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_correction(angle=1.0471975511965976)[source]#

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

equivalent_well_radius(delta_x)[source]#

Calculate the equivalent Peaceman well-block radius for square cells.

Parameters:

delta_x (ArrayLike) -- Square-cell side length.

Returns:

Result produced by the operation.

Return type:

np.ndarray

cell_size(radii)[source]#

Calculate square-cell side lengths from equivalent well-block radii.

Parameters:

radii (ArrayLike) -- Cell radii.

Returns:

Result produced by the operation.

Return type:

ArrayLike

peaceman_matrix_WI(k_h, r_e, r_w)[source]#
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

peaceman_WI(k_h, r_e, r_w, rho, mu)[source]#

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)

two_phase_peaceman_WI(k_h, r_e, r_w, rho_1, mu_1, k_r1, rho_2, mu_2, k_r2)[source]#

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)

data_WI(q, p_w, p_gb, well_type='injector')[source]#

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.

co2brinepvt(pressure, temperature, phase_property, phase, OPM)[source]#

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 | Path) -- (str | pathlib.Path): Path to OPM installation.

  • phase_property (Literal['density', 'viscosity'])

Returns:

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

Return type:

quantity (float)

hydrostatic_fluid(rho, height, gravity=9.8067)[source]#

Calculate the pressure due to hydrostatic fluid.

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

\[P = rho * g * h,\]

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
hydrostatic_gas(reference_pressure, height, temperature, molecule_mass, gravity=9.8067)[source]#

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:

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

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
area_squaredcircle(radius, sidelength)[source]#

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