Non-intrusive model reduction for a 3D unstructured meshcontrol volume finite element reservoir model and its

application to fluvial channels

D. Xiaoa,c,∗, F. Fanga, C.C. Paina, I.M. Navonb, P. Salinasa, A. Muggeridgea

aApplied Modelling and Computation Group,Department of Earth Science and Engineering,Imperial College London,

Prince Consort Road, London, SW7 2BP, UK.URL:http://amcg.ese.imperial.ac.ukbDepartment of Scientific Computing, Florida State University,Tallahassee, FL, 32306-4120, USA

cChina University of Geosciences, Wuhan, 430074, China

Abstract

A non-intrusive model reduction computational method using hypersurfaces represen-tation has been developed for reservoir simulation and further applied to 3D fluvialchannel problems in this work. This is achieved by a combination of a radial basisfunction (RBF) multi-dimensional interpolation method and proper orthogonal decom-position (POD). The advantage of the method is that it is generic and non-intrusive, thatis, it does not require modifications to the original complexsource code, for example,a 3D unstructured mesh control volume finite element (CVFEM)reservoir model usedhere.

The capability of this non-intrusive reduced order model (NIROM) based on hy-persurfaces representation has been numerically illustrated in a horizontally layeredporous media case, and then further applied to a 3D complex fluvial channel case. Bycomparing the results of the NIROM against the solutions obtained from the high fi-delity full model, it is shown that this NIROM results in a large reduction in the CPUcomputation cost while much of the details are captured.

Keywords: RBF, POD, reservoir, hypersurface, 3D fluvial channel

1. Introduction

The simulations of reservoir are very important and have a wide range of applica-tions, from ground-water production to radioactive wasterand the extraction of oil andgas from the subsurface. 3D reservoir modelling provides more details for multiphaseflows in porous media. However, the computational cost of 3D reservoir simulationsis intensive. The non-intrusive reduced order modelling technique presented here iscapable of resolving 3D reservoir modelling problems whileavoiding the high compu-tational cost.

∗Corresponding authorEmail address:dh.xiao@imperial.ac.uk (D. Xiao)

Preprint submitted to Elsevier February 22, 2016

The reduced order modelling technique has been shown to be a powerful capabil-ity of representing the large dynamical systems using only afew number of reducedorder basis functions. Among the model reduction methods, the POD approach is themost widely used. This method extracts the most energetic parts of the system throughsnapshots method, and then constructs optimal basis functions. The POD method andits variants have been successfully applied to numerous research fields. In geophys-ical fluid dynamics it is referred to as empirical orthogonalfunctions (EOF) [1], insignal analysis and pattern recognition it is termed as Karhunen-Love method [2] andin statistics it is called the principal component analysis(PCA) method [3]. The PODtechnique has also been applied to ocean models[4, 5, 6, 7, 8], air pollution modeling[9], data assimilation [10, 11, 12, 13, 14] and mesh optimization [15].

POD in combination with the Galerkin projection method is aneffective method forderiving a reduced order model (ROM). By projecting the 3D dynamical system ontoa reduced space which is constructed based on the optimal basis functions, the com-putational efficiency can be enhanced by several orders of magnitude. However, thismethod suffers from numerical instability [16]. Various methods have been proposedto overcome or improving the stability issue of the POD/Galerkin projection method,including non-linear Petrov−Galerkin [5, 17], regularisation [18], subgrid-scale mod-elling, calibration [19, 20] and Fourier expansion [21]. Another issue that arises in theROMs is the efficient treatment of non-linear terms in the partial differential equations(PDEs). A number of efficient non-linear treatment methods have been presented, forexample, the empirical interpolation method (EIM)[22] and its discrete version discreteempirical interpolation method (DEIM) [23], residual DEIM [4], Petrov−Galerkin pro-jection method [19], Gauss−Newton with approximated tensors (GNAT) method [24]and the quadratic expansion method [25, 26]. However, these methods are still depen-dent on the governing equations of the full physical system.In most cases the sourcecode describing the full physical system has to be modified inorder to form the reducedorder model.

To circumvent these shortcomings, more recently, non-intrusive methods have beenintroduced into ROMs, which do not require the knowledge of the governing equationsand the original source code [27]. Chenet al. proposed a non-intrusive model reductionmethod based on black-box stencil interpolation method andmachine learning method[27]. Waltonet al.proposed a non-intrusive reduced order technique for unsteady fluidflow using RBF interpolation and POD [28]. Audouzeet al.proposed a non-intrusivereduced order modelling approach for nonlinear parametrized time-dependent PDEsbased on a two-level POD method. This method is verified and validated using Burg-ers equation and convection-diffusion-reaction problems [29, 30]. Xiao et al.presentedthree non-intrusive reduced order methods for Navier-Stokes equations using hypersur-faces representation. The hypersurfaces are established by POD and RBF interpolation,Smolyak sparse grid and Taylor series expansion method [31, 32]. In addition, the hy-persurface method based on NIROM has been successfully applied to fluid-structureinteraction problems [33].

Recently, reduced order methods (e.g, POD, POD/DEIM, trajectory piecewise lin-earisation and bilinear approximation techniques) have been applied to reservoir mod-elling [34, 35, 36, 37, 38, 39, 40, 41]. Heijn et al. [34] and Cardosoet al. [35, 36]first developed POD reduced order models for reservoir simulation. Chaturantabutet

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al. [37], Yang et al. [38] and Yoonet al. [39] further introduced DEIM into modelreduction for non-linear flows [37]. Again, these reduced order methods are intrusiveand equations/codes dependent. There are very few studies in non-intrusive reducedorder modelling in porous media flows and 3D fluvial channel problems. Klie first pro-posed a non-intrusive model reduction approach based on a three-layer neural networkcombined with POD and DEIM to predict the production of oil and gas reservoirs[41], where the RBF neural network is used for developing learning functions frominput-output relationships. In this work, we used RBF as an interpolation method forconstructing the time-dependent POD ROM.

In this work we extend the hypersurfaces based NIROM [31] to 3D reservoir mod-elling, and applied to a fluvial channel problem. The noveltyof this work lies in thehypersurfaces representation of 3D reservoir modelling ona reduced space under theframework of the Imperial College Finite Element ReservoirSimulator (IC-FERST).In this approach, solutions to the full fidelity 3D reservoirmodel are recorded usingthe snapshot methods, and from these snapshots POD bases aregenerated that opti-mally represent the 3D reservoir simulation. The RBF interpolation method is thenused to form a set of hypersurfaces (interpolation functions) that approximate the time-dependent ROM. After obtaining hypersurfaces, the solution of ROM at the currenttime level can be calculated by inputting POD coefficients of earlier time levels intothe hypersurfaces. The capabilities of results from the newNIROM have been as-sessed by two 3D reservoir simulation test cases.

The structure of the paper is as follows: section2 presents the governing equationsof the 3D reservoir model; section3 presents the reduced order modelling method usinghypersurfaces representation; section4 illustrates the methodology derived by meansof two numerical examples. The illustration consists of twotest problems where ahorizontally layered porous media test case and a 3D fluvial channel case are resolved.Finally in section5, the conclusion is presented.

2. Governing equations of 3D reservoir modelling

The governing equations used in the underlying 3D reservoirmodel are given inthis section. The darcy’s law for immiscible multiphase flowin porous media has theform:

qα = −Krα

µαK (∇pα − suα) , (1)

whereqα is theαth phase Darcy velocity. TheKrα is the relative permeability of theαth phase, and it is a function that is denoted byKrα (Sα) corresponding to the phasesaturation variableSα. pα is the pressure of theαth phase, which may include capillarypressure.K is the absolute permeability tensor of the porous medium.µα andsuα arethe phase dynamic viscosity and source term respectively, which may include gravity.

A saturation-weighted Darcy velocity is introduced into the equation (1) and de-fined as

vα =qαSα, (2)

3

then equation (1) can be rewritten as follows:

uα = σαvα = −∇pα + suα, σ

α= µαSα

(KrαK)−1 (3)

whereuα denotes the force per unit volume, which is defined asσαvα and used as a

prognostic variable in this approach. Theσα

represents the implicit linearisation of the

viscous frictional forces.The saturation equation can be written as:

φ∂Sα∂t+ ∇ · (vαSα) = scty,α, (4)

whereφ denotes the porosity. Thet is time andscty,α is a source term of theαth phase.Finally, equation (4) is bounded by the constraint:

Nα∑

α=1

Sα = 1, (5)

whereNα denotes the number of phases.

2.1. Discretisation of the governing equations

The Discretisation of the above equations (1)-(5) at time leveln can be written in ageneral form:

Anvvn = sn

v, Anppn = sn

p, AnSSn = sn

S (6)

wherevn = (vn1, . . . , v

nα, . . . , v

nNα

)T , pn = (pn1, . . . , p

nα, . . . , p

nNα

)T andSn = (Sn1, . . . ,S

nα, . . . ,S

nNα

)T.vnα, pn

α and Snα are velocity, pressure and saturation vectors of phaseα, and vn

α =

(vα,1, . . . , vα,N)T , pnα = (Pα,1, . . . ,Pα,N)T , Sn

α = (Sα,1, . . . ,Sα,N )T , andN is the num-ber of nodes.

3. Model reduction for 3D reservoir modelling

In this section, the process of deriving a non-intrusive ROM(NIROM) for 3D reser-voir model is described.

3.1. Proper Orthogonal Decomposition (POD) formulation

POD is a technique used to find a set of optimal basis functionsfrom the snapshotsof solutions obtained from the original model. The optimal POD basis functions arethen used to formulate a reduced dynamical system that contains the main features ofthe flow. Due to the optimality of convergence in terms of kinetic energy of the PODbasis functions, dominant components of a large dimensional process can be capturedwith only a small number of basese.g., 10− 100.

In this work, we find a set of basis functions for each phase (α) of the variables:velocity vα, pressurepα and saturationSα. At time level n, those variables can beexpressed:

vnα = vα + ΦT

α,vvnr,α, pn

α = pα + ΦTα,ppn

r,α, Snα = Sα + ΦT

α,SSnr,α, α ∈ {1, 2, . . .Nα} (7)

4

wherevα, pα and Sα are the mean of the ensemble of snapshots for the variablesvnα, pn

α and Snα respectively,Φα,v = (Φα,v,1, . . . ,Φα,v,M), Φα,p = (Φα,p,1, . . . ,Φα,p,M),

Φα,S = (Φα,S,1, . . . ,Φα,S,M) are the POD bases forvnα, pn

α andSnα respectively, which are

extracted through truncated singular value decomposition, andM is the number of PODbases used in the POD model.vr,α, pr,α andSr,α denote phaseα ’s POD coefficientsof velocity, pressure and saturation respectively.vn

r,α = (vnr,α,1, . . . , v

nr,α, j, . . . , v

nr,α,M)T ,

pnr,α = (pn

r,α,1, . . . , pnr,α, j, . . . , p

nr,α,M)T andSn

r,α = (Snr,α,1, . . . ,S

nr,α, j, . . . ,S

nr,α,M)T .

Projecting equation (6) onto the reduced space, yields:

ΦTα,vA

nα,vΦα,vv

nα = sn

v, ΦTα,pAn

α,pΦα,ppnα = sn

p, ΦTα,SAn

α,SΦα,SSnα = sn

S (8)

The ROM for solving the POD coefficientsvnr,α, j, pn

r,α, j andSnr,α, j (wherej ∈ {1, 2, . . .M},

α ∈ {1, 2, . . .Nα}) at time leveln can be written in the general form:

vnr,α, j = fα,v, j(vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα , pn−1r,Nα ,S

n−1r,Nα , ),

pnr,α, j = fα,v, j(vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα,Sn−1

r,Nα, ),

Snr,α, j = fα,v, j(vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα,Sn−1

r,Nα, ), (9)

subject to the initial condition

v0r,α, j = ((v0

α, j − vα),Φα, j), p0r,α, j = ((p0

α, j − pα),Φα, j), S0r,α, j = ((S0

α, j − Sα),Φα, j),(10)

where (vn−1r,1 , p

n−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα,Sn−1

r,Nα, ) denotes a complete

set of POD coefficients for solution fields velocitiesv, pressuresp and saturationsS ofall phases at time stepn− 1 (n ∈ {1, 2, . . . ,Nt}), Nt is the number of time levels in thecomputational simulation.

3.2. Hypersurfaces based on RBF interpolation

The RBF is an efficient method for interpolation problems. In this work, the RBFinterpolation method is used to construct a set of hypersurfaces for reduced order 3Dreservoir model. The theory of RBF interpolation is briefly reviewed in this section.The RBF interpolation method constructs a approximation function using the form of,

f (x) =N∑

i=1

wi φ(‖x − xi‖), (11)

where the interpolation function (hypersurface)f (x) is represented as a linear com-bination of N RBFs (φ). N denotes the number of RBFs. Each RBF is associatedwith a different centerxi , and weighted by a coefficient wi . x denotes a data pointin multidimensional space and it consists of a complete set of POD coefficients forsolution fields of all phases such as velocity and pressure, saturation. In this workx = (vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα,Sn−1

r,Nα, ). ‖x − xi‖ is a scalar

distance defined by theL2 norm.

5

The weight coefficientswi are determined by ensuring that the interpolation func-tion values f (x) will match the given data y exactly. This is achieved by enforcingf (x) = y, which produces a linear equation

Bw= y, (12)

where

B =

φ (‖x1 − x1‖2) φ (‖x1 − x2‖2) · · ·φ (‖x1 − xn‖2)φ (‖x2 − x1‖2) φ (‖x2 − x2‖2) · · ·φ (‖x2 − xn‖2)

......

...

φ (‖xn − x1‖2) φ (‖xn − x2‖2) · · ·φ (‖xn − xn‖2)

, (13)

w = [w1,w2, ...,wn]T , y = [y1, y2, ..., yn]T (14)

The weight coefficientsw j are then determined by solving the linear system (12) Aw=y. How to define an appropriate RBFφ is also important, a number of most well-known choices forφ are listed in table1, wherer > 0 is a radius andσ > 0 is ashape parameter. The standard RBFs have two major types: Infinitely smooth RBFs

Table 1: some well-known RBFsType Name of functions Definition

Gaussian (GA) φ(r) = e−(r/σ)2

Type I Multi-Quadratic φ(r) =√

r2 + σ2

Inverse Multi-Quadratic φ(r) = 1√r2+σ2

Inverse Quadratic φ(r) = 1r2+σ2

Type II Thin Plate Spline φ(r) = r2log r

and Infinitely smooth (except at centres) RBFs. [42, 43].Type I: Infinitely smooth RBFs are infinitely differentiable and dependent on the

shape parameterσ.Type II : Infinitely smooth (except at centres) RBFs are independenton shape pa-

rameter and are not differentiable infinitely.The infinitely smooth RBFs ensure the matrices A in equation12non-singular and

symmetric [44]. The basis functions of Infinitely smooth (except at centres) RBFs arecomparatively less accurate than that of Infinitely smooth RBFs [42].

3.3. Hypersurfaces representing the reduced system of the 3D reservoir model

In this section the procedure of forming a set of hypersurfaces for 3D reservoirmodel in a reduced space is described. This is achieved by using Gaussian RBF in-terpolation method to construct the ROM in (9). By applying the RBF method, a setof hypersurfacesf n

α,v, j , f nα,p, j and f n

α,S, j for each POD coefficientvnr,α, j, pn

r,α, j andSnr,α, j

6

( j ∈ {1, 2, . . . ,M}, α ∈ {1, 2, · · · ,Nα}) may be approximately represented by the inter-polation functions below:

vnr,α, j = fα,v, j(vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα,Sn−1

r,Nα, ) =

N∑

i=1

wα,v,i, j ∗ φ(r i),

pnr,α, j = fα,p, j(vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα , pn−1r,Nα ,S

n−1r,Nα , ) =

N∑

i=1

wα,p,i, j ∗ φ(r i),

Snr,α, j = fα,S, j(vn−1

r,1 , pn−1r,1 ,S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α ,S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα,Sn−1

r,Nα, ) =

N∑

i=1

wα,S,i, j ∗ φ(r i), (15)

whereφ(r i) is the RBF whose values depend on the distance from a collection ofcentre points,(vn−1

r,1 , pn−1r,1 , S

n−1r,1 , · · · , v

n−1r,α , p

n−1r,α , S

n−1r,α , · · · , vn−1

r,Nα, pn−1

r,Nα, Sn−1

r,Nα, ) (where i ∈

1, 2, . . . ,N andα ∈ {1, 2, . . .Nα}) and weighted bywα,v,i , wα,p,i andwα,S,i . In this work,the Gaussian RBF are chosen:

φ(r i) = e−(r i/σ)2= e−(‖(vn−1

r ,pn−1r ,S

n−1r )−(vr,i ,pr,i ,Sr,i )‖/σ)2

(16)

wherevr denotes the POD coefficients of velocity including all phases, andvr =

(vr,1, · · · , vr,α, · · · , vr,Nα). This apples to pressure vectorpn−1r and saturation vector

Sn−1r . r i is a radius or the distance defined by theL2 norm,σ > 0 is a shape pa-

rameter. (vr,i, pr,i, Sr,i) denotes a center point. The weighting coefficientswα,v,i , wα,p,iandwα,S,i are determined so as to ensure that the interpolation function values at thecollection data point (vr,k, pr,k,Sr,k) match the given datafα,v,k, fα,p,k and fα,S,k. Thiscan be expressed by,

Bwα,v,i, j = fα,v,i, j, Bwα,p,i, j = fα,p,i, j, Bwα,S,i, j = fα,S,i, j, i ∈ {1, 2, . . . ,N}, (17)

where

• wα,v,i = (wα,v,i,k)Tk=1,...,N, wα,p,i = (wα,p,i,k)T

k=1,...,N andwα,S,i = (wα,S,i,k)Tk=1,...,N,

• fα,v,i = (wα,v,i,k)Tk=1,...,N, fα,p,i = (wα,p,i,k)T

k=1,...,N andfα,S,i = (wα,S,i,k)Tk=1,...,N,

• B is the interpolation matrix of elementsBk,l = φ(r k,l),

• k, l ∈ {1, 2, . . . ,N}, N is the number of data points.

The weighting coefficientswα,v,i, j, wα,p,i, j andwα,S,i, j are then determined by solvingthe linear system (17).

3.4. Summary of constructing the NIROM for the 3D reservoir model

In this section, the algorithm of constructing the hypersurfaces and obtaining re-sults using the NIROM is summarized, which includes the offline process and onlineprocess. The offline process consists of obtaining the basis functions and constructingthe hypersurfaces.

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Algorithm 1: NIROM for 3D reservoir model

(1) Offline calculation:

(a) Obtaining the POD bases for 3D reservoir model

i. Generate the snapshots at time leveln = 1, . . . ,Nt by solving the 3Dreservoir model (6);

ii. Construct the POD basesΦv, Φp andΦS using the SVD method;

(b) Construct a set of hypersurfaces

i. Calculate the functional valuesfv,i,k, fp,i,k and fS,i,k at the data point(vr,k, pr,k,Sr,k) through the solution from the full models, wherek ∈ {1, 2, . . .N};

ii. Find the weightswv,i , wp,i andwS,i by solving (17) such that thehypersurfacesfv,i, fp,i andfS,i pass through the data points;

(2) Online calculation: The hypersurfaces in (15) denotes a 3αM-dimensional hypersurface. Once a set of interpolation functionsfα,v, j , fα,p, j and fα,S, j areconstructed, they are then used to estimate thejth POD coefficient of phase (α)vn

r,α, j, pnr,α, j andSn

r,α, j at time leveln.

Initializationv0r, j , p0

r, j andS0r, j ;

for n = 1 to Nt dofor j = 1 to M do

for α = 1 to Nα do

(i) Inputs: a complete set of POD coefficients for solution fieldsv, p andS at timestepn− 1:

vn−1r = (vn−1

r,α, j)Tj=1,...,M.α=1,...,Nα , pn

α = (pn−1r,α, j)

Tj=1,...,M.α=1,...,Nα , Sn

α = (Sn−1r,α, j)

Tj=1,...,M.α=1,...,Nα ,

(ii) Outputs: Estimate the POD coefficientvnr,α, j, pn

r,α, j andSnr,α, j at current time step

n using the hypersurfaces (15);

vnr,α, j = fv,α, j(vn−1

r , pn−1r ,S

n−1r ),

pnr,α, j = fp,α, j(vn−1

r , pn−1r ,S

n−1r ),

Snr,α, j = fS,α, j(vn−1

r , pn−1r ,S

n−1r ),

endforendforObtain the solution of variablesvn

α, pnα andSn

α in (4) by projectingvnr,α, j, pn

r,α, jandSn

r,α, j onto the full space (see (7)).

vnα = vα + ΦT

v,αvnr,α, pn

α = pα + ΦTp,αp

nr,α, Sn

α = Sα + ΦTS,αS

nr,α,

endfor

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4. Numerical Examples

4.1. Introduction of an unstructured mesh 3D reservoir model

The NIROM has been implemented under the framework of an advanced 3D un-structured mesh reservoir model (IC-FERST). A novel control volume finite elementmethod (CVFEM) is used to obtain the high-order fluxes on CV boundaries which arelimited to yield bounded fields (e.g., positive saturations ). This method is combinedwith a novel family of FE pairs, originally introduced for geophysical fluid dynamicsapplications. In particular, theP2DG − P1DG element pair (quadratic discontinuouspolynomial FE basis function for velocity (P2DG) and linear discontinuous polynomialFE basis function for pressure,P1DG), is used to accurately represent sharp saturationchanges between heterogeneous domains, see [45, 46].

The water-flooding is a widely known technique in oil and gas reservoir engineer-ing. It increases the production from oil reservoirs through injecting water into thereservoir. As illustrated in figure1, the water is injected into the reservoir to increasethe reservoir pressure, the oil is then displaced toward theproduction well. This phe-nomenon is also referred to the immiscible displacement in porous media. In thissection, the capability of the NIROM developed for 3D unstructured mesh reservoirmodelling has been numerically illustrated in a horizontally layered porous media case,and then further applied to a 3D complex fluvial channel case.

4.2. Case 1: horizontally layered porous media case

The first case for numerical illustration of the method proposed in this paper is ahorizontally layered porous media case. This test case is dimensionless and for sim-plicity no gravity has been considered. This problem domainis consist of a rectangleof non-dimensional size 1× 0.2. The domain is divided into two identical areas witha permeability of 4 within the top half part and 1 on the bottomhalf part. The outletboundary has a dimensionless pressure of 0, the whole domainis initially saturatedwith the non-wetting phase and the wetting phase at the irreducible saturation. Thewetting phase is injected over the inlet boundary with a dimensionless velocity of 1.The viscosity ratio of the phases is 1. The Brooks-Corey model for the relative perme-ability, with an exponent of 2 and an end-point relative permeability of 1, is consideredfor both phases. The porosity is homogeneous and equal to 0.2. The immobile fractionof the wetting phase is set to 0.2 and 0.3 for the non-wetting phase.

The problem was resolved with a mesh of 984 nodes during the simulation timeperiod [0, 0.02]. Fifty snapshots were taken from the pre-computed solution at regu-larly spaced time intervals∆t = 0.0002 and from these POD bases are generated forthe solution variablesv, p,S.

The first 18 POD bases are presented in figure2. As shown in the figure, the firstfour POD bases capture most of flow features while the 5th-18th POD bases capture thedetails of small scale flow structures. Figure3 shows the singular eigenvalues in orderof decreasing magnitude. In general, the more POD bases and snapshots are chosen,the better the energy is represented. There is a trade-off between the accuracy and theCPU time. In this work, 18 POD bases with 50 snapshots are chosen resulting in 92%of ’energy’ being captured.

9

Figure4 shows the saturation solutions of the horizontally layeredporous mediaproblem at time instancest = 0.01 andt = 0.02, as calculated using the full andNIROMs. It can be seen that both model solutions are in good agreement with eachother. The NIROM performs well in capturing the saturation shock-front.

Figure5 shows the saturation solution at a particular position (0.026937, 0.16246).It is noted the results from the NIROM using 6 and 12 POD bases become oscillatoryafter t = 10. By increasing the number of POD bases from 6 to 18, the NIROMbecomes stable and exhibits an overall good agreement with the full modelling.

The ability of the NIROM is further highlighted in figure6, which presents thesaturation solution along a line parallel to the x-axis. We can see the NIROM has alarge error near the shock-front when using 6 POD bases. Thiscan be significantlyimproved as the number of POD bases increases. Using 18 POD bases, the error ofsaturation solutions is decreased by 50%− 97% in comparison to that using only 6POD bases, and the shock-front is captured well.

To further validate the quality of the NIROM, the corresponding error estimationof the POD ROM was carried out in this work. The accuracy of NIROM was assessed.The correlation coefficient of solutions between the full and NIROMs is computed foreach time step, and is defined for given expected valuesSn

full and Sn and standarddeviationsσSn

f ullandσSn,

corr(Snfull ,S

n)n =cov(Sn

full ,Sn)

σSnf ullσSn

=E(Sn

full − σSnf ull

)(Sn − σSn)

σSnf ullσSn

. (18)

whereE denotes mathematical expectation,covdenotes covariance,σ denotes standarddeviation. The measured error is given by the root mean square error (RMSE) which iscalculated for each time stepn by,

RMS En =

√

∑Ni=1(Sn

full,i − Sni )2

N. (19)

In this expressionSnfull,i andSn

i denote the full and NIROM solutions at the nodei,respectively, andN represents number of nodes on the full mesh.

The RMSE and correlation coefficient of saturation solutions between the full andNIROMs are presented in figure7. With an increase in the number of POD bases, theRMSE in the saturation results decreases by about 50% while the correlation increasesup to 98%.

10

Figure 1: Water-flooding technique for oil production.

(a) The 1st POD bases (b) The 2nd POD bases

(c) The 3rd POD bases (d) The 4th POD bases

(e) The 5th POD bases (f) The 8th POD bases

(g) The 12th POD bases (h) The 18th POD bases

Figure 2: Case 1: the figures displayed the first 18 the POD bases functions of the horizontally layeredporous media problem.

11

0 3 6 9 12 15 18NUMBER OF POD BASES

0

5

10

15

20

25

SIN

GU

LAR

EIG

EN

VA

LUE

S

Figure 3: Case 1: the figure shows the singular eigenvalues inorder of decreasing magnitude.

(a) full model, t = 0.01 (b) full model, t = 0.02

(c) NIROM 6 POD bases,t = 0.01 (d) NIROM 6 POD bases,t = 0.02

(e) NIROM 12 POD bases,t = 0.01 (f) NIROM 12 POD bases,t = 0.02

(g) NIROM 18 POD bases,t = 0.01 (h) NIROM 18 POD bases,t = 0.02

Figure 4: Case 1: the figures displayed above show the saturation solutions of the horizontally layered porousmedia problem at time instances 0.01 and 0.02 (where 6, 12 and 18 POD bases are chosen with 50 snapshots).The permeability on the top half part is 4, and the bottom halfpart is 1.

12

0 10 20 30 40 500.6

0.65

0.7

Timestep

Saturation

full model 6 POD bases

12 POD bases 18 POD bases

Figure 5: Case 1: the graph shows the solution saturations predicted by the full model and the NIROM at aposition (0.026937, 0.16246) (where 6, 12 and 18 POD bases are chosen with 50 snapshots)

(a) The cross section locations

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

Distance along the X direction

Saturation

full model

6 POD bases

12 POD bases

18 POD bases

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

Distance along the X direction

Saturation

full model

6 POD bases

12 POD bases

18 POD bases

(b) Saturation at the cross section along the top line(c) Saturation at the cross section along the bottom line

Figure 6: Case 1: Saturation along lines parallel to the x axis.

13

0 10 20 30 40Timestep

0

0.01

0.02

0.03

0.04

0.05

0.06

RM

SE

06 POD Bases12 POD Bases18 POD Bases

10 20 30 40Timestep

0.5

0.6

0.7

0.8

0.9

1co

rrel

atio

n co

effic

ient

06 POD bases12 POD Bases18 POD Bases

(a) RMSE (b) Correlation

Figure 7: Case 1: The graph shows the RMSE and correlation coefficient of solutions between the full andNIROMs.

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4.3. Case 2: 3D fluvial channel case

The second case is a 3D fluvial channel case. There is a set of channels in thecomputational domain which is composed of 31776 nodes-see figure8. The domainis initially filled with immovable water and movable oil and the saturations of oil andwater areSoil = 0.8 andSwater = 0.2 respectively. The water is then injected intothe computational domain from the right side at constant pressure. The flow passesthrough the channels from right side to the left side. All other sides are treated asbarriers to flows except for the inlet side (right boundary inthe computational domain)with a pressure of 55× 106 and outlet side (left boundary) with a pressure of 2× 106.The homogeneous porosity is set to beφ=0.2 and the saturations of the residual oil andirreducible water are set to be 0.2. The viscosities of the residual oil and irreduciblewater are set to be 0.004 and 0.001 respectively. The simulation was run during thesimulation period [0, 86400000] with a time step size of 864000. 100 snapshots ofsolutions were taken at regularly spaced time intervals∆t = 864000 for each solutionvariable.

Figure10shows the first 36 leading POD bases functions of saturation.As shown inthe figure, these leading POD bases capture the dominant characteristics of solutions.The POD bases corresponding to small eigenvalues, for example, the 30th and 36th

POD bases, contain small scale flow features. Figure9 shows the singular eigenvaluesin order of decreasing magnitude. As shown in the figure, the first 6 POD bases almostcaptured all of the total energy (96.83%).

Evaluation of accuracy of the POD model was carried out through comparison ofPOD solutions with those from the full model. Figure11shows the saturation solutionsobtained from the high fidelity full model and NIROM with 6 and36 POD bases at timeinstances 43200000 and 69120000 seconds. It shows clearly that the results of NIROMare close to that of high fidelity full model. The solutions ofsaturation at a particularpoint (figure12(a)) in the computational domain are presented in figure12. The figureagain shows that the accuracy of solution can be improved by increasing the number ofPOD bases functions to 36 and the NIROM using 36 POD bases is ingood agreementwith the high fidelity full model.

The error of saturation solutions between the high fidelity full model and the NIROMswith 6 and 36 POD bases are presented in figure13.

15

(a) water channel profile (top to bottom) (b) water channel profile (bottom to top)

Figure 8: Case 2: the figures displayed above show the water channel profile.

0 5 10 15 20 25 30 35 40NUMBER OF POD BASES

0

500

1000

1500

2000

SIN

GU

LAR

EIG

EN

VA

LUE

S

Figure 9: Case 2: the figure shows the singular eigenvalues inorder of decreasing magnitude.

16

(a) The 1st POD bases (b) The 2nd POD bases

(c) The 12th POD bases (d) The 24th POD bases

(e)The 30th POD bases (f) The 36th POD bases

Figure 10: Case 2: the figures displayed the first 36 the POD bases functions of the 3D fluvial channelproblem.

17

(a) full model, t = 43200000 (b) full model, t = 69120000

(c) NIROM (6 POD bases),t = 43200000 (d) NIROM (6 POD bases),t = 69120000

(e) NIROM (36 POD bases),t = 43200000 (f) NIROM (36 POD bases),t = 69120000

Figure 11: Case 2: the figures displayed above show the saturation of full model and NIROM of the 3Dfluvial channel problem at time instances 43200000 and 69120000 seconds using 6 and 36 POD bases.

0 50 100 150 200Timesteps

0.77

0.775

0.78

0.785

0.79

0.795

0.8

Sat

urat

ion

Full model6 POD bases36 POD bases

(a) location of the point (b) saturation values of the point

Figure 12: Case 2: the figures displayed above show the valuesof saturation at a particular point in themesh-see figure(a).

18

(a) error of NIROM with 6 POD bases,t = 43200000 (b) error of NIROM with 6 POD bases,t = 69120000

(c) error of NIROM with 36 POD bases,t = 43200000 (d) error of NIROM with 36 POD bases,t = 69120000

Figure 13: Case 2: the figures displayed above show the error of saturation between the full model andNIROM of the 3D fluvial channel problem at time instances 43200000 and 69120000 seconds using 6 and36 POD bases.

19

4.4. Efficiency of the NIROM

Table2 shows a comparison of the online CPU time required for running the fullmodel and NIROM. The simulations were performed on 12 cores machine of an In-tel(R) Xeon(R) X5680 processor with 3.3GHz and 48GB RAM. Thetest cases wererun in serial, which means only one core was used when simulating. Note that theoffline CPU time required for constructing the POD bases and the interpolation func-tion fv,i,k, fp,i,k and fS,i,k (see algorithm 1) is not listed here. The online CPU time forrunning the NIROM includes:

• interpolation for calculating the POD coefficientsvnr, j, pn

r, j andSnr, j (see equation

(15));

• projecting thevnr, j, pn

r, j andSnr, j onto the full space (see equation (7)).

It can be seen that the online CPU time required for running the NIROM is consid-erably less than that for the full model and is reduced by a factor of 2500. It is worthnoting that as the number of nodes increases the CPU time required for the full modelincreases rapidly while the CPU time for the NIROM almost remains the same.

Table 2: Comparison of the online CPU time (dimensionless) required for running the full model andNIROMs during one time step.

Cases Model assembling and projection interpolation totalsolving

Full model 0.81605 0 0 0.81605Case 1 NIROM 0 0.0003 0.0001 0.00040

Full model 98.3998 0 0 98.3998Case 2 NIROM 0 0.0003 0.0001 0.00040

5. Conclusion

A NIROM based on hypersurfaces representation has recentlybeen developed forIC-FERST which has the capabilities of using (1) anisotropic unstructured meshes toresolve fine scale flow feature; and (2) a novel control volumefinite element method toresolve the high-order flux flows on CV boundaries. In this work the NIROM has beenfurther applied to a fluvial channel problem.

A RBF interpolation method is used to form a multi-dimensional interpolationfunction (hyper surface) that represents the solution of the 3D reservoir model withinthe reduced space. The non-intrusive approach used here to construct the NIROM isgeneric and does not require any information of the originalsource code or the modelequations. It can be applied to any software or commercial codes. In addition, it avoidsthe instability of existing Galerkin POD ROMs [28], the results might be smoothed byRBF [47].

The capabilities of the newly developed NIROM are illustrated in two test cases:horizontally layered porous media case and a 3D fluvial channel case. A comparisonbetween the full and NIROM results are made. An error analysis was also carried out

20

for the validation and accuracy assessment of the NIROM. It is shown that the NIROMexhibits an overall good agreement with the high fidelity full model. An increase inthe number of POD bases leads to an improvement in the accuracy of the NIROM.The saturation shock-front can be captured with relativelyfew POD basis functions, 18POD basis function (figure (h) of4) in the examples.

In comparison to the full model, without compromising the accuracy of results theCPU time required for the NIROM can be reduced by a factor of 2500. It is worth ofmentioning that for 3D large scale reservoir simulation, anincrease in the number ofnodes used in the computational domain will result in a largeincrease of the CPU timein the full simulation, but has very little effect on that of the NIROM. Future work willinvestigate the effects of applying this new approach to more complex 3D reservoirsimulation cases.

Acknowledgment

This work was carried out under funding from the UK’s NaturalEnvironment Re-search Council (projects NER/A/S/2003/00595,NE/C52101X/1 and NE/C51829X/1),the Engineering and Physical Sciences Research Council (GR/R60898, EP/I00405X/1and EP/J002011/1), and the Imperial College High Performance Computing Service.The authors would like to thank Pan Yang from AMCG, Imperial College London forhelping to set up the test cases. Dr. Xiao acknowledges the support of NSFC grant11502241 and China postdoctoral science foundation grant (2014M562087). Prof.Pain and Dr. Salinas would like to thank EXXON for helping to fund the new unstruc-tured mesh model used here; Prof. Pain, Prof. Muggeridge andDr. Fang are gratefulfor the support of BP on reduced order modelling under the LAMBSID project.

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