""" Patchy cement model ==================== """ # %% import numpy as np import matplotlib.pyplot as plt plt.rcParams['font.size']=14 plt.rcParams['font.family']='arial' plt.rcParams['axes.labelpad'] = 10.0 #plt.rcParams["font.weight"] = "bold" # %% # import the module from rockphypy import EM, GM # %% # # Patchy cement model (PCM) is proposed by Avseth (2016) and mainly applied to high porosity cemented clean sandstone. *Patchy* means the sandstone is weakly to intermediately cemented. The microstructure of the represented as an effective medium comprising a mixture of two end member: cemented sandstones (where all grain contacts are cemented) and loose, unconsolidated sands. # The cemented sandstone can be modeled using the # Dvorkin-Nur model, whereas the loose sand end member can be # modeled using the Hertz-Mindlin (or Walton) contact theory. The effective dry-rock moduli for # a patchy cemented high-porosity end member then can be formulated as follows: # # - Assuming stiff isotropic mixture according to the Hashin-Shtrikman upper bound, # # .. math:: # K_{patchy}=K_{cem}+\frac{(1-f)}{\left(K_{unc}-K_{cem}\right)^{-1}+f\left(K_{cem}+\frac{4}{3} \mu_{cem}\right)^{-1}} # # # .. math:: # \mu_{patchy}=\mu_{cem}+\frac{(1-f)}{\left(\mu_{unс}-\mu_{cem}\right)^{-1}+2 f\left(\frac{K_{cem}+2 \mu_{cem }}{5 \mu_{cem}\left(K_{cem}+\frac{4}{3} \mu_{cem}\right)}\right)} # # # - Assuming soft isotropic mixture according to the Hashin-Shtrikman lower bound # # .. math:: # K_{patchy}=K_{unc}+\frac{f}{\left(K_{cem}-K_{unc}\right)^{-1}+(1-f)\left(K_{unc}+\frac{4}{3} \mu_{unc}\right)^{-1}} # # # .. math:: # \mu_{patchy}=\mu_{unc}+\frac{f}{\left(\mu_{cem}-\mu_{unc}\right)^{-1}+2 (1-f)\left(\frac{K_{unc}+2 \mu_{unc}}{5 \mu_{unc}\left(K_{unc}+\frac{4}{3} \mu_{unc}\right)}\right)} # # where :math:`K_{cem}` and :math:`K_{unc}` are the dry-rock bulk moduli of cemented rock and unconsolidated rock, respectively; :math:`\mu_{cem}` and :math:`\mu_{unc}` are the ditto dry-rock shear moduli; and :math:`f` is the volume fraction of cemented rock in the binary mixture of cemented and unconsolidated rock of the patchy cemented rock. # # The ``GM.pcm`` is the implementation of the patchy cement model provided by ``rockphypy``. # # Example # ^^^^^^^ # Rock-physics modeling of patchy cemented sandstone, for which the connected patchy cement and disconnected patchy cement are modeled. # # %% # specify model parameters Dqz, Kqz, Gqz = 2.65, 36, 42 ## grain density, bulk and shear modulus Dsh, Ksh, Gsh = 2.7, 21, 7 # shale/clay density, bulk and shear modulus Dc,Kc, Gc =2.65, 36, 42 # cement density, bulk and shear modulus vsh=0 _,_,K0=EM.VRH(np.array([vsh,1-vsh]),np.array([Ksh,Kqz])) # clay fraction can be considered. _,_,G0=EM.VRH(np.array([vsh,1-vsh]),np.array([Gsh,Gqz])) phic=0.4 # critical porosity f=np.linspace(0,1,6) # volume fraction of the stiff phase in the binary mixture. sigma=10 phi=np.linspace(1e-6,phic,100) Cn=6 v_cem=0.1 v_ci=0.111 scheme=2 f_=0 #reduce shear factor # %% fig=plt.figure(figsize=(6,6)) plt.xlabel('Porosity',labelpad=10) plt.ylabel(r'K_{\rm dry} (GPa)',labelpad=10) plt.xlim(0,0.4) plt.xticks([0, 0.1, 0.2, 0.3,0.4], ['0', '0.1', '0.2', '0.3','0.4']) #plt.yticks([0, 10, 20, 30,40], ['0', '1', '2', '3','4']) plt.ylim(0,40) #plt.title('Connected patchy cement') for i,val in enumerate(f): if val==0: kwargs = {'color':"orange", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } elif val==1: kwargs = {'color':"darkgreen", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } else: kwargs = {'color':"grey", # for edge color 'linewidth':2, # line width of spot 'linestyle':'--', # line style of spot } Kdry,Gdry=GM.pcm(val,sigma, K0,G0,phi, phic,v_cem,v_ci, Kc,Gc,Cn=Cn, mode='stiff',scheme=scheme,f_=f_) plt.plot(phi,Kdry,**kwargs) plt.scatter(0.25,30,s=4000, c='darkgreen') plt.scatter(0.25,30,s=1700, c='orange') plt.title('Connected patchy cement model') # %% fig=plt.figure(figsize=(6,6)) plt.xlabel('Porosity',labelpad=10) plt.ylabel(r'K_{\rm dry} (GPa)',labelpad=10) plt.xlim(0,0.4) plt.xticks([0, 0.1, 0.2, 0.3,0.4], ['0', '0.1', '0.2', '0.3','0.4']) plt.ylim(0,40) #plt.title('Disconnected patchy cement') for i,val in enumerate(f): if val==0: kwargs = {'color':"orange", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } elif val==1: kwargs = {'color':"darkgreen", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } else: kwargs = {'color':"grey", # for edge color0 'linewidth':2, # line width of spot 'linestyle':'--', # line style of spot } Kdry,Gdry=GM.pcm(val,sigma, K0,G0,phi, phic,v_cem,v_ci, Kc,Gc,Cn=Cn,mode='soft',scheme=scheme,f_=f_) plt.plot(phi,Kdry,**kwargs) # plot HS coating relation animation plt.scatter(0.25,30,s=4000, c='orange') plt.scatter(0.25,30,s=1700, c='darkgreen') plt.title('Disconnected patchy cement model') # %% # Estimate stress sensitivity of sandstone using PCM # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # By varing the effective stress at given porosity, the stress sensitivities of patchy cement sandstone can be modeled. # # %% sigma=np.linspace(1e-7,20,100) # varing effective stress phi=0.36 # fix porosity fig=plt.figure(figsize=(6,6)) plt.xlabel('Effective stress (MPa)') plt.ylabel(r'K_{\rm dry} (GPa)') plt.xticks([0, 5, 10, 15,20], ['0', '5', '10', '15','20']) #xticks([0,5,10,15,20]) #ax1.set_xticklabels(['0', '5','10','15','20']) plt.xlim(0,20) plt.ylim(0,8) #plt.subplot(121) plt.title('Connected patchy cement\n\phi=0.36') for i,val in enumerate(f): if val==0: kwargs = {'color':"orange", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } elif val==1: kwargs = {'color':"darkgreen", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } # elif val==0.8: # kwargs = {'color':"red", # for edge color # 'linewidth':3, # line width of spot # 'linestyle':'--', # line style of spot # } else: kwargs = {'color':"grey", # for edge color 'linewidth':2, # line width of spot 'linestyle':'--', # line style of spot } Kdry,Gdry=GM.pcm(val,sigma, K0,G0,phi, phic,v_cem,v_ci, Kc,Gc,Cn=Cn,mode='stiff',scheme=scheme,f_=f_) plt.plot(sigma,Kdry,**kwargs) plt.text(10,Kdry[60]+0.1,'{:.1f}'.format(val)) # %% fig=plt.figure(figsize=(6,6)) plt.xlabel('Effective stress (MPa)') plt.ylabel(r'K_{\rm dry} (GPa)') plt.xticks([0, 5, 10, 15,20], ['0', '5', '10', '15','20']) plt.xlim(0,20) plt.ylim(0,8) plt.title('Disconnected patchy cement\n\phi=0.36') for i,val in enumerate(f): if val==0: kwargs = {'color':"orange", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } elif val==1: kwargs = {'color':"darkgreen", # for edge color 'linewidth':3, # line width of spot 'linestyle':'-', # line style of spot } # elif val==0.8: # kwargs = {'color':"darkblue", # for edge color # 'linewidth':3, # line width of spot # 'linestyle':'--', # line style of spot # } else: kwargs = {'color':"grey", # for edge color 'linewidth':2, # line width of spot 'linestyle':'--', # line style of spot } Kdry,Gdry=GM.pcm(val,sigma, K0,G0,phi, phic,v_cem,v_ci, Kc,Gc,Cn=Cn,mode='soft',scheme=scheme,f_=f_) plt.plot(sigma,Kdry,**kwargs) plt.text(10,Kdry[60]+0.06,'{:.1f}'.format(val)) # %% # **Reference** # # - Avseth, P., Skjei, N. and Mavko, G., 2016. Rock-physics modeling of stress sensitivity and 4D time shifts in patchy cemented sandstones—Application to the Visund Field, North Sea. The Leading Edge, 35(10), pp.868-878. # # - Yu, J., Duffaut, K., and Avseth, P. 2023, Stress sensitivity of elastic moduli in high-porosity-cemented sandstone — Heuristic models and experimental data, Geophysics, 88(4) #