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frac_gen.py
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frac_gen.py
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import numpy as np
import scipy.stats as stats
def fit(pts, edges, frac, family, ks_size=100, p_val_min=0.05):
"""
Compute the distribution from a set of fracture families for the length and angle.
Parameters:
pts: list of points
edges: list of edges as point ids
frac: fracture identification number for each edge
family: family identification number for each edge
ks_size: (optional) sample size for the Kolmogorov-Simirnov test, default 100
p_val_min: (optional) minimum p-value to validate the goodness of the fitting
Return:
dist_l: for each family distribution for the fracture length
dist_a: for each family distribution for the fracture angle
Note:
1) so far this implementation does not take care of the family
"""
dist_l = fit_length_distribution(
pts, edges, frac, family, ks_size=ks_size, p_val_min=p_val_min
)
dist_a = fit_angle_distribution(
pts, edges, frac, family, ks_size=ks_size, p_val_min=p_val_min
)
return dist_l, dist_a
def fit_length_distribution(
pts, edges, frac=None, family=None, ks_size=100, p_val_min=0.05
):
"""
Compute the distribution from a set of fracture families for the length and angle.
Parameters:
pts: list of points
edges: list of edges as point ids
frac: fracture identification number for each edge
family: family identification number for each edge
ks_size: (optional) sample size for the Kolmogorov-Simirnov test, default 100
p_val_min: (optional) minimum p-value to validate the goodness of the fitting
Return:
dist_l: for each family distribution for the fracture length
Note:
1) so far this implementation does not take care of the family
"""
if frac is None:
frac = np.arange(edges.shape[1])
if family is None:
family = np.zeros(edges.shape[1])
# fit the lenght distribution
dist = np.array([stats.expon, stats.lognorm])
# fit the possible lenght distributions
l = length(pts, edges, frac)
dist_fit = np.array([d.fit(l, floc=0) for d in dist])
# determine which is the best distribution with a Kolmogorov-Smirnov test
ks = lambda d, p: stats.ks_2samp(l, d.rvs(*p, size=ks_size))[1]
p_val = np.array([ks(d, p) for d, p in zip(dist, dist_fit)])
best_fit = np.argmax(p_val)
if p_val[best_fit] < p_val_min:
raise ValueError("p-value not satisfactory for length fit")
# collect the data
dist_l = {
"dist": dist[best_fit],
"param": dist_fit[best_fit],
"p_val": p_val[best_fit],
}
return dist_l
def fit_angle_distribution(
pts, edges, frac=None, family=None, ks_size=100, p_val_min=0.05
):
"""
Compute the distribution from a set of fracture families for the length and angle.
Parameters:
pts: list of points
edges: list of edges as point ids
frac: fracture identification number for each edge
family: family identification number for each edge
ks_size: (optional) sample size for the Kolmogorov-Simirnov test, default 100
p_val_min: (optional) minimum p-value to validate the goodness of the fitting
Return:
dist_a: for each family distribution for the fracture angle
Note:
1) so far this implementation does not take care of the family
2) the angle should be divided in two categories, since we have conjugate fractures
"""
if frac is None:
frac = np.arange(edges.shape[1])
if family is None:
family = np.zeros(edges.shape[1])
# start the computation for the angles
dist = stats.vonmises
a = angle(pts, edges, frac)
dist_fit = dist.fit(a, fscale=1)
# check the goodness of the fit with Kolmogorov-Smirnov test
p_val = stats.ks_2samp(a, dist.rvs(*dist_fit, size=ks_size))[1]
if p_val < p_val_min:
raise ValueError("p-value not satisfactory for angle fit")
# collect the data
dist_a = {"dist": dist, "param": dist_fit, "p_val": p_val}
return dist_a
def generate(pts, edges, frac, dist_l, dist_a):
num_frac = np.unique(frac).size
# generate lenght and angle
l = generate_from_distribution(num_frac, dist_l)
a = generate_from_distribution(num_frac, dist_a)
# first compute the fracture centres and then generate them
avg = lambda e0, e1: 0.5 * (pts[:, e0] + pts[:, e1])
pts_c = np.array([avg(e[0], e[1]) for e in edges.T]).T
# compute the mean centre based on the fracture id
mean_c = lambda f: np.mean(pts_c[:, np.isin(frac, f)], axis=1)
mean_c = np.array([mean_c(f) for f in np.unique(frac)]).T
dist_c = stats.uniform.rvs
c = dist_c(np.amin(mean_c, axis=1), np.amax(mean_c, axis=1), (num_frac, 2)).T
return fracture_from_center_angle_length(c, l, a)
def fracture_from_center_angle_length(c, l, a):
# generate the new set of pts and edges
num_frac = l.size
pts_n = np.empty((2, l.size * 2))
delta = 0.5 * l * np.array([np.cos(a), np.sin(a)])
for i in np.arange(num_frac):
pts_n[:, 2 * i] = c[:, i] + delta[:, i]
pts_n[:, 2 * i + 1] = c[:, i] - delta[:, i]
edges_n = np.array([2 * np.arange(num_frac), 2 * np.arange(1, num_frac + 1) - 1])
return pts_n, edges_n
def generate_from_distribution(num_fracs, dist_a):
if isinstance(dist_a["param"], dict):
return dist_a["dist"].rvs(**dist_a["param"], size=num_fracs)
else:
return dist_a["dist"].rvs(*dist_a["param"], size=num_fracs)
def length(pts, edges, frac):
"""
Compute the total length of the fractures, based on the fracture id.
The output array has length as unique(frac) and ordered from the lower index
to the higher.
Parameters:
pts: list of points
edges: list of edges as point ids
frac: fracture identification number for each edge
Return:
length: total length for each fracture
"""
# compute the length for each segment
norm = lambda e0, e1: np.linalg.norm(pts[:, e0] - pts[:, e1])
l = np.array([norm(e[0], e[1]) for e in edges.T])
# compute the total length based on the fracture id
tot_l = lambda f: np.sum(l[np.isin(frac, f)])
return np.array([tot_l(f) for f in np.unique(frac)])
def angle(pts, edges, frac):
"""
Compute the mean angle of the fractures, based on the fracture id.
The output array has length as unique(frac) and ordered from the lower index
to the higher.
Parameters:
pts: list of points
edges: list of edges as point ids
frac: fracture identification number for each edge
Return:
angle: mean angle for each fracture
"""
# compute the angle for each segment
alpha = lambda e0, e1: np.arctan2(pts[1, e0] - pts[1, e1], pts[0, e0] - pts[0, e1])
a = np.array([alpha(e[0], e[1]) for e in edges.T])
# compute the mean angle based on the fracture id
mean_alpha = lambda f: np.mean(a[np.isin(frac, f)])
mean_a = np.array([mean_alpha(f) for f in np.unique(frac)])
# we want only angles in (0, pi)
mask = mean_a < 0
mean_a[mask] = np.pi - np.abs(mean_a[mask])
mean_a[mean_a > np.pi] -= np.pi
return mean_a
def count_center_point_densities(p, e, domain, nx=10, ny=10, **kwargs):
""" Divide the domain into boxes, count the number of fracture centers
contained within each box.
Parameters:
p (np.array, 2 x n): Point coordinates of the fractures
e (np.array, 2 x n): Connections between the coordinates
domain (dictionary): Description of the simulation domain. Should
contain fields xmin, xmax, ymin, ymax.
nx, ny (int, optional): Number of boxes in x and y direction. Defaults
to 10.
Returns:
np.array (nx x ny): Number of centers within each box
"""
p = np.atleast_2d(p)
# Special treatment when the point array is empty
if p.shape[1] == 0:
if p.shape[0] == 1:
return np.zeros(nx)
else: # p.shape[0] == 2
return np.zeros((nx, ny))
pc = _compute_center(p, e)
if p.shape[0] == 1:
x0, dx = _decompose_domain(domain, nx, ny)
num_occ = np.zeros(nx)
for i in range(nx):
hit = np.logical_and.reduce(
[pc[0] > (x0 + i * dx), pc[0] <= (x0 + (i + 1) * dx)]
)
num_occ[i] = hit.sum()
return num_occ.astype(np.int)
elif p.shape[0] == 2:
x0, y0, dx, dy = _decompose_domain(domain, nx, ny)
num_occ = np.zeros((nx, ny))
# Can probably do this more vectorized, but for now, a for loop will suffice
for i in range(nx):
for j in range(ny):
hit = np.logical_and.reduce(
[
pc[0] > (x0 + i * dx),
pc[0] < (x0 + (i + 1) * dx),
pc[1] > (y0 + j * dy),
pc[1] < (y0 + (j + 1) * dy),
]
)
num_occ[i, j] = hit.sum()
return num_occ
else:
raise ValueError("Have not yet implemented 3D geometries")
def define_centers_by_boxes(domain, intensity, distribution="poisson"):
""" Define center points of fractures, intended used in a marked point
process.
The domain is assumed decomposed into a set of boxes, and fracture points
will be allocated within each box, according to the specified distribution
and intensity.
A tacit assumption is that the domain and intensity map corresponds to
values used in and computed by count_center_point_densities. If this is
not the case, scaling errors of the densities will arise. This should not
be difficult to generalize, but there is no time right now.
The implementation closely follows y Xu and Dowd:
A new computer code for discrete fracture network modelling
Computers and Geosciences, 2010
Parameters:
domain (dictionary): Description of the simulation domain. Should
contain fields xmin, xmax, ymin, ymax.
intensity (np.array, nx x ny): Intensity map, mean values for fracture
density in each of the boxes the domain will be split into.
distribution (str, default): Specify which distribution is followed.
For now a placeholder value, only 'poisson' is allowed.
Returns:
np.array (2 x n): Coordinates of the fracture centers.
Raises:
ValueError if distribution does not equal poisson.
"""
if distribution != "poisson":
return ValueError("Only Poisson point processes have been implemented")
nx, ny = intensity.shape
num_boxes = intensity.size
max_intensity = intensity.max()
x0, y0, dx, dy = _decompose_domain(domain, nx, ny)
# It is assumed that the intensities are computed relative to boxes of the
# same size that are assigned in here
area_of_box = 1
pts = np.empty(num_boxes, dtype=np.object)
# First generate the full set of points with maximum intensity
counter = 0
for i in range(nx):
for j in range(ny):
num_p_loc = stats.poisson(max_intensity * area_of_box).rvs(1)[0]
p_loc = np.random.rand(2, num_p_loc)
p_loc[0] = x0 + i * dx + p_loc[0] * dx
p_loc[1] = y0 + j * dy + p_loc[1] * dy
pts[counter] = p_loc
counter += 1
# Next, carry out a thinning process, which is really only necessary if the intensity is non-uniform
# See Xu and Dowd Computers and Geosciences 2010, section 3.2 for a description
counter = 0
for i in range(nx):
for j in range(ny):
p_loc = pts[counter]
threshold = np.random.rand(p_loc.shape[1])
delete = np.where(intensity[i, j] / max_intensity < threshold)[0]
pts[counter] = np.delete(p_loc, delete, axis=1)
counter += 1
return np.array(
[pts[i][:, j] for i in range(pts.size) for j in range(pts[i].shape[1])]
).T
def _compute_center(p, edges):
# first compute the fracture centres and then generate them
avg = lambda e0, e1: 0.5 * (np.atleast_2d(p)[:, e0] + np.atleast_2d(p)[:, e1])
pts_c = np.array([avg(e[0], e[1]) for e in edges.T]).T
return pts_c
def _decompose_domain(domain, nx, ny=None):
x0 = domain["xmin"]
dx = (domain["xmax"] - domain["xmin"]) / nx
if "ymin" in domain.keys() and "ymax" in domain.keys():
y0 = domain["ymin"]
dy = (domain["ymax"] - domain["ymin"]) / ny
return x0, y0, dx, dy
else:
return x0, dx