pyopmnearwell.utils.formulas module

Provide useful mathematical formulas for reservoir modelling.

TODO: The typing is off in this module. Instead of returning an ArrayLike, most functions should return an np.ndarray, however that does not work when a float is passed as then an np.float64 or similar is returned. Not sure how to fix this.

pyopmnearwell.utils.formulas.area_squaredcircle(radius: ArrayLike, sidelength: ArrayLike) → ArrayLike

Calculate the area that lies both inside a circle and inside a square centered at the origin.

Parameters:

  • radius (ArrayLike) – The radius of the circle.

  • sidelength (ArrayLike) – The sidelength of the square.

Returns:

The calculated area.

Return type:

ArrayLike

Raises:

ValueError – If radius or sidelength contain a negative value.

Examples

>>> area_squaredcircle(3.0, 3.0)
9.0  # Area of a sqare with sidelength 3.0

>>> area_squaredcircle(2.0, 4.0)
12.566370614359172  # Area of a circle with radius 2.0

>>> area_squaredcircle(2.0, 3.0)
6.283185307179586  # Area inside a  2.0
pyopmnearwell.utils.formulas.cell_size(radii: ArrayLike) → ArrayLike

Calculate the size of a quadratic cell for a given equivalent well block radius.

Parameters:

delta_x (ArrayLike) – _description_

Returns:

_description_

Return type:

(ArrayLike)

pyopmnearwell.utils.formulas.co2brinepvt(pressure: float, temperature: float, phase_property: Literal['density', 'viscosity'], phase: Literal['CO2', 'water'], OPM: str | Path) → float

Call OPM’s co2brinepvt to calculate density/viscosity.

Parameters:

  • pressure (float) – Unit: [Pa].

  • temperature (float) – Unit: [K].

  • property (Literal["density", "viscosity"]) – Phase property to return.

  • phase (Literal["CO2", "water"]) – Phase of interest.

  • OPM – (str | pathlib.Path): Path to OPM installation.

Returns:

Density (unit: [kg/m^3]) or viscosity (unit: [Pa*s])

Return type:

quantity (float)

pyopmnearwell.utils.formulas.data_WI(q: ArrayLike, p_w: ArrayLike, p_gb: ArrayLike, well_type: Literal['injector', 'producer'] = 'injector') → ArrayLike

Compute the well productivity index from given data.

\[WI = \frac{q}{p_w - p_{gb}\]
Parameters:

  • q (ArrayLike) – Injection rate. Unit: [m^3/s].

  • p_w (ArrayLike) – Bottom hole pressure. Unit: [Pa].

  • p_gb (ArrayLike) – Grid block pressure. Unit: [Pa].

  • well_type (Literal["injector", "producer], optional) – Calculate WI for injector

  • "injector". (or producer. Defaults to)

Returns:

\(WI\). Unit: [m^4*s/kg].

Return type:

WI (ArrayLike)

Raises:

ValueError – If bottom hole pressure and grid block pressure are equal for any point.

pyopmnearwell.utils.formulas.equivalent_well_radius(delta_x: ArrayLike) → ndarray

Calculate the equivalent well block radius for a given quadratic cell size.

Parameters:

delta_x (ArrayLike) – _description_

Returns:

_description_

Return type:

(ArrayLike)

pyopmnearwell.utils.formulas.hydrostatic_fluid(rho: ArrayLike, height: ArrayLike, gravity: ArrayLike = 9.8067) → ArrayLike

Calculate the pressure due to hydrostatic fluid.

This function calculates the pressure due to hydrostatic fluid using the formula:

..math:

P = rho * g * h,

where \(P\) is the pressure, \(rho\) is the fluid density, \(g\) is the acceleration due to gravity, and \(h\) is the height.

Parameters:

  • rho (ArrayLike) – Fluid density. Unit [kg/m^3].

  • height (ArrayLike) – Height of the fluid column. Unit [m].

  • gravity (ArrayLike, optional) – Acceleration due to gravity. Unit [m/s^2]. Defaults to units.GRAVITATIONAL_ACCELERATION, i.e., \(\approx 9.8\).

Returns:

Hydrostatic pressure at the given height. Unit [Pa].

Return type:

ArrayLike

Examples

>>> hydrostatic_fluid(1000, 10)
98067.0
pyopmnearwell.utils.formulas.hydrostatic_gas(reference_pressure: ArrayLike, height: ArrayLike, temperature: ArrayLike, molecule_mass: ArrayLike, gravity: ArrayLike = 9.8067) → ArrayLike

Calculate the pressure due to a column of gas in hydrostatic equilibrium.

This function calculates the pressure due to a column of gas in hydrostatic equilibrium using the ideal gas law and the hydrostatic equilibrium formula. The formula is:

..math:

P = P_0 * e^((-m * g * h) / (R^* * T)),

where \(P is the pressure,\) \(P_0 is the reference pressure,\) \(m is the molecular mass of the gas,\) \(g is the acceleration due to gravity,\) \(h is the height,\) \(R^* is the universal gas constant, and\) \(T is the temperature.\)

Parameters:

  • reference_pressure (ArrayLike) – Reference pressure at height 0. Unit [Pa].

  • height (ArrayLike) – Height of the gas column. Unit [m].

  • temperature (ArrayLike) – Temperature. Unit [K].

  • molecule_mass (ArrayLike) – Molecular mass of the gas. Unit [kg/mol].

  • gravity (ArrayLike, optional) – Acceleration due to gravity. Unit [m/s^2].

  • units.GRAVITATIONAL_ACCELERATION (Defaults to)

  • i.e.

  • 9.8`. (ca.)

Returns:

Hydrostatic pressure at the given height. Unit [Pa].

Return type:

ArrayLike

Examples

>>> hydrostatic_gas(101325, 5000, 300, 0.029, 9.81)
75819.90376642203
pyopmnearwell.utils.formulas.peaceman_WI(k_h: ArrayLike, r_e: ArrayLike, r_w: ArrayLike, rho: ArrayLike, mu: ArrayLike) → ArrayLike

Compute the well productivity index (adjusted for density and viscosity) from the Peaceman well model.

\[WI = \frac{2 \pi h \mathbf{k} \frac{\mu}{\rho}}{\ln(r_e/r_w)}\]
Parameters:

  • k_h (ArrayLike) – Permeability times the cell thickness. Unit: [m^3].

  • r_e (ArrayLike) – Equivalent well-block radius. Needs to have the same unit as r_w.

  • r_w (ArrayLike) – Wellbore radius. Needs to have the same unit as r_e.

  • rho (ArrayLike) – Density. Unit: [kg/m^3].

  • mu (ArrayLike) – Viscosity. Unit: [Pa*s].

Returns:

\(WI\). Unit: [m*s].

Return type:

WI (ArrayLike)

pyopmnearwell.utils.formulas.peaceman_matrix_WI(k_h: ArrayLike, r_e: ArrayLike, r_w: ArrayLike) → ArrayLike
Compute the well productivity index (without taking into account density and

viscosity) from the Peaceman well model.

\[WI = \frac{2 \pi h \mathbf{k}}{\ln(r_e/r_w)}\]
Parameters:

  • k_h (ArrayLike) – Permeability times the cell thickness. Unit: [m^3].

  • r_e (ArrayLike) – Equivalent well-block radius. Needs to have the same unit as r_w.

  • r_w (ArrayLike) – Wellbore radius. Needs to have the same unit as r_e.

Returns:

\(WI\). Unit: [m^3].

Return type:

WI (ArrayLike)

Raises:

  • ValueError – If r_w is zero for any point.

  • ValueError – If either r_e or r_w contains a negative value

pyopmnearwell.utils.formulas.pyopmnearwell_correction(angle: ArrayLike = 1.0471975511965976) → ArrayLike

Calculate a correction factor that scales cell volume, equivalent radius etc. from a 2D triangle grid to a radial grid.

In pyopmnearwell, when using the cake grid, a 2D triangle grid is used. As the scaling for cell volumes between a radial and a triangular grid is linear, the radial solution can be recovered. This function calculates the ratio

\[r_{radial_grid} / x_{triangle_grid},\]

where \(x_{triangle_grid}\) is the altitude of the triangle, s.t. the solution on the triangle grid is equal to the solution on a radial grid with the given cell radii.

Parameters:

angle (ArrayLike) – Angle between both sides of the triangle grid. Default is \(\pi/3\).

Returns:

\(r_{radial_grid} / x_{triangle_grid}\)

Return type:

ArrayLike

pyopmnearwell.utils.formulas.two_phase_peaceman_WI(k_h: ArrayLike, r_e: ArrayLike, r_w: ArrayLike, rho_1: ArrayLike, mu_1: ArrayLike, k_r1: ArrayLike, rho_2: ArrayLike, mu_2: ArrayLike, k_r2: ArrayLike) → ArrayLike

Compute the well productivity index (adjusted for density, viscosity and relative permeabilties) from the Ptwo-phase eaceman well model.

Cf. [A. Yapparova, B. Lamy-Chappuis, S. W. Scott, and T. Driesner, “A Peaceman-type well model for the 3D Control Volume Finite Element Method and numerical simulations of supercritical geothermal resource utilization” 2022, doi: 10.1016/j.geothermics.2022.102516.]

\[WI = \frac{2 \pi h \mathbf{k}}{\ln(r_e/r_w)} \left(\frac{k_{r,1}}{\mu_1} + \frac{k_{r,2}}{\mu_2})\right)\]

Note

The result is only accurate for fine grid sizes.

Parameters:

  • k_h (ArrayLike) – Permeability times the cell thickness. Unit: [m^3].

  • r_e (ArrayLike) – Equivalent well-block radius. Needs to have the same unit as r_w.

  • r_w (ArrayLike) – Wellbore radius. Needs to have the same unit as r_e.

  • rho_1 (ArrayLike) – Density of phase 1. Unit: [kg/m^3].

  • mu_1 (ArrayLike) – Viscosity of phase 1. Unit: [Pa*s].

  • k_r1 (ArrayLike) – Relative permeability of phase 1.

  • rho_2 (ArrayLike) – Density of phase 2. Unit: [kg/m^3].

  • mu_2 (ArrayLike) – Viscosity of phase 2. Unit: [Pa*s].

  • k_r2 (ArrayLike) – Relative permeability of phase 2.

Returns:

\(WI\). Unit: [m*s].

Return type:

WI (ArrayLike)