Institute of Fluid Mechanics. and Computer Applications in Civil Engineering University of Hannover. Prof. Dr.-Ing. W. Zielke.

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1 Institute of Fluid Mechanics and Computer Applications in Civil Engineering University of Hannover Prof. Dr.-Ing. W. Zielke ROCKFLOW Tutorial E Version for RockFlow Software developers: René Kaiser Strömungsmodell 0, Adaption Abderrahmane Habbar Reaktives Transportmodell 10097, Olaf Kolditz Wärmetransportmodell Carsten Thorenz Mehrphasenströmungsmodell Martin Kohlmeier THM plus by A. Ahmari Hanover, April 2006

2 Rock Flow Tutorial Tutorial E Multidirectional two-phase flow simulation in porous media Tutorial E: Tutorial Name prerequisites Index E1 Oil / Water Simulation I (The Buckley-Leverett problem) Tutorial B Introduction Multiphase Flow Equations Oil/water relative permeabilities and capillary pressure Illustration of the displacement of oil by water E2 Oil / Water Simulation II (The five spot problem) Tutorial E1 Introduction Well modeling in reservoir simulation Reservoir simulation in 2- and 3- dimension Multidirectional two-phase flow analysis in porous media Evaluation and comparison of results

3 Rock Flow Tutorial AION 2D RockFlow Tutorial E1 OIL-WATER SIMULATION unidirectional two-phase flow (The Buckley-Leverett problem) Numerical simulation of the displacement of two immiscible fluids (water and oil) in porous media Presentation of an originally analytical solution proposed in Buckley & Leverett for the oil-water simulation Analysis of the influence of capillary pressure and relative permeability in the two-phase flow modeling Illustration of the displacement of oil by water # Model Keywords: #PROJECT #MODEL #TIME, #NUMERICS_PRESSURE #NUMERICS_SATURATION, #RENAMBER #LINEAR_SOLVER_PROPERTIES_PRESSURE #LINEAR_SOLVER_PROPERTIES_SATURATION #LINEAR_SOLVER_PROPERTIES_CONCENTRATION #OUTPUT, #OUTPUT_X #NONLINEAR_SOLVER_PROPERTIES_PRESSURE #NONLINEAR_SOLVER_PROPERTIES_SATURATION #INITIAL_CONDITIONS_PRESSURE #INITIAL_CONDITIONS_SATURATION #BOUNDARY_CONDITIONS_PRESSURE #BOUNDARY_CONDITIONS_SATURATION #BOUNDARY_CONDITIONS_CONCENTRATION #SINK_VOLUME_FLUID_MIXTURE #REFERENCE_CONDITIONS, #FLUID_PROPERTIES #SOIL_PROPERTIES, #COMPONENT_PROPERTIES, #CURVES #STOP In this Tutorial we describe a method for simulation unidirectional two-phase flow (water and oil) and displacement of one fluid (oil) by another fluid (water). For this goal the classic benchmark (Buckley-Leverett problem) is used, which has been widely used to test the handling of self-sharpening fronts in multiphase flow simulation programs (e.g. Huyakorn 1978; Helmig, 1993). In this example the initial oil filling is extracted on one side and water is entering the system on the other side. Thus we try to clarify the influence of the consideration of water-oil capillary pressure and oil-water relative permeability of two-phase flow modeling in porous media. We present that capillary pressure between the two phases (oil and water) smoothes the front.

4 Rock Flow Tutorial Preface: The aim of this example is to describe the relative permeability and capillary pressure (k r, p c ) effect in two phase flow simulation in porous media and also to define the necessary parameters for this problem. This chapter is composed of the following sections: Overview of the multiphase flow equations. Description of the mandatory and necessary keywords and sub-keywords as well as program-inputs and parameters. Analysis the influence of the considering of oil-water relative permeabilities and capillary pressure in two phase flow simulation. Illustration of the results. Table 1: The parameters used in multidirectional two-phase flow analysis Symbol and Meaning Value Unit oil viscosity, µ o 0.01 Pa s water viscosity, µ w Pa s oil standard density, ρ o kg/m³ water standard density, ρ w kg/m³ permeability (vertical), k v 1.0e-011 m² permeability (horizontal), k h 1.0e-011 m² porosity, n 20 % model Brooks-Corey BC-parameter, λ 2 - entry pressure, p d 10e04 Pa residual saturation of oil, s ro residual saturation of water, s rw 0.2 -

5 Rock Flow Tutorial Analytical solution Multiphase Flow Equations One of the solutions to the unidirectional two-phase flow is proposed by Buckley & Leverett [1, 2, 4, 6] which will be presented here. The multiphase flow equations [1,2] for one-dimensional, horizontal flow in a layer of constant cross sectional area as consisting of a continuity equation for each fluid phase flowing (E1-1) and the corresponding Darcy equations for each phase (E1-2) are: ( ρ lul ) = ( nρ l Sl ) (E1-1) x t u l kkrl = µ l p l x (E1-2) where: p cog = p g p pcow = po p l = o, w, g S l l= o, w, g = 1 o w Flow equations for the two phases with the following assumptions The fluid phases: oil and water only Substituting Darcy's equations Black oil fluid descriptions into the continuity equations Including production/injection terms in the equations are: kkro x µ ob o po x q o ns = t Bo o (E1-3) kkrw x µ wb w p x w q w ns = t B w w (E1-4) where: p S = p w o o + S w p =1 cow

6 Rock Flow Tutorial and: p = density, kg/m3 µ = viscosity, Pa s S = saturation, - P = pressure, Pa q = flow rate, m 3 /d n = porosity, - B = formation volume factor Capillary pressure The capillary pressure [3,4] between two phases is calculated as follows: (the Laplace equation) see Figure 2: p c = + (E1-5) 12 p2 p1 p 1 1 4σ 12 cosα = σ 12 + (E1-6) rx r y d c12 = Where: σ 12 interfacial tension, r main curvature radii of the meniscus rx y Equation E1-6 shows that an increase of the saturation of the non-wetting phase (fluid 2) must lead to an increase of the capillary pressure. If the saturation of the wetting phase decrease, the wetting fluid retreats to smaller pores or fracture apertures. That results in the following capillary pressure-saturation relation: p c = p c ( S w ) (E1-7) The capillary pressure is often formulated as a function of the effective saturation. For Brooks-Corey (BC) model (See also Figure 6) : p 1/ λ p c ( Sw ) = pd ( Se ) c p d (E1-8) Where effective saturation (S e ) is defined as: Sw Srw Se = Srw Sw 1 Srn 1 S S rw rn (E1-9) and λ Brooks-Corey (BC)-parameter [-] S wr residual water saturation [-] p d entry pressure [Pa]

7 Rock Flow Tutorial Relative permeability If there is more than one fluid phase in a pore space or fractures at the same time, the presence of one of these phases disturbs the flow behaviour of the other phase and vice versa. That is, by decrease of saturation of the wetting phase, the cross-sectional area available flow also decreases. In this example we use the Buckley-Leverett solution in the case of relative permeabilitysaturation relationship according to Brooks-Corey model as follow (see also Figure 7) [3]: 2+ 3λ S w S rw λ k rw = ( ) (E1-10) 1 S S rw rn Figure 1: Interfacial tension and wetting angle [4] Figure 2: Interface element und capillary tube [4]

8 Rock Flow Tutorial Numerical solution Input data The input data in RockFlow consists of two files: *.RFI und *.RFD, which are mentioned below RFI- file buckley2d.rfi The model domain is a 2-D inclined plane. Thus, the gravitational terms in the numerical model can be activated. The grid is discretized by 100 rectangular elements with 202 nodes (Figure 3 and Figure 4) Z Y X Figure 3: Geometry of the grid used in this example (Buckley- Leverett problem) Figure 4: Discretization of the grid with two- dimensional plane elements

9 Rock Flow Tutorial RFD- file Buckley2d.rfd In this section only the keywords and the parameters will be described, which deviate from the last preceding examples or are particularly important to understand the example (see also Tutorial B). #Model #MODEL 1 ; simulation flag ; model identifier 0 ; flow model flag 0 ; convection model flag 0 ; chemical model flag 0 ; transport phase of multiphase model 1 ; simulation optimizer flag 0 ; material groups 0 ; phases 0 ; components 0 ; adaptive mesh refinement flag 0 ; chain_reaction_model 0 ; heat_reaction_model 0 ; saturation_calculation_method 0 ; mobile immobile model flag ; ; New sub keywords: $NUMBER_OF_PHASES 2 $NUMBER_OF_GROUPS 1 $NUMBER_OF_COMPONENTS 0 The Keyword "# MODEL" commands the execution mode of the program. The computation model is selected with "model identifier". The choice of the model "10699" (multiphase flow model) enables us the modeling of displacement of one fluid (oil) by another fluid (water) in porous media. In this example the following sub keywords are used: Sub keyword $NUMBER_OF_PHASES specifies the number of moving fluid phases. In this case it will be entered 2 (two different phases: water and oil). Sub keyword $NUMBER_OF_GROUPS specifies the number of material groups. In this example there is only one material group. Sub keyword $NUMBER_OF_COMPONENTS specifies the number of components within the fluids. In this example there are no components in fluids

10 Rock Flow Tutorial #TIME #TIME ; final simulation time 2000 ; maximum time step number 0 ; time step control 2000 ; time step number ; time step length The time stepping is specified using the keyword #TIME. The temporal discretization takes place in 2000 time steps with a length from 3,6 h. It amounts to an entire simulation time over 300 days. #OUTPUT #OUTPUT 0 ; files 1 ; geometry 1 ; initial condition 0 ; format 1 ; numbering 3 ; type ; parameters As described before the keyword #OUTPUT controls the output or nodal results. The keyword specifies output data in the RFO-data. By type 3 an output operation of the computation results will be taken place after indicated time interval under "parameter" ( after 1 day). Thus all computation results will be output. #OUTPUT_EX #OUTPUT_EX 13 ; type buckley2d_01.plt ; name 2 ; geo_type ; node coordinate (start) ; node coordinate (end) 0.05 ; radius 2 ; mode: output by steps 1 ; method: output every specified step 0 ; data output method (dom): no separation 0 ; number of variables 0 ; all of variables ; 13 ; type buckley2d_02.plt ; name 2 ; geo_type ; node coordinate (start) ; node coordinate (end) 0.05 ; radius 2 ; mode: output by steps 1 ; method: output every specified step 0 ; data output method (dom): no separation 0 ; number of variables 0 ; all of variables Keyword #OUTPUT_EX specifies output data in the RFO-data. With type 13 the node values will be specified along geometric object (in this example along the line (geo_type=2) Figure 23 ). The results are readable in Tecplot (*.plt- data). With mode 2 and number of variable 0 all of variables in PLT-data will be written.

11 Rock Flow Tutorial The Keyword #NUMERICS specifies numerical parameters for the FEM of the corresponding PDE. For the multiphase model (model 10699) the #NUMERICS keywords are no longer valid and must be replaced. In this example it is replaced by the keywords #NUMERICS_PRESSURE and #NUMERICS_SATURATION. #NUMERICS_PRESSURE #NUMERICS_SATURATION #NUMERICS_PRESSURE $METHOD 1 $TIMECOLLOCATION &GLOBAL 0 1. $UPWINDING $MASS_LUMPING 0 ; #NUMERICS_SATURATION $METHOD 1 $UPWINDING $MASS_LUMPING 1 $TIMECOLLOCATION &GLOBAL ; Finite element calculation ; specifies the time collocation. ; specifies the main time collection ; upwinding of Gaussian integration points ; masslumping reduce wiggles ; Finite element calculation ; upwinding of Gaussian integration points ; masslumping reduce wiggles ; specifies the time collocation. ; specifies the main time collection The Keyword #NUMERICS_PRESSURE contains sub-keywords and subsubkeywords which are explained in the following: $MASS_LUMPING: The keywords are followed by an integer value 1 which specifies that masslumping must be used to reduce wiggles. This is necessary for multiphase flow calculations or transport calculations with very steep fronts. $TIMECOLLOCATION: The time collocation value can be chosen globally for the regarded partial differential equation and differently for certain parts. These are distinguished by optional subsubkeywords (labeled with &) and in this case with the &GLOBAL : &GLOBAL: Specifies the main time collocation $UPWINDING: The keyword is followed by an integer value which specifies the unwinding scheme. #LINEAR_SOLVER_PROPERTIES_PRESSURE #LINEAR_SOLVER_PROPERTIES_SATIRATION #LINEAR_SOLVER_PROPERTIES_PRESSURE 2 ; method 0 ; norm 100 ; preconditioning 0 ; maximum iterations -1 ; repeating 6 ; criterion 1.0e-010 ; absolute error 0 ; kind 4 ; matrix storage technique ; #LINEAR_SOLVER_PROPERTIES_SATURAZION 2 ; method 0 ; norm 100 ; preconditioning -1 ; maximum iterations 0 ; repeating 6 ; criterion 1.0e-010 ; absolute error 0 ; kind 4 ; matrix storage technique

12 Rock Flow Tutorial #LINEAR_SOLVER_PROPERTIES_CONCENTRATION #LINEAR_SOLVER_PROPERTIES_CONCENTRATION 2 ; method 0 ; norm 100 ; preconditioning 0 ; maximum iterations -1 ; repeating 6 ; criterion 1.0e-010 ; absolute error 0 ; kind 4 ; matrix storage technique The keywords #LINEAR_SOLVER_PROPERTIES_* are used to specify properties of linear equation solvers. #NONLINEAR_SOLVER_PROPERTIES_PRESSURE #NONLINEAR_SOLVER_PROPERTIES_SATIRATION #NONLINEAR_SOLVER_PROPERTIES_PRESSURE 1 ; method 10 ; maximum iterations 3 ; criterium 1.0e-003 ; maximum iterations 0.0 ; absolute eps 1 ; relative eps 0.0 ; adaptive eps 0 ; time control ; #NONLINEAR_SOLVER_PROPERTIES_SATURAZION 1 ; method 10 ; maximum iterations 3 ; criterium 1.0e-002 ; maximum iterations 0.0 ; absolute eps 1 ; relative eps 0.0 ; adaptive eps 0 ; time control Non-linear solver is developed to solve non-linear PDEs. For a multiphase flow non-linear sets of equations must be solved numerically. The appropriate control parameters are given by Keyword #NONLINEAR_SOLVER_PROPERTIES_PRESSURE and #NONLINEAR_SOLVER_PROPERTIES_SATIRATION, respectively.

13 Rock Flow Tutorial #INITIAL_CONDITIONS_PRESSURE #INITIAL_CONDITIONS_PRESSURE 3 ; type 0 ; mode 0 ; begin_node 100 ; end_node 1 ; step_nodes e+005 ; values e+005 ; values ; 3 ; type 0 ; mode 101 ; begin_node 200 ; end_node 1 ; step_nodes 1.0e+005 ; values 1.0e+005 ; values #INITIAL_CONDITIONS_SATURATION #INITIAL_CONDITIONS_SATURATION ; phase_0 (oil) 0 ; type 0 ; mode 0.8 ; value ; #INITIAL_CONDITIONS_SATURATION ; phase_1 (water) 0 ; type 0 ; mode 0.2 ; value ; Initial conditions in this example can be specified with the following keywords: #INITIAL_CONDITIONS_PRESSURE #INITIAL_CONDITIONS_SATURATION The initial saturations (in this case the saturation of oil present in porous media) is the result of an absorption process at the time of oil accumulation [ ] (phase_o=oil; saturation value= 0.8 and phase_1=water; saturation value = 0.2 at the time t= t 0 =0). Thus, with "type" "0" for initialization of saturation for all nodes same values (phase_0= oil: 80% and phase_1= water: 20%) are assigned (see also Figure 5). The pressure allocation (initial condition) is also defined at the time t= t 0 =0. With type 3 a linear value distribution between two given nodes (left side: first node=0, second node=100, pressure value = e+005 Pa, right side: first node=101, second node=200; pressure value = 1.0e+005 Pa) is specified. The initial conditions for corresponding phases (in this case for #INITIAL_CONDITIONS_SATURATION ) are specified by repeatedly use of the keywords.

14 Rock Flow Tutorial Figure 5: assignment of the initial condition and boundary condition #BOUNDARY_CONDITIONS_PRESSURE #BOUNDARY_CONDITIONS_PRESSURE 13 ; type 0 ; mode 0 ; curve 2 ; count of point 0 ; node_1 1.0e+005 ; (gravity on) ; value_1 (gravity off) 101 ; node_1 1.0e+005 ; value_ ; epsilon #BOUNDARY_CONDITIONS_SATURATION #BOUNDARY_CONDITIONS_SATURATION ; phase_0 (oil) 13 ; type 0 ; mode 0 ; curve 2 ; count of point 0 ; node_1 0.2 ; value_1 101 ; node_2 0.2 ; value_ ; epsilon ; #BOUNDARY_CONDITIONS_SATURATION ; phase_1 (water) 13 ; type 0 ; mode 0 ; curve 2 ; count of point 0 ; node_1 0.8 ; value_1 101 ; node_2 0.8 ; value_ ; epsilon The boundary conditions for multiphase flow are the same as for one phase flow, but have to be specified for each phase. The keywords #BOUNDARY_CONDITIONS_* are available to specify boundary conditions. In this example pressure and saturation must be defined on the boundary. With type 13 a linear value distribution along polygon/nodes (eps) between two given nodes is specified. With the keyword #BOUNDARY_CONDITIONS_PRESSURE on left side (on nodes 1 and 101 ) a water pressure amounting to 100 KPa is defined and with BOUNDARY_CONDITIONS_SATURATION (also on nodes 1 and 101 ) a saturation as boundary condition amounting to 80% for water (phase_1) and 0.2 for oil (phase_0) respectively (see also Figure 5).

15 Rock Flow Tutorial #SINK_VOLUME_FLUID_MIXTURE #SINK_VOLUME_FLUID_MIXTURE 0 ; type 0 ; mode 0 ; curve 100 ; node 4.000e-006 ; value ; 0 ; type 0 ; mode 0 ; curve 201 ; node 4.000e-006 ; value The keyword #SINK_VOLUME_FLUID_MIXTURE simulates sink term for multiphase flow. With type 0 at the nodes 100 and 201 (see Figure 4 and Figure 5) an outlet of 4.000e- 006 m3/s (= value ) for both phases (phase_0= oil, phase_1= water) is simulated. #REFERENCE_CONDITIONS #REFERENCE_CONDITIONS 0.0 ;9.8 ; gravity on ; gravity off ; initial temperature ; initial pressure The reference condition is defined with keyword #REFERENCE_CONDITIONS. At first the gravitation field to the simplification is off. Afterwards we turn it on, in order to examine its influence of the water flow process. #FLUID_PROPERTIES #FLUID_PROPERTIES ; phase_0 (oil) 0 ; density function 900 ; parameter 0 ; viscosity function ; parameter 0.0 ; real gas factor 0.0 ; heat capacity 0.0 ; heat conductivity ; #FLUID_PROPERTIES ; phase_1 (water) 0 ; density function, 1000 ; parameter 0 ; viscosity function ; parameter 0.0 ; real gas factor 0.0 ; heat capacity 0.0 ; heat conductivity With the Keyword #FLUID_PROPERTIES the fluid parameters, in particular density and viscosity of the fluids are specified (see also Table 1). By dual use of the keyword the data for two phases (phase_0 = oil, and phase_1 = water) are specified. We chose the viscosity and density of water phase µ w = Pa s and ρ w = 1000 kg/m 3 respectively and the viscosity and density of oil phase: µ o = 0.01 Pa s and ρ o = 900 kg/m 3 respectively.

16 Rock Flow Tutorial #SOIL_PROPERTIES #SOIL_PROPERTIES 1 ; dimension 1.0 ; area 0 ; porosity model 2.0e-001 ; porosity 1.0 ; tortuosity 0 ; mobile immobile model 0 ; litho logical component 0 ; maximum sorption model 0 ; non-linear flow parameter 0.0 ; storativity 0 ; permeability model 0 ; permeability tensor 1.0e-011 ; permeabilities ; ; k_rel-s function 6.0 ; type (specified by curve) 0.2 ; residual water saturation 0.8 ; maximal water saturation 2.0 ; exponent 0.0 ; not used! 0.0 ; not used! 0.0 ; not used! ; ; p_c-s function 1.0 ; capillary pressure (specified by curve) 1.0 ; curve number (curve 1) 0.0 ; not used! 0.0 ; not used! 0.0 ; not used! 0.0 ; not used! ; ; mass dispersion parameters, not used! ; heat dispersion parameters, not used! ; rock density, heat capacity, not used! ; heat conductivity parameters, not used! Medium properties are specified with the Keyword #SOIL_PROPERTIES". (see Tutorial A, B, C and D). As described before we try in this example to clarify the influence of the consideration of relative permeability and capillary pressure, which are functions of water saturation. In this example the relative permeability is specified using type 6.0 (Brooks/Corey-model) [3]. As our selected numerical method the relative permeability equation is given by: S w 4 k rw = ( ) (E2-1) of the Brooks-Corey model (Brooks & Corey 1964) with isotropic medium permeability of K=10-11 m 2 and a uniform porosity n=0.2. (see also relative permeability and Figure 7)

17 Rock Flow Tutorial The capillary pressure is specified by type 1.0, i.e. by a curve. Curve 1 specifies the relation between the wetting phase (phase_1=water) saturation and the capillary pressure using the function: 1/ 2 p c ( S w ) = pd ( Se ) (E2-2) (See Figure 6) #CURVES #CURVES ; CURVES 1 (Brooks-Corey model) The keyword #CURVES specifies the capillary pressure curve (curve 1 specifies the relation between the wetting phase (phase_1=water) saturation and the capillary pressure) (Figure 6).

18 Rock Flow Tutorial capillary pressure [Pa] ,0 0,2 0,4 0,6 0,8 1,0 water saturation [-] Figure 6: Capillary pressure curve (Brooks- Corey model; curve 1) Figure 7: Relative permeability for two phases (oil and water) versus water saturation

19 Rock Flow Tutorial Output data RFO- file Buckley_2d.rfo Structure of the RFO file: see Tutorial A. PLT- file *_2D.plt In the following Figures the result of Simulation are represented, which are provided as *.plt-files and processed by Tecplot. We consider two cases: Case 1: Without capillary pressure: The results of case 1 to the Buckley-Leverett problem are shown in Figure 9 to Figure 11. In this application we neglect the capillary pressure, in order to prove its influence on the two phase flow modeling. In this case there is no capillary force. Therefore a wetting process via the wetting phase (water) in the range of front i. e. at the interface of two domains (water/oil) does not take place (see Figure 8). Figure 8: Spreading at the interface of two domains (without capillary pressure)

20 Rock Flow Tutorial (a) (a) (b) (b) (c) (c) (d) (d) water saturation [-] oil saturation [-] Figure 9: Water and oil saturation (capillary pressure and the relative permeability is specified) at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) (a) (b) (c) (d) pressure [kpa] Figure 10: Water pressure with considering of the capillary pressure: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d)

21 Rock Flow Tutorial time= 0.0 s time= 100 days time= 200 days time= 300 days saturation [-] distance [m] Figure 11: water saturation profile (The capillary pressure is neglected and the relative permeabilities are specified by curves. Number of element=100) at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) Figure 12 : Reference solution for Buckley-Leverett with zero capillary pressure [7]

22 Rock Flow Tutorial Figure 11 shows water saturation profile (water saturation versus distance) at 100, 200 and 300 days using Brooks-Corey model to considering the capillary pressure. Figure 12 shows that by a reference solution computed [7] with parameters in Table 2. The results show a good agreement. Table 2: The parameters used in reference solution [7] Symbol and Meaning Value Unit domain, Ω (0, 300) m oil viscosity, µ o 0.01 Pa s water viscosity, µ w Pa s total velocity, u 9*10e-07 m/s Permeability (vertical), k v 1.0e-011 m² Permeability (horizontal), k h 1.0e-011 m² Porosity, n 20 % model Brooks-Corey BC-parameter, λ 2 - entry pressure, p d 10e04 Pa residual saturation of oil, s ro residual saturation of water, s rw boundary condition water saturation, s w S w (0)=1-S rn - water saturation, s w S w (300)= S rw - Initial condition water saturation, s w s w0 =s rw - Case 2: Including capillary pressure: Figure 13 to Figure 15 show simulation considering of the capillary pressure. The effect of water and oil saturation are displayed in Figure 13 and the water pressure in Figure 14. Figure 15 shows also the water saturation profile (water saturation versus distance) at 100, 200 and 300 days. In this case we specify the constitutive relations of the capillary pressure and relative permeability according to the Brooks-Corey model (BC-model) (see capillary pressure, #SOIL_PROPERTIES, Figure 6, Figure 7 and also #CURVES).

23 Rock Flow Tutorial (a) (a) (b) (b) (c) (c) (d) (d) water saturation [-] oil saturation [-] Figure 13: Water and oil simulation with considering of relative permeability and capillary pressure at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) (a) (b) (c) (d) pressure [kpa] Figure 14: Water pressure with considering of relative permeability and capillary pressure: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d)

24 Rock Flow Tutorial time= 0.0 s time= 100 days time= 200 days time= 300 days saturation [-] distance [m] Figure 15: Final water saturation profile (The capillary pressure and Relative permeabilities are specified. Number of element=100) at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) Figure 16 : Reference solution for Buckley-Leverett with capillary pressure [7]

25 Rock Flow Tutorial Figure 16 shows the final water saturation profile by the reference solution [7] computed with capillary pressure. The parameters, which are used in this modeling, are also available in Table with capillary pressure without capillary pressure 0.7 saturation [-] distance [m] Figure 17: Comparison of simulation cases at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) (a) with considering of capillary pressure (b) capillary pressure is neglected Finally, we give a comparison between two cases in Figure 17 and a zoom of front at 200 days in Figure 18. That demonstrates two clearly different slopes at the front from two different cases (Case 1: Without capillary pressure and Case 2: With capillary pressure) as shown in these figures. That means, with capillary pressure the graph becomes more smoothly shape at front. By considering of the capillary pressure the capillary pressure saturation relations can be formulated as follows (see also E1-8): p p = p S ) (E2-3) o w cow ( w The equation describes the capillary pressure and the wetting process at the interfaces oil/water i. e. at the front (see Figure 19).

26 Rock Flow Tutorial with capillary pressure without capillary pressure 0.5 saturation [-] distance [m] Figure 18: Front at t=200 days. Figure 19: Spreading at the interface of two domains (with capillary pressure)

27 Rock Flow Tutorial Now we activate the gravitational acceleration (see #REFERENCE_CONDITIONS and #BOUNDARY_CONDITIONS_PRESSURE), in order to examine its influence on the computation. (a) (a) (b) (b) (c) (c) (d) (d) water saturation [-] oil saturation [-] Figure 20: Water and oil saturation with consideration of the gravity (g= 9,8; the capillary pressure and Relative permeabilities are specified): t = 0 s (a), t = 100 days (b), t = 200 days (c), und t = 300 days (d) Figure 20 to Figure 23 show the results of consideration of gravity filed. As the Figures show the water, in contrast to the last case, pulls itself downward because of gravity and its greater specific weight compared with the specific weight of oil. Therefore the water flows through more under the oil (compare with Figure 13). Figure 24 shows us a comparison of the water saturation profile along the line L 1 and L 2 (see Figure 23).

28 Rock Flow Tutorial (a) (b) (c) (d) pressure [kpa] Figure 21: Water pressure after activation of the gravity (the capillary pressure is specified): t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) Z Y X oil saturation [-] Figure 22: Graphical illustration of the oil saturation field with consideration of the gravity (g= 9,81) at t=300 days ( the capillary pressure is specified) Figure 23: Grid system and positions of the lines the lines L 1 and L 2 used in Figure 24 (see also #OUTPUT_EX)

29 Rock Flow Tutorial L1-L1 L2-L2 0.7 saturation [-] distance [m] Figure 24: Water saturation profile along the lines L 1 and L 2 (see also Figure 23) with consideration of the gravity (g= 9,8). The capillary pressure is specified.

30 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) 4D RockFlow Tutorial E2 OIL-WATER SIMULATION (The five spot problem) two phase flow Numerical Reservoir simulation Multidirectional two-phase flow analysis in porous media illustration the evolution of the water/oil ratio (WOR) for the two-phase flow simulation and comparison with some references # Model Keywords: #PROJECT #MODEL #TIME, #NUMERICS_PRESSURE #NUMERICS_SATURATION, #RENAMBER #LINEAR_SOLVER_PROPERTIES_PRESSURE #LINEAR_SOLVER_PROPERTIES_SATURATION #LINEAR_SOLVER_PROPERTIES_CONCENTRATION #OUTPUT, #OUTPUT_X #NONLINEAR_SOLVER_PROPERTIES_PRESSURE #NONLINEAR_SOLVER_PROPERTIES_SATURATION #INITIAL_CONDITIONS_PRESSURE #INITIAL_CONDITIONS_SATURATION #BOUNDARY_CONDITIONS_PRESSURE #BOUNDARY_CONDITIONS_SATURATION #BOUNDARY_CONDITIONS_CONCENTRATION # SOURCE_VOLUME_FLUID_PHASE #REFERENCE_CONDITIONS, #FLUID_PROPERTIES #SOIL_PROPERTIES, #COMPONENT_PROPERTIES, #CURVES #STOP In this tutorial we solve unidirectional two-phase flow (water and oil) in porous media. We simulate the displacement of a non-wetting phase (oil) by a wetting phase (water) used fivespot pattern in 2- and 3- dimension, in which the injection well is uniformly surrounded by four of the Production wells (Figure 25).

31 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) Figure 25: Five spot domain and well arrangement 300 PW y[m] IW x[m] Figure 26: Finite element grid and well arrangement for quarter five spot simulations

32 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) Table 3: The parameters used in Multidirectional two-phase flow analysis Symbol and Meaning Value Unit oil viscosity, µ o 0.01 Pa s water viscosity, µ w Pa s oil standard density, ρ o kg/m³ water standard density, ρ w kg/m³ permeability (vertical), k v 1.0e-011 m² permeability (horizontal), k h 1.0e-011 m² porosity, n 20 [%] model Brooks-Corey BC-parameter, λ 2 - entry pressure, p d 10e04 Pa residual saturation of oil, s ro residual saturation of water, s rw Preface: This Tutorial describes a simulation of the unidirectional two-phase flow (water and oil) in porous media formulated with one source and one sink according to the classical five-spot problem given in Figure 25. Four wells with the diameter of 10 m are placed in each corner of the pattern to represent oil production wells (PW) and a water injection well (diameter also 10 m) is located in the middle of field (IW). Now we consider a quarter of this pattern given in Figure 26. The radial sub-grids will defined on the corner points to represent flow near the vertical wells (Figure 27). In analogy to the Buckley-Leverett problem we also compute the displacement of oil (nonwetting phase) by water (wetting phase) with consideration of capillary pressure. The gravitational effects remain neglected. Finally we will show a three-dimensional simulation, in order to compare the results with the two-dimensional case.

33 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) Numerical solution Input data RFI- file Five_spot_2d.rfi The modeling takes place in two dimensions. The quadrant of five-spot flood network element is separated in D-square element with overall 1756 knots (Figure 26 and Figure 27) y[m] y 0 IW x x[m] Figure 27: The radial sub grids on the corner points (injection well) RFD file Five_spot_2d.rfd In this section only the keywords will be described, which diverge from last preceding examples or are particularly important to understand the example (see tutorial A, B and E1).

34 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) #OUTPUT_EX #OUTPUT_EX 13 ; type 5-spot_01.PLT ; name 2 ; geo_type ; node coordinate (start) ; node coordinate (end) 0.05 ; radius 2 ; mode: output by steps 1 ; method: output every specified step 0 ; data output method (dom): no separation 0 ; number of variables 0 ; all of variables Keyword #OUTPUT_EX specifies output data in the RFO-data. With type 13 the node values will be specified along geometric object (in this example along the diagonal cut of the five-spot pattern (line L in Figure 28 ). The results are readable in Tecplot (*.plt- data). With mode 2 and number of variable 0 all of variables in PLT-data will be written. L PW IW L Figure 28: Grid system and the position of diagonal cut of the five-spot pattern (the line L specified in #OUTPUT_EX )

35 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) #INITIAL_CONDITIONS_PRESSURE #INITIAL_CONDITIONS_SATURATION #INITIAL_CONDITIONS_CONCENTRATION #INITIAL_CONDITIONS_PRESSURE 0 ; type 0 ; mode 0.0 ; value ; #INITIAL_CONDITIONS_SATURATION ; phase_0 (oil) 0 ; type 0 ; mode 0.85 ; value ; #INITIAL_CONDITIONS_SATURATION ; phase_1 (water) 0 ; type 0 ; mode 0.15 ; value ; #INITIAL_CONDITIONS_CONCENTRATION 0 ; type 0 ; mode 0.0 ; value #BOUNDARY_CONDITIONS_PRESSURE #BOUNDARY_CONDITIONS_PRESSURE 5 ; type 0 ; mode 0 ; curve 17 ; count of point ; points ; values The keyword #BOUNDARY_CONDITIONS_PRESSURE specifies the pressure as boundary condition. #BOUNDARY_CONDITIONS_FREE_OUTFLOW #BOUNDARY_CONDITIONS_FREE_OUTFLOW 5 ; type 0 ; mode 0 ; curve 17 ; count of point ; points ; values The keyword #BOUNDARY_CONDITIONS_FREE_OUTFLOW specifies open boundaries by multiphase flow conditions. In the iterative process the pressure will be set to zero, which is the reference pressure for the transition between full and partially saturated conditions, if thereby an outflow is enabled, otherwise the boundary is regarded as impermeable.

36 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) #SOURCE_VOLUME_FLUID_PHASE ; Gas-phase #SOURCE_VOLUME_FLUID_PHASE 6 ; type 0 ; mode 0 ; curve 17 ; count of point ; node coordinate ; epsilon 0.0 ; value ; ; wetting-phase (water) #SOURCE_VOLUME_FLUID_PHASE 6 ; type 0 ; mode 0 ; curve 17; count of point ; node coordinate ; epsilon 1.2e-05 ; value With keyword #SOURCE_VOLUME_FLUID_PHASE the source terms are specified, in this application volume fluid, for corresponding phases and components. With type 6 the nodal sources at a polygon are given by node coordinates (eps).

37 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) Output data RFO- file five_spot_2d.rfo Structure of the RFO file: see Tutorial A. PLT- file *_2d.plt In the following figures the result of simulation are represented, which are provided as *.pltfiles and processed by Tecplot at the production well (PW) at the injection well (IW) Water-Oil Ratio [-] time [days] Figure 29: Water-Oil Ratio at the production well (PW) and injection well (IW) Figure 29 shows water-oil ratio at the production (PW) and injection well (IW) versus time. It becomes clear that the water injection rate at injection well in first 100 days (see also Figure 28) and at the production well in finally 20 days strongly sharp.

38 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) time= 0.0 s time= 100 days time= 200 days time= 300 days saturation [-] distance [m] Figure 30: water saturation profile and front along diagonal cut the five-spot pattern (2D-Dimention modeling) Figure 31: Reference solution for five-spot problem [7]

39 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) Table 4: The parameters used in reference solution [7] Symbol and Meaning Value Unit domain, Ω (0, 300) 2 m oil viscosity, µ o 0.01 Pa s water viscosity, µ w Pa s permeability (vertical), k v 1.0e-011 m² permeability (horizontal), k h 1.0e-011 m² porosity, n 20 % model Brooks-Corey BC-parameter, λ 2 - entry pressure, p d 10e04 Pa residual saturation of oil, s ro residual saturation of water, s rw boundary condition outlet pressure, p out 10e05 Pa the average flux, (u.n e ) in h -1 *1.05e-04 m/s Initial condition water saturation, s w s w0 =s rw Figure 32: Fives pot domain used in reference solution [7] The results of saturation profile along the diagonal cut the five-spot pattern for the homogeneous computation at 100, 200 and 300 days by RockFlow are shown in Figure 30. Figure 31 shows that (water saturation versus diagonal distance) at 100, 200 and 300 days using Brooks-Corey model along diagonal cut the five-spot pattern by a reference solution [7] computed with parameters in Table 4 and geometry in Figure 32. This test shows also that the results are similar when using the same parameters and the same initial and boundary conditions

40 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) (a) (c) Z Y (b) pressure [kpa] (d) X Figure 33: Water pressure and front in five spot calculated at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) The water pressure and water saturation profile calculated at 100, 200 and 300 days are shown in Figure 33 and Figure 34. These correspond to the 2D-mesh with D-square element with 1756 nodes ( see Figure 28). In order to compare the 2D- and 3D-simulation in RockFlow we present the results of a 3Dsimulation shown in Figure 35 to Figure 37. These show an exact match between the 2Dand 3D-dimensional modeling ( compare with Figure 30, Figure 33 also Figure 1 ).

41 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) (a) (c) Z Y (b) saturation [-] (d) X Figure 34 : Water saturation and front in five spot calculated at the time: t = 0 s (a), t = 100 days (b), t =200 days (c), und t = 300 days (d) time= 0.0 s time= 100 days time= 200 days time= 300 days saturation [-] distance [m] Figure 35: water saturation profile and front along diagonal cut the five-spot pattern (3D-Dimention modeling)

42 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) Z X Y Figure 36: Water pressure profile in five spot. 3D-dimensional simulation calculated at 200 days Z X Y Figure 37: Water saturation profile in five spot. 3D-dimensional simulation calculated at 200 days

43 Rock Flow Tutorial E2 oil / water simulation (The five spot problem) References [1] Mattax, C.C. and Kyte, R.L.: 1990, Reservoir Simulation, Monograph Series, SPE, Richardson, TX. [2] Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied Science Publishers LTD, London [3] Brooks, R. H.; 1964, Corey A. T.: Hydrilogy papers clorado state university fort Collins, hydraulic properties of porous media, Colorado. [4] Helmig, R.: 1997, Multiphase Flow and Transport Processes in the Subsurface - A Contribution to the Modeling of Hydrosystems. Springer-Verlag. [5] Muskat, M.: 1937, The flow of homogeneous fluids through porous media. [6] Carsten, Th.: 2001, Institute of Fluid Mechanics and Computer Applications in Civil Engineering University of Hanover, Model adaptive simulation of multiphase and density driven flow in fractured and porous media. [7] Riviere, B.; Bastian, P.: 2004, Discontinuous Galerkin Methods for Two-Phase Flow in Porous Media.