Institute of Fluid Mechanics and Computer Applications in Civil Engineering University of Hannover Prof. Dr.-Ing. W. Zielke ROCKFLOW Tutorial E Version 0.1.0 for RockFlow 3.8.68 Software developers: René Kaiser Strömungsmodell „0“, Adaption Abderrahmane Habbar Reaktives Transportmodell „10097“, „10095“ Olaf Kolditz Wärmetransportmodell „10093“ Carsten Thorenz Mehrphasenströmungsmodell „10699“ Martin Kohlmeier THM plus „18460“ by A. Ahmari Hanover, April 2006
Transcript
Institute of Fluid Mechanics and Computer Applications in Civil Engineering
University of Hannover
Prof. Dr.-Ing. W. Zielke
ROCKFLOW
Tutorial E
Version 0.1.0
for RockFlow 3.8.68
Software developers:
René Kaiser Strömungsmodell „0“, Adaption Abderrahmane Habbar Reaktives Transportmodell „10097“, „10095“
Olaf Kolditz Wärmetransportmodell „10093“ Carsten Thorenz Mehrphasenströmungsmodell „10699“ Martin Kohlmeier THMplus „18460“
by
A. Ahmari
Hanover, April 2006
Rock Flow Tutorial - 2 - 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)
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.
Rock Flow Tutorial - 4 - E1 – oil / water simulation (The Buckley-Leverett problem)
Preface:
The aim of this example is to describe the relative permeability and capillary pressure (kr, pc) 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, µo 0.01 Pa s
water viscosity, µw 0.001 Pa s
oil standard density, ρo 900.0 kg/m³ water standard density, ρw 1000.0 kg/m³ permeability (vertical) , kv 1.0e-011 m²
permeability (horizontal) , kh 1.0e-011 m² porosity, n 20 %
model Brooks-Corey BC-parameter, λ 2 -
entry pressure, pd 10e04 Pa residual saturation of oil, sro 0.15 -
residual saturation of water, srw 0.2 -
Rock Flow Tutorial - 5 - E1 – oil / water simulation (The Buckley-Leverett problem)
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:
( ) ( llll Snt
ux
ρρ∂∂
=∂∂
− ) (E1-1)
xpkk
u l
l
rll ∂
∂−=
µ (E1-2)
where:
ogcog ppp −=
wocow ppp −= gwol ,,=
1,,
=∑= gwol
lS
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:
Rock Flow Tutorial - 6 - E1 – oil / water simulation (The Buckley-Leverett problem) and: p = density, kg/m3 µ = viscosity, Pa·s S = saturation, - P = pressure, Pa q = flow rate, m3/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:
1212 pppc += (E1-5)
drrp
yxc
ασσ
cos411 121212 =⎟
⎟⎠
⎞⎜⎜⎝
⎛+= (E1-6)
Where: 12σ interfacial tension , main curvature radii of the meniscus xr yr 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:
)( wcc Spp = (E1-7) The capillary pressure is often formulated as a function of the effective saturation. For Brooks-Corey (BC) model (See also Figure 6) :
λ/1)()( −= edwc SpSp (E1-8) dc pp ≥ Where effective saturation (Se) is defined as:
rnrw
rwwe SS
SSS−−
−=
1 (E1-9) rnwrw SSS −≤≤ 1
and λ Brooks-Corey (BC)-parameter [-] Swr residual water saturation [-] pd entry pressure [Pa]
Rock Flow Tutorial - 7 - E1 – oil / water simulation (The Buckley-Leverett problem) 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 permeability-saturation relationship according to Brooks-Corey model as follow (see also Figure 7) [3]:
λλ32
)1
(+
−−−
=rnrw
rwwrw SS
SSk (E1-10)
Figure 1: Interfacial tension and wetting angle [4]
Figure 2: Interface element und capillary tube [4]
Rock Flow Tutorial - 8 - E1 – oil / water simulation (The Buckley-Leverett problem)
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).
0 2 4 6
0
2
4
6
0
50
100
150
200
250
300
0
2
4
6
0
50
100
150
200
250
3000 2 4 6
YX
Z
Figure 3: Geometry of the grid used in this example (Buckley- Leverett problem)
Figure 4: Discretization of the grid with two- dimensional plane elements
Rock Flow Tutorial - 9 - E1 – oil / water simulation (The Buckley-Leverett problem)
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 10699 ; 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
Rock Flow Tutorial - 10 - E1 – oil / water simulation (The Buckley-Leverett problem)
#TIME #TIME 25920000 ; final simulation time 2000 ; maximum time step number 0 ; time step control 2000 ; time step number 12960 ; 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.
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 0.0 0.0 0.0 ; node coordinate (start) 305.0 0.0 0.0 ; 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 0.0 7.07 7.07 ; node coordinate (start) 305.0 7.07 7.07 ; 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.
Rock Flow Tutorial - 11 - E1 – oil / water simulation (The Buckley-Leverett problem) 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 ; Finite element calculation $TIMECOLLOCATION ; specifies the time collocation. &GLOBAL 0 1. ; specifies the main time collection $UPWINDING 1 1.0 ; upwinding of Gaussian integration points $MASS_LUMPING 0 ; masslumping reduce wiggles ; #NUMERICS_SATURATION $METHOD 1 ; Finite element calculation $UPWINDING 1 1.0 ; upwinding of Gaussian integration points $MASS_LUMPING 1 ; masslumping reduce wiggles $TIMECOLLOCATION ; specifies the time collocation. &GLOBAL 0 1.0 ; 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.
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.
Rock Flow Tutorial - 13 - E1 – oil / water simulation (The Buckley-Leverett problem)
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= t0=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= t0=0. With “type” “3” a linear value distribution between two given nodes (left side: first node=0, second node=100, pressure value = 1.693567e+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.
#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_2 0.01 ; 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_2 0.01 ; 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).
Rock Flow Tutorial - 15 - E1 – oil / water simulation (The Buckley-Leverett problem)
#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 293.0 ; initial temperature 101325.0 ; 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 0.002 ; 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 0.001 ; 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 µw= 0.001 Pa s and ρw= 1000 kg/m3 respectively and the viscosity and density of oil phase: µo= 0.01 Pa s and ρo= 900 kg/m3 respectively.
Rock Flow Tutorial - 16 - E1 – oil / water simulation (The Buckley-Leverett problem)
#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! ; 0.0 0.0 ; mass dispersion parameters, not used! 0.0 0.0 ; heat dispersion parameters, not used! 0.0 0.0 ; rock density, heat capacity, not used! 0.0 0.0 ; 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:
4)65.0
2.0( −= w
rwSk (E2-1)
of the Brooks-Corey model (Brooks & Corey 1964) with isotropic medium permeability of K=10-11 m2 and a uniform porosity n=0.2. (see also relative permeability and Figure 7)
Rock Flow Tutorial - 17 - E1 – oil / water simulation (The Buckley-Leverett problem)
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:
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).
Rock Flow Tutorial - 18 - E1 – oil / water simulation (The Buckley-Leverett problem)
Figure 7: Relative permeability for two phases (oil and water) versus water saturation
Rock Flow Tutorial - 19 - E1 – oil / water simulation (The Buckley-Leverett problem)
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)
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)
-50 -40 -30 -20 -10 0 10 20 30 40 50
pressure [kPa]
(b)
(a)
(d)
(c)
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)
Rock Flow Tutorial - 21 - E1 – oil / water simulation (The Buckley-Leverett problem)
distance [m]
satu
ratio
n[-]
0 100 200 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time= 300 days
time= 0.0 s
time= 100 daystime= 200 days
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]
Rock Flow Tutorial - 22 - E1 – oil / water simulation (The Buckley-Leverett problem) 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, µo 0.01 Pa s
water viscosity, µw 0.001 Pa s
total velocity, u 9*10e-07 m/s Permeability (vertical) , kv 1.0e-011 m²
Permeability (horizontal) , kh 1.0e-011 m² Porosity, n 20 %
model Brooks-Corey BC-parameter, λ 2 -
entry pressure, pd 10e04 Pa residual saturation of oil, sro 0.15 -
residual saturation of water, srw 0.2 - boundary condition
water saturation, sw Sw(0)=1-Srn - water saturation, sw Sw(300)= Srw -
Initial condition water saturation, sw sw0=srw -
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).
Rock Flow Tutorial - 23 - E1 – oil / water simulation (The Buckley-Leverett problem)
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)
-50 -40 -30 -20 -10 0 10 20 30 40 50
pressure [kPa]
(b)
(a)
(d)
(c)
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)
Rock Flow Tutorial - 24 - E1 – oil / water simulation (The Buckley-Leverett problem)
distance [m]
satu
ratio
n[-]
0 100 200 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time= 300 days
time= 0.0 s
time= 100 daystime= 200 days
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]
Rock Flow Tutorial - 25 - E1 – oil / water simulation (The Buckley-Leverett problem) 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 2.
distance [m]
satu
ratio
n[-]
0 100 200 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
with capillary pressurewithout capillary pressure
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):
( wcowwo Sppp =− ) (E2-3) The equation describes the capillary pressure and the wetting process at the interfaces oil/water i. e. at the front (see Figure 19).
Rock Flow Tutorial - 26 - E1 – oil / water simulation (The Buckley-Leverett problem)
distance [m]
satu
ratio
n[-]
100 150 200 2500.1
0.2
0.3
0.4
0.5
0.6 without capillary pressurewith capillary pressure
Figure 18: Front at t=200 days.
Figure 19: Spreading at the interface of two domains (with capillary pressure)
Rock Flow Tutorial - 27 - E1 – oil / water simulation (The Buckley-Leverett problem) Now we activate the gravitational acceleration (see #REFERENCE_CONDITIONS and #BOUNDARY_CONDITIONS_PRESSURE), in order to examine its influence on the computation.
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 L1 and L2 (see Figure 23).
Rock Flow Tutorial - 28 - E1 – oil / water simulation (The Buckley-Leverett problem)
-50 -40 -30 -20 -10 0 10 20 30 40 50
pressure [kPa]
(b)
(a)
(d)
(c)
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)
X
Y
Z
0.800.730.670.600.530.470.400.330.270.20
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 L1 and L2 used in Figure 24 (see
Rock Flow Tutorial - 29 - E1 – oil / water simulation (The Buckley-Leverett problem)
distance [m]
satu
ratio
n[-]
0 100 200 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L1-L1L2-L2
Figure 24: Water saturation profile along the lines L1 and L2 (see also Figure 23) with
consideration of the gravity (g= 9,8). The capillary pressure is specified.
Rock Flow Tutorial - 30 - 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
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 five-spot pattern in 2- and 3- dimension, in which the injection well is uniformly surrounded by four of the Production wells (Figure 25).
Rock Flow Tutorial - 31 - E2 – oil / water simulation (The five spot problem)
Figure 26: Finite element grid and well arrangement for quarter five spot simulations
Rock Flow Tutorial - 32 - 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, µo 0.01 Pa s
water viscosity, µw 0.001 Pa s oil standard density, ρo 900.0 kg/m³
water standard density, ρw 1000.0 kg/m³ permeability (vertical) , kv 1.0e-011 m²
permeability (horizontal) , kh 1.0e-011 m² porosity, n 20 [%]
model Brooks-Corey BC-parameter, λ 2 -
entry pressure, pd 10e04 Pa residual saturation of oil, sro 0.15 -
residual saturation of water, srw 0.2 -
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 (non-wetting 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.
Rock Flow Tutorial - 33 - 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 1663 2D-square element with overall 1756 knots (Figure 26 and Figure 27)
x [m]
y[m
]
0 5 10 15
0
5
10
15
1671 1688
1663
1681
1680
IW
1672
1655
16701668 1669
1683
1682
Figure 27: The radial
RFD – file • Five_spot_2d In this section only examples or are part
x
y
sub grids on the corner points (injection well)
.rfd
the keywords will be described, which diverge from last preceding icularly important to understand the example (see tutorial A, B and E1).
Rock Flow Tutorial - 34 - E2 – oil / water simulation (The five spot problem)
#OUTPUT_EX #OUTPUT_EX 13 ; type 5-spot_01.PLT ; name 2 ; geo_type 7.0710678119 7.0710678119 0.0 ; node coordinate (start) 292.9289321881 292.928932181 0.0 ; 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.
PW
IW
L
L
Figure 28: Grid system and the position of diagonal cut of the five-spot pattern (the line L
specified in “#OUTPUT_EX”)
Rock Flow Tutorial - 35 - 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
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.
Rock Flow Tutorial - 36 - E2 – oil / water simulation (The five spot problem)
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).
Rock Flow Tutorial - 37 - 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 *.plt-files and processed by Tecplot.
time [days]
Wat
er-O
ilR
atio
[-]
0 100 200 3000
0.5
1
1.5
2
2.5
3
3.5at the production well (PW)at the injection well (IW)
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.
Rock Flow Tutorial - 38 - E2 – oil / water simulation (The five spot problem)
distance [m]
satu
ratio
n[-]
0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time= 300 days
time= 0.0 s
time= 100 daystime= 200 days
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]
Rock Flow Tutorial - 39 - 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, µo 0.01 Pa s
water viscosity, µw 0.001 Pa s
permeability (vertical) , kv 1.0e-011 m² permeability (horizontal) , kh 1.0e-011 m²
porosity, n 20 % model Brooks-Corey
BC-parameter, λ 2 - entry pressure, pd 10e04 Pa
residual saturation of oil, sro 0.15 - residual saturation of water, srw 0.2 -
boundary condition outlet pressure, pout 10e05 Pa
the average flux, (u.ne)in h-1*1.05e-04 m/s Initial condition
water saturation, sw sw0=srw
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 Figure30. 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
Rock Flow Tutorial - 40 - E2 – oil / water simulation (The five spot problem)
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 1663 2D-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 3D-simulation shown in Figure 35 to Figure 37. These show an exact match between the 2D- and 3D-dimensional modeling ( compare with Figure 30, Figure 33 also Figure 1 ).
Rock Flow Tutorial - 41 - E2 – oil / water simulation (The five spot problem)
Richardson, TX. [2] Aziz, K. and Settari, A.: 1979. 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.
