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.

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

Args:

radius (ArrayLike): The radius of the circle. sidelength (ArrayLike): The sidelength of the square.

Returns:

ArrayLike: The calculated area.

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

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

Args:: delta_x (ArrayLike): _description_
Returns:: (ArrayLike): _description_

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.

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

Compute the well productivity index from given data.

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

Args:

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 or producer. Defaults to "injector".

Returns:

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

Raises:

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

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

Args:: delta_x (ArrayLike): _description_
Returns:: (ArrayLike): _description_

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.

Args:

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:

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

Examples:

>>> hydrostatic_fluid(1000, 10)
98067.0

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

Args:

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]. Defaults to units.GRAVITATIONAL_ACCELERATION, i.e., \(\approx 9.8\).

Returns:

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

Examples:

>>> hydrostatic_gas(101325, 5000, 300, 0.029, 9.81)
75819.90376642203

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

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

Args:

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

Compute the well productivity index (without taking into account density and: viscosity) from the Peaceman well model.

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

Args:

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

Raises:

ValueError: If r_w is zero for any point. ValueError: If either r_e or r_w contains a negative value

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.

Args:

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

Returns:

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

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.

Args:

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