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laplace_hankel_solutions.f90
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!
! Copyright (c) 2012-2022 Kristopher L. Kuhlman (klkuhlm at sandia dot gov)
!
! Permission is hereby granted, free of charge, to any person obtaining a copy
! of this software and associated documentation files (the "Software"), to deal
! in the Software without restriction, including without limitation the rights
! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
! copies of the Software, and to permit persons to whom the Software is
! furnished to do so, subject to the following conditions:
!
! The above copyright notice and this permission notice shall be included in
! all copies or substantial portions of the Software.
!
! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
! IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
! AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
! OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
! THE SOFTWARE.
!
module laplace_hankel_solutions
implicit none
private
public :: lap_hank_soln
contains
function lap_hank_soln(a,rD,np,nz,w,f,s,lap) result(fp)
use constants, only : DP, EP, MAXEXP
use types, only : well, formation, invLaplace, solution
use time, only : lapTime
use utility, only : operator(.X.)
implicit none
real(EP), intent(in) :: a
real(DP), intent(in) :: rD
integer, intent(in) :: np,nz
type(invLaplace), intent(in) :: lap
type(solution), intent(in) :: s
type(well), intent(in) :: w
type(formation), intent(in) :: f
complex(EP), dimension(np,nz) :: fp
complex(EP), allocatable :: eta(:), xi(:), udp(:,:)
intrinsic :: bessel_j0
select case(s%model)
case(0)
! Theis (fully penetrating confined)
fp(1:np,1:nz) = theis(a,lap%p,nz)
case(1)
! Hantush (partially penetrating confined)
fp(1:np,1:nz) = hantush(a,s%zD,s,lap%p,f,w)
case(2)
! Hantush-style solution with wellbore storage
fp(1:np,1:nz) = hantushstorage(a,s%zD,s,lap%p,f,w)
case(3:5)
! Moench hybrid water table condition made a special case
! Malama 2011 partial penetrating model (Neuman 74 when beta=0)
allocate(eta(np),xi(np),udp(np,nz+1))
eta(1:np) = sqrt((lap%p(:) + a**2)/f%kappa)
xi(1:np) = eta(:)*f%alphaD/lap%p(:)
if (s%model == 3) then
! Moench model -> just alters alphaD
xi(1:np) = xi(:)*f%MoenchM/sum(1.0_EP/(1.0_EP + (lap%p .X. 1.0_EP/f%MoenchGamma)),dim=2)
end if
if (s%model == 4) then
udp(1:np,1:nz) = theis(a,lap%p,nz)
udp(1:np,nz+1) = udp(1:np,nz)
else
udp(1:np,1:nz+1) = hantush(a,[s%zD,1.0_DP],s,lap%p,f,w)
end if
where (spread(real(eta) < MAXEXP,2,nz))
fp(1:np,1:nz) = udp(:,1:nz) - spread(udp(:,nz+1),2,nz)* &
& cosh(eta .X. s%zD)/spread((1.0_EP + f%beta*eta*xi)*&
& cosh(eta) + xi*sinh(eta),2,nz)
elsewhere
fp(1:np,1:nz) = udp(:,1:nz) - spread(udp(:,nz+1),2,nz)* &
& exp(eta .X. (s%zD - 1.0_EP))/&
& spread(1.0_EP + f%beta*eta*xi + xi,2,nz)
end where
deallocate(eta,xi,udp)
case(6)
! Mishra/Neuman 2010 model
select case(s%MNtype)
case(0)
#ifdef USE_ARB_LIBRARY
! naive implementation of paper (often doesnt work)
fp(1:np,1:nz) = mishraNeuman2010(a,s%zD,s,lap%p,f,w)
#else
stop 'ERROR: must compile code with arb dependency (and '//&
&'USE_ARB_LIBRARY preprocessor flag) for this Mishra/Neumann 2010 model, type 0'
#endif
case(1)
! Malama (finiteness condition @ land surface) version
! but doesn't handle partial penetration or wellbore storage
fp(1:np,1:nz) = mishraNeumanMalama(a,s%zD,lap%p,f)
case(2)
! finite difference solution of ODE in vadose zone
fp(1:np,1:nz) = mishraNeuman2010FD(a,s%zD,s,lap%p,f,w)
end select
end select
! apply common Hankel-transform and time-behavior factors
fp(1:np,1:nz) = a*bessel_j0(a*rD)*fp*spread(lapTime(lap),2,nz)
end function lap_hank_soln
function theis(a,p,nz) result(udf)
use constants, only : EP
real(EP), intent(in) :: a
complex(EP), dimension(:), intent(in) :: p
integer, intent(in) :: nz
complex(EP), dimension(size(p),nz) :: udf
udf(:,:) = spread(2.0_EP/(p(:) + a**2),dim=2,ncopies=nz)
end function theis
function hantush(a,zD,sol,p,f,w) result(udp)
! implemented in the form given in Malama, Kuhlman & Barrash 2008
use constants, only : DP, EP
use types, only : well, formation, invLaplace, solution
use time, only : lapTime
use utility, only : operator(.X.)
implicit none
real(EP), intent(in) :: a
real(DP), dimension(:), intent(in) :: zD
complex(EP), dimension(:), intent(in) :: p
type(solution), intent(in) :: sol
type(well), intent(in) :: w
type(formation), intent(in) :: f
complex(EP), dimension(size(p),size(zD)) :: udp
complex(EP), dimension(size(p)) :: eta
complex(EP), dimension(3,size(p),size(zD)) :: g
complex(EP), dimension(2,size(p),size(zD)) :: ff
integer, dimension(size(p),size(zD)) :: zLay
real(EP) :: dD1, lD1
integer :: np,nz
nz = size(zD) ! # z vals requested of this function
np = size(p)
dD1 = 1.0_EP - w%dD
lD1 = 1.0_EP - w%lD
! are # zvals requested same as given in program input?
if(sol%nz == nz) then
! called directly
zLay(1:np,1:nz) = spread(sol%zLay(:),1,np)
else
! called from other method, requiring solution
! at the water table for boundary conditions
! (top boundary condition is always layer 3)
zLay(1:np,1:nz) = spread([sol%zLay(:),3],1,np)
end if
eta(1:np) = sqrt((p(:) + a**2)/f%kappa)
! above well screen
g(1,1:np,1:nz) = cosh(eta(:) .X. (dD1 - zD(1:nz)))
ff(1,1:np,1:nz) = spread(sinh(eta(:)*w%dD),2,nz)
ff(2,1:np,1:nz) = spread(sinh(eta(:)*lD1),2,nz)
g(2,1:np,1:nz) = (ff(1,:,:)*cosh(eta(:) .X. zD(:)) + ff(2,:,:)*&
& cosh(eta(:) .X. (1.0_EP - zD)))/spread(sinh(eta(:)),2,nz)
! below well screen
g(3,1:np,1:nz) = spread(exp(-eta(:)*lD1) - &
& (ff(1,:,1) + exp(-eta(:))*ff(2,:,1))/sinh(eta(:)),2,nz)
where(zLay == 1)
! below well screen (0 <= zD <= 1-lD)
udp(1:np,1:nz) = g(3,:,:)*cosh(eta(:) .X. zD(:))
elsewhere
where (zLay == 2)
! next to well screen (1-lD < zD < 1-dD)
udp(1:np,1:nz) = 1.0_EP - g(2,:,:)
elsewhere
! zLay == 3
! above well screen (1-lD <= zD <= 1)
udp(1:np,1:nz) = g(1,:,:) - g(2,:,:)
end where
end where
udp(1:np,1:nz) = udp(:,:)*theis(a,p,nz)/w%bD
end function hantush
function hantushstorage(a,zD,sol,p,f,w) result(u)
use constants, only : DP, EP, PI
use types, only : well, formation, invLaplace, solution
use time, only : lapTime
use utility, only : operator(.X.)
use cbessel, only : cbesk ! Amos routine
implicit none
real(EP), intent(in) :: a
real(DP), dimension(:), intent(in) :: zD
complex(EP), dimension(:), intent(in) :: p
type(solution), intent(in) :: sol
type(well), intent(in) :: w
type(formation), intent(in) :: f
complex(EP), dimension(size(p),size(zD)) :: u, uDp
complex(DP), dimension(0:1,size(p)) :: K
complex(EP), dimension(size(p)) :: eta, xi, A0, uDf
complex(EP), dimension(3,size(p),size(zD)) :: ff
complex(EP), dimension(2,size(p),size(zD)) :: g
real(EP) :: dD1, lD1
real(DP) :: CDw, tDb
integer :: np,nz,i
! things related to Amos library
integer(4), parameter :: kode = 1, num = 2
integer(4) :: nzero, ierr
integer, dimension(size(p),size(zD)) :: zLay
np = size(p)
nz = size(zd)
dD1 = 1.0_EP - w%dD
lD1 = 1.0_EP - w%lD
! are # zvals requested same as given in program input?
if(sol%nz == nz) then
zLay(1:np,1:nz) = spread(sol%zLay(:),1,np)
else
zLay(1:np,1:nz) = spread([sol%zLay(:),3],1,np)
end if
! dimensionless pumping wellbore storage coefficient
CDw = w%rDw**2/(2.0_DP*(w%l - w%d)*f%Ss)
! dimensionless well delay time
tDb = PI*sol%rDwobs**2/(sol%sF*f%Ss)
xi(1:np) = w%rDw*sqrt(p(:))
eta(1:np) = sqrt((p(:) + a**2)/f%kappa)
do i=1,np
call cbesk(z=cmplx(xi(i),kind=DP),fnu=0.0_DP,kode=kode,&
& n=num,cy=K(0:1,i),nz=nzero,ierr=ierr)
if (ierr > 0 .and. ierr /= 3) then
print *, 'ERROR: CBESK z=',xi(i),&
&' i,ierr,nz:',i,ierr,nzero
end if
end do
A0(1:np) = 2.0_EP/(p(:)*CDw*K(0,:) + xi(:)*K(1,:))
uDf(1:np) = A0(:)/((p(:) + a**2)*(p(:)*tDb + 1.0_EP)) !!! <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
where (zLay == 3)
g(1,1:np,1:nz) = cosh(eta(:) .X. (dD1 - zd(:)))
end where
where (zLay == 1 .or. zLay == 2)
ff(1,1:np,1:nz) = spread(sinh(eta(:)*w%dD),2,nz)
ff(2,1:np,1:nz) = spread(sinh(eta(:)*lD1),2,nz)
end where
where (zLay == 2 .or. zLay == 3)
g(2,1:np,1:nz) = (ff(1,:,:)*cosh(eta(:) .X. zd(:)) + &
& ff(2,:,:)*cosh(eta(:) .X. (1.0_EP - zd(:))))/&
& spread(sinh(eta(:)),2,nz)
end where
where (zLay == 1)
ff(3,1:np,1:nz) = spread(exp(-eta(:)*lD1) - &
& (ff(1,:,1) + exp(-eta(:))*ff(2,:,1))/sinh(eta(:)),2,nz)
end where
where (zLay == 3) !!(spread(zD > 1-w%dD,2,nz))
uDp(1:np,1:nz) = g(1,:,:) - g(2,:,:)
elsewhere
where (zLay == 1) !!(spread(zD < (1.0-w%lD),2,nz))
uDp(1:np,1:nz) = ff(3,:,:)*cosh(eta(:) .X. zd(:))
elsewhere
! zLay == 2
uDp(1:np,1:nz) = 1.0_EP - g(2,:,:)
end where
end where
u(1:np,1:nz) = spread(uDf(:)/w%bD,2,nz)*uDp(:,:)
end function hantushstorage
#ifdef USE_ARB_LIBRARY
function mishraNeuman2010(a,zD,s,p,f,w) result(sD)
use constants, only : DP, EP, QP
use types, only : well, formation, solution
implicit none
! https://gcc.gnu.org/onlinedocs/gfortran/ISO_005fC_005fBINDING.html#ISO_005fC_005fBINDING
interface
function arb_J(nu,z) bind(c,name="arb_J") result(J)
use, intrinsic :: iso_c_binding, only : C_FLOAT128_COMPLEX, C_FLOAT128
complex(C_FLOAT128_COMPLEX), intent(in), value :: z
real(C_FLOAT128), intent(in), value :: nu
complex(C_FLOAT128_COMPLEX) :: J
end function arb_J
end interface
interface
function arb_Y(nu,z) bind(c,name="arb_Y") result(Y)
use, intrinsic :: iso_c_binding, only : C_FLOAT128_COMPLEX, C_FLOAT128
complex(C_FLOAT128_COMPLEX), intent(in), value :: z
real(C_FLOAT128), intent(in), value :: nu
complex(C_FLOAT128_COMPLEX) :: Y
end function arb_Y
end interface
real(EP), intent(in) :: a
real(DP), dimension(:), intent(in) :: zD
complex(EP), dimension(:), intent(in) :: p
type(solution), intent(in) :: s
type(well), intent(in) :: w
type(formation), intent(in) :: f
complex(EP), dimension(size(p),size(zD)) :: sD, sU
complex(EP), dimension(size(p),size(zD)+1) :: sH
integer :: np, nz
real(QP) :: nu, nu1, C
complex(QP), dimension(size(p)) :: phi, arg2
complex(QP), parameter :: EYE = (0.0_QP, 1.0_QP)
complex(QP) :: arg1
complex(QP), dimension(size(p)) :: BD, chi, q
complex(QP), dimension(size(p)) :: eta
complex(QP), dimension(size(p),0:1) :: J,Y
integer :: i
np = size(p)
nz = size(zD)
eta(1:np) = sqrt((a**2 + p(1:np))/f%kappa)
sH(1:np,1:nz+1) = hantush(a,[zD,1.0_DP],s,p,f,w)
! <between D7 & D8> ! <<<<<< <1/rD**2 ?>
BD(1:np) = p(:)*f%acD*f%Sy/(f%Ss*f%b*f%kappa)*exp(f%akD*(f%psikD - f%psiaD))
C = (a**2)/f%kappa ! <<<<<< <also y**2, except for r**2/r_D**2 factor?>
! <between D8 & D9>
phi(1:np) = EYE*2.0_QP/f%lambdaD*sqrt(BD)*exp(f%lambdaD*f%usLD/2.0_QP) ! phi(L_D)
nu = sqrt((f%akD**2 + 4.0_QP*C)/f%lambdaD**2) ! <n>
nu1 = nu + 1.0_QP
do i= 1,np
J(i,0) = arb_J(nu, phi(i))
J(i,1) = arb_J(nu1, phi(i))
Y(i,0) = arb_Y(nu, phi(i))
Y(i,1) = arb_Y(nu1, phi(i))
end do
arg1 = f%akD + nu*f%lambdaD
arg2(1:np) = f%lambdaD*phi(1:np)
! <Eqns D11 and 25>
chi(1:np) = -(arg1*J(:,0) - arg2(:)*J(:,1))/(arg1*Y(:,0) - arg2(:)*Y(:,1))
! phi(0)
phi(1:np) = EYE*2.0_QP/f%lambdaD*sqrt(BD)
do i= 1,np
J(i,0) = arb_J(nu, phi(i))
J(i,1) = arb_J(nu1, phi(i))
Y(i,0) = arb_Y(nu, phi(i))
Y(i,1) = arb_Y(nu1, phi(i))
end do
! <Eqns D13 and 26>
q(1:np) = arg1/2.0_QP - EYE*sqrt(BD)*(J(:,1) + chi(:)*Y(:,1))/(J(:,0) + chi(:)*Y(:,0))
! <Eqns C17 and 23> ((but C20 and C21 have sinh -> cosh??))
sU(1:np,1:nz) = -spread(sH(1:np,nz+1)/ &
& (cosh(eta(:)) - eta(:)/q(:)*sinh(eta(:))),2,nz)* &
& cosh(spread(eta(:),2,nz)*spread(zD(:),1,np))
! <Eqns B1 and 19>
sD(1:np,1:nz) = sH(:,1:nz) + sU(:,:)
end function mishraNeuman2010
! endif for USE_ARB_LIBRARY
#endif
function mishraNeumanMalama(a,zD,p,f) result(sD)
use constants, only : DP, EP
use types, only : well, formation, solution
use utility, only : operator(.X.)
implicit none
real(EP), intent(in) :: a
real(DP), dimension(:), intent(in) :: zD
complex(EP), dimension(:), intent(in) :: p
type(formation), intent(in) :: f
complex(EP), dimension(size(p),size(zD)) :: sD
integer :: np, nz
real(EP) :: beta0, vartheta, phiDa, phiDk, u0
complex(EP), dimension(size(p)) :: eta, etasq, eta1, u, v, Delta0
np = size(p)
nz = size(zD)
beta0 = f%ak*f%b ! assume ac == ak in Malama formulation
phiDa = f%psia/f%b
phiDk = f%psik/f%b
vartheta = beta0*f%Sy/(f%Ss*f%b)*exp(-beta0*(phiDa - phiDk))
eta1(1:np) = sqrt((p(:)*vartheta + a**2)/f%kappa)
u0 = beta0/2.0
v(1:np) = sqrt(1.0 + (eta1(:)/u0)**2)
u(1:np) = u0*(1.0 - v(:))
etasq(1:np) = (p(:) + a**2)/f%kappa
eta(1:np) = sqrt(etasq(:))
Delta0(1:np) = eta(:)*sinh(eta(:)) - u(:)*cosh(eta(:))
! saturated zone solution
sD(1:np,1:nz) = 2.0/spread(f%kappa*etasq,2,nz)* &
& (1.0 + spread(u/Delta0,2,nz)*cosh(eta .X. zD))
end function mishraNeumanMalama
function mishraNeuman2010FD(a,zD,s,p,f,w) result(sD)
use constants, only : DP, EP
use types, only : well, formation, solution
use utility, only : operator(.X.), solve_tridiag
implicit none
real(EP), intent(in) :: a
real(DP), dimension(:), intent(in) :: zD
complex(EP), dimension(:), intent(in) :: p
type(solution), intent(in) :: s
type(well), intent(in) :: w
type(formation), intent(in) :: f
! finite difference matrix related
! a,b,c are sub,main,super diagonals of FD matrix
! sigma is result (A1 + solution in vadose zone)
! v is right-hand side
complex(EP), dimension(size(p),s%order) :: aa,b,c,sigma,v
complex(EP), dimension(size(p),size(zD)) :: sD
complex(EP), dimension(size(p),size(zD)+1) :: sH
integer :: np, nz, j, n
integer, dimension(s%order) :: ii
complex(EP), dimension(size(p)) :: eta, cc
real(EP) :: h, invhsq, B2
real(EP), dimension(0:3) :: beta
complex(EP), dimension(size(p),s%order) :: omega, B1
integer, parameter :: NPRINT = 2
n = s%order
h = f%usLD/real(n-1,EP) ! F.D. grid spacing
invhsq = 1.0_EP/h**2
np = size(p)
nz = size(zD)
! integers 0:n-1
forall(j=1:n) ii(j) = j-1
beta(0) = f%ac*f%Sy/f%Ss
beta(1) = -f%lambdaD ! b*(ac-ak) (-1* definition in driver_io.f90)
beta(2) = f%ak*f%b1 ! ak*(psi_a - psi_k)
beta(3) = f%akD ! ak*b
eta(1:np) = sqrt((a**2 + p(1:np))/f%kappa)
sH(1:np,1:nz+1) = hantush(a,[zD,1.0_DP],s,p,f,w)
B1(1:np,1:n) = spread(p(:)*beta(0)*exp(-beta(2))/f%kappa,2,n)
B2 = (a**2)/f%kappa
omega(1:np,1:n) = B1(:,:)*spread(exp(-beta(1)*ii(:)*h),1,np) + B2
! main diagonal (first and last entries are different)
cc(1:np) = beta(3)/h - invhsq - omega(1:np,1)
b(1:np,1) = 0.5_EP*(exp( eta(:))*(cc(:) - eta(:)/h) + &
& exp(-eta(:))*(cc(:) + eta(:)/h))
b(1:np,2:n) = beta(3)/h - 2.0*invhsq - omega(1:np,2:n)
b(1:np,n) = b(1:np,n) + invhsq - beta(3)/h
! super-diagonal (last entry (n) is undefined)
c(1:np,1:n-1) = invhsq - beta(3)/h
! sub-diagonal (first entry 1 is undefined, second entry is different)
aa(1:np,2:n) = invhsq
aa(1:np,2) = aa(1:np,2)*cosh(eta(:))
! right-hand side (all zero but first and second rows)
v(1:np,3:n) = 0.0_EP
v(1:np,2) = -invhsq*sH(1:np,nz+1)
v(1:np,1) = -cc(1:np)*sH(1:np,nz+1)
! sigma(1:np,1) is A1, the constant needed in
! solution for saturated domain
call solve_tridiag(aa,b,c,v,sigma)
! @ early times sigma_1 is underflowing while cosh(eta*z) is overflowing;
! this should allow the solution to proceed (~Hantush)
where(spread(abs(sigma(1:np,1)) > tiny(1.0_EP),2,nz))
sD(1:np,1:nz) = sH(1:np,1:nz) + spread(sigma(:,1),2,nz)*&
& cosh(eta(1:np) .X. s%zD(1:nz))
elsewhere
sD(1:np,1:nz) = sH(1:np,1:nz)
end where
if (s%quiet > 1) then
write(*,999) ' sD:',sD(1:NPRINT,1), ' A1:',sigma(1:NPRINT,1),&
& ' a01:',aa(1:NPRINT,1),' b01:',b(1:NPRINT,1),' c01:',c(1:NPRINT,1)&
& ,' v01:',v(1:NPRINT,1),' omega01:',omega(1:NPRINT,1)
do j=2,n
write(*,998) &
&' '//&
&' a',j,':'&
&,aa(1:NPRINT,j),' b',j,':',b(1:NPRINT,j),' c',j,':',c(1:NPRINT,j)&
& ,' v',j,':',v(1:NPRINT,j),' omega',j,':',omega(1:NPRINT,j)
end do
end if
998 format(5(A,I2.2,A,2('(',ES11.2E4,',',ES11.2E4,')')))
999 format(7(A,2('(',ES11.2E4,',',ES11.2E4,')')))
end function mishraNeuman2010FD
end module laplace_hankel_solutions