Analytical Modeling of Oil Production from a Matrix Block by FreeFall Gravity Drainage Mechanism

Corresponding Author: Salman Ghorbani Zadeh

Coauthors: Muhammad Shojaadini, Mahdi Matin, Mohammad Hossein Ghazanfari

PhD Student, Institute of Petroleum Engineering, Tehran University

Chemical and Petroleum Engineering Department, Sharif University of Technology

Corresponding Author Email: salman.ghorbanizadeh@gmail.com

Abstract

Free fall gravity drainage is one of the most effective mechanisms ofproducing the oil from the matrix blocks in the gas invaded zone of thenaturally fractured reservoirs. Although several analytical models havebeen proposed to characterize this mechanism in a matrix block, butthese models suffer from the practical or theoretical viewpoints. In thisstudy, a new analytical model was presented to predict the oil productionrate versus time from a homogeneous matrix block under free fall gravitydrainage mechanism. The model was developed by considering thebalance between involved forces in the gas-oil gravity drainage process.By simplifying the equation derived from the momentum balance, adifferential equation was obtained. To develop new analytical model, theobtained differential equation was analytically solved by using theLambert W function. The prediction of derived model was tested againstthe results of numerical analysis and conventional simulation as well asexperimental data in the core and micromodel scale. The validationresults show that the model predictions match with the data setsexcellently.

Keywords: Matrix Block, Free Fall Gravity Drainage, AnalyticalModeling, Gas Invaded Zone, Naturally Fractured Reservoir

. IntroductionAfter beginning the oil production from a typical naturally fracturedreservoir, a new zone named gas invaded zone or secondary gas capwas formed because of gas cap expansion or liberated gas migrationfrom the oil zone toward the top of the reservoir. The matrix blocks whichare in this zone are partially or totally immerse in the gas, and containconsiderable volume of the oil. The most effective mechanism ofproducing this volume of oil is the free fall gravity drainage. Severalauthors have presented different analytical models to characterize theperformance of the gas-oil gravity drainage mechanism in a matrix block.Darcy () formulated the flow of water in a vertical column of porousmedia only under influence of the gravity force; his formulation is appliedonly for the steady-state flow cases. Cardwell et al. () considered aheterogeneous vertical column of porous media. They justified the non-uniform liquid saturation distribution through the porous media bypresenting an unsteady-state liquid drainage model. They divided theporous media to the two saturated and unsaturated regions. Bysubtracting the velocities in these two distinct zones, they could find thegas-oil interface velocity. They did not consider the effect of capillaryforce in the unsaturated zone. Also some quantities which are used intheir model, such as liquid relative permeability in the unsaturated zone,need the empirical equations to obtain which are complex and uncertain.

Nenniger et al. () derived the basic differential equations for theprediction of saturation-time curves for the drainage of packed beds inthe gravitational fields. They presented an approximate series solutionfor these equations. They worked based on film flow theory and considera moving demarcator. Dykstra () extended the Cardwell model tothe oil reservoirs; he improved the relative permeability-saturationrelation by entering residual oil saturation in the equations. He modifiedthe Cardwell model by some simplification to achieve more accurateresults, but some terms still remains uncertain and empirically obtained.Hagoort () developed a new method based on the centrifugal gas-oildisplacement in the small cores for the accurate measurement of the oilrelative permeability. He did this, because it was shown that the accurateobtaining oil relative permeability is a key factor to increase the accuracyof existing models, such as Cardwell et al. and Dykstra models. Vangolf() considered the gas-oil displacement in a vertical matrix blocktotally or partially immerse in the gas. He assumed that the side walls ofthe matrix block are impermeable, the gas enters the matrix from the topof that and displaces the oil toward the bottom of the matrix. Hedeveloped his model to obtain the gas-oil interface velocity by usingDarcy formula and considering equal velocities for the oil and gas. Saidi() presented a differential equation for the oil saturation distributionabove the gas-oil interface in the matrix block. He could solve and

integrate that equation only for the stationary case, when a steady statecondition is reached. Pavone et al. () developed an analyticalsolution for their presented non-linear governing equation for the gassaturation during oil drainage. To solve the governing equation, theylinearized the governing equation by assuming straight-line permeabilitycurve and logarithmic capillary pressure curve. They also conductedsome experiments to validate their model; the analytical solution wasused twice for early and later times, respectively, to match the same setof experimental data. Bech et al. () obtained a diffusivity equation bywriting the flow equations and considering the dual porosity model forsimulating the gravity drainage in the fractured reservoirs. Luan ()studied gravity drainage mechanism in a stack of blocks from bothphysical and mathematical viewpoints. He solved the governing equationgiven by Hagoort analytically and numerically. However, he used thesame boundary condition as that utilized by Pavone et al. Schechter etal. () divided the matrix block to the two distinct zones similar to theCardwells study. They considered the Darcy velocity for the saturatedzone and both the bulk and film oil residual saturation for theunsaturated zone. They generated a differential equation by equaling theDarcy velocity in the saturated zone and the oil volume change throughthe whole matrix block. They solved this equation numerically and useduncertain film flow terms in their solution. Xu () developed an

unsteady-state model to characterize the gravity-driven flow in anangular capillary tube as a fracture. He wrote mass conservation lawsfor the corner flow and inserted steady state relation for the velocityformula in that. By solving derived equation, he could present a formulafor the velocity above and at the gas-liquid interface. However heneglected effect of capillary in his formulation. Donato et al. ()performed the analytical and numerical analysis of the gas-oil gravitydrainage. They presented three distinct analytical models to characterizethe gravity drainage mechanism, ) gravity is the main driving force and) both gravity and capillary effects are important. In the second case itis possible that gravity dominated drainage or capillary dominateddrainage occurs. They did not suggest a clear criterion to use eachmodel for practical purposes. Li et al. () assumed that the free fallgravity drainage is similar to the spontaneous imbibition; therefore theyapplied the model suggested by Aronofsky et al. () in their study tomatch the oil production by the gravity drainage. They claimed that initialoil production rate, entry capillary pressure, and average residual oilsaturation can be estimated by using their oil recovery model. Howeverthey did not present any theoretical support to justify their model. Theseexisting models suffer from the practical or theoretical viewpoints, whichhave mentioned briefly at above material. In this study, a new analyticalmodel for the free fall gravity drainage in a homogeneous matrix block is

presented, which considers the effect of capillary force completely andcan be used for the all conditions: gravity force dominant, capillary forcedominant or when both gravity and capillary forces are important.. Analytical Modeling

To clarify the process which has been modeled in this paper, first aschematic of a typical fractured reservoir after beginning the oilproduction was plotted. Then we focused on the gas invade zone andalso on the gas-oil contact where the oil saturated matrix blocks aretotally or partially immerse in the gas. After that a matrix block which issurrounded by the fractures was considered and finally the free fallgravity drainage was modeled in this matrix (See Fig. ). As you see inFig. , H is the height of matrix block and hm, hf are the heights of oilcolumn in the matrix and fracture respectively. The fluid flow direction isdownward, therefore downward direction was considered as the positivedirection. There are three important forces which govern the free fallgravity drainage in a matrix block; gravity, capillary and friction forces.We formulated these three forces by using equivalent pressurestatement correspond to each force.

Gravity Force (hydrostatic pressure): this is due to the differencebetween the oil and gas density and also difference between theheight of oil column in the matrix and fracture.

)( Capillary Force (capillary pressure): capillary pressure in the matrix

can be defined by Laplace equation as below:

)(

Let now assume that the matrix is represented by the bundle of capillarytubes, by comparing the Darcy law and Hagen-Poiseuille equation toreplace Rm, we can get:

)(

Friction Force (viscous pressure loss): This is due to the frictionbetween the rock wall and viscous fluid flow in the porous media.Assuming laminar and incompressible fluid flow in matrix block andusing Darcys law for the oil velocity, the viscous pressure loss canbe obtained as follow:

)( Substituting the velocity in Eq. () by derivative of height with respect tothe time gives:

)( After formulating the active forces in free fall gravity drainage, themomentum balance of oil flow correspond to these forces must be

written to develop the desired differential equation. It has been knownthat in this zone the capillary and friction forces have resistant roleswhile the gravity acts as a driving force. Therefore the momentumbalance inside the matrix block can be written as follow:

)( + = It is desired for us to find oil recovery and oil production rate from matrixblock, therefore we placed derivative of height with respect to the time inone sides and all other terms in other side.

)( = + All terms in right side of Eq. () are constant and depend on the rock andfluid properties except hm, hf. So we can define two constant parameterswhich can be obtained for each special rock and fluid sample;

)(= , = Substituting equation () in equation () gives:

)( = + Eq. () is a differential equation which characterizes the free fall gravitydrainage mechanism in a matrix block. Parameter a, b are

representatives for the gravity and capillary forces respectively, and theeffect of viscous pressure loss has been covered in the both parametersa, b through viscosity term. Eq. () has been solved to develop ananalytical model, but before solving this equation we considered threedifferent assumptions for the height of oil column in the fracture; ) Thereisnt any oil in the fracture and the matrix blocks are totally immerse inthe gas, the most of the blocks in the gas invaded zone (above the gas-oil contact) are in this condition. ) The height of the oil column in thefracture is constant. This occurred in the gas-oil contact when the rate ofoil production from the oil zone is equal to the aquifer or gravity drainagecompensation rate. ) The height of oil column in the fracture is variable.This occurred in the gas-oil contact when the rate of oil production fromthe oil zone is not equal to the aquifer expansion or gravity drainagerate. All three assumptions can be occurred in the real reservoirconditions; however the first one is the most probable case. Thereforefirst assumption was considered to solve Eq. ();

)( = + To solve Eq. (), first the cognate parameters were placed in the sameside of equation and then integration applied for both sides of theequation. For the initial condition (t=hm=H) the following solution wasdeveloped:

)(= ( + ) = + ( + )Eq. () is an implicit expression for hm, but it is desirable to find anexplicit expression, such as hm=f (t). There are several methods, Zhmudet al. (), to estimate an explicit solution for hm (t), but to obtain amore accurate solution a new approach, Lambert W function, has beenfollowed, similar to that was used by Fries et al. () for analyticalmodeling of capillary rise. Using this approach the implicit expression inEq. () was converted to an explicit expression as Eq. (). More detailof this conversion are given in Fries et al. paper () and its onlinesupplementary material.

)( ( ) = + ( + ) ( )For practical purpose, Eq. () which is the height of oil column versustime in the matrix block must be converted to the oil recovery and oilproduction rate versus time. If one assumes that in the initial condition,the matrix block is completely saturated with oil, then the oil recoveryformula can be obtained as follow:

)(( ) = ( ) = ( )

For homogeneous matrix, the residual oil saturation above the gas-oilinterface is uniform or it is constant with respect to the height; thereforeEq. () presents more realistic by multiplying (-Sor). Substituting Eq.() in Eq. () results in a new model for oil recovery in free fall gravitydrainage from a matrix block,

)(( ) = ( ) + ( + ) ( )After some rearrangements and manipulations by introducing someconstant parameters Eq. () changed to a shorter form as follows,

)(= , = , = , = Substituting the introduced parameters, Eq. (), in Eq. () gives:

)(( ) = ( ){ [+ ( )]}Substituting Eq. () in Eq. (), multiplying by effective cross sectionarea (A ) and using introduced parameters, Eq. (), a new expressionis developed for oil production rate,

)(( ) = ( ) ( )[+ ( )]This is must be clarified that the flow of gas in the matrix block wasneglected, so the viscous pressure loss due to the flow of gas was not

consider. It is because of assuming gas-oil replacement instead of gas-oil displacement. However the effect of flow of gas in the matrix block isnegligible for the free fall gravity drainage.

. Model Validation

In this section, the prediction of developed analytical model, Eq. (),has been validated by using three distinct methods: numerical analysis,conventional simulation and experimental data. The explanation of eachmethod and associated results are presented as follows.

.. Numerical Analysis

For checking the accuracy of the applied approach to obtain the explicitsolution presented in Eq. (), a computer code was developed tonumerically solve differential Eq. (). The code is written based onimplicit finite difference method. Three samples with different rock andfluid properties were considered to compare the results of numericalsolution with the analytical model predictions. The properties of threedifferent samples are given in Table . The results shown in Fig. revealthat the explicit solution, presented in Eq. (), can be considered as anexact solution for Eq. ().

.. Experimental Study

A few works in laboratory scale have been investigated the free fallgravity drainage in one matrix block. In the most of these works, severaleffects have been simultaneously studied; therefore their results cannotbe helpful for our purpose. However, two experiments were selected tovalidate the predictions of proposed model in the core and micromodelscale. The analytical model was tested against the experimental resultsin the core scale obtained by Pedrera et al. (), using a completelyhomogeneous m long core sample of mm diameter, positionedvertically and had a permeability of md and a porosity of %. Thecase selected for this study was the strongly water-wet sample with awettability index of . and an initial water saturation of %. The resultsof the comparison between Pedrera experiment and analytical model,Eq. (), is shown in Fig. . It is obvious from the figure that there is anexcellent consistency between the experimental data and analyticalmodel predictions. In addition to Pedrera work, a micromodel scaleexperiment was conducted to insure the validity of our model. Themicromodel was constructed by using laser technology and fusingprocess as described by Mohammdi et al. (). It consists of a matrixblock which was surrounded by the fractures in the two sides of that(See Fig. ). The constructed homogeneous -D glass micromodel was

saturated with crude oil and positioned vertically to record the free fallgravity drainage data through photo capturing. The oil of fractures drainsvery quickly due to high permeability ( Darcy). A high resolutiondigital camera, Nikon D, was used to photograph the micromodel andAdobe Photoshop CS software along with a developed computer codewas used to analyze the micromodel photographs. The model and crudeoil properties are presented in Table . The experiment was carried outat room temperature and pressure. The result of the comparisonbetween micromodel experimental data and analytical modelpredictions, Eq. (), is shown in Fig. . Again, there is an excellentmatch between the experimental data and analytical model predictions.

.. Conventional Simulator

In this part, in order to check the validity of the developed model,numerical simulations were performed using the commercial simulatorECLIPSE. Cartesian grid system was used with dimensions (See Fig. ). The matrix was represented by a cube with dimensions ft ft ft. the threshold height of the matrix block was taken ft.The rock was considered to be homogeneous. The porosity of matrixwas taken . and the matrix was assumed to be isotropic (kx= ky=kz=md). The outermost grid blocks properties was chosen very high(porosity and permeability kx=ky=kz=mD) to represent the

fracture. All grid blocks outside the matrix region had unit gas saturation.Since the model is developed for homogenous system, capillarypressure was assumed uniform at different oil saturation. Relativepermeability for oil was assumed zero above the gas-oil interface wherebelow the front its value was set to unity. The properties of the oil usedin the simulator are viscosity . cp, specific gravity ., and interfacialtension . kg/s. As one can see in Fig. , there is an excellentconsistency between the simulation results and model predictions.. ConclusionIn this study, we presented a new method based on balances of involvedforces in free fall gravity drainage, and developed a new analytic modelfor prediction of produced oil rate in a homogeneous matrix block underthis mechanism. The Lambert W function was used to analytically solvethe derived differential equation. The presented model considers theeffect of capillary force completely and can be used for the all conditions:gravity force dominant, capillary force dominant or both gravity andcapillary forces are important. The prediction of developed modelmatched excellently with the data sets in the core and micromodel scale,also its prediction is in reliable agreement with the results of numericalanalysis and conventional simulation models. Extending of the results toa heterogeneous porous media is the authors ongoing research subject.

Nomenclature: difference density between oil and gasgravity acceleration term height of oil column in the matrix height of oil column in the fractureinterfacial tension between oil and gascontact angle between oil and gas on rock surfaceequivalent mean radius of the matrix blockmean oil velocity in the matrix blockabsolute permeability of the matrix blockeffective porosity of the matrix blockheight of the matrix block( ) matrix block oil recovery versus time( ) matrix block oil production rate versus time

Table-: Rock and fluid characteristics used in the numerical analysis(Matlab program)

Sample Sample Sample Property. Darcy. Darcy DarcyPermeability

%%%Porosity. cp. cp cpOil viscosity...Oil Specific Gravity

. kg/s. kg/s. kg/sGas-Oil InterfacialTension m m mMatrix Block Height

Table-: Micromodel and fluid characteristics used in the gravitydrainage experiment

DarcyPermeability%Porosity

. cpOil viscosity.Oil Specific Gravity

kg/s.Gas-Oil Interfacial Tension. mMatrix Block Height

%Residual Oil Saturation

ReferencesAronofsky, J. S., Masse, L., Natanson, S. G. . A Model for theMechanism of Oil Recovery from the Porous Matrix due to WaterInvasion in Fractured Reservoirs. Trans. AIME. ,.Bech, N., Jensen, O. K., and Nielsen, B. . Modeling of Gravity-Imbibition and Gravity-Drainage Processes. SPE Res. Eng. J. , -.Cardwell, W. T., Parsons, R. L. . Gravity Drainage Theory. Trans.AIME., , .Darcy , Amyx, W. J., Bass, M.D., Whiting, L.R. . Textbook ofPetroleum Reservoir Engineering. McGraw-Hill Edition Press: Texas; pp-.Donato, G., Tavassoli, Z., and Blunt, M. . Analytical and NumericalAnalysis of Oil Recovery by Gravity Drainage. J. Pet. Sci. Eng. , -.Dykstra, H. J. . The Prediction of Oil Recovey by Gravity Drainage.Pet. Technol., , SPE , DOI: ./-PA.Fries, N., Dreyer, M. J. . An Analytic Solution of Capillary RiseRestrained by Gravity. J. Colloid Interface Sci., , .Hagoort, J. . Oil Recovey by Gravity Drainage. SPE J., .SPE , DOI: ./-PA.Li, K., Horne, R. N. . Modeling of Oil Production by GravityDrainage. J. Pet. Sci. Eng. , -.Luan, Z. . Some Theoretical Aspects of Gravity Drainage inNaturally Fractured Reservoirs. SPE , DOI: ./-MS.Mohammadi, S., Maghzi, A., Ghazanfari, M.H., Masihi, M., Mohebbi, A.,and Kharrat, R. . One the Control of Glass MicromodelCharacteristics. Accepted in Energy Sources, Part A.Nenniger, E., Storrow, J.A. . Drainage of Packed Beads inGravitational and Centrifugal-Force Fields. AIChE., (), .

Pavone, D., Bruzzi, P., Verre, R. . Gravity Drainage at LowInterracial Tension. th European Symposium on Enhanced OilRecovery.Pedrera, B., Bertin, H., Hamon, G., Augustin, A. . Wettability Effecton Oil Relative Permeability during a Gravity Drainage. SPE . DOI:./.Saidi, A.M. . Textbook of Reservoir Engineering of FracturedReservoirs; TOTAL Edition Press: Paris; pp -.Schechter, D. S., Guo, B. . Mathematical Modeling of GravityDrainage after Gas Injection into Fractured Reservoirs. SPE . DOI:./-MS.Van, Golf-Racht. . Textbook of Fractured Reservoir Engineering;Elsevier Scientific Publication; pp -.Xu, J-Q. . Modeling unsteady-state gravity-driven flow in porousmedia .J. Pet. Sci. Eng. , -.Zhmud, B. V., Tiberg, F., Hallstensson, K. . Dynamics of CapillaryRise. J. Colloid Interface Sci., , .

Figure-: Schematic of a typical fractured reservoir after beginning the oilproduction and a matrix block in the gas invaded zone

Figure-: Schematic of a matrix block which is surrounded by thefractures, oil drains in both matrix and fracture due to the gravity force

Figure-: Comparison of the oil recovery calculated by the analyticalsolution to that calculated by numerical solution

Figure-: Comparison of the oil recovery calculated by the presentedanalytical model to the experimental data in core scale (Pedrera et al.,).

Figure-: Photo of micromodel which is used in free fall gravity drainageexperiment

Figure-: Comparison of the oil recovery calculated by the presentedanalytical model to the experimental data in micromodel scale

Figure-: Schematic of a matrix block which is surrounded by thefractures used in Eclipse software to simulate free fall gravity drainge \

Figure-: Comparison of the oil recovery calculated by the presentedanalytical model to the simulation data using Eclipse software