Journal of Petroleum Science and Engineering 145 (2016) 404–422

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Journal of Petroleum Science and Engineering

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Estimation of permeability of naturally fractured reservoirs by pressuretransient analysis: An innovative reservoir characterization and flowsimulation

Reda Abdelazim a,c,n, Sheik S. Rahman b,n

a School of Petroleum Engineering, American University of Ras Al Khaimah, UAEb School of Petroleum Engineering, The University of New South Wales, Sydney, NSW 2052, Australiac Cairo University, Egypt

a r t i c l e i n f o

Article history:Received 18 November 2015Received in revised form22 March 2016Accepted 23 May 2016Available online 25 May 2016

Keywords:Well testNaturally fractured reservoirsFinite element modelDiscrete fracture

x.doi.org/10.1016/j.petrol.2016.05.02705/& 2016 Elsevier B.V. All rights reserved.

esponding author.ail addresses: reda.abdelazim@aurak.ac.ae (R.hman@unsw.edu.au (S.S. Rahman).

a b s t r a c t

Fluid flow processes in naturally fractured reservoirs are strongly controlled by fracture intensity, theirorientation, and interconnectivity. Therefore, knowledge of fracture properties plays a critical role inreservoir management. Developing a detailed description of subsurface fracture map is challenging forgeothermal and petroleum industry due to a number of reasons: It is often difficult to obtain sufficienthard data such as borehole images and core description. Also, there is a general lack of quality of in-terpretation of soft data, such as well logs, seismic attributes, and tectonic history etc. for fracture in-terpretation. To overcome the short comings the industry has been relying heavily on geo-statisticalanalysis of both hard and soft data.

In this paper we used well test data (dynamic data) to reduce the level of uncertainty of the existingmethods for generating fracture map through an innovative inversion technique. The major inversiontechniques include simulated annealing, sequential successive linear estimator, and gradient andstreamline based method. In this paper we used gradient based method which utilises adjoint equation.The inversion is carried out in a number of steps. First we analyse hard and soft data to generatecharacteristic fracture properties such as fracture density and fractal dimension. Next we use geo-sta-tistical technique to spatially distribute fractures. Then with use of gradient based technique, we optimisethe fracture attributes through different realization. Next we use an innovative simulation of fluid flowthrough discrete fractures in 3D fractured porous media and estimated pressure and pressure derivatives.The pressure and pressure derivatives are compared with well test data taken from a typical fracturedbasement reservoir located in offshore Vietnam to determine percentage error.

The results showed that the simulated pressure change and pressure derivatives match well with welltest data. This has allowed us to capture the complex subsurface fracture pattern. This vital informationcan help the operator design an effective reservoir management plan.

& 2016 Elsevier B.V. All rights reserved.

1. Introduction

Naturally fractured reservoirs host more than 50% of the worldremaining hydrocarbon reserves (Gutmanis et al., 2015). Me-chanism of fluid flow through such reservoirs is not well under-stood (Gang and Kelkar, 2006) and this is mainly because thesereservoirs comprise of two mediums of diverse properties: rockmatrix and fractures. Naturally fractured reservoirs have beenclassified according to the relative contribution of the matrix and

Abdelazim),

fractures to the total fluid production (Nelson, 2001). Accordingly,the fractured reservoirs are classed as: Type 1 - fractures provideessential porosity and permeability, Type 2 - fractures provideessential permeability, and Type 3 - fractures provide permeabilityassistance. Fractures are mechanical breaks, which arise fromstress concentrations around flaws, heterogeneities, and physicaldiscontinuities. These characteristics proprieties affect fluid flowand mechanical stability of the reservoir. A comprehensive un-derstanding of fluid flow mechanism in fracture-matrix system isan essential requirement for estimation of production potential ofand selection of appropriate production methods for these re-servoirs. Simulation of fluid flow for estimation of productionpotential of these reservoirs, however, is extremely challenging

MatrixFractur

Matrix block Fractur(a) (b)

Fig. 1. Dual porosity model description of a naturally fractured reservoir: (a) actual reservoir and (b) sugar cube reservoir model.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 405

due to (a) complex fracture geometry (b) lack of information onfracture properties, which introduce a high level of uncertaintyassociated production estimation (Narr, 1996; Ortega et al., 2006).

The main objective of this paper is to present a methodologyfor characterizing the naturally fractured reservoirs in order tomaximize the benefits of geomechanical control on predictingfracture properties, therefore intensity, geometry and orientationof fractures and their effect on fluid recovery. Characterization offracture properties has been carried out by integrating informationfrom two different sources; static data (seismic, well logs, coredescription, borehole images, tectonic history, geological struc-tures etc.) and dynamic data (well test data and wells productionhistory). Several methods have been used for generation subsur-face fracture map based on static data. Most notable methods areAbdelazim and Rahman et al. (2014), Doonechaly and Rahman(2012), and Will et al. (2005). Among these, the paper by Doo-nechaly and Rahman (2012) describes a comprehensive metho-dology of generating subsurface fracture map. In this approach thetwo important parameters, fracture density and fractal dimensionare estimated form a number of data sources including tectonichistory reservoir structure, seismic attributes, well logs, boreholeimages and core description. Often it has been observed that alldata sources, such as core description and borehole images are notalways available for every well drilled in an area. Again cores andborehole images are taken from a section of the wellbore. In orderto overcome discontinuity of the data in both horizontal andvertical directions a back propagation neural network is used tocreate a non-linear relationship of attributes of different datasources (for example, attributes of seismic, borehole sonic, Gamaray etc.) with the fractal dimension and fracture density. In thenext step a nested neuro-stochastic simulation (sequential Gaus-sian approach) is used to generate subsurface fracture map basedon fracture density and fractal dimension. In the final step simu-lated annealing is used to generate an optimised fracture map.This approach can be further enhanced integrating the fractureattributes from the outcrop if available. Most of the geologicalmodels of fractured reservoirs generated based on static dataalone sometimes failed to reproduce wells production history(Ouenes et al., 2004; Efendiev et al., 2005). This is mainly becausethe hydrodynamic properties of fracture network are not con-sidered in the statistical analysis.

Since 1970s, a significant progress has been achieved to gen-erate a consistent methodology to characterize naturally fracturedhydrocarbon and geothermal reservoirs by utilising well test and

production data. This was achieved in two steps: First the staticdata is used to statistically generate subsurface fracture map withfracture intensity orientations and geometries in different rea-lisations. Then inversion techniques are used to converse the si-mulated pressure data with that of well test data. These techni-ques include stochastic algorithms, gradient based and streamlinebased techniques (Chen et al., 1974, Chavent et al., 1975, Oliver,1994, Landa et al., 2000, Zhang and Reynolds, 2002, Gang andKelkar, 2006, Oliver and Chen, 2011; Azim, 2015).

Gradient based methods require an optimisation algorithm (e.g.Quasi-Newton, Conjugate Gradient, and Levenberg-Marquardt)and these algorithms need the derivative of the production re-sponses with respect to the changes of the reservoir parameters.Chen et al. (1974) and Chavent et al. (1975) developed an efficienttechnique for derivatives calculation based on the adjointequation (model) applied to single phase flow problems. Extensionof these adjoint models for multiphase flow problems were in-troduced by Yang et al. (1988), Bissell (1994), and Zhang andReynolds (2002). These methods tend to converge quickly, butsubsurface heterogeneity mapping for fractured reservoirs bygradient base methods are simply inappropriate because they dealwith block based permeability tensors. Stochastic simulation isanother technique for generation of multiple realisations of thereservoir fracture property, rather than simply estimation of themean value of the property. The most common stochastic methodsapplied for petroleum and geothermal engineering problems aresimulated annealing (Gupta et al., 1994) and genetic algorithms(Goldberg and Holland, 1988; Carter and Ballester, 2004). In thesemethods, gradients are not required and instead evaluation of theforward simulation model is used. The disadvantage of thesemethods is that they require numerous simulation runs for con-vergence (Wu et al., 2002 and Liu and Oliver, 2004) which isconsidered computationally exhaustive for large-scale applica-tions. Streamline based methods have been presented by Vascoet al. (1999) and Agarwal and Blunt (2003). These methods havesignificant advantages over simulated annealing due to two mainreasons. First, it is extremely efficient in resolving complex het-erogeneous reservoir because the solution methodology is fasterthan conventional methods. Second, the sensitivity coefficients arecomputed with only one single simulation run. The limitation ofthis method is that the model equations do not honour multipointgeostatistics.

Central to all inversion algorithms is the forward modelling/numerical simulation of fluid flow through matrix and fractures.

Fig. 2. Diagram of fractures radius and dip angle for the generated subsurface fracture map of the studied reservoir from a typical fractured basement reservoir in southernVietnam.

Fig. 3. 3D fracture map generated using object based model. Fractures are dis-tributed stochastically with different radius, dip and azimuth angles using fractureintensity value of 0.1 m�1.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422406

Currently, three major approaches are used to simulate fluid flowthrough naturally fractured reservoirs which include: continuum,dual porosity/dual permeability, and flow through discrete frac-ture approaches. In single continuum approach fractured mediumis divided into a number of representative volumes and bulkmacroscopic values of reservoir properties varying from point-to-point are averaged over the volume which is often known asblocked based permeability tensors (Lough et al., 1998; Park et al.,2002; Gupta et al., 2001; Sarkar et al., 2004; Teimoori et al., 2005;Fahad, 2013). Gupta et al. (2001) considered matrix with no per-meability and simulated fluid flow through fractures only. Parket al. (2002) considered matrix and fractures as a continuoussystem and ignored the fluid flow at fracture matrix interface.While Lough et al. (1998) calculated the effective permeabilitytensors by considering flow through matrix and fractures withregular fracture pattern. Park et al. (2002), Gupta et al. (2001),

Sudipata et al. (2004), and Teimoori et al. (2005) on the other handpresent comprehensive methodologies for estimating perme-ability tensor for arbitrarily oriented and interconnected fracturesystems.

In the dual continuum approach, the reservoir is divided intotwo major parts: fractures and matrix. According to Warren andRoot (1963), fractured reservoirs are often visualized as a set ofstacked sugar cubes (see Fig. 1b). In this approach, the fracturesprovide the main flow paths, while the matrix acts as a source offluid. The fluid transfer between the fractures and the matrix isdefined based on transfer functions. Duguid and Lee (1977), Pruess(1985), Gilman (1986) and Bourbiaux et al. (1999) introduced arange of different matrix/fracture transfer functions to simulatethe fluid flow in large scales. Following this, a significant numberof studies have been carried out both in analytical and numericalframeworks using dual porosity approach (Choi et al., 1997; Prideand Berryman, 2003; Gong et al., 2008).

Landereau et al. (2001) proposed a new technique for simu-lating flow in fractured porous media like rock masses. The pro-posed technique have been created to separate the contributionsfrom the rock matrix and from the fracture systems (which arealso treated as an equivalent continua), the so-called dual (double)porosity models, dual (double) permeability models and dual(double) continuum models, in order to simplify the complexity inthe fracture–matrix interaction behaviour and to partially considerthe size effects caused mainly by the fractures.

It has been shown that the advective transport component in morepermeable matrix blocks (e.g., porous sandstones) results in additionalsolute transfer between fractures and matrix, which is not typicallytaken into account by dual‐porosity concepts (Cortis and Birkholzer,2008). Therefore, Noetinger and Estebenet (2000) proposed a newtechnique called continuous time random walk (CTRW) method. Thismethod is an alternative to the classical dual‐porosity models formodelling and upscaling flow and/or solute transport in fractured rockmasses. Originally developed to describe electron hopping in hetero-geneous physical systems (Scher and Lax, 1973).

One of the applications of classical dual porosity approach hasbeen the analysis of pressure transient data from fractured porousmedia. The concept was presented by Warren and Root (1963). Inthis approach, the fluid flow from the matrix to the fractures isassumed to be pseudo steady state which means that the pressureat every point in the matrix elements decreases at the same rate

Fig. 4. Description of how the reservoir domain is discretised for hybrid methodology (a) the reservoir domain is divided into a number of grid blocks without considering oflong fractures (b) 3D block based permeability tensor for short to medium fractures (c) the block based permeability tensors are distributed to the corresponding tetrahedralelements (matrix porous media) and long fractures are discretised explicitly.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 407

(the reservoir is assumed to be homogeneous). Warren and Root(1963) developed an analytical model for unsteady state flowwhich allowed production analysts to analyse the pressure buildup data in some details. In this analysis two parameters are used todescribe the behaviour of the fractured porous media. Firstly thestorativity ratio (ω) was introduced to describe the fluid capaci-tance in fractures as:

ωϕ

ϕ ϕ=

+ ( )

c

c c 12 2

2 2 1 1

where ϕ1 is the matrix porosity, ϕ2 is the secondary porosity,c2 isthe total compressibility in the secondary system (facture) and c1 is

the total compressibility in the primary system (matrix). The otherparameter (λ) represents the interporosity flow coefficient relatedto degree of system heterogeneity and is expressed as:

λα

=( )

−k r

k 2

w12

2

where k1 is the matrix permeability, k̅2 is the effective (average)permeability of the heterogeneous medium, α is the matrix shapefactor and rw is the wellbore radius. This interporosity flow coef-ficient parameter is dimensionless and describes the ability of thefluid to flow from the matrix to the fracture network. Also, it givesan indication about how rapidly the matrix contributes to the

X

ZY

100 m

100

m

Fig. 5. Generated fractures for the studied reservoir (the plot shows that most ofthe fractures direction toward x- axis).

Fig. 6. Polar diagram for directional permeability: a) permeability tensor for, kxx

Table 1Reservoir inputs parameters for validation of permeability tensor calculations.

Rock and reservoir properties

Reservoir dimensions 100 m�100 m�200 mMatrix permeability 0.01 mDMatrix porosity 0.002Initial reservoir pressure 34.9 MPa (5,063 psia)Injection pressure (injection case) 54.9 MPa (7963.65 psia)Fracture propertiesFracture aperture 0.1 mmFracture Intensity 0.12 (m2/m3)Fractal dimension, D 0.125Fluid propertiesViscosity 1.38 cPCompressibility 1.0E�06 psi�1

Density 1000 kg/m3

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422408

production process. In order to simplify the solution, Warren andRoot (1963) ignored the wellbore storage effects by assuming thatthe well bore is a line source. Mavor and Ley (1979) extended theanalytical solution of Warren and Root (1963) by including thewellbore storage effects by incorporating a finite wellbore radius.Bourdet et al. (1983) used Mavor and Ley's (1979) analytical model

−yy b) permeability tensor for −kxx zz , and c) permeability tensor for −kyy zz .

Table 2Reservoir inputs data for a typical fractured basement.

Parameter Value

Reservoir dimensions 500 m�500 m�250 m Vertical well is partiallypenetrated the formation thickness (90 m)

Matrix permeability 0.0095 mDMatrix porosity 2%Fracture aperture 7.06�10�3 mmInitial fracture intensity 0.15 m�1

Initial reservoir pressure 34.9 MPa (5063 psia)Injection pressure (injec-tion case)

54.9 MPa (7963.65 psia)

Fluid viscosity 1.38 cPFluid compressibility 10�8 MPa�1

Production time beforeshut in (tp)

72 h

Production flow rate beforeshut in

5571bbl/d

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 409

and presented the first pressure derivative plot for analysingpseudo steady state flow transfer in a dual porosity system.

Kazemi (1969) presented a numerical approach, as an alter-native to Warren and Root's (1963) model. Kazemi (1969) studiedthe transient flow condition between the matrix and fractures byrepresenting the reservoir as an equivalent system of horizontalfractures embedded between matrix layers. Bourdet and Gringar-ten (1980) and Bourdet et al. (1984) presented an analytical so-lution for the Kazemi's (1969) transient model to include thewellbore storage effects and skin. Following the work of Bourdetet al. (1984) and Braester (1984) studied the effect of the matrixblock size on the pressure derivative for transient flow and ob-served that the matrix block size has no effect on the behaviour ofthe pressure change and its derivatives.

The main drawbacks of the aforementioned approaches are:(1) the fluid distribution within the matrix blocks remains con-stant during the simulation period, (2) the model can be applied toonly interconnected fractures and is applicable to a small numberof large scale fractures in flow simulation.

In discrete fracture approach, each fracture surface is re-presented by individual attributes of permeability and aperture.The size, spatial location, orientation and intensity of these sur-faces have a substantial importance and are accounted for duringthe generation of discrete fracture network. The fluid flow equa-tions through the fracture network and matrix system are solvedusing both analytical and numerical methods, e.g. finite elementmethod (Karimi-Fard et al., 2004); finite volume method (Reich-enberger et al., 2006), mixed finite element and finite volumemethod (Geiger et al., 2004). The limitation of this approach is thatan exhaustive computation time is required which prevents itsapplication to simulation of fluid flow in large number of frac-tures. Doonechaly and Sheik (2013) extended the work of Teimooriet al. (2005) by introducing hybrid of single continuum and flowthrough discrete fractures (fractures that are greater than the setthreshold value. This has allowed the authors to simulate flowthrough hundreds of long fractures that are greater than 100 m(Abdelazim and Rahman et al., 2014).

Noetinger (2014) and Noetinger (2015) they simulated a single-phase fluid flow inside a fractured medium. The authors considerthe rock surrounding the fractures as impervious and model theflow by Darcy's law. The medium is approximated by a DiscreteFracture Network (DFN) in which the fractures are assumed tohave a negligible thickness with respect to the other dimensions,and are represented as planar polygons intersecting each other inthree dimensional space, with an equivalent bi-dimensional con-ductivity obtained by averaging the tri-dimensional one along thenegligible dimension.

Wei et al. (1998) developed a 3D numerical model in order tosimulate pressure transient through fracture/matrix using a dualporosity model. This simulation allows the determination of frac-ture transmissivities and the quantification of fracture pore vo-lume. The authors showed that dual porosity model failed to de-scribe the behaviour of fluid flow through fractured system inmany cases. Carlson (2003) was using specific transfer functions tosimulate the flow transfer from fractures to matrix. It was assumedthat the fractures provide the main flow conduit and matrix acts asa source/sink to the fractures. Kamal et al. (2005) reviewed thelimitation of traditional well testing, and presented a new methodcalled Numerical Well Testing to incorporate the wealth informa-tion contained in the transient behaviour of the tests. In this study,reservoir geological factors (as reservoir heterogeneities, compli-cated boundary, multi-phase flow and production history) areconsidered. This method is applied for conventional reservoirs(not fractured reservoirs) using different field cases to match thewell test data and field production history. Basquet et al. (2005)used a homogenization method (note: we need to explainhomogenization in the light of flow simulation). The homo-genization method used in different steps to simulate pressuretransient through fractured system. The first step is subdividingthe fractured reservoir into two scales: local scale and global scale.The local scale is for a single matrix surrounded by fractures andlarge scale is the grid block length. In the next step of homo-genization process, the flow equations (mass balance and Darcy'sequation) are described on the local scale (matrix). Then, the dif-ferentiation operator in the flow equations is split into a global andlocal scale terms for solving the flow equations numerically. Thesplitting procedure is only apply for fracture equation.

Will et al. (2005) presented a method to integrate seismic an-isotropy attributes with reservoir-performance data (well test andproduction data) for characterization of naturally fractured re-servoir through the use of block based permeability tensors andEclipse 100 as a forward model.

Recently, there are many studies revolving the use of pressuretransient data for characterising naturally fractured reservoir throughinversion of well test data. Morton (2012) presented two new tech-niques used to calibrate with the inversion of well-test data by in-tegrating a reservoir flow simulation model and an inversion techni-que. In this approach a boundary element method is used to simulatefluid flow and study the pressure transient behaviour of the reservoirwith arbitrary distributed vertical fractures. Kuchuk and Biryukov(2012) presented an analytical solution and thus can be easily ex-tended to incorporate other reservoir features such as sealing or leakyfaults, domains with altered petro-physical properties in order tounderstand pressure behaviour of continuously and discretely frac-tured reservoirs. The author showed that Warren and Root's (1963)dual-porosity model is not adequate for pressure transient well-testinterpretations and the pressure behaviour of discretely fractured re-servoirs shows many different flow regimes depending on fracturedistribution, its intensity and conductivity.

In this paper, pressure transient data from a fractured basementreservoir offshore Vietnam has been used to evaluate the fracture mapwhich is generated by statistical analysis of field data (static data) asper Doonechaly and Rahman (2012). This is carried out in two inver-sion steps. In the first inversion step, the reservoir is divided into anumber of grid blocks and the block based permeability tensors areestimated by considering all fractures (small to large fractures) that areintersected by the blocks. Fluid flow is simulated (forward modellingby single continuum approach, therefore the permeability tensors) toestimate change in pressure and pressure derivatives. The simulatedpressure data is compared with that obtained from well test to esti-mate error. The gradient based technique is utilised to repeat theforward modelling for different realisations of block based fracturepermeability (permeability tensors) until the error is reduced to

Fig. 7. Schematic representation of reservoir pressure (Top) after (a) 1 year (left) and (b) 10 years (right) of water injection and fluid velocity (bottom) after (c) 1 year (left)and (d) 10 years (right) of water injection with Pinj¼54.9 MPa, Δp¼41.14 MPa, and pi¼34.6 MPa. Velocity contour map in (m/s) and pressure contour map in (psi).

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422410

minimum. The optimised permeability tensors are then correlated tofracture properties of the corresponding blocks. In the next inversionstep, different subsurface fracture maps are realised based on thecorrelation and the forward modelling carried out by using singlecontinuum and discrete fracture approach, which was developed by,Gholizadeh and Abdelazim et al. (2015) until an optimised fracturemap is obtained.

1.1. Fracture data analysis

Fracture data analysis is the first step in reservoir character-ization process. The analysis consist of the determination of typesof fractures or fracture parameters that control the distribution

and quality of flow zones. Borehole images and production dataare used to identify a set of variables such as dip, azimuth, aper-ture, or density that controlling hydrocarbon flow. Fracture in-dicators such as production rates are combined with boreholeimages to flag the flow contributing fracture zones. This techniquehas been used successfully in fractured basement reservoirs(Tandom et al., 1999; Luthi, 2005). The fracture sets are definedbased on fractures dip, length, and azimuth. The initial data offractures length and dip angles ranging from 9 m to 60 m and 70°to 90° respectively and the fracture aperture ranges from0.004 mm to 0.04 mm. Once the fracture set has been identified, itis used in the form of a fracture intensity curve. Generation ofintensity logs from fracture data is essential in the data analysis

Fig. 8. Measured and simulated shut in pressure at well location with using thesubsurface fracture map presented in Fig. 3, fractal dimension¼1.25, matrixpermeability¼0.0095 md, Pinitial¼5063 psi and production time before shutin¼72 h. This plot is produced based on single continuum approach.

Generate different fracture realizations

Flow simulation to estimate pressure change and pressure derivative

Compare the pressure change and pressure derivatives with field

measured data to estimate the error

Generate the subsurface fracture map using the initial

fracture realisation

Estimate the block based permeability tensor by using single continuum approach

Error is less than the predefined

threshold

Finish the process

Modify the fracture attribute

through the generated

realizations

Optimize the initial block based permeability tensors to minimize the error between the simulated data and well

test data through an Inversion technique (Gradient based method)

Yes

No

Fig. 9. The different steps used in optimising the subsurface fracture map.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 411

because the intensity curve describes the density of fractures perunit length and provides an input fracture distribution to the 3Dfluid flow simulation model.

The Object based simulation is used to generate 3D subsurfacefracture map of a typical fractured basement reservoir in offshoreVietnam (Gholizadeh and Sheik, 2012) using the calculated

fracture intensity. Fig. 2 shows a diagram for dip angle and radiusof the generated fractures for the first fracture realization. Thefractures are generated stochastically using Gaussian stochasticsimulation in which each fracture feature is generated based onthe random realization and it continues until the total fractureintensity and fractal dimension of the studied area are met. Thesubsurface fracture map of the area (which include top, middleand bottom zone) around the tested well is generated by using thecalculated fracture intensity of 0.4 m�1. Fig. 3 presents initialrealization of the generated subsurface fracture map.

1.2. Simulation of fluid flow in fractured reservoirs

In this paper, a hybrid methodology, which combines the singlecontinuum and the discrete fracture approach in a poroelasticframework, is utilised to simulate fluid flow. A typical fracturedbasement reservoir is used for flow simulation. In the proposedmethodology small to medium fractures (fractures (l o40 m, 4500fractures) along with their original properties (orientations andlocations) are considered as part of the matrix (in the form ofpermeability tensor, which can be contributing to local verticalheterogeneity) and long fractures (l Z40 m, 2000 fractures) alongwith their original properties (orientations and locations) areconsidered as discrete fractures. For this purpose a threshold valuefor fracture length is defined. Fractures with the length smallerthan the threshold value are used to generate the grid basedpermeability tensor (3D). Fractures with length longer than thethreshold value, are explicitly discretised in the domain by usingthe tetrahedral elements. Single phase flow is then simulated bycoupling permeability tensors and flow through discrete fracturesin a poroelastic framework. The threshold length has been selecteddepends on the effect of different fracture length on the reservoirperformance. A preliminary run was performed to predict fluidrecovery from the studied reservoir using all fracture as a discretefractures (using our novel multiphase fluid flow simulator infractured reservoirs, Azim, 2015). Then we started to remove shortfractures in length and equivalent permeability tensors have beenused with trying to match the oil recovery with the first run. Thisloop is repeated by removing short fractures in length until a poormatch occurred (between oil recovery resulted from the first runand oil recovery by using equivalent permeability tensors). Thispoor matching gave a clue about the minimum fracture thresholdlength in which we have to maintain to preserve the matchingprocess.

1.3. Simulation of block based permeability tensor

First, the reservoir is divided into a number of grid blocks (1875grid blocks) with dimension of (20 m�20 m�30 m) (see Fig. 4a)and short to medium fractures, that cut these blocks, are used tocalculate permeability tensors (see Fig. 4b). The grid block (do-main) along with small and medium fractures as defined by thethreshold value is discretised using tetrahedral elements in 3Ddomain for matrix and by triangular elements in 2D domain forfractures. Once the block based permeability tensors (3D) arecalculated the reservoir domain along with long fractures arediscretised by tetrahedral elements for matrix as well as triangularelements for fractures as shows in Fig. 4c. The used grid blockswere built by using in house developed mesh generator and thegrid block size was selected arbitrarily to be less than the mini-mum discrete fracture length (40 m) to honour the reservoir het-erogeneity. The used method for calculating permeability tensor inthis paper has been validated in Azim (2015).

In this paper, the grid based permeability tensor includes diagonaland off diagonal terms which are calculated by using three dimen-sional flow equations given by Darcy's law and continuity equation. In

Fig. 10. Initial block based RMS (Root Mean Square) permeability tensor (a) 3D block based permeability tensor map of the reservoir, (b) well cross section with permeabilitytensor map. Wellbore trajectory is marked by vertical line.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422412

this method, pressure and flux boundary conditions that have beenpresented by Durlofsky (1991) are used along the block boundaries.The major drawback of this method is that it assumes no flux inter-change between matrix and fractures. Nevertheless this approach isuseful for rocks with negligible permeability or where the matrixcontribution is known to be very small (like in fractured basementreservoirs). A well comparison has been made by Anupam (2010) tocompare the effective permeabilities obtained from the percolationanalysis to the actual flow based upscaled permeability in case of thematrix contribution is very low. The comparison results show a goodagreement between the two methods for computing effective

Fig. 11. Optimised block based RMS (Root Mean Square) permeability tensor (a) 3D bloshow the changes in permeability values near well bore location after optimisation. We

permeability, and this confirm that we can use the flow based up-scaled permeability along with Durlofsky (1991) periodic boundaryconditions in our study. The governing flow equations and theirdiscretisation in a finite element formulation are presented inAppendix A.

1.4. Validation of permeability tensor methodology

The main aim of this part is to calculate the permeability tensorfor the generated subsurface map in Fig. 5 and 6 using Oda (1980)method and our developed numerical simulator as well to check

ck based permeability tensor map after optimisation and (b) well cross section tollbore trajectory is marked by vertical line.

Fig. 14. Measured and simulated shut in pressure after inversion at wellbore lo-cation with using the optimised subsurface fracture map presented in Fig. 3, fractaldimension¼1.25, matrix permeability¼0.0095 md, Pinitial¼5063 psi and produc-tion time before shut in¼72 h. This plot is produced based on hybrid approach.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 413

its reliability. Input reservoir parameters that have been usedduring the calculations process are provided in Table 1. The frac-ture aperture is assumed to be 0.1 mm with fracture radius ran-ging from 40 m to 100 m. The number of fractures in the entirereservoir region is about 2000 fractures. The permeability tensor iscalculated by considering the reservoir is one block and the resultsof the numerical simulator are compared against the Oda's ana-lytical solution.

Using the analytical equations derived by Oda (1985), the cal-culated permeability tensors are as follow:

=

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥

k k k

k k k

k k k

4.246 0.6418 0.01670.6418 3.2708 0.008170.0167 0.00817 7.3243 3

xx xy xz

yx yy yz

zx zy zz

The results of permeability tensor calculations using the de-veloped numerical simulator are as follow:

Fig. 12. Plot of fracture intensity versus mean square permeability.

Fig. 13. (a) 3D optimised fracture map generated using the optimised block based permeability tensors and (b) initial 3D fracture map generated using object based model.

Fig. 15. Pressure change and pressure derivatives after inversion at wellbore lo-cation with using the optimised subsurface fracture map presented in Fig. 13, fractaldimension¼1.25, matrix permeability¼0.0095 md, Pinitial¼5063 psi and produc-tion time before shut in¼72 h. This plot is produced based on hybrid approach.

Fig. 16. Block based fracture intensity map for the entire reservoir region obtained by using the optimised block based permeability tensors. Fracture intensity blocks lessthan 0.165 m�1 are removed from the fracture intensity map.

Fig. 17. Close view of fracture intensity map around the wellbore. The producing well is partially penetrated the formation thickness and intersects with blocks have a lowfracture intensity value. Fracture blocks intensity less than 0.165 m�1 are removed from the fracture intensity map.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422414

=

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥

k k k

k k k

k k k

4.195 0.7218 0.02350.7218 3.458 0.006250.0235 0.00625 7.135 4

xx xy xz

yx yy yz

zx zy zz

The permeability tensors values from Oda's analytical and nu-merical solutions are in a good agreement.

1.5. The hybrid approach

Single phase fluid flow in a typical fractured basement re-servoir is simulated by coupling 3D permeability tensors with flowthrough discrete fractures. Long fractures (lZ40 m) along with

their original properties (orientations and locations) are dis-cretised explicitly within the reservoir domain. The reservoir andfluid properties used in fluid flow simulation are presented inTable 2. Results of velocity and pressure fields for the case of in-jection are presented in Fig. 7. Change in reservoir pressure andfluid velocity due to water injection is shown in Fig. 7(a), (b), (c),and (d) after 1 and 10 years respectively. As can be seen from thesefigures, the permeability of the discrete fractures has a significanteffect on pressure diffusion. Fluid is moving fast through the in-terconnected fractures and after 10 years most of the reservoir isswept by injected fluid. The reservoir pressure and velocity issignificantly high around the injection well and with continuousfluid circulation for 10 years, pressure and fluid velocity reach to a

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 415

quasi-steady state and the variation of reservoir pressure and fluidvelocity across the reservoir remain roughly constant.

1.6. Inversion of well test data

In this section, well test data (dynamic data) are used to reducethe level of uncertainty associated with generation of subsurfacefracture map by statistical analysis of field data (static data). In thecontext of inversion of well test data flow simulation by hybridapproach (forward simulation) plays a critical part as it traces theflow path and consequent pressure changes along the fractures,thus allowing interpretation of well tests data for characterisingfracture properties. In view of this the simulated pressure changesfor a typical fractured basement reservoir from Vietnam offshore iscompared with that of the well test data (see Fig. 8). The resultsshow that although the pressure change in both cases shows asimilar trend there is a large discrepancy between the simulatedpressure data and that of the well test data. This discrepancy ismainly due to the fracture map generated based statistical analysisfield data which has failed to mimic well production history. Thereare numerous techniques available in the literature as discussed inthe introduction section which can be used to reduce the un-certainty associated with the statistical analysis. In this paper, aninnovative gradient based inversion of well test data is utilised.Few papers are focused on history matching process in fracturedreservoirs by using optimisation technique. Among these, studiesby Suzuki et al. (2005), Cui and Kelkar (2005) and Gang and Kelkar,(2006) are noteworthy. In these papers the authors used a gradientbased optimisation technique to reduce the error between theestimated pressure data and the well test data. In their optimisa-tion approach the authors used dual porosity approach and Eclipse100 to perform the forward numerical flow simulation. Gang andKelkar (2006) proposed adjusting the fracture permeability in-stead of the grid block effective permeability during the historymatching. Other approaches have used fracture intensity as ahistory matching parameter (Suzuki et al., 2005; Cui and Kelkar,2005), where the effective permeability of a grid block was com-puted from the fracture intensity A similar method in which cellproperties were computed based on the number of fractures thatcross each cell was presented by Hu and Jenni (2005). In theirwork, they proposed a generalization of the gradual deformationmethod for history matching object-based models. The methodwas applied to a fracture network, where fractures were modelledusing an object-based approach (Cacas et al., 2001). The historymatching was conducted by moving the fracture location alongtrajectories defined by an algorithm for migrating general Poissonpoint patterns. Dual-media approaches have also been used withthe gradual deformation technique for history matching fracturedreservoirs (Jenni et al., 2004). While Verscheure et al. (2012)proposed a newmethodology to perform the history matching of afractured reservoir model by adapting the sub-seismic faultproperties and positions.

The scale description of the gradient based optimisation tech-nique is presented in Appendix B. The main purpose of this in-version technique is to change a parameter (such as the perme-ability) until the objective function is achieved, that is the differ-ence between the simulated and field observed data (pressure) isminimised. The gradient based optimisation technique is carriedout in three steps: (a) fluid flow simulation by the forward nu-merical model, (b) adjoint system formulation, and (c) the sensi-tivity coefficient formulation. The derivation of forward numericalmodel along with adjoint system and the sensitivity coefficientformulations are described in details in Appendix B.

2. Results

In Fig. 9, a flow chart describing different steps used to optimisethe subsurface fracture map is presented. The estimated blockbased permeability tensors (which include all fractures, mall tolarge fractures) for the subsurface fracture map (as presented inFig. 5) are presented in Fig. 10. After the initial inversion (first stepinversion), the optimised block based permeability tensors of thereservoir are presented in Fig. 11. A convergence criterion is ap-plied in the first inversion step to achieve minimum error betweenthe simulated and measured well test data. It is evident from theresults that the region around the wellbore is mostly affected bythe permeability changes during the first step of inversion. This isbecause, the pressure build up test is carried out on a singlewellbore and as a consequence of this pressure changes only takeplace around the wellbore. It can also be observed from thesefigures that the change in permeability is directional. This meansthat the fracture flow properties, such as the fracture geometry,aperture, connectivity and orientation have an effect not only onthe change in magnitude of permeability also the direction ofpermeability.

In the second inversion step, a relationship between the frac-ture intensity and the optimised permeability for each block isestablished. Fig. 12 shows the fracture intensity versus block basedpermeability tensor. Based on the plot (see Fig. 12), Eq. (5) for bestfit is obtained (see below):

= * − ( )−⎡

⎣⎢⎤⎦⎥FI K0.0104 0.002 5RMS

where FI is the fracture intensity and−KRMS is the optimised root

mean squared permeability of the block. Eq. (5) gives block basedfracture intensity which is then used to generate different rea-lisations of subsurface fracture map. The forward numericalmodelling with the Hybrid approach is used in the 3rd inversionstep to simulate fluid flow through different subsurface fracturenetworks. The simulated change in pressure and pressure deriva-tives are compared with that from the well test data for eachrealization until the error is minimised. The optimised discretefracture map of the reservoir with minimum error is presented inFig. 13. The simulated and measured pressure, change in pressureand pressure derivatives for the optimised subsurface fracturemap are presented in Figs. 14 and 15 respectively. In Fig. 15, thebuild-up test data presented by Farag et al. (2010) are juxtaposedwith that produced in this study. From the plot it can be seen thatthe optimised data produced in this study and that by Farag et al.(2010) match quite well with the well test data. Although the re-sults of pressure data from both studies show a good match withthe well test data, it is interesting to note that it is possible to showchanges in permeability (directional), pressure and velocity profilenear the wellbore region during the process of optimisation (aspresented as part of this study). This means that the optimisationprocess not only changes the permeability, also it changes fractureproperties, therefore the fractures orientation and distributionaround the tested well.

3. Discussion

From the pressure derivative plot (see Fig. 15) three distinctflow regimes can be identified: the early time, the mid time andlate time flow regimes. In the early time, flow regime is a sphericalmarked by a negative slope (negative slope of the pressure deri-vative curve) which is due to partial penetration of the reservoir(90 m open hole out of 250 m) by the producer. Due to the partialpenetration, flow to wellbore changes from radial to spherical at

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422416

the base of the well. According to Brons and Marting (1961) andOdeh and Babu (1990) the deviation from radial to spherical flowis due to restricted fluid entry to the wellbore which contributes toan additional pressure drop near the wellbore and can be inter-preted as skin factor. This flow regime occurred due to using ofpartially penetrated well (90 m out of 250 m, total formationthickness). The mid time flow regime is a short radial flow markedin the pressure derivative curve as a flat trend. Noteworthy is thatboth early and mid-time flow regimes behave in the same way asthat of a porous media. This can be explained by the fact that thewell is hardly penetrated by any fracture or medium to highfracture intensity region (see Figs. 16 and 17). The late time flowregime is a linear flow as recognized by a positive half-slope of thederivative curve which is caused by the fluid flow in discretefractures. The geometry of linear flow regime consists of strictlyparallel flow vectors. The parameters that can be calculated usingthis flow regime are the permeability of the formation in the di-rection of the flow vectors and the flow area normal to the flowvectors. In case of the fractured reservoir, the slope of the straightline fitting the data on the derivative plot can be used to de-termine the fracture length (Bourdet, 2002).

4. Conclusion

In this paper, the governing equations of single phase fluid floware derived from mass and momentum balance equations. Theseequations are used to simulate flow in discrete fractures andporous matrix medium. The single phase fluid flow is used as aforward fluid flow model in history matching of well test data bygradient based optimising method. Pressure change and pressurederivatives for different fracture patterns, which are generated indifferent realisations using an object based model, are estimatedand compared with the well test data. This process is continueduntil the difference between the simulated pressure data and welltest data is reduced to minimum.

The results show that, the simulated well test data from thefirst realization of fractures vary significantly (42%) from the ac-tual data. The inversion process has allowed us to modify the si-mulated pressure data by changing the fracture parametersthrough different realisations which in turn changed the flowproperties of fracture. This means that, had the 3D simulation offluid flow in discrete fractures along with the inversion techniquenot been used to match the pressure transient data, it would nothave been possible to generate a realistic image of the subsurfacefracture map, which includes fracture distribution, orientation,and connectivity between fractures. In addition, the workflowpresented in this study can effectively traces fluid flow in fracturesaround the wellbore and overcome the limitations of conventionalwell test analysis presented by Farag et al. (2010). Thus, in afractured reservoir well test data serves as critical information toevaluate fracture properties near wellbore region. It is, thereforesuggested to have multiple well test data (interference well test)which can allow field wide optimisation of fracture properties.

Appendix A: Calculations of 3D permeability tensors

Three-dimensional flow equations used for permeability-tensorcalculations are given by Darcy's law and continuity equations asfollow:

= −

∂∂∂∂∂∂

=→⎯→⎯

∇

( )

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

vv

v

k k k

k k k

k k k

pxpypz

k p.

A.1

x

y

z

xx xy xz

yx yy yz

zx zy zz

∇ = ( )v. 0 A.2

where v is the fluid velocity vector, ⃗ ⃗k is the permeability tensorand ∇p is the local pressure gradient. The grid based permeabilitytensors for short to medium fractures are calculated by using finiteelement technique. Each fracture is represented as a sandwichedelement (triangular element) between three dimensional tetra-hedral elements representing the matrix porous media as shownin Fig. A.1. Single phase fluid flow governing equation throughmatrix porous medium can be written as follows:

μ∇

→→

∇ + =

( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟p Q.

k. 0

A.3H

μ= −

→→

∇

( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟v p

k.

A.4

where ⃑ ⃑k is a full permeability tensor, μ is the fluid viscosity, Q H isfluid source/sink term represents the fluid exchange betweenmatrix and fracture, or fluid extraction (injection) from thewellbore.

Single phase fluid flow governing equation through discretefractures can be expresses as:

μ∇ ∇ + + =

( )+ −

⎛⎝⎜

⎞⎠⎟

kp q q. 0.0

A.5

ff

where +q andq- are the leakage fluxes across the boundary inter-faces, ∇̅ is the divergence operator in local coordinates system(Watanabe et al., 2010), and Pf is the pressure inside the fracture.The permeability of the fracture can be expressed by a parallelplate concept (cubic law) as shown in Eq. (A.6). It has been as-sumed that the fracture surfaces are parallel and the fluid-flowthrough a single discrete fracture is laminar (Snow, 1969).

= ( )kb12 A.6f

2

where b is the fracture aperture, and kf is the fracture perme-ability. As a result of superimposing fracture/matrix system, thefracture/matrix interface is directly treated by using the finiteelement technique (see Fig. A.1).

Finite element method formulation

The weighted-residual method is used to derive the weak for-mulation of the governing equation of fluid flow through a frac-tured system. Standard Galerkin method is applied to discretisethe weak forms with appropriate boundary conditions (Zimmer-man and Bodvarsson, 2000).

Eqs. (A.3) and (A.5) are written separately for matrix porousmedium and discrete fractures in finite element formulation. Thematrix is discretised using 3D tetrahedral elements and fracturesare discretised by using 2D triangle elements. If FEQ representsthe flow equation, the integral form can be written as follow:

Fig. A.1. Formulation of a full matrix in 3D space. Fracture is represented as (Sandwich elements) and matrix elements (Elements 1&2) assembled to be used during thesimulation process.

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 417

∫ ∫ ∫ ∫ ∫ ∫Ω Ω Ω= + ¯

( )Ω Ω Ω

FEQ d FEQ d FEQ d

A.7m f

m f

where Ω̅f represents the fracture part of the domain as a 2D entity,and Ωm represents matrix domain and Ω is the entire domain. 2Dintegral fracture equation is multiplied by the fracture aperture bfor consistency of the integral form.

The formulation of the governing equation for hydraulic pro-cess through matrix porous medium can be written as:

μ∇

→→

∇ + =

( )

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

kp Q. 0.0

A.8H

Introducing weak formulation, Eq. (A.8) is expanded as follow:

∫ ∫

μ μ μ

μ μ μ

μ μ μ

Γ×

∂∂

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

∂∂

+ ∂∂

+ ∂∂

+ =

( )

Ω Γ

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟

w

xk p

x

k py

k pz

y

k px

k py

k pz

zk p

x

k py

k pz

w qd 0.0

A.9

xx xy xz

yx yy yz

zx zy zz

T

where (w¼w(x, y, z)) is a trial function. The weak formulation fordiscrete fractures of Eq. (A.9) is expanded as follow:

∫ ∫ ∫μΓ Γ∇ × ∇ + × + × =

( )Ω Γ Γ

+ −⎛⎝⎜

⎞⎠⎟w

kp w q d w q d. 0.0

A.10

ff

f f f

Galerkin finite element method

Three basic steps in finite element approximation are (1) domaindiscretisation by finite elements, (2) discretisation of integral (weak)formulation and (3) shape functions generation of field variables un-knowns. Galerkin finite element method is used to discretise the in-tegral weak forms of Eqs. (A.9) and (A.10). Unknown variables areapproximated by using appropriate shape functions.

= × ( )p N p A.11p

where Np the pressure is shape function and p is the vector of nodalvalues of the unknown variable.

Finite element formulation of the governing equation for hy-draulic process for matrix and fracture system can be written in amatrix form:

⎯→⎯⎯ →⎯⎯Δ→ =

→( )M p f A.12

⎯→⎯⎯ →⎯⎯=

⎯ →⎯⎯⎯ →⎯⎯⎯+

⎯→⎯⎯ →⎯⎯( )M M M A.13f m

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422418

∫ ∫μΩ

μΓ

⎯→⎯⎯ →⎯⎯= ∇

→⎯→⎯

∇ + ∇×

∇( )Ω Γ

M Nk

N d Nk b

N dA.14p

mpm

pf f

pfT T

∫ Γ→

= − −⎯ →⎯⎯⎯ →⎯⎯⎯

× → −⎯ →⎯⎯⎯⎯ →⎯⎯⎯⎯

× →( )Γ

− −f N Q d M p M p A.15p

mH f

im

i1 1T f m

where m is referring to matrix porous medium and f to fracturenetwork.

Boundary conditions

Pressure and flux boundary conditions that have been pre-sented by Durlofsky (1991) are used for calculations of perme-ability tensor in fractured porous medium. These conditions arelisted in Eq. (A.16) to Eq. (A.21) and Fig. A.2.

( ) ( )= = = − ∂ ∂ ( )p y x z p y x z G on D and D, 0, , 1, A.163 4

( ) ( )= ⎯→⎯ = − = → ∂ ∂ ( )u y x z n u y x z n on D and D, 0, . , 1, . A.173 4 3 4

( ) ( )= = = − ∂ ∂ ( )p y x z p y x z G on D and D0, , 1, , A.181 2

( ) ( )= ⎯→⎯ = − = → ∂ ∂ ( )u y x z n u y x z n on D and D0, , . 1, , . A.191 2 1 2

( ) ( )= = = − ∂ ∂ ( )p y x z p y x z G on D and D, , 0 , , 1 A.205 6

( ) ( )= ⎯→⎯ = − = → ∂ ∂ ( )u y x z n u y x z n on D and D, , 0 . , , 1 . A.215 6 5 6

where n is the outward normal vector at the boundaries, ∂D is thegrid block boundary and G is the pressure gradient. Specifying azero pressure gradient along the y-, z- faces ( =∂

∂0p

y, and =∂

∂0p

z)

and solving Eq. (A.12) with using the abovementioned periodicboundary conditions, the average velocity along x-,y- and z- di-rections can be calculated as follow:

∫= −⎯ →⎯

( )∂v v n dxdz. 3

A.22xD3

Fig. A.2. : Periodic boundary conditions used for over a grid block.

∫= − ⎯→⎯( )∂

v v n dydz.A.23y

D1

1

∫= − ⎯→⎯( )∂

v v n dxdy.A.24z

D5

5

Eq. (A.1) can be expressed in an explicit form as follow:

= − ∂∂

− ∂∂

− ∂∂ ( )

v kpx

kpy

kpz A.25x xx xy xz

= − ∂∂

− ∂∂

− ∂∂ ( )

v kpx

kpy

kpz A.26y yx yy yz

= − ∂∂

− ∂∂

− ∂∂ ( )

v kpx

kpy

kpz A.27z zx zy zz

where vx, vy and vz are velocities in x, y, and z directions respec-tively. Since pressure gradient through x direction is assumed andvx, vy and vz are known, then kxx, kxy and kxz are easily determined.The rest of permeability tensor elements can be calculated byusing the same boundary conditions as abovementioned but inother directions.

Appendix B: Derivations of gradient based method equations

Adjoint system formulation

To formulate Adjoint system of equations, the single phase flowequation is discretised in the following from:

( ) ( )γ φ∇ ∇ − ∇ ( ) − = ∂∂ ( )k p D x y z q x y z t cpt

. , , , , , B.1t

( )( ) = − ( )+ + +F p k k k R p p, , , B.2m

l lx y z i j k i j k

li j kl1 1

, , , ,1

, ,

where

ϕ=

Δ ( )R

c

t B.3i j ktl, ,

( ) = − ( )+ + +F p k k k A A, , , B.4m

l lx y z m

lml1 1 1

( ) ( )φ = ( )+ + +A p R p, B.5m

l l l1 1 1

( ) ( )ϕ = ( )A p R p, B.6ml l l

wherelrefers to the time step and m refers to control variables

( )φ k k k, , ,x y z . Fm is independent of porosity and only depends on

permeability tensor, while +Aml 1and Am

l are porosity- dependentand permeability-independent. According to these factors thefollowing equations hold:

Travel time shift

Three different ways used to represent the production datamisfit, namely, the “amplitude misfit”, “travel time misfit”, and“generalized travel time misfit”.

The “travel time misfit” is defined as where the misfit is obtained

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 419

by lining up the observed and the predicted data at a reference timesuch as the breakthrough or the first arrival time. The travel timemisfit has major advantages compared to the amplitude during in-version, first it has a quasi-linear properties compared to the high nonlinearity of the amplitude as a result the travel time inversion is robustand converge rapidly even if the initial model is far away from thesolution. Second, it is computationally efficient because the number oftravel-time is equal to the number of wells, regardless of the numberof data points. This leads to considerable savings in computationaltime during the minimization (Cheng et al., 2003).

∇ = ∇ = ∇ = ( )+ + +⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦A A A 0 B.7k m

lk m

lk m

l1 1 1x y z

∇ = ∇ = ∇ = ( )⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦A A A 0 B.8k m

lk m

lk m

lx y z

Now, the generalized travel time shift at each well can be de-noted by ( )g p k k k, , ,x y z subject to a set of constraints equations in

the form of finite difference equations.In order to see how the perturbations of kx, ky, and kz at each grid

block affect the generalized travel time shit at each well, a two di-mensional vector of Lagrange multipliers is used to adjoin g with thefluid flow equations to from the augmented objective function J asfollow:

λ λ λ λ= ( )⎡⎣ ⎤⎦................ B.9

l lMl

1 2

( )∑ λ= + − +( )=

−+ + +J g F A A

B.10l

Ll

ml

ml

ml

0

11 1 1

The total differential equation of Eq. (B.10) is performed withrespect to the state variables pressure and control variables,

( )k k k, ,x y z

( )( ) ( )

( ) ( )( ) ( )

∑

∑

λ

λ φ

φ

= + ∇ −

+

∇ + ∇

+∇ − ∇

+∇ + ∇

+

( )

φ

φ

=

−

=

−+

+ +

+ +

⎡⎣ ⎤⎦⎛

⎝

⎜⎜⎜⎜⎜

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

dJ dg F A dp

F dk F dk

F dk A d

A d A dp

BT

B.11

l

Ll

p ml

ml

l

Ll

k ml

x k ml

y

k ml

z ml

ml

p ml l

L

1

1

0

11

1 1

1 1

l

x y

z

l

where BTL is calculated from the following equation:

( )λ= ∇ − ( )+ ⎡⎣ ⎤⎦BT F A dp B.12

L lp m

LmL1

l

Initially, the pressure values at each grid block is known andthe variation of control variables will not have any effect on it,thus,

= ( )dp 0 B.130

Therefore, any term multiplied by dp0had to be neglected fromEq. (B.13).

Also, by taking total differentiation of g with respect to stateand control variables, the equation will be as follow:

(∑

ϕ

= ∇ ) + ∇ + ∇ + ∇

+ ∇ ( )ϕ

=⎪ ⎪

⎪ ⎪⎧⎨⎩

⎫⎬⎭

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦

dg g dp g dk g dk g dk

g d B.14

l

L

p k x k y k y1

lx y y

Substituting Eq. (B.14) into Eq. (B.11);

{ }{

}} }

( ) ( )

( ) ( )( ) ( ) ( ) ( )

∑ ∑

∑

λ λ

λϕ

= + ∇ − + ∇ + ∇

∇ + ∇ + ∇ + ∇

+ ∇ + ∇ − ∇ + ∇ + ∇+

ϕ ϕ ϕ

=

−

=

−+

=

−+

+ +

+ +

⎜ ⎟

⎜

⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎛⎝

⎞⎠

⎫⎬⎭⎛⎝⎜

⎡⎣⎢

⎤⎦⎥

⎛

⎝

⎜⎜⎜⎜⎜

⎡

⎣

⎢⎢⎢⎢

⎡⎣ ⎤⎦⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎡⎣ ⎤⎦ ⎛⎝ ⎡⎣ ⎤⎦

⎤

⎦

⎥⎥⎥⎥B.15

dJ dg F A dp g dp A dp

F g dk F g dk

F g dk A A g d

BT

l

Ll

pl ml

ml

pll

Ll

pl ml l

l

Ll

kx ml

kx x ky ml

ky y

kz ml

kz z ml

ml

L

1

1

0

11

0

11

1 1

1 1

Eq. (B.15) shows how the changes in ( )k k k, ,x y z affect thepressure at each grid block at each time step.

The objective now is to find equations that relate the sensitivityof (g) function with respect to control variables (permeability) andto remove the dependency of J function from pressure. Adjointvariables were selected to fulfil this objective. The result is thefollowing adjoint equations:

(( ) ( )∑ ∑λ λ∇ − + ∇ = − ∇( )=

−+

⎪⎪

⎪

⎪⎧⎨⎩

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎫⎬⎭

F A dp A dp gB.16

lp m

lml

l

Ll

p ml l

p0

11

l l l

Similarly, to remove the dependency of pressure change at theend of simulation run, dpl has to be removed from the system ofequations, thus, the following condition is set as follow:

λ = ( )0 B.17L

Therefore, from Eq. (B.12) and Eq. (B.16).

= ( )BT 0 B.18L

From the Eq. (B.17), and Eq. (B.18), the change in the augmentedfunction j in Eq. (B.15) will be:

{ }{

}} }

( ) ( )( ) ( ) ( ) ( )

∑ λϕ

=∇ + ∇ + ∇ + ∇

+ ∇ + ∇ − ∇ + ∇ + ∇ϕ ϕ ϕ=

−+

+ +

+ +⎜

⎛

⎝

⎜⎜⎜⎜⎜

⎡

⎣

⎢⎢⎢⎢

⎡⎣ ⎤⎦⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎡⎣ ⎤⎦ ⎛⎝ ⎡⎣ ⎤⎦

⎤

⎦

⎥⎥⎥⎥B.19

dJF g dk F g dk

F g dk A A g dl

Ll

kx ml

kx x ky ml

ky y

kz ml

kz z ml

ml0

11

1 1

1 1

In this study, the (g) function is given by:

( )∑= −( )=

⎡⎣⎢⎢

⎤⎦⎥⎥g

nt t

1

B.20dj i

n

obs ji

cal ji

1, ,

dj

where ndj is the number of data point for well j, (i) is the data pointindex at time ti. The gradient of the scalar function g in the adjointsystem of equations is given by using the gradient of Eq. (B.21) asfollow:

∇ = ∇ Δ˜ =∂Δ˜

∂∂Δ˜

∂∂Δ˜

∂ ( )

⎡⎣⎢⎢

⎤⎦⎥⎥g t

t

P

t

P

t

P B.21p p j

j j j

M1 2l l

At grid blocks of producing wells, the partial derivative of thegeneralized travel time with respect to pressure is given by:

( )∑∂Δ˜

∂= ∂

∂− ≅ −

∂˜

∂ ( )=

⎡⎣⎢⎢

⎤⎦⎥⎥

t

P n Pt t

n

t

p

1.

1

B.22

jl

djl

i

n

obs ji

cal ji

dj

cal jl

1, ,

,dj

Sensitivity Coefficients Formulation

In adjoint formulation section, the adjoint variable at each gridblocks has been calculated. This section describes the calculationsprocess for sensitivity coefficient by using the knowns adjointvariables. The sensitivity coefficients calculation equations are gi-ven as follow:

( ) ( )∑ ∑ λ∇ = ∇ − ∇( )= =

−+ +J F g

B.23k

T

m o l

L

k ml

ml

kT

0

11 1

x x x

( ) ( )∑ ∑ λ∇ = ∇ − ∇( )= =

−+ +J F g

B.24k

T

m o l

L

k ml

ml

k

T

0

11 1

y y y

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422420

( ) ( )∑ ∑ λ∇ = ∇ − ∇( )= =

−+ +J F g

B.25k

T

m o l

L

k ml

ml

kT

0

11 1

z z z

Implementation of Finite Element Method (FEM) for governingequations of Poroelasticity for Single Phase Fluid Flow.

This appendix details the FEM formulation of the Poroelasticproblem.

Assuming off-diagonal components of permeability tensor arezero,

=→→ ⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟k

k

k

k

0 0

0 0

0 0

xx

yy

zz

ϕ αμ μ

μ

∂∂

− ∂∂

∂∂

∂∂

∂∂

= ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂ ( )

→⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

cpt t x y z

ux

k px y

k py

zk p

z B. 26

tx y

z

Multiplying both sides of Eq. (B.26) by a trial function, w andintegrating over the domain, Ω yields:

∫ ∫

∫

ϕ Ω α Ω

μ μ μΩ

∂∂

− ∂∂

∂∂

∂∂

∂∂

= ∂∂

∂∂

+ ∂∂

+ ∂∂ ( )

Ω Ω

Ω

→⎜ ⎟⎛⎝

⎞⎠

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟⎞⎠⎟⎟

wcpt

d wt x y z

u d

wx

k px

k

yk

zd

B. 27

t

x y z

Using the Green formulae, Eq. (B.2) becomes:

∫ ∫

∫ ∫

∮

∫

ϕ Ωμ

Ω

μΩ

μΩ

μ μ μΓ

α Ω

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

∂∂

∂∂

= ∂∂

+ ∂∂

+ ∂∂

+

∂∂

∂∂

∂∂

∂∂ ( )

Ω Ω

Ω Ω

Ω

→

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

wcpt

dwx

k px

d

wy

k py

dwz

k pz

d

wk p

xn

k py

nk p

zn d

wt x y z

u dB. 28

tx

y z

r

xx

yy

zz

where ‘Γ’ is the boundary and ‘n’ is the outward normal to theboundary. Finite difference method is used to discretise the termsincluding differentiation with respect to time. Permeability andporosity remain constant with change in time. After rearranging,Eq. (B.28) becomes:

∫ ∫

∫ ∫

∫

ϕΔ

Ωμ

Ω

μΩ

μΩ

α Ω

− + ∂∂

∂∂

+

∂∂

∂∂

+ ∂∂

∂∂

= ∂∂

∂∂

∂∂

→ − →

Δ ( )

Ω Ω

Ω Ω

Ω

− −− −

− −

−

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

wcp p

td

wx

k px

d

wy

k pd

wz

k pz

d

wx y z

u u

td

B. 29

ti i

i i

ixi i

yi i

y

zi i

i i

i

1 11 1

1 1

1

In which ‘i’ and ‘i�1′ are current and previous times respec-tively. Using Galerkin approach (Zienkiewicz and Taylor, 2000k),Eq. (B.29) becomes:

∫ ∫

∫ ( )

ϕ Ωμ μ μ

Ω

α Ω

→ → →−

→

Δ+

∂→

∂∂→

∂+

∂→

∂∂→

∂+

∂→

∂∂→

∂→

→ ∂∂

∂∂

∂∂

→−

→

Δ=

Ω Ω

Ω

− −

−

−

−

−

−

−

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎤

⎦

⎥⎥⎥⎥⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟ B. 30

c N N dP P

t

k N

x

N

x

k N

y

N

yk N

z

N

zd P

Nx y z

dU U

t0

ti

pT

p

i i

ixi

ipT

p yi

ip

Tp z

i

ipT

p i

pT

i i

i

1 1

1

1

1

1

1

1

where:

( )→= ⋯ ( )P p p p B. 31

T

n1 2

( )⎯→⎯= ⋯ ( )N N N Nn B. 32p

T

1 2

⎯→⎯=

( )

⎡⎣⎢

⎤⎦⎥N

N N

N N

0 0

0 0 B. 33u

n

1 2

1

( )→= ⋯ ( )U u u u u u B. 34

Tx y z x yn1 1 1 2

where ‘n’ is the total number of nodes. After rearrangement, Eq.(B.30) can be written as:

→⎯→⎯ → − + Δ⎯→⎯⎯ →⎯⎯ → −

→⎯→⎯ →− =

→

( )− −⎜ ⎟⎛

⎝⎞⎠

⎛⎝⎜

⎞⎠⎟L p p t H p Q U U 0

B. 35

i i i i ii1 1

where:

∑→⎯→⎯=

⎯→⎯⎯ →⎯⎯

( )=

L LB. 36e

nee

1

in which ‘e’ denotes element, ‘ne’ the number of elements and

∫ ϕ Ω⎯→⎯⎯ →⎯⎯

=⎯→⎯ ⎯→⎯

( )Ω

− −⎛⎝⎜

⎞⎠⎟L c N N d

B. 37e

ti e

p

T

p1 i 1

In which;

∑⎯→⎯⎯ →⎯⎯=

⎯ →⎯⎯⎯ →⎯⎯⎯

( )=

H HB. 38e

nee

1

∫μ μ

μΩ

⎯ →⎯⎯⎯ →⎯⎯⎯

=∂⎯ →⎯⎯

∂∂⎯ →⎯⎯

∂+

∂⎯ →⎯⎯

∂∂⎯ →⎯⎯

∂

+∂⎯ →⎯⎯

∂∂⎯ →⎯⎯

∂( )

Ω

−

−

−

−

−

−

⎛

⎝

⎜⎜⎜⎞

⎠

⎟⎟⎟

H

k N

x

N

x

k N

y

N

y

k N

z

N

zd

B. 39

e

ex

e

e

ep

T

pe

ye

e

ep

T

pe

ze

e

ep

T

pe

i

i

i

i

i

i

1

1

1

1

1

1

∑→⎯→⎯=

⎯→⎯⎯ →⎯⎯

( )=

Q QB. 40e

nee

1

where:

∫α Ω=⎯→⎯ ∂

∂+

∂∂ ( )Ω

⎛⎝⎜

⎞⎠⎟Qe N

Nx

Nx

dB. 41

e p

e eu

eu

T

Eq. (B.35) in an incremental form can be written as:

(→⎯→⎯

+ Δ⎯→⎯⎯ →⎯⎯

)Δ −→⎯→⎯

Δ⎯ →⎯⎯⎯

=→

( )L t H P Q U f B. 42i i i

1

Δ⎯ →⎯⎯

=⎯→⎯

−⎯ →⎯⎯⎯⎯

Δ⎯ →⎯⎯⎯

=⎯→⎯

−⎯ →⎯⎯⎯⎯

→= − Δ

⎯→⎯⎯ →⎯⎯ ⎯ →⎯⎯⎯⎯( )

− −

−

where

P P P U U U and

f t H P

, :

B. 43

i i i i

ei i

1 1

11

Gaussian quadrature is applied for integration, get:

R. Abdelazim, S.S. Rahman / Journal of Petroleum Science and Engineering 145 (2016) 404–422 421

∫ σ Ω⎯→⎯ ⎯→⎯⎯ →⎯⎯

Δ⎯ →⎯⎯

=→

( )Ω

⎛

⎝⎜⎜

⎞

⎠⎟⎟W S d 0

B. 44

TT

where:

⎯→⎯=

( )( )( ) ( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

W

W x y z

W x y z

W x y z

, ,

, ,

, , B. 45

1

2

3

And w1 and w2 are trial functions. Using Green's identity oneobtains:

∫ ∫σ Γ σ Ω⎯→⎯ ⎯→⎯⎯ →⎯⎯

Δ⎯ →⎯⎯

− Δ⎯ →⎯⎯ →⎯→⎯ ⎯→⎯

=→

( )Γ Ω

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟W M d S W d 0

B. 46

TT

T

Where:

⎯ →⎯⎯⎯ →⎯⎯⎯=

( )

⎡⎣⎢

⎤⎦⎥M

nn

n

nn

n00 0

0 B. 47T x

y

y

x

z

z

Rearrange Eq. (B.46), one gets:

∫ ∫

∫

Ω α Ω

σ Γ

→⎯→⎯ ⎯→⎯ ⎯→⎯⎯ →⎯⎯ →⎯→⎯Δ⎯ →⎯⎯⎯

+→⎯→⎯ ⎯→⎯

Δ⎯ →⎯⎯

=⎯ →⎯⎯⎯ →⎯⎯⎯

Δ⎯ →⎯⎯

( )

Ω Ω

Γ

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

S W D S U d S W P d

W M dB. 48

T

e

i i

T Ti

Using Galerkin method Eq. (B.48) yields:

∫

∫ ∫

Ω

α Ω σ Γ

→⎯→⎯ ⎯→⎯ ⎯→⎯⎯ →⎯⎯ →⎯→⎯ ⎯→⎯Δ⎯ →⎯⎯⎯

+∂⎯→⎯

∂∂⎯→⎯

∂⎯→⎯

Δ⎯ →⎯⎯

=⎯→⎯ ⎯→⎯⎯ →⎯⎯

Δ⎯ →⎯⎯

( )

Ω

Ω Γ

⎛

⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞

⎠⎟⎟

⎛

⎝⎜⎜⎜

⎛

⎝⎜⎜

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎞

⎠⎟⎟

⎞

⎠⎟⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

S N D S N d U

NX

NX

N d P N M d

B. 49

u

T

e u

i

u uP

Ti

u T

Ti

Or in a compact form as follow:

→⎯→⎯Δ

→+

⎯ →⎯⎯ →⎯⎯Δ

→=

→( )K U Q P f B. 50

T2

∑→⎯→⎯=

⎯ →⎯⎯ →⎯⎯

( )=

K KB. 51e

nee

1

∫ Ω⎯ →⎯⎯ →⎯⎯

=→⎯→⎯ ⎯→⎯ ⎯→⎯⎯ →⎯⎯ →⎯→⎯ ⎯→⎯

( )Ω

⎛⎝⎜⎜

⎞⎠⎟⎟K S N D S N d

B. 52e

u

eT

e u

e

e

∑→=

→

( )=

f fB. 53e

ne e

21

2

∫ σ Γ→

=⎯→⎯ ⎯ →⎯⎯⎯ →⎯⎯⎯

Δ⎯ →⎯⎯

( )Γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟f N M d

B. 54u

Ti

2 e

Eq. (B.42) and Eq. (B.50) are the final finite element equationsto be simultaneously solved as a system of linear equations whichis as follows:

→⎯→⎯+ Δ

⎯→⎯⎯ →⎯⎯−

⎯ →⎯⎯ →⎯⎯

→⎯→⎯

→⎯→⎯Δ

→

Δ→

=

→

→( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

L t H

Q

Q

K

P

U

f

fB. 55T

i

i

1

2

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