-phase flow equations
$k(\varphi)$	component of the Darcy velocity along $\mathbf{n}$ at a unit pressure gradient along $\mathbf{n}$ , written as a permeability, mDSpecial topics in upscaling
$\mathbf{L}$	matrix of second differences multiplied by the diffusivity, 1/dFinite differences: from equations to computation
$L$	length of the block along the flow, mPermeability upscaling for single-phase flow distance over which the flow changes noticeably: between wells, the width of the displacement front, the zone of radial inflow near a well, mSpecial topics in upscaling
$l_f,\ k_f$	width of the fault zone and its permeability; only the ratio $l_f/k_f$ enters the formula, m, mDInterblock flow and transmissibility
$L_x,\ L_y,\ L_z$	dimensions of the matrix block — the distances between fractures in the three directions; if there are no fractures in one direction, we leave out that term, mTypes of reservoir simulation model
$L,\ H$	length of the cell along the flow, and thickness of the layers between which the saturation equalises, mTwo-phase upscaling: pseudo-functions
$\Delta x_l,\ \Delta x_m$	sizes of the cells across this face, mPermeability upscaling for single-phase flow
$\mathbf{m}$	the set of model parameters being adjusted, —History matching
$M$	misfit function — the mean square of the normalised discrepancy, —History matching number of stored numbers of the Cholesky decomposition (for the LU decomposition it is twice as many: $\mathbf{L}$ and $\mathbf{U}$ ), —Solving systems of linear equations in simulators
$\mathbf{M}$	the preconditioner: a rough copy of $\mathbf{A}$ , with which a system is cheap to solve, in the units of $\mathbf{A}$ Solving systems of linear equations in simulators
$m_i$	mass of component $i$ in the cell, kgTypes of reservoir simulation model
$m_i^{n}$	amount of fluid (a phase or a component) in cell $i$ at step $n$ ; for oil in a black oil model it is convenient to express it in standard cubic metres (the volume the oil takes at surface conditions): $V_{p,i}\,S_{o,i}/B_{o,i}$ , that is, pore volume × oil saturation ÷ the formation volume factor, which turns a reservoir volume into a surface volume, kg or standard m³What a reservoir simulation model is and why it is needed
$m_j^{\text{prior}}$	prior estimate of the $j$ -th parameter — before the history is taken into account, units of the parameterHistory matching
$m_{ki}$	multiplier of row $k$ ; the multipliers form the lower triangular matrix $\mathbf{L}$ of the decomposition, —Solving systems of linear equations in simulators
$\Delta t_{\max}$	the largest step at which the explicit scheme is stable, dPressure equation for single-phase flow
$n$	number of unknowns, —Solving systems of linear equations in simulators
$\mathbf{n}$	unit vector of this direction, $(\cos\varphi, \sin\varphi)$ , —Special topics in upscaling
$N$	number of measurements, —History matching number of rings, —Grids in reservoir simulation models number of cells between the injection and production wells, —Grids in reservoir simulation models
$\Delta t^{n},\ \Delta t^{n+1}$	previous and next steps, dIMPES and fully implicit schemes
$N_i,\ r_i$	number of tubes of class $i$ per unit of cross-section area, and their radius, —, µmPore-scale flow modelling
$N_o,\ N_w,\ G$	amounts of the components “oil”, “water” and “gas” in the cell, m³ at standard conditionsTypes of reservoir simulation model
$n_w,\ n_o$	Corey exponents for water and oil, —IMPES and fully implicit schemes Corey exponents: the slopes of straight lines through the origin, in logarithms, —Petrophysical input for simulation: relative permeability and capillary pressure
$n_x,\ n_y$	number of cells along the flow and across it, —Permeability upscaling for single-phase flow
$N_x,\ N_y$	numbers of cells along the axes by which the numbering goes (fastest along $x$ ), —Solving systems of linear equations in simulators
$n,\ n + 1$	numbers of the moments of time: the start and the end of the step, —Pressure equation for single-phase flow
$n,\ n + 1,\ \Delta t$	start and end of the step, time step, —, —, dTwo-phase flow equations
$\mathrm{NTG}$	net-to-gross ratio — the share of the cell thickness taken by reservoir rock, fractionBuilding a simulation model: input data and results
$\tilde q_o,\ \tilde q_w$	withdrawals of the phases per unit reservoir volume at standard conditions, 1/sTwo-phase flow equations
$O(\Delta x^m)$	terms of order $\Delta x^m$ : when the step is halved, they decrease by about $2^m$ times, —Finite differences: from equations to computation
$p$	pressure in the “tank”, the same throughout the volume, MPaWhat a reservoir simulation model is and why it is needed pressure in the cell, bar (MPa)Building a simulation model: input data and results observed order of convergence: when the grid is refined twofold, the error falls by a factor of $2^{p}$ , —Grids in reservoir simulation models pressure; if gravity is taken into account, the potential $p - \rho g D$ with constant $\rho$ , PaPressure equation for single-phase flow
$\mathbf{p}^{(\nu)}$	search direction, in the units of the unknownsSolving systems of linear equations in simulators
$p_0$	pressure of the cell with the well, calculated by the simulator, MPaWell model in a grid block
$p_c$	capillary pressure, MPaPetrophysical input for simulation: relative permeability and capillary pressure
$P_c$	common capillary pressure of the cell, MPaTwo-phase upscaling: pseudo-functions
$p_{c,j}(S_w)$	capillary pressure curve of layer (lamina) $j$ . For one rock type it is $p_{c,j} = J(S_w)\,\sigma\cos\theta\sqrt{\phi_j/k_j}$ in consistent units (Leverett J-function), MPaTwo-phase upscaling: pseudo-functions
$p_c(0)$	capillary pressure at the base: the parameter that sets the average saturation of the cross-section, PaTwo-phase upscaling: pseudo-functions
$p_c(S_w)$	capillary pressure; in a water-wet rock it is positive and falls as $S_w$ rises, MPaTwo-phase flow equations
$p_c(z)$	capillary pressure $p_o - p_w$ at depth $z$ , MPaBuilding a simulation model: input data and results
$p_{cL}$	capillary pressure at the inner face, PaPetrophysical input for simulation: relative permeability and capillary pressure
$p_f$	pressure at the common face of the cells, MPaInterblock flow and transmissibility
$p_{f,i}$	pressure of the phase in the fine cell, MPaTwo-phase upscaling: pseudo-functions
$p_i$	initial reservoir pressure, MPaWhat a reservoir simulation model is and why it is needed
$p_i^{n},\ p_i^{n+1}$	pressure in cell $i$ at the start and at the end of the step, MPaPressure equation for single-phase flow pressure at node (cell) $i$ at the start and at the end of the step, MPaFinite differences: from equations to computation oil pressure in the cell at the start and at the end of the step, MPaIMPES and fully implicit schemes
$p_i^{n},\ p_j^{n}$	pressures of the cell and its neighbours at the start of the step (known), MPaPressure equation for single-phase flow pressures at the start of the step, MPaSolving systems of linear equations in simulators
$p_{i-1}^{n+1},\ p_i^{n+1},\ p_{i+1}^{n+1}$	unknown pressures of the cell and of its neighbours at the end of the step, MPaPressure equation for single-phase flow
$p_i,\ p_{i\pm1}$	values of the function at node $i$ and at the neighbouring nodes, MPaFinite differences: from equations to computation
$p_i,\ p_j$	pressures at the cell centres, PaWhat a reservoir simulation model is and why it is needed pressures in the pores, PaPore-scale flow modelling
$p_j^{n+1} - p_i^{n+1}$	pressure difference at the end of the step, MPaIMPES and fully implicit schemes
$p_l$	pressure of the cell of layer $l$ that the well passes through, MPaWell model in a grid block
$p_l,\ p_m$	pressure in the fine cell $l$ and in the neighbouring cell $m$ , MPaPermeability upscaling for single-phase flow
$p_m,\ p_f$	pressures in the matrix cell and in the fracture cell, PaTypes of reservoir simulation model
$p^{n}$	pressure at step $n$ (known), MPaWhat a reservoir simulation model is and why it is needed
$p^{n},\ p^{n+1}$	pressure at the start and at the end of the step, MPaWhat a reservoir simulation model is and why it is needed pressures at the start and at the end of the step, MPaFinite differences: from equations to computation
$p^{n+1}$	pressure at the next step (to be found), MPaWhat a reservoir simulation model is and why it is needed
$p_o,\ p_c$	oil pressure and capillary pressure, PaTwo-phase flow equations
$p_o(z)$	oil pressure at depth $z$ — the pressure of the cell in the simulator's reports, MPaBuilding a simulation model: input data and results
$p_{\text{ref}}$	reference pressure — the pressure to which the specified porosity refers, bar (MPa in the text of the lesson)Building a simulation model: input data and results
$p_{\text{ref}},\ z_{\text{ref}}$	the pressure at the reference depth and the reference depth itself (from formation tester measurements), MPa, mBuilding a simulation model: input data and results
$p_w = p_o - p_c(S_w)$	water pressure in the cell, MPaTwo-phase flow equations
$p_{wf}$	bottomhole flowing pressure of the well (the pressure at the bottom of the flowing well), constant, MPaWhat a reservoir simulation model is and why it is needed bottomhole flowing pressure, MPaWhat a reservoir simulation model is and why it is needed bottomhole pressure (at the reference depth of the well), MPaWell model in a grid block common bottomhole pressure (referred to one depth), MPaWell model in a grid block
$p_\alpha,\ \rho_\alpha,\ \mu_\alpha$	pressure, density and viscosity of the phase at reservoir conditions, Pa, kg/m³, Pa·sTwo-phase flow equations
$p,\ q$	orders in space and in time: for the pressure equation $p$ = 2; $q$ = 1 for the implicit scheme and 2 for the Crank–Nicolson scheme, —Finite differences: from equations to computation
$p'_i,\ p''_i,\ p'''_i,\ p^{(4)}_i$	first to fourth derivatives with respect to $x$ at node $i$ , MPa/m, MPa/m², …Finite differences: from equations to computation derivatives at node $i$ , MPa/m, MPa/m², …Finite differences: from equations to computation
$p(x, y, z, t)$	pressure, the only unknown, PaPressure equation for single-phase flow
$q$	well rate at reservoir conditions, from the fine-grid calculation, m³/dPermeability upscaling for single-phase flow injection rate of the well at reservoir conditions, m³/dSpecial topics in upscaling
$Q$	total rate through the block at reservoir conditions, m³/sPermeability upscaling for single-phase flow rate between them in the fine-grid calculation, at reservoir conditions, m³/dPermeability upscaling for single-phase flow
$Q_{\text{acc}}$	the part of the production of the new well that the neighbouring wells would have produced later — acceleration of production, m³Reservoir simulation in practice: case studies
$q_f$	rate outward through the boundary face $f$ of a fine cell (an inflow has a minus sign), m³/sPermeability upscaling for single-phase flow rate of the phase through the face between the blocks: the sum of the flows of the fine cells of this face, at reservoir conditions, m³/dTwo-phase upscaling: pseudo-functions
$Q_g$	capacity for gas treatment and injection, m³/d (standard)Reservoir simulation in practice: case studies
$q_i$	withdrawal by a well from cell $i$ at reservoir conditions (negative for injection), m³/dPressure equation for single-phase flow withdrawal by a well from the cell at reservoir conditions, m³/dFinite differences: from equations to computation withdrawal by a well at reservoir conditions (injection with a minus sign), m³/dSolving systems of linear equations in simulators
$Q_i$	total flow entering the cell from the neighbours and wells, at reservoir conditions, m³/dIMPES and fully implicit schemes
$q_{i-1/2},\ q_{i+1/2}$	volumetric flow through the left and right faces, positive in the direction of the $x$ axis, m³/sPressure equation for single-phase flow
$q_{ij}$	volumetric rate from cell $j$ into cell $i$ at reservoir conditions, m³/sWhat a reservoir simulation model is and why it is needed rate through the throat between pores $i$ and $j$ , m³/sPore-scale flow modelling
$q_l,\ q$	rate from layer $l$ and the given rate of the well at standard conditions, m³/dWell model in a grid block
$q_{mf}$	volumetric rate from the matrix into the fractures within a single cell, at reservoir conditions, m³/sTypes of reservoir simulation model
$q_o$	oil rate of the field, m³/dReservoir simulation in practice: case studies
$q_{\text{pot}}$	the potential of the wells — the oil rate they would deliver without the limit on gas, m³/dReservoir simulation in practice: case studies
$q_s$	rate at standard conditions; negative for injection, m³/dWell model in a grid block
$q_{t,i}$	withdrawal of liquid by a well at reservoir conditions (negative for injection), m³/dIMPES and fully implicit schemes
$q_{w,i}$	withdrawal by a well from the cell at reservoir conditions (negative for injection), m³/sPressure equation for single-phase flow withdrawal of water by a well at standard conditions; the share of water in it follows from the mobilities at the start of the step, m³/dIMPES and fully implicit schemes
$Q_{\text{well}}$	the new well's own production, m³Reservoir simulation in practice: case studies
$Q_{\text{with}},\ Q_{\text{without}}$	production of the sector over the same period in the runs with and without the intervention; $Q_{\text{without}}$ is the baseline production, m³Reservoir simulation in practice: case studies
$q_{\alpha,i}$	withdrawal of the phase from the cell by a well at standard conditions (negative for injection), m³/dTwo-phase flow equations withdrawal of the phase by a well at standard conditions, m³/dTwo-phase flow equations
$q_{\alpha,i}^{n+1}$	withdrawal of the phase by a well, calculated from the pressure and the mobilities at the end of the step, m³/dIMPES and fully implicit schemes
$r$	mesh Fourier number: the time step in units of the time that the pressure needs to level out over one cell, —Finite differences: from equations to computation mesh Fourier number $\eta\Delta t/\Delta x^2$ , —Finite differences: from equations to computation distance from the axis of the well, mSpecial topics in upscaling
$R$	producing gas–oil ratio, m³/m³Reservoir simulation in practice: case studies
$\mathbf{R}$	vector of the residuals of all equations, m³/dIMPES and fully implicit schemes
$\mathbf{r}^{(\nu)}$	residual of the approximation $\mathbf{x}^{(\nu)}$ (by how much the equations are still not satisfied), in the units of the right-hand sideSolving systems of linear equations in simulators residual at iteration $\nu$ , in the units of the right-hand sideSolving systems of linear equations in simulators
$R_\infty$	estimate of the result on an infinitely fine grid, %Grids in reservoir simulation models
$r_0$	equivalent radius of the cell: the distance at which the pressure of radial flow equals the cell pressure, mWell model in a grid block equivalent radius of the cell, mWell model in a grid block
$r_0,\ r_w,\ h,\ S$	equivalent radius, well radius, thickness, skin, m, m, m, —Well model in a grid block
$r_1,\ r_2$	distances from the axis to the inner and outer faces of the sample, mPetrophysical input for simulation: relative permeability and capillary pressure
$r_c$	smallest radius of the tubes filled with oil at the capillary pressure $p_c$ : $r_c = 2\sigma\cos\theta/p_c$ , µmPore-scale flow modelling
$r_e$	outer radius of the model, mGrids in reservoir simulation models
$r_f$	radius of the boundary between the rings, mInterblock flow and transmissibility
$r_i$	radius of the $i$ -th boundary between rings; $r_0 = r_w$ , $r_N = r_e$ , mGrids in reservoir simulation models ratio of the saturation differences downstream and upstream, a measure of how smooth the profile is, —Numerical dispersion
$r_i,\ r_{i+1}$	radii of the calculation points of the rings, mInterblock flow and transmissibility
$r_{ij},\ \ell_{ij}$	radius and length of the throat, mPore-scale flow modelling
$R_N$	result of the calculation (for example, oil recovery) on a grid with $N$ cells in the direction being refined; $R_{N/2}$ and $R_{2N}$ are the same on a grid twice as coarse and twice as fine, %Grids in reservoir simulation models
$R_s$	solution gas–oil ratio of the oil (the gas dissolved in a unit volume of oil) at the pressure in the cell, m³/m³Types of reservoir simulation model solution gas–oil ratio: standard cubic metres of gas dissolved in 1 m³ of stock-tank oil, m³/m³Building a simulation model: input data and results
$r_w$	wellbore radius, mGrids in reservoir simulation models
$r_w,\ S$	radius of the well and skin factor, m, —Well model in a grid block
$R_{\alpha,i}$	residual of the balance of phase $\alpha$ in cell $i$ : how much the current approximation breaks the balance, m³/d (standard conditions)IMPES and fully implicit schemes residual of the balance of phase $\alpha$ in cell $i$ after the iteration, m³/d (standard conditions)IMPES and fully implicit schemes
$r,\ R$	radii of the throat and of the pore (for a square throat $r$ is the radius of the inscribed circle), mPore-scale flow modelling
$R,\ R_{\text{exact}}$	result of the calculation (pressure, rate, breakthrough time) and its exact value, in units of the resultFinite differences: from equations to computation
$\hat k_{rf}^{\parallel},\ \hat k_{rf}^{\perp}$	pseudo relative permeability of phase $f$ along the layers and across them, —Two-phase upscaling: pseudo-functions
$\hat k_{rf,x}$	pseudo relative permeability of phase $f$ (oil or water) for flow along $x$ , —Two-phase upscaling: pseudo-functions
$S$	storage capacity of a cell (the same for all cells), m³/MPaSolving systems of linear equations in simulators
$S_i$	storage capacity of the cell: the volume of fluid that it gives up when the pressure falls by 1 MPa, m³/MPaPressure equation for single-phase flow storage capacity of the cell $V_{p,i}c_t$ , m³/MPaSolving systems of linear equations in simulators
$S_i,\ q_i,\ \Delta t,\ p_i^{n}$	as in the equation of a cell, m³/MPa, m³/d, d, MPaPressure equation for single-phase flow
$S_i,\ a_{ij}$	storage capacity of the cell and the coefficients of its links, m³/MPa, m³/(d·MPa)Pressure equation for single-phase flow storage capacity of the cell and the link coefficients with the neighbours, m³/MPa, m³/(d·MPa)Finite differences: from equations to computation
$S_i,\ a_{ij},\ q_i,\ \Delta t$	as in the equation of a cell, m³/MPa, m³/(d·MPa), m³/d, dPressure equation for single-phase flow
$S_{i+1/2}$	saturation at the face between cells $i$ and $i + 1$ , used to find the water fraction in the flow between them (the flow goes from $i$ to $i + 1$ ), —Numerical dispersion
$s_j$	uncertainty of the prior estimate, units of the parameterHistory matching
$S_j$	saturation of phase $j$ — gas, oil or water, fractionTypes of reservoir simulation model
$S_n$	normalised water saturation $(S_w - S_{wi})/(1 - S_{wi} - S_{or})$ , —IMPES and fully implicit schemes normalised water saturation of the sample (Corey model), —Petrophysical input for simulation: relative permeability and capillary pressure
$S_o,\ k_{ro},\ k_{rw}$	oil saturation and the relative permeabilities of oil and water, —Pore-scale flow modelling
$S_o,\ S_w$	saturations, —Two-phase flow equations
$S_o,\ S_w,\ S_g$	oil, water and free-gas saturations (the shares of the pore volume that each fluid fills); they add up to one, fractionTypes of reservoir simulation model
$S_{oi}^{},\ S_{or}^{}$	initial and residual (trapped) oil saturation as fractions of the pore volume not occupied by irreducible water, —Petrophysical input for simulation: relative permeability and capillary pressure
$S_{or,\max}^{*}$	trapped oil at the largest initial oil saturation $S_{oi,\max}^{}$ ; in the oil zone $S_{oi,\max}^{} = 1$ , —Petrophysical input for simulation: relative permeability and capillary pressure
$S_w$	water saturation, the unknown, —Two-phase flow equations
$S_w,\ S_t$	water saturation of the cell and the corresponding saturation of the table, —Petrophysical input for simulation: relative permeability and capillary pressure
$S_w,\ B_w$	water saturation and formation volume factor of water; $B_w$ at the end of the step is taken at the new pressure, —, m³/m³IMPES and fully implicit schemes
$S_{wi},\ S_{or}$	irreducible water saturation and residual oil saturation, —IMPES and fully implicit schemes end-point saturations of the cell, —Petrophysical input for simulation: relative permeability and capillary pressure
$S_{wi},\ S_{or},\ k_{rw}',\ k_{ro}'$	end points of the sample itself, —Petrophysical input for simulation: relative permeability and capillary pressure
$S_{wi,t},\ S_{or,t}$	end-point saturations of the table of the rock type, —Petrophysical input for simulation: relative permeability and capillary pressure
$S_{wirr}$	irreducible water saturation of the sample or of the cell, —Petrophysical input for simulation: relative permeability and capillary pressure
$S,\ p$	saturations and pressure of the cell from which the mobility is calculated, —, MPaInterblock flow and transmissibility
$S = V_p c_t$	storage capacity of the cell, $V_p = A\,\Delta x\,\phi$ , m³/MPaFinite differences: from equations to computation
$t$	time since the start of withdrawal, dWhat a reservoir simulation model is and why it is needed
$\tilde q_t$	withdrawal of liquid at reservoir conditions per unit reservoir volume, $B_o\tilde q_o + B_w\tilde q_w$ , 1/sTwo-phase flow equations
$T$	absolute temperature: $T\,[\text{K}] = t\,[{}^{\circ}\text{C}] + 273.15$ , KTypes of reservoir simulation model
$T^{*}_{I,II}$	transmissibility of the link between the coarse cells $I$ and $II$ , mD·mPermeability upscaling for single-phase flow
$T_b,\ p_b$	transmissibility from the cell centre to the face of the block, and the pressure set on that face, mD·m, MPaPermeability upscaling for single-phase flow
$t_c$	time for capillary forces to even out the saturation over the thickness $H$ , about $\phi\,\Delta S\,\mu_oH^2/(k_vk_{ro}'\Delta p_c)$ , sTwo-phase upscaling: pseudo-functions
$t_i,\ t_j$	transmissibilities of the half-cells, mD·mInterblock flow and transmissibility
$T_{ij}$	transmissibility of the connection between the cells (how easily fluid passes between them; this is its geometric part, without viscosity), m³What a reservoir simulation model is and why it is needed transmissibility of the connection: the geometry of the cells and their absolute permeability, without the fluid properties, mD·mInterblock flow and transmissibility transmissibility of the connection: the flow between the centres equals $0.0853\,T_{ij}(p_i - p_j)/\mu$ , mD·mInterblock flow and transmissibility transmissibility of the connection (geometric part): $A/\big(\Delta x_i/(2k_i) + \Delta x_j/(2k_j)\big)$ , mD·mPressure equation for single-phase flow transmissibility of the connection (the geometric part, including the absolute permeability), as in the single-phase lesson, mD·mTwo-phase flow equations transmissibility of the connection (the geometric part with the absolute permeability), mD·mIMPES and fully implicit schemes
$T_{lm}$	transmissibility of the link between neighbouring cells: two half-cells in series, mD·mPermeability upscaling for single-phase flow
$t_x$	time for the flow to cross the cell length $L$ , about $\phi\,\Delta S\,L/u$ , sTwo-phase upscaling: pseudo-functions
$T_z,\ T_x$	transmissibilities of the connection with the neighbour above (below) and with the neighbour at the side, mD·mInterblock flow and transmissibility
$\Delta S_{\text{target}},\ \Delta p_{\text{target}}$	target changes per step, the accuracy settings, —, MPaIMPES and fully implicit schemes
$u$	Darcy velocity (the rate per unit area) at the distance $r$ from the axis of the well, m/dSpecial topics in upscaling
$\mathbf{u}_t$	total Darcy velocity of oil and water, m/sTwo-phase flow equations
$\mathbf{u}_\alpha$	Darcy velocity vector of the phase at reservoir conditions, m/dTwo-phase flow equations
$u_{\alpha,i\pm1/2}$	Darcy velocity of the phase through the face (its flow rate per unit area), positive in the direction of the $x$ axis, m/dTwo-phase flow equations
$v$	velocity of the front — the Darcy velocity divided by the porosity, m/sGrids in reservoir simulation models
$V$	volume of the block, m³Permeability upscaling for single-phase flow
$V_b$	bulk volume of the cell (of the rock), m³Types of reservoir simulation model
$v_i$	share of the block volume occupied by the $i$ -th fine cell ( $\sum v_i = 1$ ), —Permeability upscaling for single-phase flow
$V_i$	volume of the cell (rock together with pores), $A\,\Delta x_i$ , m³Pressure equation for single-phase flow rock volume of the $i$ -th fine cell, m³Permeability upscaling for single-phase flow
$V_i\phi_i$	pore volume of the fine cell, m³Two-phase upscaling: pseudo-functions
$V_i,\ A$	bulk volume of the cell (including the rock) and area of its face, m³, m²Two-phase flow equations
$V_i,\ \phi_i,\ S_{w,i}$	volume, porosity and water saturation of the $i$ -th fine cell, m³, fraction, fractionTwo-phase upscaling: pseudo-functions
$V_p$	pore volume of the “tank”, m³What a reservoir simulation model is and why it is needed pore volume of the cell: its volume multiplied by the porosity, m³Types of reservoir simulation model pore volume of the cell, m³Types of reservoir simulation model pore volume of the cell at the reference pressure, m³Building a simulation model: input data and results
$V_{p,i}$	pore volume of the cell, m³Pressure equation for single-phase flow pore volume of the cell; $V_{p,i}c_{t,i}^{n}$ is its storage capacity at the saturations at the start of the step, as in the single-phase lesson, m³IMPES and fully implicit schemes
$V_{p,i},\ T_{ij},\ \Delta t$	pore volume of the cell, transmissibility of the connection, step, m³, mD·m, dIMPES and fully implicit schemes
$V_p(p_{\text{ref}})$	pore volume calculated from the porosity of the cell, m³Building a simulation model: input data and results
$W$	number of multiply–add operations, —Solving systems of linear equations in simulators
$W_i$	withdrawal by a well from cell $i$ (negative for injection), the sameWhat a reservoir simulation model is and why it is needed
$W_i,\ N_{pD}$	water injected and oil produced, in pore volumes of the sample ( $W_i$ is the same as $W_{iD}$ of the lesson Buckley–Leverett theory and the Welge method), —Petrophysical input for simulation: relative permeability and capillary pressure
$W_{iD}$	injected volume as a fraction of the pore volume of the model, —Grids in reservoir simulation models
$W_{iD,1}$	the volume injected, in pore volumes of the cross-section, when the tracer arrives through the fastest layer, —Special topics in upscaling
$w_k$	weight of the measurement; 1 by default, changed to balance kinds of data with different numbers of points, —History matching
$\tilde q_w^{\text{res}}$	withdrawal of water per unit reservoir volume at reservoir conditions, 1/sTwo-phase flow equations
$\bar S_w,\ S_{w2}$	average water saturation of the sample ( $S_{wi} + N_{pD}$ ) and the saturation at the outlet face ( $S_{we}$ of the Buckley–Leverett lesson), —Petrophysical input for simulation: relative permeability and capillary pressure
$\mathrm{WI}$	well index, the geometric part without the viscosity, mD·mWell model in a grid block
$\mathrm{WI}^{*}$	well index of the coarse cell, mD·mPermeability upscaling for single-phase flow
$\mathrm{WI}_l$	well index in layer $l$ , mD·mWell model in a grid block
$\mathbf{x}$	vector of the unknowns at the end of the step $(p_{o,1},\ S_{w,1},\ \ldots,\ p_{o,N},\ S_{w,N})$ , MPa, —IMPES and fully implicit schemes
$x_f,\ y_f$	coordinates of the centre of this face, mPermeability upscaling for single-phase flow
$x_i^{(\nu)}$	approximation of the unknown $i$ at iteration $\nu$ , in the units of the unknownsSolving systems of linear equations in simulators
$z$	height above the base of the reservoir, mTwo-phase upscaling: pseudo-functions
$z_{\text{FWL}}$	depth of the free water level — the contact depth in the input data, with zero capillary pressure at it, mBuilding a simulation model: input data and results
$\alpha$	ratio of the radii of neighbouring boundaries, —Grids in reservoir simulation models phase: $o$ is oil, $w$ is water, —Two-phase flow equations
$\tilde q_\alpha$	withdrawal of the phase per unit reservoir volume at standard conditions; non-zero only in cells with wells, 1/dTwo-phase flow equations
$\alpha_\nu,\ \beta_\nu$	the length of the step along the direction and the coefficient of the new direction, —Solving systems of linear equations in simulators
$\gamma_\alpha,\ D$	specific weight of the phase $\rho_\alpha g$ (in MPa/m it equals $0.00981\,\rho_\alpha$ with $\rho_\alpha$ in g/cm³) and the depth of the cell centre, measured downwards, MPa/m, mInterblock flow and transmissibility
$\Delta$	size of the cell in the same direction, mSpecial topics in upscaling
$\delta_i,\ \delta_j$	changes of the pressure of the cell and its neighbours over the step, the unknowns, MPaSolving systems of linear equations in simulators
$\delta\mathbf{x}$	correction of the unknowns at the iteration, MPa, —IMPES and fully implicit schemes
$\Delta\rho$	density difference of the phases, kg/m³Petrophysical input for simulation: relative permeability and capillary pressure
$\Delta\rho\,g$	difference of the hydrostatic gradients of water and oil, Pa/mTwo-phase upscaling: pseudo-functions
$\Delta\bar\Phi_f$	difference of the phase potential (the phase pressure corrected for depth) between the centres of blocks I and II, MPaTwo-phase upscaling: pseudo-functions
$\varepsilon$	how many times the error must be reduced (the equation uses the reciprocal of that number), —Solving systems of linear equations in simulators tolerance: the relative residual at which the iterations are stopped, —Solving systems of linear equations in simulators
$\varepsilon_{MB}$	tolerance on the material balance error of the phase over the whole model, —IMPES and fully implicit schemes
$\varepsilon_R$	tolerance on the largest normalised residual, a fraction of the pore volume of the cell, —IMPES and fully implicit schemes
$\eta$	hydraulic diffusivity, $0.0853\,k/(\phi\mu c_t)$ , m²/dFinite differences: from equations to computation
$\eta = 0.0853\,k/(\phi\mu c_t)$	hydraulic diffusivity (how fast pressure changes spread through the formation); $\eta/\Delta x^2 = a/S$ , m²/dFinite differences: from equations to computation
$\eta_x,\ \eta_y,\ \eta_z$	hydraulic diffusivity along the axes (with the permeability $k_x$ , $k_y$ , $k_z$ ), m²/dPressure equation for single-phase flow
$\theta$	angle of the sector; a full circle is $2\pi$ , radInterblock flow and transmissibility angle between the first principal axis and the $x$ axis of the grid, °Interblock flow and transmissibility
$\theta_a$	advancing contact angle, measured through the water (contact angle), °Pore-scale flow modelling
$\kappa$	condition number $\lambda_{\max}(\mathbf{A})/\lambda_{\min}(\mathbf{A})$ , the ratio of the largest to the smallest eigenvalue; it shows how sensitive the solution is to errors, —Solving systems of linear equations in simulators condition number of the matrix, —Solving systems of linear equations in simulators condition number, —Solving systems of linear equations in simulators
$\lambda$	size of the heterogeneity in the given direction: the correlation length of permeability, mSpecial topics in upscaling
$\lambda_{\min},\ \lambda_{\max}$	smallest and largest eigenvalues, m³/(d·MPa)Solving systems of linear equations in simulators
$\lambda_o f_w\,\|\dd p_c/\dd S_w\|$	coefficient of the capillary “diffusion” of saturation, m²/sTwo-phase flow equations
$\lambda_o f_w = \lambda_o\lambda_w/\lambda_t$	common factor of the capillary and gravity flows; it is small if only one phase can move, m²/(Pa·s)Two-phase flow equations
$\lambda_o,\ \lambda_w$	mobilities of oil and water, functions of $S_w$ , m²/(Pa·s)Two-phase flow equations
$\lambda_t$	total mobility, a function of water saturation, m²/(Pa·s)Two-phase flow equations
$\lambda_{t,ij}^{n}$	total mobility at the face per unit permeability, $(k_{ro}/\mu_o)_\uparrow + (k_{rw}/\mu_w)_\uparrow$ : each phase is taken from its own upstream cell, 1/(mPa·s)IMPES and fully implicit schemes
$\lambda_\alpha$	mobility of the phase; where permeability depends on direction (an anisotropic reservoir), the permeability tensor (a set of permeabilities for the different directions) replaces $k$ , m²/(Pa·s)Two-phase flow equations
$\lambda_{\alpha\uparrow}$	mobility of the phase at the face, taken from the cell upstream of this phase (see the section below), 1/(mPa·s)Interblock flow and transmissibility mobility of phase $\alpha$ at the face between cells $i$ and $j$ , 1/(mPa·s)Interblock flow and transmissibility
$\mu$	fluid viscosity (the resistance of the fluid to flow), Pa·sWhat a reservoir simulation model is and why it is needed dynamic viscosity of the oil, mPa·sTypes of reservoir simulation model fluid viscosity, Pa·sTypes of reservoir simulation model viscosity of the fluid, Pa·sPressure equation for single-phase flow viscosity, mPa·sPermeability upscaling for single-phase flow
$\mu,\ B$	viscosity and formation volume factor of the fluid, mPa·s, m³/m³Well model in a grid block
$\mu_f$	viscosity of the phase, mPa·sTwo-phase upscaling: pseudo-functions
$\mu_o,\ u,\ k_{ro}'$	oil viscosity, Darcy velocity, end-point relative permeability of oil, Pa·s, m/s, —Two-phase upscaling: pseudo-functions
$\mu_w,\ \mu_o$	viscosities of water and oil at the conditions of the experiment, mPa·sPetrophysical input for simulation: relative permeability and capillary pressure
$\nu$	iteration number; the initial approximation $\mathbf{x}^{(0)}$ is the values at the start of the step or a forecast from the previous steps, —IMPES and fully implicit schemes number of iterations needed to reduce the error by a factor of $1/\varepsilon$ , —Solving systems of linear equations in simulators
$\xi$	phase shift of the harmonic per cell, $0 < \xi \le \pi$ ; the wavelength is $2\pi\Delta x/\xi$ , radFinite differences: from equations to computation
$\rho,\ \phi$	density of the fluid at reservoir conditions and porosity, kg/m³, —Pressure equation for single-phase flow
$\rho(\mathbf{G})$	spectral radius of the iteration matrix of the method, —Solving systems of linear equations in simulators
$\rho_fg$	hydrostatic gradient of the phase, $0.00981\rho_f$ with the density in g/cm³, MPa/mTwo-phase upscaling: pseudo-functions
$\rho_J,\ \rho_{GS},\ \rho_{SOR}$	spectral radii of the Jacobi, Gauss–Seidel and over-relaxation methods, —Solving systems of linear equations in simulators
$\rho_o$	density of the oil at reservoir conditions, kg/m³Building a simulation model: input data and results
$\rho_o,\ \rho_w$	densities of oil and water in the reservoir, g/cm³Building a simulation model: input data and results
$\rho_o,\ \rho_w,\ g,\ D$	phase densities, acceleration due to gravity, depth, kg/m³, m/s², mTwo-phase flow equations
$\rho_{o,\text{sc}},\ \rho_{g,\text{sc}}$	densities of the oil and the gas at standard conditions — the density records of the input deck, kg/m³Building a simulation model: input data and results
$\sigma$	shape factor of the matrix blocks, m⁻²Types of reservoir simulation model weight of the end of the step: 0 is the explicit scheme, 1 is the implicit scheme, ½ is the Crank–Nicolson scheme, —Finite differences: from equations to computation weight of the scheme, —Finite differences: from equations to computation oil–water interfacial tension, N/mPore-scale flow modelling
$\sigma,\ \eta,\ \Delta t,\ \Delta x$	weight, diffusivity, steps, —, m²/d, d, mFinite differences: from equations to computation
$\sigma,\ \theta$	interfacial tension and the receding contact angle (the contact angle while oil pushes water out, measured through the water), N/m, °Pore-scale flow modelling
$\sigma\cos\theta$	adhesion tension of the pair of fluids, mN/mPetrophysical input for simulation: relative permeability and capillary pressure
$\sigma_k$	measurement error (standard deviation, the typical size of the random error), including the inaccuracy of tying the measurement to the model, units of the measurementHistory matching error of the $k$ -th measurement, units of the measurementHistory matching
$\tau$	characteristic depletion time (the time scale of the pressure decline), dWhat a reservoir simulation model is and why it is needed
$\tau,\ p_{wf}$	as in the exact solution, d, MPaWhat a reservoir simulation model is and why it is needed
$\tau_i$	characteristic time of the cell: the time in which the cell would equalise its pressure with its neighbours if the flows did not change, dPressure equation for single-phase flow
$\phi$	porosity of the reservoir rock, fractionBuilding a simulation model: input data and results
$\varphi$	direction: the angle from the $x$ axis to the $y$ axis, °Special topics in upscaling
$\bar\phi,\ \bar S_w$	porosity and water saturation of the coarse cell, fractionPermeability upscaling for single-phase flow
$\phi,\ \mu,\ c_t$	porosity, viscosity, total compressibility (constants), —, Pa·s, 1/PaPressure equation for single-phase flow
$\phi,\ S_\alpha,\ B_\alpha$	porosity, saturation of the phase, formation volume factor of the phase, —, —, m³/m³Two-phase flow equations porosity, saturation and formation volume factor, —, —, m³/m³Two-phase flow equations
$\phi_i,\ S_{w,i}$	porosity and water saturation of the fine cell, fractionPermeability upscaling for single-phase flow
$\phi_j,\ k_j,\ h_j$	porosity, permeability and thickness of layer $j$ ; layer 1 has the largest $k/\phi$ , fraction, mD, mSpecial topics in upscaling
$\Phi_\alpha$	phase potential at the cell centre: the phase pressure converted to zero depth, which accounts for the weight of the fluid, MPaInterblock flow and transmissibility phase potential $p_\alpha - \gamma_\alpha D$ ; the water pressure is $p_w = p_o - p_c(S_w)$ , MPaIMPES and fully implicit schemes
$\Phi_\alpha,\ \gamma_\alpha$	potential of the phase and its specific weight $\rho_\alpha g$ (weight per unit volume), MPa, MPa/mTwo-phase flow equations
$\Phi_{\alpha,i},\ \Phi_{\alpha,j}$	potentials of the phase at the cell centres, MPaInterblock flow and transmissibility
$\Psi(r)$	slope limiter; $\Psi_{\text{vL}}$ is the van Leer limiter, —Numerical dispersion
$\psi_i^{n}$	truncation error: the left side of the scheme minus the right side when the exact solution is substituted, MPa/dFinite differences: from equations to computation
$\omega$	over-relaxation parameter, $1 < \omega < 2$ , —Solving systems of linear equations in simulators angular velocity, $2\pi n/60$ at $n$ revolutions per minute, rad/sPetrophysical input for simulation: relative permeability and capillary pressure
$\omega_{opt}$	optimal over-relaxation parameter, —Solving systems of linear equations in simulators