ressim: reservoir simulation in python

Reservoir simulation in Python. It currently supports 2D rectangular grids and isotropic permeability.

Modules

• ressim.py: main module containing classes to define the grid, and to model and solve the pressure and saturation equations.

• utils.py: module containing useful function definitions, e.g. linear and quadratic mobility functions, fractional flow function, etc.

Usage

See main_*.py for examples.

ressim.py

A module for reservoir simulation in Python.

Grid

Grid(self, nx, ny, lx=1.0, ly=1.0)

A class to define a rectangular grid. Given nx, ny, lx, ly, it maintains an update of attributes vol, dx, dy, ncell, shape.

Attributes

• nx, ny: int Grid resolution.

• lx, ly: float Grid physical dimensions. Default is lx=1.0, ly=1.0, i.e. unit square.

• vol: float Cell volume.

• dx, dy: float Cell dimensions.

• ncell: int Number of cells.

• shape: int Grid shape, i.e. (ny, nx).

PressureEquation

PressureEquation(self, grid=None, q=None, k=None, diri=None, lamb_fn=None, s=None)

A class to model and solve the pressure equation,

with no-flow boundary conditions (closed reservoir).

Attributes

• grid: Grid Simulation grid.

• q: ndarray, shape (ny, nx) | (ny*nx,) Integrated source term.

• k: ndarray, shape (ny, nx) Permeability

• diri: list of (int, float) tuples Dirichlet boundary conditions, e.g. [(i1, val1), (i2, val2), ...] means pressure values val1 at cell i1, val2 at cell i2, etc. Defaults to [(ny*nx/2, 0.0)], i.e. zero pressure at center of the grid.

• lamb_fn: callable Total mobility function lamb_fn(s)

• s: ndarray, (ny, nx) | (ny*nx,) Water saturation

• p: ndarray, (ny, nx) Pressure

• v: dict of ndarray

  • 'x' : ndarray, shape (ny, nx+1) Flux in x-direction
  • 'y' : ndarray, shape (ny+1, nx) Flux in y-direction

Methods

• step(): Solve the pressure equation to obtain pressure and flux. Update self.p and self.v.

• solve(mat, q): Method to solve the system of linear equations. Default is scipy.sparse.linalg.spsolve(mat, q). You can override this method to use a different solver.

SaturationEquation

SaturationEquation(self, grid=None, q=None, phi=None, s=None, f_fn=None, v=None, df_fn=None)

A class to model and solve the (water) saturation equation under water injection,

Attributes

• grid: Grid Simulation grid

• q: ndarray, (ny, nx) | (ny*nx,) Integrated source term.

• phi: ndarray, (ny, nx) | (ny*nx,) Porosity

• f_fn: callable Water fractional flow function f_fn(s)

• v: dict of ndarray

  • 'x' : ndarray, (ny, nx+1) Flux in x-direction
  • 'y' : ndarray, (ny+1, nx) Flux in y-direction

• df_fn: callable (optional) Derivative (element-wise) of water fractional flow function df_fn(s). It is used to compute the jacobian of the residual function. If None, the jacobian is approximated by the solver (which can be slow).

• s : ndarray, (ny, nx) | (ny*nx,) Water saturation

Methods

• step(dt): Solve saturation forward in time by dt. Update self.s.

• solve(residual, s0, residual_jac=None): Method to perform the minimization of the residual. Default is scipy.optimize.nonlin.nonlin_solve(residual, s0, jacobian=residual_jac). If residual_jac is None, defaults to 'krylov'. You can override this method to use a different solver.

utils.py

Useful functions for reservoir simulation tasks.

linear_mobility

linear_mobility(s, mu_w, mu_o, s_wir, s_oir, deriv=False)

Function to compute water and oil mobility with a linear model.

Parameters

• s: ndarray, (ny, nx) | (ny*nx,) Water saturation

• mu_w: float Viscosity of water

• mu_o: float Viscosity of oil

• s_wir: float Irreducible water saturation

• s_oir: float Irreducible oil saturation

• deriv: bool If True, also return derivatives

Returns

if deriv=False:

• lamb_w, lamb_o: ndarray, (ny, nx) | (ny*nx,)
  • lamb_w: water mobility
  • lamb_o: oil mobility

if deriv=True:

• lamb_w, lamb_o, dlamb_w, dlamb_o: ndarray, (ny, nx) | (ny*nx,)
  • lamb_w: water mobility
  • lamb_o: oil mobility
  • dlamb_w: derivative of water mobility
  • dlamb_o: derivative of oil mobility

quadratic_mobility

quadratic_mobility(s, mu_w, mu_o, s_wir, s_oir, deriv=False)

Function to compute water and oil mobility with a quadratic model.

Parameters

• s: ndarray, (ny, nx) | (ny*nx,) Water saturation

• mu_w: float Viscosity of water

• mu_o: float Viscosity of oil

• s_wir: float Irreducible water saturation

• s_oir: float Irreducible oil saturation

• deriv: bool If True, also return derivatives

Returns

if deriv=False:

• lamb_w, lamb_o: ndarray, (ny, nx) | (ny*nx,)
  • lamb_w: water mobility
  • lamb_o: oil mobility

if deriv=True:

• lamb_w, lamb_o, dlamb_w, dlamb_o: ndarray, (ny, nx) | (ny*nx,)
  • lamb_w: water mobility
  • lamb_o: oil mobility
  • dlamb_w: derivative of water mobility
  • dlamb_o: derivative of oil mobility

f_fn

f_fn(s, mobi_fn)

Water fractional flow function.

Parameters

• s: ndarray, (ny, nx) | (ny*nx,) Water saturation

• mobi_fn: callable Mobility function lamb_w, lamb_o = mobi_fn(s) where:

  • lamb_w: water mobility
  • lamb_o: oil mobility

Returns

• ndarray, (ny, nx) | (ny*nx,) Fractional flow

df_fn

df_fn(s, mobi_fn)

Derivative function (element-wise) of water fractional flow.

Parameters

• s: ndarray, (ny, nx) | (ny*nx,) Water saturation

• mobi_fn: callable Mobility function lamb_w, lamb_o, dlamb_w, dlamb_o = mobi_fn(s, deriv=True) where:

  • lamb_w: water mobility
  • lamb_o: oil mobility
  • dlamb_w: derivative of water mobility
  • dlamb_o: derivative of oil mobility

Returns

• ndarray, (ny, nx) | (ny*nx,) Fractional flow derivative

lamb_fn

lamb_fn(s, mobi_fn)

Total mobility function.

Parameters

• s: ndarray, (ny, nx) | (ny*nx,) Water saturation

• mobi_fn: callable Mobility function lamb_w, lamb_o = mobi_fn(s) where:

  • lamb_w: water mobility
  • lamb_o: oil mobility

Returns

• ndarray, (ny, nx) | (ny*nx,) Total mobility

References

Aarnes J.E., Gimse T., Lie KA. (2007) An Introduction to the Numerics of Flow in Porous Media using Matlab. In: Hasle G., Lie KA., Quak E. (eds) Geometric Modelling, Numerical Simulation, and Optimization. Springer, Berlin, Heidelberg