def test_constant(self):
assert_allclose(mu_constant(np.array([5,10,20,50,100]),
np.array([1.5, 2, 1, 4, 8]),
np.array([1.6, 1, 5, 0.4, 4])),
[1.159679143, 1.578343528,
2.253865374, 2.335174298,
10.07573309])
def test_ideal(self):
assert_allclose(mu_constant(np.array([5,10,20,50,100]),
np.array([1,1,1,1,1]),
np.array([5,10,20,50,100])),
[0.936497825, 1.578343528,
2.253865374, 3.163688441,
3.855655749])
def luetal2010(z,
t,
rc,
re,
H=1,
rs=None,
ks=None,
kv=1.0,
kvc=1.0,
kh=1.0,
khc=1.0,
mvs=0.1,
mvc=0.1,
gamw=10,
utop=1,
ubot=1,
nterms=100):
"""Composite foundation considering radial and vert flows in the column
An implementation of [1]_.
Features:
- Single layer, soil properties constant over time.
- Instant load linear with depth.
- Vertical and radial flow in in soil
- Vertical and radial flow in drain/column
- Different stiffness in column and soil
- Pore pressure in soil, column and averaged at depth
- Average pore pressure over whole profile vs time
- Settlement of whole profile vs time
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time.
rc : float
Drain radius
re : float
Drain influence radius.
H : float, optional
Drainage path length. Default H=1.
rs : float, optional
Drain influence radius. Default rs=None i.e. no smear zone.
ks : float, optional
Smear zone permeability. Default ks=None, i.e. no smear zone.
kv, kvc : float, optional
Vertical coefficient of permeability in soil and column.
Default kv= kvc = 1
kh, khc : float, optional
Horizontal coefficient of permeability in soil and column.
default kh = khc = 1
mvs, mvc : float, optional
Volume compressibility in soil and column. default mvs=mvc=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
utop, ubot : float, optional
Initial pore water pressure at top and bottom of soil.
Default utop=ubot=1.
nterms : int, optional
Number of terms to use in solution. Default=100.
Returns
-------
por, pors, porc : 2d array of float
Pore pressure at depth and time overall, in soil, in column.
por is an array of size (len(z), len(t)).
avp : 2d array of float
Average overall pore pressure of whole soil profile.
settle : 2d array of float
Settlement of whole layer.
References
----------
.. [1] Lu, Meng-Meng, Kang-He Xie, and Biao Guo. 2010. 'Consolidation
Theory for a Composite Foundation Considering Radial and Vertical
Flows within the Column and the Variation of Soil Permeability
within the Disturbed Soil Zone'. Canadian Geotechnical
Journal 47 (2): 207-17. doi:10.1139/T09-086.
"""
t = np.atleast_1d(t)
z = np.atleast_1d(z)
# ch = kh / mv / gamw
n = re / rc
Y = mvs / mvc
# Th = ch * t / 4 / re**2
if ks is None or rs is None:
mu0 = smear_zones.mu_ideal(n)
else:
s = rs / rc
kap = kh / ks
mu0 = smear_zones.mu_constant(n, s, kap)
M = ((2 * np.arange(nterms) + 1) / 2 * np.pi)
betm_numer = n**2 * mu0 / 2 * kvc / kh
betm_numer += (n**2 - 1) / 8 * kvc / khc
betm_numer *= (rc / H)**2 * M**2
betm_numer += n**2 - 1 + kvc / kv
betm_numer *= kv / mvs * (n**2 - 1 + Y)
betm_denom = n**2 * mu0 / 2 * kv / kh
betm_denom += (n**2 - 1) / 8 * kv / khc
betm_denom *= (n**2 - 1) * kvc / kv + 1
betm_denom += n**4 / M**2 * (H / rc)**2
betm_denom *= gamw * rc**2
betm = betm_numer / betm_denom
C = re**2 * mu0 / 2 / kh
C += rc**2 * (n**2 - 1) / 8 / khc
C *= gamw * ((n**2 - 1) * kvc + kv) * mvs
C /= (n**2 - 1 + Y) * (kvc - kv)
D = re**2 * mu0 / 2 / kh
D += rc**2 * (n**2 - 1) / 8 / khc
D *= kv * kvc / (kvc - kv)
minus1 = np.ones_like(M)
minus1[1::2] = -1
por = (2 / M[None, None, :] * (utop - minus1[None, None, :] *
(utop - ubot) / M[None, None, :]) *
np.sin(M[None, None, :] * z[:, None, None] / H) *
np.exp(-betm[None, None, :] * t[None, :, None]))
pors = por * (1 +
(C * H**2 * betm[None, None, :] - D * M[None, None, :]**2) /
(H**2 * (n**2 - 1)))
porc = por * (1 - C * betm[None, None, :] + D * M[None, None, :]**2 / H**2)
por = np.sum(por, axis=2)
pors = np.sum(pors, axis=2)
porc = np.sum(porc, axis=2)
avp = (2 / M[None, :]**2 * (utop - minus1[None, :] *
(utop - ubot) / M[None, :]) *
np.exp(-betm[None, :] * t[:, None]))
avp = np.sum(avp, axis=1)
settle = (utop + ubot) / 2 - avp
settle *= H * mvs * n**2 / (n**2 - 1 + Y)
return por, pors, porc, avp, settle
# mvs=0.1
# mvc=mvs/200
# gamw=10
# utop=1
# ubot=1
# nterms=100
# z= np.linspace(0,H,50)
# t = np.array([0.1, 0.8, 2.0])
# t = np.logspace(-2,0.5, 30)
por, pors, porc, avp, settle = luetal2010(z, t, rc, re, H, rs, ks, kv, kvc,
kh, khc, mvs, mvc, gamw, utop,
ubot, nterms)
if 1:
mu = smear_zones.mu_constant(re / rc, rs / rc, kh / ks)
print('mu', mu)
print('\npor', repr(por))
print('\npors', repr(pors))
print('\nporc', repr(porc))
print('\navp', repr(avp))
print('\nset', repr(settle))
labels = [
'Overall pore pressure', 'column pore pressure', 'Soil pore pressure'
]
for p, lab in zip([por, porc, pors], labels):
fig = plt.figure()
ax = fig.add_subplot(1, 1, 1)
for j, t_ in enumerate(t):
ax.plot(p[:, j], z, marker='o', label="t={0:.3g}".format(t[j]))
def luetal2010(z, t, rc, re, H=1, rs=None, ks=None,
kv=1.0, kvc=1.0, kh=1.0, khc=1.0,
mvs=0.1, mvc=0.1, gamw=10, utop=1, ubot=1, nterms=100):
"""Composite foundation considering radial and vert flows in the column
An implementation of [1]_.
Features:
- Single layer, soil properties constant over time.
- Instant load linear with depth.
- Vertical and radial flow in in soil
- Vertical and radial flow in drain/column
- Different stiffness in column and soil
- Pore pressure in soil, column and averaged at depth
- Average pore pressure over whole profile vs time
- Settlement of whole profile vs time
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time.
rc : float
Drain radius
re : float
Drain influence radius.
H : float, optional
Drainage path length. Default H=1.
rs : float, optional
Drain influence radius. Default rs=None i.e. no smear zone.
ks : float, optional
Smear zone permeability. Default ks=None, i.e. no smear zone.
kv, kvc : float, optional
Vertical coefficient of permeability in soil and column.
Default kv= kvc = 1
kh, khc : float, optional
Horizontal coefficient of permeability in soil and column.
default kh = khc = 1
mvs, mvc : float, optional
Volume compressibility in soil and column. default mvs=mvc=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
utop, ubot : float, optional
Initial pore water pressure at top and bottom of soil.
Default utop=ubot=1.
nterms : int, optional
Number of terms to use in solution. Default=100.
Returns
-------
por, pors, porc : 2d array of float
Pore pressure at depth and time overall, in soil, in column.
por is an array of size (len(z), len(t)).
avp : 2d array of float
Average overall pore pressure of whole soil profile.
settle : 2d array of float
Settlement of whole layer.
References
----------
.. [1] Lu, Meng-Meng, Kang-He Xie, and Biao Guo. 2010. 'Consolidation
Theory for a Composite Foundation Considering Radial and Vertical
Flows within the Column and the Variation of Soil Permeability
within the Disturbed Soil Zone'. Canadian Geotechnical
Journal 47 (2): 207-17. doi:10.1139/T09-086.
"""
t = np.atleast_1d(t)
z = np.atleast_1d(z)
# ch = kh / mv / gamw
n = re / rc
Y = mvs / mvc
# Th = ch * t / 4 / re**2
if ks is None or rs is None:
mu0 = smear_zones.mu_ideal(n)
else:
s = rs / rc
kap = kh / ks
mu0 = smear_zones.mu_constant(n, s, kap)
M = ((2 * np.arange(nterms) + 1) / 2 * np.pi)
betm_numer = n**2 * mu0 / 2 * kvc / kh
betm_numer += (n**2 - 1) / 8 *kvc / khc
betm_numer *= (rc / H) ** 2 * M**2
betm_numer += n**2 - 1 + kvc/kv
betm_numer *= kv / mvs * (n**2 - 1 + Y)
betm_denom = n**2 * mu0 / 2 * kv / kh
betm_denom += (n**2 - 1) / 8 * kv / khc
betm_denom *= (n**2 - 1) * kvc / kv + 1
betm_denom += n**4 / M**2 * (H / rc)**2
betm_denom *= gamw * rc**2
betm = betm_numer / betm_denom
C = re**2 * mu0 / 2 / kh
C += rc**2 * (n**2 - 1)/ 8 / khc
C *= gamw * ((n**2 - 1)*kvc + kv) * mvs
C /= (n**2 - 1 + Y) * (kvc - kv)
D = re**2 * mu0 / 2 / kh
D += rc**2 * (n**2 - 1)/ 8 / khc
D *= kv * kvc / (kvc - kv)
minus1 = np.ones_like(M)
minus1[1::2] = -1
por = (2 / M[None, None, :] *
(utop - minus1[None, None, :] * (utop - ubot)/ M[None, None, :]) *
np.sin(M[None, None, :] * z[:, None, None]/H) *
np.exp(-betm[None, None, :] * t[None, :, None]))
pors = por * (1 + (C * H**2 * betm[None, None, :] - D * M[None, None, :]**2) /
(H**2 * (n**2 - 1)))
porc = por * (1 - C * betm[None, None, :]
+ D * M[None, None, :]**2 / H**2)
por = np.sum(por, axis=2)
pors = np.sum(pors, axis=2)
porc = np.sum(porc, axis=2)
avp = (2 / M[None, :]**2 *
(utop - minus1[None, :] * (utop-ubot)/ M[None, :]) *
np.exp(-betm[None, :] * t[:, None]))
avp = np.sum(avp, axis=1)
settle = (utop + ubot)/2 - avp
settle *= H * mvs * n**2 / (n**2 - 1 + Y)
return por, pors, porc, avp, settle
def dengetal2013(z, t, rw, re, A1=1, A2=0, A3=0, H=1, rs=None, ks=None,
kw0=1e10, kh=1, mv=0.1, gamw=10, ui=1):
"""Radial consolidation with depth and time dependent well resistance
Implementation of [1]_.
Features:
- Radial flow to drain (no vertical flow in soil)
- Vertical flow in drain.
- Linear depth variation of drain permeability in time.
- Exponential decrease in drain peremability with time.
- Uses approximate (like Hansbo) method to solve pore pressure in drain.
- Radially averaged pore pressure at depth and time in soil.
Drain pereability is kw = kw0 * (A1 - A2 * z / H) * exp(-A3 * t)
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time.
rw : float
Drain radius.
re : float
Drain influence radius.
A1 : float, optional
Parameter controlling depth dependance of well resistance.
Default A1=1.
A2 : float, optional
Parameter controlling depth dependance of well resistance.
Default A2=0.
A3 : float, optional
Parameter controlling time dependance of well resistance.
Default A3=0.
H : float, optional
Drainage path length. Default H=1.
rs : float, optional
Drain influence radius. Default rs=None i.e. no smear zone.
ks : float, optional
Smear zone permeability. Default ks=None, i.e. no smear zone.
kw0 : float, optional
Initial well permeability. Default kw0=1e10 i.e. ideal drain.
kh : float, optional
Horizontal coefficient of permeability. Default kh=1.
mv : float, optional
Volume compressibility. Default mv=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
ui : float, optional
Initial uniform pore water pressure. Default ui=1.
Returns
-------
por : 2d array of float
Pore pressure at depth and time. `por` is an array of size
(len(z), len(t)).
References
----------
.. [1] Deng, Yue-Bao, Kang-He Xie, and Meng-Meng Lu. 2013. 'Consolidation
by Vertical Drains When the Discharge Capacity Varies
with Depth and Time'. Computers and Geotechnics 48 (March): 1-8.
doi:10.1016/j.compgeo.2012.09.012.
"""
if A1 < 0:
raise ValueError("A1 must be greater than 0. "
"You have A1={}".format(A1))
if A2 > A1:
raise ValueError("A2 must be less than A1. "
"You have A1={}, A2={}".format(A1, A2))
if A3 < 0:
raise ValueError("A3 must be greater than or equal to 0. "
"You have A3={}".format(A3))
t = np.atleast_1d(t)
z = np.atleast_1d(z)
ch = kh / mv / gamw
n = re / rw
qw0 = kw0 * np.pi * rw**2
Th = ch * t / 4 / re**2
if ks is None or rs is None:
mu0 = smear_zones.mu_ideal(n)
else:
s = rs / rw
kap = kh / ks
mu0 = smear_zones.mu_constant(n, s, kap)
if A3 == 0 and A2 == 0:
# qw is constant
mus = mu0 + np.pi * kh / (qw0 * A1) * (2 * H * z - z ** 2)
por = ui * np.exp(-8 * Th[None, :] / mus[:, None])
return por
if A3 == 0 and A2 > 0:
# qw varies with depth
mu1 = A2 * z / H + (A1 - A2)*np.log(1 - A2 / A1 * z / H)
mu1 *= 2 * np.pi * kh * H**2 / (qw0 * A2**2)
mu1 += mu0
por = ui * np.exp(-8 * Th[None, :] / mu1[:, None])
return por
if A3 > 0 and A2 == 0:
# qw varies with time
a3 = A3 * 4 * re**2 / ch
alp0 = qw0 * A1 * mu0 / (np.pi * kh * (2*H*z - z**2))
por = (ui * ((1+alp0[:, None] * np.exp(-a3 * Th[None, :])) /
(1+alp0[:, None])) ** (8/(a3 * mu0)))
return por
if A3 > 0 and A2 > 0:
# qw varies with depth and time:
a3 = A3 * 4 * re**2 / ch
alp = mu0 * qw0 * A2**2 / (2 * np.pi * H**2 * kh)
alp /= A2* z / H + (A1 - A2) * np.log(1 - A2 / A1 * z / H)
por = (ui * ((1+alp[:, None] * np.exp(-a3 * Th[None, :])) /
(1+alp[:, None])) ** (8/(a3 * mu0)))
return por
# gamw=10
# utop=1
# ubot=1
# nterms=100
# z= np.linspace(0,H,50)
# t = np.array([0.1, 0.8, 2.0])
# t = np.logspace(-2,0.5, 30)
por, pors, porc, avp, settle = luetal2010(
z,t, rc, re, H, rs, ks,
kv, kvc, kh, khc,
mvs, mvc, gamw, utop, ubot, nterms)
if 1:
mu = smear_zones.mu_constant(re/rc, rs/rc, kh/ks)
print('mu', mu)
print('\npor', repr(por))
print('\npors', repr(pors))
print('\nporc', repr(porc))
print('\navp', repr(avp))
print('\nset', repr(settle))
labels = ['Overall pore pressure', 'column pore pressure', 'Soil pore pressure']
for p, lab in zip([por, porc, pors], labels):
fig=plt.figure()
ax = fig.add_subplot(1,1,1)
for j, t_ in enumerate(t):
ax.plot(p[:,j], z, marker='o', label="t={0:.3g}".format(t[j]))
ax.set_xlabel(lab)
def dengetal2014(z, t, rw, re, A3=0, H=1, rs=None, ks=None,
kw0=1e10, kh=1, mv=0.1, gamw=10, ui=1, nterms=100):
"""Radial consolidation with time dependent well resistance
An implementation of [1]_
Features:
- Single layer, soil properties constant over time.
- Instant load uniform with depth.
- Radial flow to drain (no vertical flow in soil)
- Vertical flow in drain.
- Exponential decrease in drain peremability with time.
- Uses rigorous (infintite sum ) method to solve pore pressure in drain.
- Radially averaged pore pressure at depth and time in soil.
Drain permeability is kw = kw0 * exp(-A3 * t)
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time.
rw : float
Drain radius.
re : float
Drain influence radius.
A3 : float, optional
Parameter controlling time dependance of well resistance.
Default A3=0.
H : float, optional
Drainage path length. Default H=1.
rs : float, optional
Drain influence radius. Default rs=None i.e. no smear zone.
ks : float, optional
Smear zone permeability. Default ks=None, i.e. no smear zone.
kw0 : float, optional
Initial well permeability. Default kw0=1e10 i.e. ideal drain.
kh : float, optional
Horizontal coefficient of permeability. Default kh=1.
mv : float, optional
Volume compressibility. Default mv=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
ui : float, optional
Initial uniform pore water pressure. Default ui=1.
nterms : int, optional
number of summation terms, default = 100.
Returns
-------
por : 2d array of float
Pore pressure at depth and time. `por` is an array of size
(len(z), len(t)).
References
----------
.. [1] Deng, Yue-Bao, Gan-Bin Liu, Meng-Meng Lu,
and Kang-he Xie. 'Consolidation Behavior of Soft Deposits
Considering the Variation of Prefabricated Vertical Drain
Discharge Capacity'. Computers and Geotechnics 62
(October 2014): 310-16. doi:10.1016/j.compgeo.2014.08.006.
"""
if A3 < 0:
raise ValueError("A3 must be greater than or equal to 0. "
"You have A3={}".format(A3))
t = np.atleast_1d(t)
z = np.atleast_1d(z)
ch = kh / mv / gamw
n = re / rw
qw0 = kw0 * np.pi * rw**2
Th = ch * t / 4 / re**2
if ks is None or rs is None:
mu0 = smear_zones.mu_ideal(n)
else:
s = rs / rw
kap = kh / ks
mu0 = smear_zones.mu_constant(n, s, kap)
a3 = A3 * 4 * re**2 / ch
G0 = np.pi * kh * H**2 / 4 / qw0
M = ((2 * np.arange(nterms) + 1) / 2 * np.pi)
C0 = M**2*mu0/8/G0 * n**2/ (n**2-1)
alp0 = qw0 * mu0 / (np.pi * kh * (2*H*z - z**2))
Th = Th[np.newaxis, :, np.newaxis]
M = M[np.newaxis, np.newaxis, :]
C0 = C0[np.newaxis, np.newaxis, :]
Z = (z / H)[:, np.newaxis, np.newaxis]
por = (2 / M * np.sin(M * Z) * ((1+C0 * np.exp(-a3 * Th))/
(1 + C0))**(8 / (a3 * mu0)))
por = ui * np.sum(por, axis=2)
return por
z = np.linspace(0, 20, 100)
Th = np.array([0.1, 1.0])
rw=0.035
rs = 0.175
re=0.525
gamw = 10
mv=0.2e-3
kh=2e-8
ks = kh/1.8
kw0 = 1e-3
ch = kh / mv / gamw
t = Th * 4 * re**2 / ch
mu0 = smear_zones.mu_constant(re/rw, rs/rw, kh/ks)
print('mu0', mu0)
print('et', 2/mu0/re**2)
for A2 in [0.1,1.0]:
por = dengetal2013(z, t, rw=rw, re=re, A1=1, A2=A2, A3=0, H=20,
rs=rs, ks=ks, kw0=kw0, kh=kh, mv=mv, gamw=10, ui=1)
for j, t_ in enumerate(t):
uz0 = np.exp(-8*Th[j]/mu0)
plt.plot(por[:,j], z, label="m Th={0:.3g}, A2={1:.3g}".format(Th[j], A2))
plt.plot(uz0,0, marker='o', ms=14)
for key, val in res.items():
x = val[:,0]
y = val[:,1]*20
plt.plot(x,y, label=key, marker='s', linestyle='None')
def dengetal2013(z,
t,
rw,
re,
A1=1,
A2=0,
A3=0,
H=1,
rs=None,
ks=None,
kw0=1e10,
kh=1,
mv=0.1,
gamw=10,
ui=1):
"""Radial consolidation with depth and time dependent well resistance
Implementation of [1]_.
Features:
- Radial flow to drain (no vertical flow in soil)
- Vertical flow in drain.
- Linear depth variation of drain permeability in time.
- Exponential decrease in drain peremability with time.
- Uses approximate (like Hansbo) method to solve pore pressure in drain.
- Radially averaged pore pressure at depth and time in soil.
Drain pereability is kw = kw0 * (A1 - A2 * z / H) * exp(-A3 * t)
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time.
rw : float
Drain radius.
re : float
Drain influence radius.
A1 : float, optional
Parameter controlling depth dependance of well resistance.
Default A1=1.
A2 : float, optional
Parameter controlling depth dependance of well resistance.
Default A2=0.
A3 : float, optional
Parameter controlling time dependance of well resistance.
Default A3=0.
H : float, optional
Drainage path length. Default H=1.
rs : float, optional
Drain influence radius. Default rs=None i.e. no smear zone.
ks : float, optional
Smear zone permeability. Default ks=None, i.e. no smear zone.
kw0 : float, optional
Initial well permeability. Default kw0=1e10 i.e. ideal drain.
kh : float, optional
Horizontal coefficient of permeability. Default kh=1.
mv : float, optional
Volume compressibility. Default mv=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
ui : float, optional
Initial uniform pore water pressure. Default ui=1.
Returns
-------
por : 2d array of float
Pore pressure at depth and time. `por` is an array of size
(len(z), len(t)).
References
----------
.. [1] Deng, Yue-Bao, Kang-He Xie, and Meng-Meng Lu. 2013. 'Consolidation
by Vertical Drains When the Discharge Capacity Varies
with Depth and Time'. Computers and Geotechnics 48 (March): 1-8.
doi:10.1016/j.compgeo.2012.09.012.
"""
if A1 < 0:
raise ValueError("A1 must be greater than 0. "
"You have A1={}".format(A1))
if A2 > A1:
raise ValueError("A2 must be less than A1. "
"You have A1={}, A2={}".format(A1, A2))
if A3 < 0:
raise ValueError("A3 must be greater than or equal to 0. "
"You have A3={}".format(A3))
t = np.atleast_1d(t)
z = np.atleast_1d(z)
ch = kh / mv / gamw
n = re / rw
qw0 = kw0 * np.pi * rw**2
Th = ch * t / 4 / re**2
if ks is None or rs is None:
mu0 = smear_zones.mu_ideal(n)
else:
s = rs / rw
kap = kh / ks
mu0 = smear_zones.mu_constant(n, s, kap)
if A3 == 0 and A2 == 0:
# qw is constant
mus = mu0 + np.pi * kh / (qw0 * A1) * (2 * H * z - z**2)
por = ui * np.exp(-8 * Th[None, :] / mus[:, None])
return por
if A3 == 0 and A2 > 0:
# qw varies with depth
mu1 = A2 * z / H + (A1 - A2) * np.log(1 - A2 / A1 * z / H)
mu1 *= 2 * np.pi * kh * H**2 / (qw0 * A2**2)
mu1 += mu0
por = ui * np.exp(-8 * Th[None, :] / mu1[:, None])
return por
if A3 > 0 and A2 == 0:
# qw varies with time
a3 = A3 * 4 * re**2 / ch
alp0 = qw0 * A1 * mu0 / (np.pi * kh * (2 * H * z - z**2))
por = (ui * ((1 + alp0[:, None] * np.exp(-a3 * Th[None, :])) /
(1 + alp0[:, None]))**(8 / (a3 * mu0)))
return por
if A3 > 0 and A2 > 0:
# qw varies with depth and time:
a3 = A3 * 4 * re**2 / ch
alp = mu0 * qw0 * A2**2 / (2 * np.pi * H**2 * kh)
alp /= A2 * z / H + (A1 - A2) * np.log(1 - A2 / A1 * z / H)
por = (ui * ((1 + alp[:, None] * np.exp(-a3 * Th[None, :])) /
(1 + alp[:, None]))**(8 / (a3 * mu0)))
return por
def dengetal2014(z,
t,
rw,
re,
A3=0,
H=1,
rs=None,
ks=None,
kw0=1e10,
kh=1,
mv=0.1,
gamw=10,
ui=1,
nterms=100):
"""Radial consolidation with time dependent well resistance
An implementation of [1]_
Features:
- Single layer, soil properties constant over time.
- Instant load uniform with depth.
- Radial flow to drain (no vertical flow in soil)
- Vertical flow in drain.
- Exponential decrease in drain peremability with time.
- Uses rigorous (infintite sum ) method to solve pore pressure in drain.
- Radially averaged pore pressure at depth and time in soil.
Drain permeability is kw = kw0 * exp(-A3 * t)
Parameters
----------
z : float or 1d array/list of float
Depth.
t : float or 1d array/list of float
Time.
rw : float
Drain radius.
re : float
Drain influence radius.
A3 : float, optional
Parameter controlling time dependance of well resistance.
Default A3=0.
H : float, optional
Drainage path length. Default H=1.
rs : float, optional
Drain influence radius. Default rs=None i.e. no smear zone.
ks : float, optional
Smear zone permeability. Default ks=None, i.e. no smear zone.
kw0 : float, optional
Initial well permeability. Default kw0=1e10 i.e. ideal drain.
kh : float, optional
Horizontal coefficient of permeability. Default kh=1.
mv : float, optional
Volume compressibility. Default mv=0.1.
gamw : float, optional
Unit weight of water. Default gamw=10.
ui : float, optional
Initial uniform pore water pressure. Default ui=1.
nterms : int, optional
number of summation terms, default = 100.
Returns
-------
por : 2d array of float
Pore pressure at depth and time. `por` is an array of size
(len(z), len(t)).
References
----------
.. [1] Deng, Yue-Bao, Gan-Bin Liu, Meng-Meng Lu,
and Kang-he Xie. 'Consolidation Behavior of Soft Deposits
Considering the Variation of Prefabricated Vertical Drain
Discharge Capacity'. Computers and Geotechnics 62
(October 2014): 310-16. doi:10.1016/j.compgeo.2014.08.006.
"""
if A3 < 0:
raise ValueError("A3 must be greater than or equal to 0. "
"You have A3={}".format(A3))
t = np.atleast_1d(t)
z = np.atleast_1d(z)
ch = kh / mv / gamw
n = re / rw
qw0 = kw0 * np.pi * rw**2
Th = ch * t / 4 / re**2
if ks is None or rs is None:
mu0 = smear_zones.mu_ideal(n)
else:
s = rs / rw
kap = kh / ks
mu0 = smear_zones.mu_constant(n, s, kap)
a3 = A3 * 4 * re**2 / ch
G0 = np.pi * kh * H**2 / 4 / qw0
M = ((2 * np.arange(nterms) + 1) / 2 * np.pi)
C0 = M**2 * mu0 / 8 / G0 * n**2 / (n**2 - 1)
alp0 = qw0 * mu0 / (np.pi * kh * (2 * H * z - z**2))
Th = Th[np.newaxis, :, np.newaxis]
M = M[np.newaxis, np.newaxis, :]
C0 = C0[np.newaxis, np.newaxis, :]
Z = (z / H)[:, np.newaxis, np.newaxis]
por = (2 / M * np.sin(M * Z) * ((1 + C0 * np.exp(-a3 * Th)) /
(1 + C0))**(8 / (a3 * mu0)))
por = ui * np.sum(por, axis=2)
return por
z = np.linspace(0, 20, 100)
Th = np.array([0.1, 1.0])
rw = 0.035
rs = 0.175
re = 0.525
gamw = 10
mv = 0.2e-3
kh = 2e-8
ks = kh / 1.8
kw0 = 1e-3
ch = kh / mv / gamw
t = Th * 4 * re**2 / ch
mu0 = smear_zones.mu_constant(re / rw, rs / rw, kh / ks)
print('mu0', mu0)
print('et', 2 / mu0 / re**2)
for A2 in [0.1, 1.0]:
por = dengetal2013(z,
t,
rw=rw,
re=re,
A1=1,
A2=A2,
A3=0,
H=20,
rs=rs,
ks=ks,
kw0=kw0,
kh=kh,
