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A kd-tree-accelerated hybrid data-driven/model-based approach for poroelasticity problems with multi-fidelity multi-physics data

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Abstract

We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from \citet{kirchdoerfer2016data}, we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.
Computer Methods in Applied Mechanics and Engineering manuscript No.
(will be inserted by the editor)
A kd-tree-accelerated hybrid data-driven/model-based approach for1
poroelasticity problems with multi-fidelity multi-physics data2
Bahador Bahmani ·WaiChing Sun3
4
Received: April 13, 2021/ Accepted: date5
Abstract We present a hybrid model/model-free data-driven approach to solve poroelasticity problems.6
Extending the data-driven modeling framework originated from Kirchdoerfer and Ortiz [2016], we intro-7
duce one model-free and two hybrid model-based/data-driven formulations capable of simulating the8
coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data.9
To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm ac-10
celerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid11
hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid12
solver can switch from a model-based to a model-free approach depending on the availability and the13
properties of the data. Numerical experiments are designed to verify the implementation and compare the14
performance of the proposed model to other alternatives.15
Keywords model-free poromechanics; data-driven modeling; multi-fidelity data; geomechanics16
1 Introduction17
The theory of poromechanics attempts to capture how infiltrating pore fluid interacts with the solid skele-18
ton formed by the solid constituents at the scale of the representative elementary volume where an effective19
medium can be established [Terzaghi et al.,1996,Biot,1941,Coussy,2004]. Poroelasticity is a sub-discipline20
of poromechanics problem that focuses on the path independent response of porous media. It has impor-21
tant applications across multiple disciplines including seismology [Cocco and Rice,2002,Chambon and22
Rudnicki,2001], hydraulic fracture [Detournay and Cheng,1993,Detournay,2016], petroleum engineer-23
ing, reservoir management, geological disposal [Sun,2015,Na and Sun,2017], and biomechanics modeling24
for soft tissues and bones [Cowin,1999].25
Due to the multiphysics nature of the coupled diffusion-deformation process, poromechanics models26
must combine field equations which constraints field variables to obey the balance principle with coupled27
constitutive models which characterize material laws for pore fluid and solid skeleton [Zienkiewicz et al.,28
1999,Sun,2013,Wang and Sun,2017,2019,De Bellis et al.,2017]. One important issue that affects the29
practicality, accuracy, and robustness of the poroelasticity model is the difference in fidelity of the solid30
and fluid constitutive laws. In particular, elastic responses of a variety of porous media such as sandstone31
[Renaud et al.,2013], clay [Borja et al.,1997,Bryant and Sun,2019,Na et al.,2019], sand [Cameron and32
Carter,2009] and bone [Cowin,1999] can be captured quite adequately with the existing state-of-the-art33
models [Borja,2013b] such that the error of a well-calibrated prediction is often within a few percentages.34
Hence, a simple elasticity model calibrated with simple compression or shear tests is often sufficient to35
make forecasting predictions with a narrow confidence interval.36
Corresponding author: WaiChing Sun
Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University , 614 SW Mudd, Mail
Code: 4709, New York, NY 10027 Tel.: 212-854-3143, Fax: 212-854-6267, E-mail: wsun@columbia.edu
2 Bahador Bahmani, WaiChing Sun
However, owing to the difficulty to conduct highly precise experiments and the lack of parametric37
space to characterize the hydraulic responses, a typical prediction of permeability based on the porosity-38
permeability model is expected to have a much higher variance and, in many cases, is considered accurate39
even if predicted benchmark permeability is just within the same order of magnitude [Paterson and Wong,40
2005,Sun et al.,2011a,b,Andr¨
a et al.,2013a,Sun and Wong,2018,Na and Sun,2017,Suh and Sun,2021,41
Heider et al.,2021]. In this case, a calibrated hydraulic model that minimizes the mean square error of the42
Darcy’s velocity or pressure gradient does not yield a reliable forecast due to the much wider confidence43
intervals. This disparity in the fidelity of the elasticity and hydraulic models for porous media has also44
been consistently observed in large-scale multi-research-group benchmark studies such as Andr¨
a et al.45
[2013b,a] and has become a major bottleneck for poromechanics models.46
An alternative to handle this disparity is to introduce a variational model-free approach for the porome-47
chanics problem. First introduced by Kirchdoerfer and Ortiz [2016] and later extended for the constitutive48
manifold [Iba˜
nez et al.,2017,He and Chen,2020], incorporated with digital image correlation [Leygue49
et al.,2018] and adopted to diffusion [Nguyen et al.,2020], the data-driven approach enables one to make50
predictions on the most plausible constitutive responses via a distance minimization algorithm. This dis-51
tance minimization algorithm then chooses either a data point from a material point database or a linear52
embedding manifold that minimizes the error from the conservation laws and compatibility conditions53
such that a physics simulation can be carried out without an explicitly derived constitutive law. While this54
approach holds great promise when a large amount of data is available especially for a low-dimensional55
prediction (e.g., heat flux in 2D, axial stress of a truss element), the predictions in three-dimensional space56
for arbitrary loading paths can be difficult if the amount of data is insufficient or not distributed with a57
sufficient density in the parametric space.58
Recent work, such as Leygue et al. [2018] and Karapiperis et al. [2021] have explored options to address59
the demand of data. In Leygue et al. [2018], collections of non-homogeneous strain data, and the applied60
face of a specimen, which could be obtained from digital image correction, are used to constitute database.61
This database can then be used to compute admissible stress-strain pairs from experimental data. Mean-62
while, Karapiperis et al. [2021] explores the possibility of introducing an on-the-fly sampling technique63
to generate data from sub-scale computations with no prior information needed as well as employ an64
offline goal-oriented sampling technique to incorporate high-fidelity simulations and high-resolution ex-65
periments in the material database.66
In this work, we introduce a more flexible hybrid approach where the solid constitutive law can be67
either model-based (when the fidelity of the elasticity model is sufficiently high) or model-free (when there68
are sufficient data points). Meanwhile, the hydraulic model is replaced by the model-free approach to avoid69
the usage of models with high deviations to make predictions. To cut the CPU time required to conduct70
the search for the closest data point, we introduce a k-dimensional tree search that helps organizing data71
points to accelerate the simulations. Numerical experiments are then conducted to examine and compare72
the fully model-based, the fully data-driven, and the hybrid models.73
The remaining parts of the paper will proceed as follows. We first introduce the formulations that en-74
able data-driven algorithm to replace parts or all of the constitutive laws required to generate incremental75
solution updates for the poroelasticity problems (Section 2). We then introduce a search strategy that en-76
ables us to accelerate the time used to search for the optimal data points which are often the bottleneck77
of the speed for the data-driven models (Section 3). Section 4overviews the data-driven algorithm and its78
implementation aspects. Numerical experiments are then conducted to verify the formulation and test the79
accuracy, robustness, and fidelity of the hybridized and data-driven models (Section 5) followed by a brief80
conclusion that summarized the major findings.81
As for notations and symbols, bold-faced letters denote tensors (including vectors which are rank-one82
tensors); the symbol ’·’ denotes a single contraction of adjacent indices of two tensors (e.g. a·b=aibior83
(c·d)ik =cij djk ); the symbol ‘:’ denotes a double contraction of adjacent indices of tensor of rank two or84
higher (e.g. (C:εe)ij =Ci jkl εe
kl ); the symbol ‘’ denotes a juxtaposition of two vectors (e.g. (ab)ij =aibj)85
or two symmetric second-order tensors (e.g. (αβ)ijkl =αij βkl ).86
Hybrid model/model-free poromechanics 3
2 Hybridized data-driven/model-based poromechanics problem87
This section presents formulations that solve poroelasticity problems either in a fully model-free fash-88
ion or in a hybridized model where either the solid or fluid constitutive law is replaced by a model-free89
data-driven approach. For completeness, we first review the classical poromechanics problem in which the90
solid displacement and pore pressure are the primary unknown variables. We then introduce a data-driven91
model-free algorithm to completely replace classical solid and fluid constitutive laws with the correspond-92
ing data-driven algorithm searched for the optimal data points. Finally, two hybridized formulations are93
provided in each of which either solid or fluid constitutive law follows the classical model-based approach.94
The pros and cons of each formulation for different situations (e.g., availability of data, fidelity consider-95
ation, robustness) are discussed and will be further elaborated in the numerical experiments shown in96
Section 5. Figure 1overviews the main topic of each subsection of the current section.
Solid
Fluid
Data-driven
Model-based
Constitutive law
Solid
Fluid
Fully model-based
(Sec. 2.1)
Hybrid 2
(Sec. 2.4)
Hybrid 1
(Sec. 2.3)
Fully data-driven
(Sec. 2.2)
Fig. 1: Subsections arrangement based on the constitutive assumptions for solid and fluid phases. The first
row indicates the fully model-based assumption. The second row indicates the fully data-driven assump-
tion.
97
2.1 A brief review of field equations for poroelasticity problems98
For completeness, we provide a concise review of the poroelasticity model, which consists of two ma-99
jor components, i.e., the field theory that provides the necessary constraints for the field variables in the100
space-time domain and the material laws that provides the local constitutive updates for both the solid101
skeleton and the fluid constituents. Interest readers may refer to Prevost [1985], Borja and Alarc´
on [1995],102
Zienkiewicz et al. [1999], Coussy [2004], Sun et al. [2013a,2014a], Na and Sun [2017], Na et al. [2019] for a103
more comprehensive treatment for the topic.104
For simplicity, we assume that the deformation of the solid skeleton is, infinitesimal, path independent
and elastic such that e=eeand σ=σ(ee,p) = σ(e,p). We also assume that the flow in the pore space is a
function of the pore pressure gradient such that the Darcy’s velocity q=q(xp)is a function of the pore
pressure p. In this formulation, we follow the trajectory of the solid constituent and express the balance
principles as a function of solid displacement u=usand pore pressure p. By neglecting the inertial term,
the balance of linear momentum and mass on the spatial domain over time tTread (cf. Sun et al.
[2014a]),
x·σ(e,p) + γ=0in ×T, (1)
˙
p
M+B˙
evol +x·q(xp) + s=0 in ×T, (2)
4 Bahador Bahmani, WaiChing Sun
where σis the total stress and evol =tr eis the volumetric strain. We postulate that the total stress can105
be partitioned into the effective stress of the solid skeleton and the pore fluid pressure according to the106
effective stress principle, i.e.,107
σ=σ0(e)BpIin ×T. (3)
Furthermore, γis the body force, Mis the Biot modulus, Bis the Biot coefficient, qis the Darcy’s velocity,
sis the source (sink) term. The definitions of these physical quantities are listed below.
B=1K
Ks, (4)
γ=γs+γf= (1φf)ρsg+φfρfg, (5)
M=KsKf
Kf(Bφf) + Ksφf, (6)
q=φf(vfv), (7)
where Kand Ksare the bulk moduli of the solid skeleton and the solid constituent, respectively, γsand γf
108
are the partial density of the solid and fluid constituents, while ρsand ρfare the intrinsic density of the109
solid and fluid constituents. φfis the porosity, Kfis the bulk modulus of the fluid constituent, and vfand110
vare the velocity of the fluid constituent and the solid skeleton, respectively.111
To compute the boundary value problem, the initial and boundary conditions are specified as follows.112
The initial conditions are u=u0in at t=0 and p=p0in at t=0. Meanwhile, the boundary113
conditions are u=¯uon u×T(prescribed displacement) and σ·n=¯
ton σ×T(prescribed traction),114
p=¯
pon p×T(prescribed pore pressure), and q·n=¯
qon q×T(prescribed fluid flux) where nis115
a unit normal vector pointing outward to the boundary . Furthermore, the following conditions must116
hold: σu=,σu=,qp=and qp=.117
For convenience, we first discretize the governing equations, Eq. (1) and Eq. (2), in time via the implicit
Euler scheme. Given the displacement and pore pressure at time tn, the time-discretized balance principle
within the time interval T[tn,tn+1]can be expressed as,
x·σn+1(en+1,pn+1) + γn+1=0in , (8)
pn+1
M+Bevoln+1+x·qn+1(xpn+1)t+sn+1t=pn
M+Bevolnin . (9)
To complete the initial boundary value problem, the classical model-based approach for poroelasticity118
problem required us to define material constitutive laws such that, given the strain and pore pressure119
gradient, we may obtain updated effective stress and Darcy’s velocity; the first one maps a given strain120
to the effective stress (e,σ0), another one maps a given pore pressure gradient to the Darcy’s velocity121
(xp,q). In what follows, our goal is to introduce a new formulation such that at least one of these two122
constitutive laws, (e,σ0)and (xp,q)can be replaced by the model-free data-driven approach originally123
designed for elasticity problem in Kirchdoerfer and Ortiz [2016].124
Remark 1 Notice that, for porous media with compressible constituents, both the Biot coefficient and the125
Biot modulus may depend on the bulk modulus of the skeleton and hence may evolve if the elastic response126
of the solid skeleton is nonlinear. Here we limit our scope to the types of porous media with the bulk127
modulus of both constituents significantly larger than the effective bulk modulus of the solid skeleton. As128
such, the evolution of both the Biot coefficient and the Biot modulus, Band M, are neglected [Biot,1941,129
Terzaghi et al.,1943,Cryer,1963]. This assumption and the effective stress principle together enable us to130
treat the constitutive laws for the solid deformation and fluid flow as two independent systems.131
2.2 Option 1: Pure data-driven poroelasticity132
Here, our goal is to present a new formulation constrained by the time-discretized balance principle listed133
in Eqs. (8) and (9) without employing any constitutive laws. To do so, we assume that there exists two134
databases, each have a finite number of data points for (e,σ0)and (xp,q). We further assume that there135
Hybrid model/model-free poromechanics 5
are sufficient data points distributed in the parametric space for both material laws such that a complete136
model-free approach is feasible. Notice that the data-driven model-free method developed in this research137
is categorized into non-parametric learning methods Goodfellow et al. [2016] which are basically con-138
sidered data-demanding since they have minimum assumptions about the data-driven model. This is in139
contrast to the parametric methods such as neural networks [Ghaboussi et al.,1998,Wang and Sun,2018,140
Tartakovsky et al.,2020] where there is a stronger pre-assumption about the data-driven model.141
In the following subsections, we first express the mathematical statement for the data-driven scheme142
as a double-minimization problem. Then, we provide a numerical strategy based on the fixed-point (stag-143
gered) method to solve this minimization problem in two steps: global and local minimization steps.144
Finally, we provide a numerical solution for the global minimization based on the Lagrange multiplier145
method. The local minimization step is not covered in this section and will be discussed in section 3sepa-146
rately since it has its specific treatment which is common between all the three formulations.147
2.2.1 Problem statement148
In this section, we present the fully data-driven, constitutive-model-free formulation for the poroelastic-149
ity problem. Our goal is to introduce a minimization problem that constitutes a model-free poroelastic-150
ity solver. As such, we seek solutions from data sets of material responses that weakly satisfy the time-151
discretized balance principle listed in Eqs. (8) and (9) without explicitly introducing any constitutive law152
for either the solid skeleton and the pore fluid.153
If there are an infinite number of error-free data points populating the database, then a constitutive154
manifold can be identified such that one may select elements of the manifold that satisfy the constitutive155
laws. However, material databases is rarely populated with enough data and often contains data with156
noise that makes it impractical to impose such a strict requirement. As an alternative, we follow the idea157
of Kirchdoerfer and Ortiz [2016] where we merely seek solutions that satisfy the balance principles while158
the resultant constitutive responses are the ones closest to but not necessary elements of the set of points in159
the databases. As such, we need to introduce the notion of ”distance” via an appropriate norm we selected160
for the space of admissible solution (e,σ)and (xp,q).161
In other words, we regard the balance principles as the universal law that should not be violated.162
The data-driven method is then designed to generate solutions that fulfill the balance principle while the163
material response at the integration points (e,σ)and (xp,q)are all closest to existing data points in164
the material database. The distance between an admissible response satisfied balance laws and an existing165
data point is then measured by an appropriate norm. We will express the above-mentioned statement as166
a double-minimization problem [Kirchdoerfer and Ortiz,2016,2018,He and Chen,2020,Nguyen et al.,167
2020]. For brevity, we assign a new variable for pressure gradient as r=xp.168
We then define the phase space for poroelasticity at time tn+1as all zsf
n+1= (en+1,σ0
n+1,rn+1,qn+1)
Zsf where Zsf =Zs× Zfis the product space of the solid phase space Zs=Ve×Vσ0and the fluid
phase space Zf=Vr×Vq; in which Ve,Vσ0=L2()ndim
A=1Aare spaces of real-valued symmetric 2nd
order tensor fields with square integrable components, and Vr,Vq=L2()ndim are spaces of real-valued
vector fields with square integrable components. We define the space Cmomentum
n+1as follows:
Cmomentum
n+1=(en+1(un+1),σ0
n+1,pn+1)∈ Zs×Vp| ∇x·σn+1+γn+1=0in ,un+1=¯un+1on u,
σn+1·n=¯
tn+1on σ,σn+1=σ0
n+1Bpn+1Iin ,
where un+1Vu,Vu= [H1()]ndim ,Vp=H1(), and H1denotes the Sobolev space of square-integrable
functions with square-integrable first derivative and ndim denotes the dimension of the spatial domain.
Here, strain tensor en+1(un+1)is a derived quantity of displacement vector un+1through small strain rela-
tion. All members in continuous set Cmomentum
n+1satisfy time discretized conservation of linear momentum
equation, mechanical-related boundary conditions, small deformation relation for strain tensor, and effec-
tive stress principle at time tn+1. The compatibility conditions of the strain field is automatically satisfied
since the small strain tensor is the symmetric part of the displacement field gradient. The space of admis-
6 Bahador Bahmani, WaiChing Sun
sible solutions that satisfy the conservation of mass is denoted by Cmass
n+1and defined as follows:
Cmass
n+1=n(rn+1(pn+1),qn+1,en+1(un+1)) ∈ Zf×Ve|pn+1
M+Bevoln+1+x·qn+1t+sn+1t=
pn
M+Bevolnin ,pn+1=¯
pn+1on p,qn+1·n=¯
qn+1on qo,
where the gradient of pore pressure rn+1(pn+1)is a derived quantity of pressure field herein. The admissi-169
ble solution space Ccoupled
n+1for the poroelasticity problem at time tn+1includes field variables en+1(un+1),170
σ0
n+1,rn+1(pn+1), and qn+1that satisfy all physical constraints at time tn+1. This space exists at the inter-171
section of above defined continuous sets Ccoupled
n+1=Cmomentum
n+1Cmass
n+1. The poroelasticity database at time172
tn+1is denoted by Dsf
n+1⊂ Zsf. This discrete set contains a finite number of elements that stores exper-173
imental data points corresponding to the poroelasticity constitutive laws at time tn+1. The poroelasticity174
database format and its properties will be clarified in details later.175
The data-driven solution ¯zsf
n+1∈ Zsf at time tn+1is the solution of the following double-minimization176
problem:177
¯zsf
n+1=arg
Ccoupled
n+1
min
zsf
n+1∈Dsf
n+1
min
zsf
n+1∈Ccoupled
n+1
zsf
n+1zsf
n+1
2
Zsf
, (10)
where zsf
n+1= (e
n+1,σ0
n+1,r
n+1,q
n+1)is an element of poroelasticity database,178
zsfn+1= (en+1(un+1),σ0n+1,rn+1(pn+1),qn+1)is an element of admissible poroelasticity solution space,179
and k·kZsf is a norm associated with the space Zsf which measures closeness of zsf
n+1and zsfn+1. We180
define the subtract operation on space Zsf as follows:181
zsf
n+1zsf
n+1= (e
n+1en+1,σ0
n+1σ0n+1,r
n+1rn+1,q
n+1qn+1)∈ Zsf. (11)
We define the product norm for the poroelasticity product phase space Zsf as follows:182
zsf
n+1
Zsf =q
zs
n+1
2
Zs+t
zf
n+1
2
Zf(12)
where zs
n+1= (en+1,σ0
n+1)∈ Zs,zf
n+1= (rn+1,qn+1)∈ Zfand k·kZsand k·kZfare the norms for the183
respectively solid and fluid phase spaces. The time step size tin the second term of Eq. (12) is introduced184
to make the unit consistent. We define the following norm for Zs:185
zs
n+1
2
Zs=
(en+1,σ0
n+1)
2
Zs=Z
1
2en+1:Cs:en+1+1
2σ0
n+1:Ss:σ0
n+1d, (13)
where Csand Ssare 4th order symmetric positive definite tensors. As shown in Kirchdoerfer and Ortiz186
[2016], He and Chen [2020], Nguyen et al. [2020], specific Csand Sscan be chosen to from different equiv-187
alent norms, provided that both tensors remain positive definite. The choices of Csand Ssmay affect the188
values of the norm due to the weighting but the resultant normed space is topologically identical to a189
Euclidean space.190
The weighting tensor Csshare the same unit as the elasticity tensor, i.e., Force/Length2, while the unit191
of Ssis the reciprocal of that of Cs. Both the eigenvalues and the spectral directions of these tensors affect192
the values of norms and therefore change how distance is measured and could affect the efficiency of the193
search problems (cf. Mota et al. [2016], Heider et al. [2020]). It is suggested in [Kirchdoerfer and Ortiz,194
2018,Leygue et al.,2018,He and Chen,2020,Nguyen et al.,2020] to select Ss=C1
sfor solid mechanics195
applications. On the other hand, the norm for Zfis defined as,196
zf
n+1
2
Zf=
(rn+1,qn+1)
2
Zf=Z
1
2rn+1·Cf·rn+1+1
2qn+1·Sf·qn+1d, (14)
where Cfand Sfare 2nd order symmetric positive definite tensors. Similarly, these numerical parameters197
control the importance of pressure gradient and Darcy’s velocity vectors in the norm calculations. The unit198
Hybrid model/model-free poromechanics 7
for Cfis the same as hydraulic conductivity unit ( Length4
Force×Time ). This norm, which in our case is of the unit of199
power, has been introduced in Nguyen et al. [2020] for Poisson’s equation.200
Similar to the norm equipped by the solid phase space, the implication of the choice of the specific201
weighting effect for Sfand Cfhas not been previously studied. Nguyen et al. [2020] suggests Sf=C1
f.202
The time increment tused in (12) is the scaling factor to make both terms with the same unit as energy.203
This factor could be directly included in Cfand Sf, but we preferred to be consistent with power-like204
definition of fluid phase space metric.205
To find the stationary points for the double minimization defined in Eq. (10) at time t=tn+1, a global-206
local iteration (which is our method of preference and will be described later) is needed to find both the207
admissible solution zsf
n+1and the discrete data points that minimize the distance defined by the norm in208
Eq. (12). The latter can be done by comparing every point from the material databases to identify the op-209
timized data point from Dsf
n+1for each integration point. However, searching the optimized data points210
from the entire data set can be inefficient for a large database. As such, we consider an adaptive poroe-211
lasticity database where a subset of the plausible data points are collected for each time step to constitute212
a temporal-varying material database constituted by the union of an adaptive solid effective-stress-strain213
database Ds
iand an adaptive fluid pressure-gradient-Darcy-velocity database Df
i
214
Dsf
i={(e,σ0,r,q)|(e,σ0)∈ Ds
i,(r,q)∈ Df
i}, (15)
where the subscript iindicates the snapshot taken at a discrete time step ti. For instance, an admissi-215
ble subset of data can be identified via prior knowledge (e.g., upper and lower bounds of the porosity-216
permeability relationship, correlation structures, etc.) and deductive reasoning. In our last numerical ex-217
periment, we use porosity (the ratio between the void and solid phase volume) to filter out the implausible218
data points. This treatment enables the data-driven solver to narrow down the search of possible solutions219
and therefore enhance the efficiency and reduce the memory requirement. Furthermore, the trade-off be-220
tween computational time and memory will be discussed later in Sec. 3. Note that the adaptive database221
design can also be used as a mean to incorporate an active learning algorithm that generates new data222
points on demand [Lookman et al.,2019,Wang et al.,2021]. The active learning approach as well as other223
algorithms that may identify the feasible subset of data through clustering [Liu et al.,2016,Zhang et al.,224
2019] or other techniques are not discussed in this work but will further be explored in the future.225
Here, the solid data set Ds
n+1is the set of experimental strain and effective stress pairs (e,σ0); one226
point in this data set is a bundle of strain and effective stress together corresponding to one experimental227
observation. Fluid data set Df
n+1is the set of all experimental pressure gradient and Darcy’s velocity pairs228
(r,q); one point in this data set is a bundle of pressure gradient and Darcy’s velocity together corre-229
sponding to one experimental observation. The union of separate databases for stress-strain and hydraulic230
responses is designed for practical reasons because experiments that obtain the stress-strain curves and231
the effective permeability of a specimen are often conducted separately [Bardet,1997,Paterson and Wong,232
2005,Sun and Wong,2018].233
2.2.2 Solution strategy: fixed-point iteration234
Here, we use the fixed-point method to numerically solve the double-minimization statement Eq. (10)235
associated with the fully data-driven poroelasticity problem. The use case of this method is initially shown236
by Kirchdoerfer and Ortiz [2016] for data-driven elasticity problems.237
The data-driven solution must minimize an objective, i.e., the norm defined over poroelasticity phase238
space Eq. (12), with two different sets of constraints. One set of constraints belongs to a discrete set, Dsf
n+1,239
with a finite number of members, but the other one belongs to continuous space, i.e., Ccoupled
n+1, that satisfies240
conservation laws. This minimization statement is categorized into combinatorial optimization problems241
due to the discrete nature of Dsf
n+1, making the problem NP-hard. The fixed point method, or staggered242
method, reduces complexities by proposing a sequential solution algorithm, and it has been used in many243
applications [Felippa et al.,2001,Borden et al.,2012,Hu et al.,2020]. Here, it is used to break down the244
double-minimization into two separate, simpler minimization problems. We solve for one minimization245
problem by assuming the solution for the other one is fixed. In this way, we iteratively solve a minimization246
problem and update for another one until convergence of the solution.247
8 Bahador Bahmani, WaiChing Sun
According to the fixed-point method, we assume that the optimal data points zsf
n+1are known in the248
objective function Eq. (10), therefore we just need to minimize the objective function for unknowns zsf
n+1as249
follows:250
zsf
n+1=arg min
zsf
n+1∈Ccoupled
n+1
zsf
n+1zsf
n+1
2
Zsf , given zsf
n+1∈ Dsf
n+1. (16)
We call this minimization step global minimization. The global solution field zsf
n+1is a product field consists251
of global fields as zsf
n+1= (en+1(un+1),σ0n+1,rn+1(pn+1),qn+1). The global minimization step, geometri-252
cally, project discrete points zsf
n+1onto the continuous space Ccoupled
n+1according to the defined norm in253
Eq. (12); see red dash lines in Fig. 2. In other words, this step finds solutions belong to the physical space254
Ccoupled
n+1that are closest points to the selected data points (from material space).255
In the next step of fixed-point method, we assume the solutions zsf
n+1are known, then we find optimal256
data points zsf
n+1that minimize the objective function Eq. (10) as follows:257
zsf
n+1=arg min
zsf
n+1∈Dsf
n+1
zsf
n+1zsf
n+1
2
Zsf , given zsf
n+1∈ Ccoupled
n+1. (17)
This minimization is defined over the discrete space Dsf
n+1. Since choices for zsf
n+1are finite and there258
is no constraint on data points in the database, the global objective function Eq. (17) defined over is259
minimized if the integrand is locally minimized. In this work, Gaussian quadrature is used to approximate260
the spatial integration. Let all the integration points be elements of a finite set {¯x1, ¯x2, ..., ¯xnint }, then the261
local minimization problem for an integration point ¯xareads,262
zsf
n+1(¯xa) = arg min
zsf
n+1∈Dsf
n+1
[dsf(zsf
n+1,zsfn+1(¯x))]2, given zsf
n+1(¯xa)∈ Ccoupled
n+1, (18)
where dsf(·)is a local distance function between zsf
n+1and zsf
n+1, which is defined as, Eqs. (11), (12), (13),
and (14):
dsf(zsf
n+1,zsf
n+1) = [ds(zs
n+1,zs
n+1)2+tdf(zf
n+1,zf
n+1)2]1/2, (19)
ds(zs
n+1,zs
n+1)2=1
2(e
n+1en+1):Cs:(e
n+1en+1) + 1
2(σ0
n+1σ0n+1):Ss:(σ0
n+1σ0n+1), (20)
df(zf
n+1,zf
n+1)2=1
2(r
n+1rn+1)·Cf·(r
n+1rn+1) + 1
2(q
n+1qn+1)·Sf·(q
n+1qn+1), (21)
where ds(·)and df(·)denote the local distances between solid-related and fluid-related components, re-263
spectively. The technique that solves the local minimization will be discussed in Sec. 3. Geometrically, the264
local minimization step project a point in Ccoupled
n+1onto the data set Dsf
n+1, see blue dash lines in Fig. 2. By265
minimizing the distance defined in Eq. (19), we determine points in the data set (material space) that are266
closest to the conservation laws.267
Each fixed-point iteration consists of two steps [He and Chen,2020]. First, we solve the global mini-
mization Eq. (17) to project the solution coming from material space (database) Dsf
n+1onto physical space
Ccoupled
n+1; see blue dash arrow lines in Fig. 2. Second, we solve local minimization problems to project the
most recent solutions belong to the physical space onto the material space; see red dash arrow lines Fig.
2. Fixed-point iterations continue until there is no change more than a user-defined tolerance in optimal
solutions at time step tn+1. In summary, we solve the double-minimization problem in two consecutive
steps:
Global Step: find the physical field zsf
n+1= (e,σ0,r,q)n+1for a given material field zsf
n+1∈ Dsf
n+1via:
min
zsf
n+1
zsf
n+1zsf
n+1
2
Zsf , such that zsf
n+1∈ Ccoupled
n+1.
Hybrid model/model-free poromechanics 9
min d(·)
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min || · ||Z
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Ccoupled
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D
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Physical Constraints (Conservation Law)
Material Constraints (Database)
z
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¯
z
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1
2
3
Fig. 2: Schematic representation of material zand physical ¯zresponses at a quadrature point during fixed-
point iterations. Numbers show iteration numbers. Star points present the entire database. The solid line
describes the continuous admissible solution space that respects conservation laws. Dash blue arrow lines
depict the local minimization step from the physical manifold to the material manifold for the quadrature
point. Dash red arrow lines depict the global minimization step from the material manifold to the physical
manifold. Note that the global step minimizes the norm over the entire domain (for all quadrature points).
However, the local step minimizes the distance locally at each quadrature point.
Local Step: at each local spatial point ¯xi(quadrature), find the local material point zsf
i,n+1= (e,σ0,r,q)i,n+1
for a given physical point zsfi,n+1(¯xi)∈ Ccoupled
n+1via:
arg min
zsf
i,n+1
[dsf(zsf
i,n+1,zsfi,n+1)]2, such that zsf
i,n+1∈ Dsf
n+1.
We will explain how the global and local steps can be formulated for numerical solutions in Sec. 2.2.3 and268
Sec. 3, respectively.269
2.2.3 Global minimization270
We introduce the functional associated with the global constrained optimization for the fully data-driven
poroelasticity problem. The objective function in Eq. (17) is minimized along with the set of constraints
defined in Ccoupled
n+1. As such, the trial spaces Vuand Vpfor the u/p poromechanics formulation are chosen
to strongly satisfy Dirichlet boundary conditions,
Vu=u:R3|uhH1()i3,u=¯uon u, (22)
Vp=np:R|pH1(),p=¯
pon po. (23)
The optimal solutions un+1Vu,σ0
n+1Vσ0,pn+1Vp, and qn+1Vqare the stationary points of the271
following functional:272
LDD
tot (zsf
n+1,Bn+1;zsf
n+1) = LDD
loss +LDD
momentum +LDD
mass, (24)
where LDD
loss is the original objective (loss) function, and LDD
momentum and LDD
mass are contributions from con-
straints defined in Cmomentum
n+1and Cmass
n+1, respectively. We group all the Lagrange multipliers in Bn+1=
{βmon
n+1,βσ
n+1,βmass
n+1,βq
n+1}where βmon
n+1and βσ
n+1are real-valued vector fields to weakly enforce the balance
of linear momentum and traction boundary conditions, respectively, and βmass
n+1and βq
n+1are real-valued
10 Bahador Bahmani, WaiChing Sun
scalar fields to weakly enforce conservation of mass, and normal Darcy’s velocity boundary conditions.
These terms are obtained as follows:
LDD
loss =Z(dsf(zsf
n+1,zsf
n+1))2d,(25)
LDD
momentum =Zβmom
n+1·x·(σ0
n+1Bpn+1I) + γn+1d
+Zσ
βσ
n+1·(σ0
n+1Bpn+1I·n¯
tn+1)dΓ,
(26)
LDD
mass =Zβmass
n+1(pn+1
M+Bevoln+1+x·qn+1t+sn+1tpn
MBevoln)d(27)
+Zq
βq
n+1(qn+1·n¯
qn+1)dΓ. (28)
Taking the first variation of Eq. (24), using common rules of the calculus of variations [Felippa,1994,
Nguyen et al.,2020], after applying the divergence theorem leads to
δLDD
tot =δLDD
loss +δLDD
momentum +δLDD
mass
=δLDD
u+δLDD
p+δLDD
βmom +δLDD
βmass +δLDD
σ0+δLDD
q+δLDD
βσ+δLDD
βq=0,
where each contribution is as follows:
δLDD
u=Zδun+1·e(un+1)
u:Cs:(en+1e
n+1)d+ZBδun+1·∂evol(un+1)
uβmass
n+1d,
δLDD
p=Zxδpn+1·Cf·(rn+1r
n+1)td+Z
1
Mδpn+1βmass
n+1d+ZBδpn+1xβmom
n+1:Id
Zσ
δpn+1Bβσ
n+1·ndΓZδpn+1Bβmom
n+1·ndΓ,
δLDD
βmom =Zxδβmom
n+1:(σ0
n+1Bpn+1I)d+Zδβmom
n+1·(σ0
n+1Bpn+1I)·ndΓ
+Zδβmom
n+1·γn+1d,
δLDD
βmass =Zδβmass
n+11
M(pn+1pn) + B(evoln+1evoln) + sn+1tdZxδβmass
n+1·qn+1td
+Zδβmass
n+1qn+1·ntdΓ,
δLDD
σ0=Zδσ0
n+1:Ss:(σ0
n+1σ0
n+1)dZδσ0
n+1:xβmom
n+1d+Zσ
δσ0
n+1:(βσ
n+1n)dΓ
+Zδσ0
n+1:(βmom
n+1n)dΓ,
δLDD
q=Zδqn+1·Sf·(qn+1q
n+1)tdZδqn+1· xβmass td+Zq
δqn+1·βqndΓ
+Zδqn+1·βmassntdΓ,
δLDD
βσ=Zσ
δβσ
n+1·((σ0
n+1Bpn+1I)·n¯
tn+1)dΓ,
δLDD
βq=Zq
δβq
n+1(qn+1·n¯
qn+1)dΓ.
We reduce the number of field variables by setting βσ
n+1=βmom
n+1defined on the boundary σand
βq
n+1=tβmass
n+1defined on the boundary q. After some mathematical manipulations, we obtain the
following residuals (corresponding to Euler-Lagrange equations) along with the additional restrictions on
Hybrid model/model-free poromechanics 11
fields βmom
n+1and βmass
n+1as extra boundary conditions βmom
n+1=0on uand βmass
n+1=0 on p:
Ru
n+1=Zδun+1·e(un+1)
u:Cs:(en+1e
n+1)d+ZBδun+1·∂evol(un+1)
uβmass
n+1d=0, (29)
Rp
n+1=Zxδpn+1·Cf·(rn+1r
n+1)td+Z
1
Mδpn+1βmass
n+1d
+ZBδpn+1xβmom
n+1:Id=0,
(30)
Rβmom
n+1=Zxδβmom
n+1:(σ0
n+1Bpn+1I)d+Zσ
δβmom
n+1·¯
tn+1dΓ+Zδβmom
n+1·γn+1d=0, (31)
Rβmass
n+1=Zδβmass
n+11
M(pn+1pn) + B(evoln+1evoln) + sn+1td
Zxδβmass
n+1·qn+1td+Zq
δβmass
n+1¯
qn+1tdΓ=0,
(32)
Rσ0
n+1=Zδσ0
n+1:Ss:(σ0
n+1σ0
n+1)xβmom
n+1d=0, (33)
Rq
n+1=Zδqn+1·Sf·(qn+1q
n+1)xβmasstd=0. (34)
We further reduce number of independent fields (and equations) by the local (point-wise) satisfaction of
Eqs. (33) and (34) via :
σ0
n+1=σ0
n+1+S1
s:xβmom
n+1in , (35)
qn+1=q
n+1+S1
f·xβmass
n+1in . (36)
Remark 2 The coupled system (Eqs. (29), (30), (31), and (32)) is linear even if there is any hidden non-273
linearity in the database. As such, one may simply store the LU factorization (with pivoting) of the tangen-274
tial matrix at the beginning and use the decomposition to facilitate the Gaussian elimination and therefore275
improve the efficiency by avoiding the use of a linear solver at each iteration.276
2.3 Option 2: Hybrid data-driven poroelasticity 1 (model-based solid + data-driven fluid solver)277
In this section, our goal is to introduce an alternative formulation where the fluid constitutive responses278
are determined from the data-driven approach whereas the solid constitutive responses are determined279
from a material model. This treatment is appropriate for a large variety of poroelasticity problems where280
the confidence interval for any given hydraulic model is expected to be significantly larger than the solid281
elasticity counterpart after normalization, e.g., see Fig. 3. Examples of these materials include sandstone,282
clay, rock, and biological tissues where the estimated effective permeability is often considered accurate283
if it is within the same order of the benchmark values whereas the elasticity error is expected to be much284
smaller [Paterson and Wong,2005].285
The idea is that we know an appropriate constitutive law for solid deformation, and there is a database286
Df
n+1for flow constitutive behavior which is a set of finite pairs of pressure gradient and Darcy’s velocity.287
The following derivation is not restricted to a specific solid constitutive law. The only assumption is that288
the constitutive law is derived from an energy potential in the context of hyperelasticity for the small289
deformation limit.290
Since the data-driven part is only accounted for the hydraulic constitutive law, the defined norm for the291
poroelasticity phase space Eq. (12) includes only the fluid contribution. We incorporate solid constitutive292
model as additional constraint in the set Cmomentum
n+1defined in the fully data-driven formulation, and we293
designate the new set by ¯
Cmomentum
n+1to distinguish them. The resultant problem statement for this hybrid294
option reads,295
¯zsf
n+1=arg
¯
Ccoupled
n+1
min
zf
n+1∈Df
n+1
min
zsf
n+1¯
Ccoupled
n+1
zf
n+1zf
n+1
2
Zf
, (37)
12 Bahador Bahmani, WaiChing Sun
(a)
(b)
(c)
Fig. 3: Black solid lines are model predictions reported in Andr¨
a et al. [2013a]. Red star points are ex-
perimental observations reported in Andr¨
a et al. [2013a] for Berea sandstone. The errors between model
predictions and experimental data are considerably less for shear and bulk moduli than permeability. No-
tice that the permeability is plotted in the Log scale, and so the difference between model and experiments
are even greater in real scale. These plots are reproduced from Andr¨
a et al. [2013a], see figures 2(a), 4(a),
and 6(a) in the original reference.
where ¯
Ccoupled
n+1=¯
Cmomentum
n+1∩ Cmass
n+1. Recall that zf
n+1encodes the fluid-related variables of zsf
n+1, i.e.,296
pressure gradient and Darcy’s velocity. The norm k·kZfdefined over the fluid phase space Zfis the same297
as Eq. (14). Note that the unit of the objective function in Eq. (37) is power, but it is energy for the fully data-298
driven Eq. (10). Following the same procedure described in the fully data-driven formulation, we solve the299
above double-minimization by the fixed-point method consisting global and local steps.300
For the global minimization step we have:301
LHYB1
tot (zsf
n+1,Bn+1;zf
n+1) = LHYB1
loss +LHYB1
momentum +LHYB1
mass , (38)
where LHYB1
momentum is almost the same as LDD
momentum defined in Eq. (26) with the only difference that the ef-302
fective stress term is replaced by the constitutive relation; there exits a potential ψ(e)such that σ0(e) = ∂ψ
e.303
In other words, the constitutive relation is imposed strongly (point-wise) herein. Notice that if there is a304
need to define an effective stress field as an independent field, similar to mixed formulations for elasticity305
[Washizu,1975], one could weakly impose the constitutive relation by adding its contribution through a306
tensorial Lagrange multiplier defined over the whole domain. Here, we do not intend to arrive at formu-307
lations with strain or stress fields as independent fields. Because such formulations increase the number308
of unknowns significantly, also they require solution spaces with higher regularity such as Hilbert space309
H(div, )[Arnold and Falk,1988,Korsawe et al.,2006,Teichtmeister et al.,2019,Fahrendorf et al.,2020];310
this regularity is needed to fulfill the continuity condition of normal traction between elements while tan-311
gential traction can be discontinuous. The term LHYB1
mass is exactly the same as LDD
mass defined in Eq. (28) since312
nothing related to the mass balance is changed. According to the norm defined in Eq. (14), the original313
objective (loss) function is as follows:314
LHYB1
loss =Z(df(zf
n+1,zf
n+1))2d, (39)
where the fluid distance function df(·)is defined in Eq. (21). In this hybrid formulation, the effective stress
σ0
n+1is obtained from a constitutive model, such as an hyperelastic strain-energy functional ψ(e(u)), a key
departure from the fully data-driven formulation in Eq. (24). The corresponding first variation of Eq. (38)
reads,
δLHYB1
tot =δLHYB1
loss +δLHYB1
momentum +δLHYB1
mass
=δLHYB1
u+δLHYB1
p+δLHYB1
βmom +δLHYB1
βmass +δLHYB1
q+δLHYB1
βσ+δLHYB1
βq=0, (40)
Hybrid model/model-free poromechanics 13
where:
δLHYB1
u=ZBδun+1·∂evol(un+1)
uβmass
n+1dZδun+1·xβmom :σ0(en+1)
e:e(un+1)
ud
+Zδun+1·σ0(en+1)
e:e(un+1)
u:(βmom n)dΓ
Zσ
δun+1·σ0(en+1)
e:e(un+1)
u:(βσn)dΓ,
δLHYB1
p=Zxδpn+1·Cf·(rn+1r
n+1)d+Z
1
Mδpn+1βmass
n+1d+ZBδpn+1xβmom
n+1:Id
Zσ
δpn+1Bβσ
n+1·ndΓZδpn+1Bβmom
n+1·ndΓ,
δLHYB1
q=Zδqn+1·Sf·(qn+1q
n+1)dZδqn+1·xβmasstd+Zq
δqn+1·βqndΓ
+Zδqn+1·βmassntdΓ.
The remaining terms are identical to the fully data-driven counterparts, i.e., δLHYB1
βmom =δLDD
βmom ,δLHYB1
βmass =315
δLDD
βmass ,δLHYB1
βσ=δLDD
βσ, and δLHYB1
βq=δLDD
βq. For brevity, we refer to those terms defined in the fully316
data-driven section.317
Similar to the fully data driven formulation, we first reduce the number of unknown fields by setting
βσ
n+1=βmom
n+1defined on boundary σand βq
n+1=tβmass
n+1defined on boundary q. After some
mathematical manipulations, we obtain the following residuals (corresponding to the Euler-Lagrange
equations of Eq. (38)) along with the additional restrictions on fields βmom
n+1and βmass
n+1as extra boundary
conditions βmom
n+1=0on uand βmass
n+1=0 on p:
Ru
n+1=ZBδun+1·∂evol(un+1)
uβmass
n+1dZδun+1·xβmom :σ0(en+1)
e:e(un+1)
ud=0, (41)
Rp
n+1=Zxδpn+1·Cf·(rn+1r
n+1)td+Z
1
Mδpn+1βmass
n+1d
+ZBδpn+1xβmom
n+1:Id=0,
(42)
Rβmom
n+1=Zxδβmom
n+1:(∂ψ(en+1)
eBpn+1I)d+Zσ
δβmom
n+1·¯
tn+1dΓ
+Zδβmom
n+1·γn+1d=0,
(43)
Rβmass
n+1=Zδβmass
n+11
M(pn+1pn) + B(evoln+1evoln) + sn+1td
Zxδβmass
n+1·qn+1td+Zq
δβmass
n+1¯
qn+1tdΓ=0,
(44)
Rq
n+1=Zδqn+1·Sf·(qn+1q
n+1)txβmassd=0. (45)
We further reduce number of independent fields (and equations) by the local (point-wise) satisfaction of318
Eq. (45) via:319
qn+1=q
n+1+tS1
f·xβmass
n+1in . (46)
As mentioned in the previous section, the next step in the fixed-point method is to solve the local320
minimization problem. According to Eq. (37), the local minimization for this hybrid scheme reads as follow,321
zf
n+1(¯xa) = arg min
zf
n+1∈Df
n+1
(df(zf
n+1,zf
n+1(¯xa))2, given zf
n+1(¯xa)¯
Ccoupled
n+1, (47)
14 Bahador Bahmani, WaiChing Sun
where ¯xahis an integration point in h. We will discuss how to solve the local sub-problem defined322
at each integration point in Sec. 3.323
Notice that in the data-driven formulation there are three types of computational costs: data availability324
(gathering data), global optimization (solving a system of equations), local optimization (searching inside325
a database). The local optimization part is an NP-hard problem, and its computational cost grows expo-326
nentially by increasing the database size. Therefore, if the number of unknowns (degree of freedoms) is not327
so high while the database required for data-driven schemes is large, local optimization is the dominant328
source of computational cost. As a result, the hybrid formulation could be more efficient than the fully329
data-driven counterpart if the solid behavior can be accurately captured by a constitutive law with an ac-330
ceptable standard deviation from the ground-truth. We will discuss the computational issues concerning331
the local optimization step in Sec. 3.