Research ArticleA Digital Twin for Unconventional Reservoirs: A MultiscaleModeling and Algorithm to Investigate Complex Mechanisms

Tao Zhang ,1 Yiteng Li,1 Jianchao Cai,2,3 Qingbang Meng ,4 Shuyu Sun ,1,2

and Chenguang Li5

1Computational Transport Phenomena Laboratory, Division of Physical Science and Engineering, King Abdullah University ofScience and Technology, Thuwal 23955-6900, Saudi Arabia2Institute of Geophysics and Geomatics, China University of Geosciences, Wuhan 430074, China3State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China4Key Laboratory of Tectonics and Petroleum Resources, Ministry of Education, China University of Geosciences,Wuhan 430074, China5China National Oil and Gas Exploration and Development Company Limited, China

Correspondence should be addressed to Shuyu Sun; shuyu.sun@kaust.edu.sa

Received 9 July 2020; Revised 8 September 2020; Accepted 15 September 2020; Published 2 November 2020

Academic Editor: Jinze Xu

Copyright © 2020 Tao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The special mechanisms underneath the flow and transport behaviors in unconventional reservoirs are still challenging an accurateand reliable production estimation. As an emerging approach in intelligent manufacturing, the concept of digital twin has attractedincreasing attentions due to its capability of monitoring engineering processes based on modeling and simulation in digital space.The application potential is highly expected especially for problems with complex mechanisms and high data dimensions, becausethe utilized platform in the digital twin can be easily extended to cover more mechanisms and solve highly complicated problemswith strong nonlinearity compared with experimental studies in physical space. In this paper, a digital twin is designed tonumerically model the representative mechanisms that affect the production unconventional reservoirs, such as capillarity,dynamic sorption, and injection salinity, and it incorporates multiscale algorithms to simulate and illustrate the effect of thesemechanisms on flow and transport phenomena. The preservation of physical laws among different scales is always the firstpriority, and simulation results are analyzed to verify the robustness of proposed multiscale algorithms.

1. Introduction

The successes in the commercial exploitation of unconven-tional resources, such as shale gas and tight oil, in NorthAmerica have already changed the current world energymarket, and the growing public concerns on the depletionof conventional oil and gas resources in the foreseeable futurealso stimulates more efforts in both academia and industry toinvestigate unconventional reservoirs [1, 2]. A large numberof technical studies have been carried on for a better charac-terization of oil and gas storage in either or both adsorbed orfree states and a clearer description of the rock propertiesincluding porosity and permeability [3–5]. On the other

hand, environmental issues have challenged the developmentof the unconventional oil and gas resources. As a conse-quence of hydraulic fracturing and shale gas production,groundwater pollution has become a serious issue thathaunts oil companies. Earthquake and gas explosion, theother two problems often criticized due to unconventionalreservoir production, are increasingly arising public con-cerns. In order to achieve a better balance between recoveryefficiency and environmental impacts, we need to pay moreefforts for a thorough understanding of the special mecha-nisms that control the storage, flow, and transport of uncon-ventional resources in subsurface reservoir in order to meetthe growing global energy demands in an environmentally

HindawiGeofluidsVolume 2020, Article ID 8876153, 12 pageshttps://doi.org/10.1155/2020/8876153

friendly and economical approach. For instance, the classicalviewpoint that transient linear flow dominates the flowregime for multifractured horizontal wells is challenged bythe anomalous diffusion behavior, which also enlightens usthat our understanding of the complex mechanisms and flowbehaviors in unconventional reservoirs still needs to beimproved [6].

Numerical modeling and simulation have become a pop-ular approach in the study of unconventional reservoirs,mainly due to their significant superiority on efficiency andflexibility [3]. In practice, experimental studies are limitedto time and space scale and also restricted by field or labora-tory conditions. However, these constraints can be avoidedby well-designed numerical simulation. Moreover, numericalschemes can be further verified through experimental analy-sis, and some key parameters are tuned from experimentalresults. After continual improvement and optimization, suchmathematical model is believed to be “realistic” and is reli-able to be applied to solve practical problems in much largertime and space scale in order to guide or suggest engineeringpractice. As one of the cutting-edge techniques, digital twinshave been extensively used in automatic production, predic-tive maintenance, and complete-cycle management. Digitaltwins, also known as DT, can be defined as a simulation pro-cess or simulation-based system that integrates multidisci-pline, multiphysics, and multiscale numerical methods tomake full use of physical models, sensor data, operationhistory, and other information and to complete mapping invirtual space so as to reflect the whole life cycle of a corre-sponding process or equipment in physical space [7]. Theconcept of digital twins was first proposed in the field ofadvanced manufacturing, along with the application ofstate-of-the-art information technologies in industrial pro-cesses [8, 9]. With the arrival of the big data era, the entireproduct life cycle produces plenty of data in the aspect ofdesigning, manufacturing, marketing, and service. These datacan be transferred into the digital models for simulation andanalysis, and in turn, the numerical results can provide sup-ports to improve practical manufacturing. The data obtainedin physical reality is often fragmented and isolated, but well-designed numerical schemes in virtual space can describe theunderneath physical or chemical correlations among thedata. Thus, the intelligence and efficiency of industrialmanufacturing are constantly improved to achieve intelligentproduction andmanagement. The similar idea can be appliedto reservoir simulation in which data of the reservoir geology,fluid properties, and environmental and operation condi-tions can all feed into the model in virtual space.

In this paper, we will combine several promising numer-ical models and algorithms that describe the special mecha-nisms behind the flow and transport behaviors inunconventional reservoirs in different scales as an explor-atory investigation to construct the digital twin. A thermody-namically consistent flash calculation scheme is designed toconsider the effect of capillary pressure on phase equilibria,and the dynamic sorption is included in the particle distribu-tion function to establish a delicate Lattice Bhatnagar-Gross-Krook (LBGK) scheme to simulate the shale gas flow andtransport using Lattice Boltzmann Method (LBM). Multi-

component ion exchange and double-layer expansion, bothdirectly relevant to fluid salinity, are also modeled in differentscales. Simulation results are presented to show the effective-ness of the proposed algorithms for the investigated mecha-nisms. The remainder of this paper is organized as follows.In Section 2, several important mechanisms in unconven-tional reservoirs are modeled, and the corresponding simula-tion algorithms and results are illustrated in Section 3.Conclusive remarks are provided in Section 4.

2. Design of the Digital Twin

A digital twin is expected to realize interaction and integra-tion between physical space and virtual world due to its fea-sibility on real-time synchronization, real mapping, andhigh fidelity. In order to create the virtual model of physicalentity in unconventional reservoirs, the fluid flow and trans-port process in realistic reservoir conditions are mathemati-cally described in the digital space, while data is transferredinto the twin and information is fed back from the twin.The procedures and processes in physical entity are expectedto be enabled or expanded with new capabilities based on thefeedback, which may be obtained via various approaches,such as data fusion analysis and decision iteration optimiza-tion. In this section, a digital twin for unconventional reser-voirs is designed to bridge the physical space and digitalworld, with focus on several selected mechanisms whichaffect the complete cycle of evaluation and production. Asshown in Figure 1, some processes are extracted first fromthe physical space, such as reserve prediction, recovery eval-uation, hydraulic fracturing, and environmental effect. Toprovide more effective, real-time, and intelligent improve-ment and optimization schemes to these processes, the digitaltwin is designed with three parts: models on reservoir geom-etry, Darcy’s scale fluid flow, and pore scale fluid flow. Eachpart is decomposed further with representative techniquesand research topics. It should be noted that phase equilib-rium calculations are conventionally investigated in porescale, but in [10], Darcy’s scale phase equilibrium wasstudied.

In this paper, we will start to model capillarity anddynamic sorption, both of which play significant roles inunconventional reservoirs and affect many engineering pro-cesses in practice. The widespread nanopores are often con-sidered as the main cause of significant capillary effect andconfinement effect. The strong interactions between gas mol-ecules and rock surface result into the dynamic sorption, andthe related concepts of Knudsen diffusion, Knudsen layer,and Knudsen number are introduced. The large amount ofwater needed for hydraulic fracturing is often questioned bythe local communities, and seawater or produced water afterproper treatment is expected as an alternative to save freshwater. Water salinity, one of the key factors affecting injec-tion and fracturing performance, has also attracted numer-ous attentions. In this section, thermodynamic equilibriummodeling for multicomponent fluid mixtures in unconven-tional reservoirs is used to describe phase behaviors andproperties. A delicate LBGKmodel is constructed to take intoaccount the effect of dynamic sorption on particle

2 Geofluids

distribution functions. For the two processes directly relevantto salinity, the electrical interaction process is represented bymulticomponent ion exchange equations and the electro-static forces in a double-layer expansion process needed fornumerical simulation are calculated using the DLVO-typetheory, short for Derjaguin–Landau–Verwey–Overbeektheory.

2.1. Capillarity Modeling. The presence of capillary pressurecould significantly deviate the phase behaviors of reservoirfluids in unconventional reservoirs from their bulk proper-ties. Therefore, the capillary effect has to be taken intoaccount in order to accurately estimate phase amounts andcompositions of unconventional reservoir fluids at geologicalconditions. Defined as the pressure difference between wet-ting phase and nonwetting phase, pc = pn − pw, the effect ofcapillary pressure can be concluded as moving the interfacetoward the nonwetting phase due to a positive capillary pres-sure and conversely due to a negative capillary pressure. Thework done by capillary pressure can be formulated as

dWdt

= −pndVn

dt− pw

dVw

dt= −pn

dVn

dt+ pw

dVn

dt= −pc

dVn

dt,

ð1Þ

where pc is assumed to be constant along the interfacebetween nonwetting and wetting fluids. Here, the Young-Laplace equation is used to calculate the capillary pressure as

pc =2σ cos θ

r, ð2Þ

with the interfacial tension σ, in the unit of N/M, being esti-mated by the Weinaug-Katz correlation

σ = 〠M

i=1P½ �i ni,w − ni,nð Þ

" #4: ð3Þ

2.2. Dynamic Sorption Modeling. Usually, shale gas can existas three states, including dissolved gas, adsorbed gas, and free

gas, and the dominant reserve is made up of the adsorbed gas,which has been reported in [11] that the adsorbed gas coversup to 80% of the total gas reservation. The dynamic balancebetween adsorption and desorption, as a result of shale struc-tures and fluidity, can provide helpful knowledge to designand optimize fracturing and recovery processes. Such criticalinformation is of significant importance to evaluate the reser-voir and predict the well production. A large number of stud-ies have been reported to analyze and model the dynamicsorption in unconventional reservoirs. Molecular accumula-tion, as a consequence of surface energy minimization, isoften considered the main cause of adsorption on shale sur-face, while the van der Waals force is leading the physicalsorption in potential theory. The sorption capacity is depen-dent on temperature, pressure, and other geochemical prop-erties, such as the TOC content, also known as total organiccarbon. Generally, if there are more organic matters in shale,a higher gas adsorption amount can be detected togetherwith a higher surface area, total pore volume, and porosity.Moreover, permeability, which is the key factor relevant toflow and transport in porous media, is also affected by theadsorption and desorption process in gas production. Forexample, pressure drop facilitates gas desorption from kero-gen, and on the other hand, the free gas production furtherdecreases the pore pressure. As a result, pressure differencebetween the pores and bulk matrix will reinforce the desorp-tion on the matrix surface. Plenty of isotherm models havebeen developed to mathematically describe the sorptionmechanism, among which the most commonly used one isthe Langmuir’s model

V = VLPPL + P

, ð4Þ

where V denotes adsorbate volume, P denotes pressure, andPL and VL denote the Langmuir pressure and Langmuir vol-ume, respectively. Other isotherm models, including Freun-dlich model, D-R model, BET model, and Toth model, areproposed later so that the estimations of these models getcloser to the experimental results. In this paper, the following

Reserves prediction

Recovery evaluation

Hydraulic fracturing

Environmental effect

Physical space

Information

DataPore size distribution

Pore connectivity

Rock properties

Fracture & matrix

Phase equilibrium

Lattice boltzmann method

Pore–network modeling

IMPES

Phase equilibrium

Dynamic sorption

Capillary effect

Phase field

Reservoir geometry Darcy scale fluid flow Pore scale fluid flow

Digital twin for unconventional reservoirs

Figure 1: Design of a digital twin for unconventional reservoirs.

3Geofluids

equation is selected to model the dynamic sorption balancebetween adsorption and desorption

∂V∂t

= kaC Vm − Vð Þ − kdV , ð5Þ

where ka and kd are the adsorption and desorption coeffi-cients, respectively, with the unit of S−1, and Vm denotesthe saturated adsorption capacity. The gas concentration Ccan be calculated from the deformed advection-diffusion LBscheme:

∂C∂t

+∇ · Cuð Þ = ∇ · Deff∇Cð Þ + S: ð6Þ

In the above equation, the total gas concentration at cer-tain location can be calculated as the summation of distribu-tion functions in all the directions:

C =〠gci, ð7Þ

where gci denotes the distribution function in convection-diffusion problems and the source term S; in distribution bal-ance equation, see Equation [eq: eq (6)], which represents theeffect of dynamic sorption which is determined by

S = ωiVsδt: ð8Þ

Correspondingly, the macroscopic velocity can be formu-lated as

〠 f i − z × Sið Þ = ρu, ð9Þ

where Si denotes the adsorbed amount in the site of the ithdirection and z denotes a coefficient balancing the units.

2.3. Multicomponent Ion Exchange Modeling. The process ofmulticomponent ion exchange can be modeled as (using Ca2+ as an example of divalent cations)

Clay− ⋯ Ca2+� �

+H2O↔ Clay− ⋯H+½ � + OH− + Ca2+:ð10Þ

A generalized model has been proposed in [12] for differ-ent ions, e.g., sodium, calcium, and magnesium cations,which are commonly seen in reservoir brine and rocks.

Na+ + 12

Ca − X2ð Þ↔ Na − Xð Þ + 12Ca2+,

Na+ +12

Mg − X2ð Þ↔ Na − Xð Þ + 12Mg2+:

ð11Þ

Two coefficients are presented in their study to model theion exchange selectivity as

KNa/Ca′ =ζ Na − Xð Þ m Ca2+

� �� �0:5ζ Ca − X2ð Þ½ �0:5m Na+ð Þ

×γ Ca2+� �� �0:5γ Na+ð Þ ,

KNa/Mg′ =ζ Na − Xð Þ m Mg2+

� �� �0:5ζ Mg − X2ð Þ½ �0:5m Na+ð Þ

×γ Mg2+� �� �0:5γ Na+ð Þ :

ð12Þ

The ζðMI − XÞ term in the above equations representsthe equivalent fraction of the cations on the exchanger, andMI denotes Na+, Ca2+, or Mg2+.

The wettability alteration is considered to result from ionadsorptions and corresponding surface charge change, whichcan be modeled as [13]

RCOO− −Ca − CaCO3 sð Þ + Ca2+ + SO2−4 = RCOO − Ca+ + Ca − CaCO3 sð Þ + SO2−

4 ,

ð13Þ

where SO2−4 serves as a catalyst increasing Ca2+ ion concen-

tration. A similar formula can be applied to brine containingMg2+ ions

RCOO− − Ca − CaCO3 sð Þ +Mg2+ + SO2−4 = Mg −CaCO3 sð Þ + RCOO − Ca+ + SO2−

4 :

ð14Þ

In high salinity injection, pH reversal may occur and theion exchange can be modeled as

Clay− ⋯H+½ � + Ca2+ ↔ Clay− ⋯ Ca2+� �

+ H+: ð15Þ

Dissolution and precipitation can happen with the inter-action of injected brine and rock

CaCO3 sð Þ + H+ ⇄ Ca2+ + HCO−3 ,

CaSO4 sð Þ⇄ Ca2+ + SO2−4 ,

MgCO3 sð Þ +H+ ⇄Mg2+ + HCO−3 :

ð16Þ

The HCO−3 in Equation [eqim] is also obtained from the

interaction in the aqueous phase as

CO2 + H2O⇄HCO−3 + H+,

HCO−3 ⇄ CO2−

3 + H+,

H2O⇄OH− + H+:

ð17Þ

2.4. Double-Layer Expansion Modeling. The modified DLVOtheory, which describes electrostatic forces, is commonlyused to model and calculate the double-layer expansion pro-cess [14, 15], and the results can be used in numerical simu-lation of the low salinity waterflooding process [16]. Forcecontributions can be modeled as

Π hð Þ =ΠVDW hð Þ +ΠEDL hð Þ +ΠSTR hð Þ, ð18Þ

where the disjoining pressure is a summation of contribu-tions from London van der Waals force, electrostatic force,

4 Geofluids

and structural force, all of which are functions of the wettingfilm thickness h. This formulation can be used in wettabilitycalculation, as the following augmented Young-Laplaceequation uses disjoining pressure to express the interactionequilibrium in oil/brine/rock system [15]:

pc =Π hð Þ + 2σow cos θr

, ð19Þ

where pc is capillary pressure. The contact angel θ is given by[17]

cos θ = 1 +1σow

I: ð20Þ

It can be referred from Equation [eqca] that if I > 0, thewater film is unconditionally stable due to the constant con-tact angel θ = 0∘. If the rock is water-wet, implying 90 > °θ> 0° or 0 < I < −σow, then the water film is meta-stable; oth-erwise, it is oil-wet with 180 > °θ > 90° or −σow < I < −σow.

The van der Waals forceΠVDW in Equation [eqfo] can becalculated by [18]

ΠVDW hð Þ = −A6πh3

, ð21Þ

where A is the Hamaker constant. The electrostatic double-layer force in Equation [eqfo] can be modeled using abounded estimation of the Poisson-Boltzmann equation [19]

d2ψdx2

=enbzε0εr

e− zeψ/kBTð Þ, ð22Þ

where ε0 denotes the dielectric constant of vacuum, εrdenotes the relative dielectric constant of the aqueousmedium, kB denotes the Boltzmann constant, T denotes theabsolute temperature (K), and z denotes the ion valence.An upper limit of the estimation can be established with aconstant potential assumption that attraction between theplates with the same charge but unequal potential alwaysexists

ΠEDL hð Þ = nbkBT2ψr1ψr2 cosh κhð Þ − ψ2

r1 − ψ2r2

sinh κhð Þð Þ2 !

: ð23Þ

On the contrary, if a repulsion force is assumed to existbetween the plates,

ΠEDL hð Þ = nbkBT2ψr1ψr2 cosh κhð Þ + ψ2

r1 + ψ2r2

sinh κhð Þð Þ2 !

: ð24Þ

The term ψri in the above two equations is defined as thereduced potential of ith plate, and it can be calculated by

ψri =zeζikBT

: ð25Þ

The above approximation model is called the analyticalcompression approximation (CA) model, which is suitablefor cases with low to intermediate electrostatic potential. Alinear superposition approximation (LSA) is proposed, alsocalled weak overlap approximation (WOA), as a correctanswer between the two extremes as [20]

ΠEDL hð Þ = 64nbkBT tanhζ14

� �tanh

ζ24

� �exp −κhð Þ,

ð26Þ

where ζi is the zeta potential on the ith surface of each plate.It is proved in [19] that the LSAmodel is more favorable to fitthe force measurement on surface experiments. κ in theabove models are defined as the Debye-Hückel reciprocallength and determined by the following expression:

κ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e2z2nbε0εrkBT

s: ð27Þ

The hydration force can be calculated by [21]

ΠSTR hð Þ = C1eh/ξ1 + C2e

−h/ξ2 , ð28Þ

where C is the force coefficient relevant with the boundaryconditions and h is the clay length. The hydrophobic forcenear the surface can be calculated using

ΠBR hð Þ = −Aδ6

75608rp + h

2rp + h� �7 +

6rp + h

h7

" #: ð29Þ

A more general form has been proposed in [22] to calcu-late the structural forces for all the cases.

Ystructure

= Ak exp −hhs

� �: ð30Þ

3. Simulation and Results

All the aforementioned mechanisms, such as capillarity,sorption, and salinity, have challenged the conventional con-tinuum modeling and simulation using Navier-Stokes equa-tions. Consequently, modifications and improvements haveto be introduced to account for these mechanisms so thatthe governing systems are generated in order to obey thephysical laws as well as realistic conditions. Thermodynamicequilibrium schemes are reconstructed to account for capil-lary effect and to meet the first and second laws of thermody-namics. The calculated equilibrium solutions can ensure agood initial estimate for multiphase multicomponent flowsimulation, and the new energy and entropy balance formu-lations can lead to a stable convergence while tolerating a rel-atively much larger time step. Mesoscopic numericalapproaches, representative of LBM [23] and Pore NetworkMethod [24], are widely employed in the direct simulationof flow and transport in porous media, and some simple

5Geofluids

but effective terms can be added to take such special mecha-nisms into account. The new mass and momentum conserva-tion properties can be proved rigorously, which furtherpromote such delicate numerical approaches. However, thereis still not a comprehensive model or simulation approachcovering all the effect of capillarity, sorption, and salinity. Inanother words, more investigations are expected to develop areliable and practical scheme to simulate engineering pro-cesses, and the concept of a digital twin can be a potential plat-form to cover as much mechanisms as we want. Withconservation equations governing fluid flow through porousmedia, Darcy’s scale simulation can be performed to modelthe migration and transport of oil and gas and predict the oilproduction. Pore scale simulation is conducted to investigatethe detailed mechanisms of surface interactions and to showthe correlation between the microscopic details in a singlepore, thermal equilibrium conditions, and macroscopic flowand transport properties. Mesoscopic simulations, like LBMand pore network modeling, are used as a bridge to linkbetween micromechanism and macrophenomena.

3.1. Darcy’s Scale Simulation. A mass balance equation forimmiscible incompressible oil-water two phase flow can bewritten as [25]

ϕ∂s∂t

+∂uw∂x

= 0,

ϕ∂ 1 − sð Þ

∂t+∂ uo∂x

= 0,ð31Þ

where uo and uw denote the velocity of oil phase and waterphase, respectively. The extended Darcy’s law for aqueousand oil phases can be modeled as follows if gravity is ignored:

uw = −kkrw s, γ, σsð Þ

μw

∂pw∂x

,

uo = −kkro s, γð Þ

μo

∂po∂x

,ð32Þ

where po and pw represent the pressure of oil phase and waterphase, kro and krw are the oil and water relative permeability,respectively. A commonly used model to account for theeffect of salinity on relative permeability and capillary pres-sure has been proposed in [26] as

krw = θ × kHSrw S∗ð Þ + 1 − θð Þ × kLSrw S∗ð Þ,

kro = θ × kHSro S∗ð Þ + 1 − θð Þ × kLSro S∗ð Þ,

pcow = θ × pHScow S∗ð Þ + 1 − θð Þ × PLS

cow S∗ð Þ,ð33Þ

where

θ =Sorw − SLSorw� �SHSorw − SLSorw

� � ,S∗ =

So − Sorwð Þ1 − Swr − Sorwð Þ :

ð34Þ

In the above system, pcow denotes the oil-water capillarypressure, θ is a scaling factor, and Sorw denotes the residualoil saturation after waterflooding. The superscripts HS andLS denote the high salinity water and low salinity water,respectively. It is a simple model capable of predicting theoil recovery using low salinity waterflooding at field-scalestudies or single-well tests. An obvious dependency of capil-lary pressure and relative permeability on injection salinity isexpressed in the formulas, regarding the salt species in brineonly as an additional single lumped component in the aque-ous phase. A balance between the two extreme conditions,lower salinity limit and upper salinity limit, is conductedusing the scaling factor.

A more comprehensive model is proposed in [27] whichstarts from the classical Corey equations [28]

krw = k0rw S∗wð Þnw ,kro = k0ro 1 − S∗wð Þno ,

ð35Þ

where

S∗w =Sw − Swr

1 − Swr − Sor,

Pc =cw

Sw − Swið Þ/ 1 − Swið Þð Þaw −co

1 − Sw − Sorð Þ/ 1 − Sorð Þð Þao :

ð36Þ

The first modification is assuming the residual oil satura-tion is a function of salinity only.

Sor Xcð Þ = SLSor +Xc − XLS

c

XLSc − XHS

c

SLSor − SHSor

� �: ð37Þ

An additional modification introduces the salinity effecton the end-point water relative permeability

krw Xcð Þ = kLSrw +Xc − XLS

c

XLSc − XHS

c

kLSrw − kHSrw

: ð38Þ

Next, the effect of salinity is also applied to the exponentof oil relative permeability

no Xcð Þ = nLSo +Xc − XLS

c

XLSc − XHS

c

nLSo − nHSo

� �: ð39Þ

The salinity is denoted by Xc in the above equations, andthe two thresholds of high salinity and low salinity are repre-sented by the superscript HS and LS.

6 Geofluids

A dimensionless system to model the macroscopic flowof low salinity waterflooding can be written as [29, 30]

∂s∂tD

+∂f s, γð Þ∂xD

= 0,

Ss = Sa0 − Scr γð Þð ÞAs s, γð Þ, f s, γð Þ = f s, γ, Ss γð Þð Þ,∂γs∂tD

+∂γf s, γð Þ∂xD

= 0,

1 = −λ s, γð Þ ∂P∂xD

, λ s, γð Þ = krw s, γ, Ssð Þμoμw

+ kro s, γð Þ,

ð40Þ

where the subscript D in the above system stands for dimen-sionless and s denotes the saturation. The initial condition isprovided as reservoir saturation and formation water salinity(γ) before injection [31]

tD = 0 : s = sI , γ = γI : ð41Þ

Several popular industrial software has been developed tocalculate the mechanisms related with injection salinity.PHREEQC is an industry-standard geochemistry softwarewhich has been successfully applied in the study of low salinitywaterflooding, with emphasis on the electrostatic reaction, ionexchange, and mineral dissolution [32, 33], and used to verifynew models and approaches [12]. UTCHEM simulator isanother widely accepted software in petroleum industry, devel-oped by the University of Texas at Austin, to predict the effectof injected brine with various ion compositions, and theinjected low salinity water is described using the integrated tool,UTCHEM-IPHREEQC [34]. IPHREEQC is also a state-of-artgeochemical engine, and UTCHEM-IPHREEQC is an accu-rate, robust, and flexible tool that enables to model low salinitywaterflooding and many other enhanced oil recovery tech-niques with respect to geochemistry. Later, another three-dimensional equation-of-state-based compositional simulator,also developed by the University of Texas at Austin, UTCOMP,is coupled with IPHREEQC and the effect of hydrocarboncomponents soluble in the aqueous phase on the pH bufferingand other related reactions in the oil/brine/rock system [35].

3.2. Mesoscopic Simulation. In addition to LBMmentioned inSection 2, another representative mesoscopic approach, porenetwork modeling, has also shown promising potentials insimulating the flow and transport behaviors in unconven-tional reservoirs. By constructing a porous network in whichpore bodies are connected through pore throats, such amodel could represent highly irregular structure from theperspective of topology and geometry. After selecting certaindistribution functions and key parameters to control the sizeof pore body and pore throat, the network is connected and adouble permeability media is then constructed for furtherinvestigation. As explained in [11], the two structures, porebody and pore throat, can be treated as fracture and matrix,respectively, and this body-throat connection can be easilyextended to carry on the streaming and collision process ofdistribution functions.

A network constituting of 500 × 500 pore bodies is con-structed as shown in Figure 2, where the black band repre-sents matrix and white band represent fracture. It can beeasily referred that this porous structure is generated bytwo sets of pore parameters, and the corresponding porosityand fluidity is different in these layers. The parameters of twotypes of media are listed in Table 1, and it can be stated thatthe three layers of Media 1 contain more matrix comparedwith the two layers of Media 2. Furthermore, a better mobil-ity is expected in the two layers of Media 2, and more resis-tance may occur in the three layers of Media 1.

After constructing the porous media using pore networkmodeling, the detailed mesoscopic algorithm can bedescribed as follows:

(1) Generate the optimized LBGK scheme with sorptioncoefficients, weight matrix, medium structure, andflow scenario. Determine the inclusive parameters

(2) Apply the free flow distribution function in fractures andtransport distribution function in matrix. The dynamicsorption is then calculated, while at the first iteration, thisadsorption is set to be zero. The free flow distributionfunction for fractured porous media reads as

F = −ϕν

Ku −

εFϕffiffiffiffiK

p uj ju + ϕG, ð42Þ

porous media in shale

Figure 2: A porous media generated using pore network modelwith a designed structure.

Table 1: Parameters for pore network modeling.

Parameter Media 1 Media 2 Unit

Min. pore body inscribed radius 0.0372 0.0625 mm

Max. pore body inscribed radius 0.254 0.366 mm

Mean. pore body inscribed radius 0.125 0.246 mm

Standard deviation 0.128 0.187 mm

7Geofluids

and the distribution function at equilibrium state is

f eqi = ρwi 1 +ei · uϕc2s

+uu : eiei − c2s I

� �2ϕc4s

� �: ð43Þ

(3) Use this calculated adsorption amount to update thefree flow simulation, and further calculate the diffu-sion and transport process

This two-scale LBM can be easily recovered back intoNavier-Stokes equation and advection-diffusion equation,respectively, for fracture and matrix scale LBGK scheme byChapman-Enskog expansion [36]. The effectiveness of thisalgorithm with proper modifications on the generaladvection-diffusion LBGK scheme and the coupling of scalesusing the free flow velocity can be verified with the followingexample with constant gas injection on the left boundary ofFigure 2. The adsorption distribution at different time step is

illustrated in Figure 3. It can be seen free flow is much fasterin Media 2 with more fracture, while adsorption amount ismuch larger in Media 1 with more matrix. The black colorin Figure 3 corresponds to the “fracture” structure. The resultis reasonable in both scales, and it can be concluded that moreadsorption in thematrix may be due to higher saturation sorp-tion amount or adsorption coefficient (referred fromLangmuir-type isothermal models) and can lead to smallerfree flow velocity in fractures. On the contrary, the result ofour scheme is reasonable in both media and the effect ofdynamic sorption on free flow region is illustrated. The effectof porosity in both two scales, fracture and matrix, and theeffect of sorption parameters in a Langmuir-type isothermalsorption model are all tested and analyzed. Generally, theincreasing adsorbed amount in the matrix due to the higheradsorption coefficient or saturation sorption amount willresult in a slower velocity in the free flow scale. However, ifthe increase of adsorbed gas amount is the result of largermatrix porosity, then the free flow velocity could be acceler-ated as the total resistance in the media has been decreased.

100 200 300 400 500500

450

400

350

300

250

200

150

100

50

Adsorption at t = 12001

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

500

450

400

350

300

250

200

150

100

50

Adsorption at t = 22001

100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

500

450

400

350

300

250

200

150

100

50

Adsorption at t = 76001

100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

500

450

400

350

300

250

200

150

100

50

Adsorption at t = 130001

Figure 3: Adsorption distribution at time steps 12000, 22000, 76000, and 130000.

8 Geofluids

3.3. Pore Scale Simulation. Phase equilibrium calculation isessentially needed for generating a physical-meaningful ini-tial phase distribution benefiting further multiphase flowsimulation. The following fugacity equation can be used toestablish thermodynamic equilibrium, which can be calcu-lated based on various equations of state (EOS), for example,Peng-Robinson EOS [37, 38]:

gi ≡ f ig − f io = 0, i = 1,⋯, nm: ð44Þ

Generally, volume constraint is also needed for a com-plete conservation relation, which could be described as

〠q

Nn+1q

ρn+1q

!− φn+1 = 0, q = g,w, o, ð45Þ

where N denotes the mole density, ρ denotes the mass den-sity, and φ is the constitutive equation. Unconditional stabil-ity, the most essential property determining algorithmrobustness and applicability in practice, is preserved by usingthe convex-concave splitting scheme on chemical potential.In details, the chemical potential μiðnÞ is supposed to havetwo components μconvexi ðnÞ and μconcavei ðnÞ and the counter-part splitting can be written as

μconvexi nð Þ = ∂f convex

∂ni=∂f attraction

∂ni− λ

∂f ideal

∂ni: ð46Þ

The unconditional stability of the above semi-implicitscheme has been proved in details in [37]. The evolutionequations of mole and volume can be formulated based on

Onsager’s reciprocal principle as

∂NGi

∂t= 〠

M

j=1ψi,j μGj − μLj

+ ψi,M+1 pL − pG

� �,

∂VG

∂t= 〠

M

j=1ψM+1,j μGj − μLj

+ ψM+1,M+1 pL − pG

� �:

ð47Þ

The computational efficiency and reliability require anenergy stable system consistent with the second law of ther-modynamics. Regarding the Onsager coefficient matrix Ψ,it can be divided into 4 submatrices, shown as below

Ψ =A

BT

B

C

" #: ð48Þ

Here, A = ∂ðμi,2 − μi,1Þ/∂Ni,1, B = ∂ðμi,2 − μi,1Þ/∂V1 = ∂ðp1 − p2Þ/∂Ni,1, and C = ∂ðp1 − p2Þ/∂V1. It is essential toensure the positive definition of the Onsager coefficientmatrix; otherwise, a modified Cholesky factorization will beintroduced to preserve its positive definiteness. Generally,Ψ + E should be sufficiently positive and E is added as a diag-onal matrix with suitable positive entries. This positive defi-nite property can keep the continuous increasing of entropyin the iterations, which will ensure to reach the local maxi-mum using the Newton-Raphson method. The effect of cap-illarity can be illustrated by the difference in tangent-planedistance (TPD) function and phase envelope of fluid mix-tures in porous media with various pore radius. As shownin Figure 4, the TPD function with respect to temperaturerange ½250,700�K and overall molar density range ½0, 10000�mol/m3 is plotted for an EagleFord oil in two cases eitherwith or without capillary effect. Within the specified molardensity and temperature intervals, there is a single two-phase region and the phase boundary between single-phaseand two-phase states is drawn in red and blue for the casewith and without capillary effect, respectively. It can be seenthat capillary pressure can significantly reshape the bulkphase envelope by suppressing the bubble point curve andmeanwhile expanding the dew point curve.

The effect of pore radius on the work done by capillarypressure can be explained in Figure 5. If capillary pressureis taken into account, the dew point pressure will bedecreased in the lower branch and the suppression of dewpoint pressure becomes significant as pore radius decreases.Moreover, dew point pressure increases in the upper branchand deviates more significantly from the dew point curve ofthe bulk phase where the capillary effect is negligible. Theoverall effect of the dew point pressure increasing in theupper branch and decreasing in the lower branch enlargesthe vapor-liquid region compared with bulk phase envelope.

The effect of multicomponent ion exchange on relativepermeability can be modeled as [39]

kr S, βCa, βMg

= F βCa, βMg

kHSr Sð Þ + 1 − F βCa, βMg

h ikLS Sð Þ,ð49Þ

300 350 400 450 500 550 600 650250 700

Temperature (K)

Pres

sure

(MPa

)

0

5

10

15

20

25

30

Confined (r = 10)Confined (r = 5)

Figure 4: Phase envelope for an Eagle Ford oil consideringcapillarity at different pore radii.

9Geofluids

where β is the absorbed cations and the subscripts Ca andMgrepresent the calcium and magnesium ion, respectively. Thescaling function, F, is dependent on the divalent ion adsorp-tion conditions in the precipitation and dissolution processes.

The microscopic displacement efficiency as a function oftrapping number is proposed in [40], using ionic strength (Is)calculation as

Is =12〠i

Z2i mi, ð50Þ

where zi and mi denote the charge and molarity of the fluidspecies i, respectively. The thickness of double electric layeris then determined by

κ−1 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiεrε0kBT2NAe2I

s, ð51Þ

where εr and ε0 denote the relative permeability and the per-meability of the free phase, respectively. A clay mineralmodel is built in [41] in terms of composition, structure,and charge density on the clay surface. Various salinity con-ditions, ranging from freshwater to seawater, have been con-sidered, and system wetness can be either oil-wet or water-wet in that study.

4. Conclusion and Remarks

Due to the tight formations and related special mechanismsoften met in unconventional reservoirs, there are still nonne-gligible public concerns on the economic efficiency, produc-tion safety, and environmental friendship. As an effectiveapproach to describe the flow and transport behaviors in sub-

surface reservoirs, a digital twin is designed in this paper tocover the purposes of media construction, mechanism inves-tigation, and production estimation in physical entity. Repre-sentative mechanisms, such as capillarity, sorption, andinjection salinity, have been mathematically characterizedin details, and multiscale algorithms are developed to simu-late the effect of these mechanisms. Physical reservationsand equivalence between various scales can be preserved inthe generated schemes, and several results are illustrated toprove the reliability and robustness.

More models and algorithms are expected to be includedin the digital twin in the future to construct a more compre-hensive numerical platform capable of simulating realisticengineering cases efficiently and accurately. Extensions to awider range of applications are easy to perform as long asthe physical correlations can be described numerically inthe virtual space. Field scale studies can be enabled by theusage of parallel computing, bound-preserving, reduced-space methods and many other scale coupling techniques[42]. Molecular dynamics simulation and Monte Carlo sim-ulation can also be added into this twin to lay a more solidtheoretical foundation on fundamental microscopic mecha-nisms [43]. Accelerated flash calculations using deep learningalgorithms have also been carried out by many researchers[44, 45], while the pore-scale flash calculation schemes havebeen extended to solve related engineering problems includ-ing carbon dioxide sequestration [46]. Hydraulic fracturingand rock properties are directly relevant with the exploitationof unconventional reservoirs, where numerous models havebeen developed to simulate the observations [47, 48]. Gather-ing and transportation are the connection between reservoirexploitation and market utilization, where the flow andtransport in pipelines can also be modeled to resolve theengineering problem including scaling and corrosion [49,

700600

500400

300

–15

–10

–5

0×107

TPD

min

(pa)

T (K) 20004000

60008000

1000012000

c (mol/m3)

T (k)500200 300 400 600 700

0

10

20

30

P (M

Pa)

Figure 5: Global minimum of TPD function, as well as corresponding phase envelope, as a function of the overall molar density andtemperature for an Eagle Ford oil with (red line) or without (blue line) capillarity.

10 Geofluids

50]. The next step is to find the proper connectors that linkthe many aspects of mechanisms, models, and algorithmsto establish a comprehensive digital twin.

Data Availability

Data are available on request via email sent totao.zhang.1@kaust.edu.sa.

Conflicts of Interest

The authors declare that they have no known competingfinancial interests or personal relationships that could haveappeared to influence the work reported in this paper.

Acknowledgments

The work of Tao Zhang, Yiteng Li, and Shuyu Sun was sup-ported by funding from the National Natural Scientific Foun-dation of China (Grants Nos. 51874262 and 51936001) andKing Abdullah University of Science and Technology(KAUST) through the Grant no. BAS/1/1351-01-01.

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12 Geofluids