Rosenbrock-type methods for geothermal reservoirs simulation Rosenbrock-type methods for geothermal reservoirs simulation Antoine Tambue Joint work with Inga Berre and Jan Martin Nordbotten AIMS South Africa and University of Cape Town 23 March 2016
Rosenbrock-type methods for geothermal reservoirs simulation Outline Challenge in geothermal reservoir simulation 1 Geothermal without phase change 2 Geothermal with phase change 3 4 Simulations
Rosenbrock-type methods for geothermal reservoirs simulation What is geothermal energy?
Rosenbrock-type methods for geothermal reservoirs simulation Challenge in geothermal reservoir simulation Geothermal reservoir simulation: AIMS, Challenge and research strategies 1 AIMS Predict reservoir production Optimal production strategies Understand physical processes 2 Challenge Coupled highly nonlinear physical processes Coupled processes on multiple scales Heterogeneous environments Working in fixed-grid with phase change 3 Our goal Propose an alternative efficient, stable and accurate time stepping methods where Newton iterations are no required at every time step as in standard implicit methods mostly used currently in reservoir simulation.
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Geothermal with one phase flow 1 Energy Equation ∂ T s ( 1 − φ ) ρ s c ps = ( 1 − φ ) ∇ · ( k s ∇ T s ) + ( 1 − φ ) q s + he ( T f − T s ) ∂ t ∂ T f φρ f c pf ∂ t = φ ∇ · ( k f ∇ T f ) − ∇ · ( ρ f c pf v T f ) + φ q f + he ( T s − T f ) 2 Darcy’s Law v = − K µ ( ∇ p − ρ f g ) , (2) 3 Mass balance equation ∂φρ f = −∇ · ( v ρ f ) + Q f , (3) ∂ t
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Geothermal with one phase flow 1 State functions µ, ρ f , C p f , α f , β f 2 Slightly compressible rock and compressible fluid φ = φ 0 ( 1 + α b ( p − p 0 )) α f = − 1 ∂ρ f β f = 1 ∂ρ f (4) , ∂ p . ρ f ∂ T f ρ f 3 Model equations ∂ T s ( 1 − φ ) ρ s c ps = ( 1 − φ ) ∇ · ( k s ∇ T s ) + ( 1 − φ ) q s + he ( T f − T s ) ∂ t ∂ T f φρ f c pf ∂ t = φ ∇ · ( k f ∇ T f ) − ∇ · ( ρ f c pf v T f ) + φ q f + he ( T s − T f ) ∂ T f ∂ t + ρ f ( φβ f + φ 0 α b ) ∂ p � ρ f K � − φρ f α f ∂ t = ∇ · µ ( ∇ p − ρ f g ) + Q f
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Finite volume for space discrete Keys features of the method 1 Integrate each equations over each control volume Ω i . 2 Use the divergence theorem to convert the volume integral into the surface integral in all divergence terms. 3 Use two-point flux approximations for diffusion heat and flow fluxes Semi-discrete system after space discretization dT h = G ( T h , p h , t ) , dt ( φα f )( T h f , p h ) dp h = G 3 ( p h , T h f , p h ) · G 2 ( T h s , T h f , t ) + f , p h , t ) , (6) ( φβ f + φ 0 α b )( T h dt G ( T h , p h , t ) = ( G 1 ( T h s , T h f , t ) , G 2 ( T h s , T h f , p h , t )) T , f ) T ≈ ( T s , T f ) T . T h = ( T h s , T h
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Rosenbrock-Type methods: Construction Motivation When the equations are non-linear, implicit equations can in general be solved only by iteration. This is a severe drawback, as it adds to the problem of stability, that of convergence of the iterative process. An alternative, which avoids this difficulty, is ......., (H.H. Rosenbrock 1962/63 Consider the following ODEs y ′ = f ( y ) The corresponding diagonally implicit Runge-Kutta method is given by i − 1 s � � k i = hf ( y n + a i , j k j + a i , i k i ) , y n + 1 = y n + b i k i (7) j = 1 i = 1
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Rosenbrock-Type methods: Construction Linearization i − 1 � k i = hf ( g i ) + f ′ ( g i ) a i , i k i , g i = y n + a i , j k j + a i , i k i . (8) j = 1 The equation (8) can be interpreted as the application of one Newton iteration to each stage of previous RK method. No continuation of iterating until convergence, a new class of methods are deduced with judicious choice of coefficients a i , j to ensure their convergence, their stability and the accuracy. The s-stage Rosenborck methods is given by i − 1 i s � � � k i = hf ( y n + a i , j k j ) + hf ′ ( y n ) γ i , j k j , y n + 1 = y n + b i k i . (9) j = 1 j = 1 i = 1 Difference with RK, extra coefficients γ i , j are needed.
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Rosenbrock-Type methods: Embedded approximations To control the local errors and adaptivity purposes, cheaper and stable scheme is needed, the corresponding embedded approximation associated to Rosenbrock-Type methods is given by s � ˆ y 1 n + 1 = y n + b i k i . (10) i = 1 For Rosenbrock -type method of order p , the coefficients ˆ b i are determined using the consistency conditions such that the embedded approximation is order p − 1. The the embedded approximation is always more stable that the associated scheme and the local error is estimated as err = norm ( y n − y 1 n ) .
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal without phase change Application to geothemal model The second order scheme ROS2(1) and the third order scheme denoted ROS3p are used. We solve sequentially the following systems dT h = G ( T h , p h , t ) dt (11) T h ( 0 ) , p h ( 0 ) given , and ( φα f )( T h dp h f , p h ) = G 3 ( p h , T h f , p h ) · G 2 ( T h s , T h f , t ) + f , p h , t ) ( φβ f + φ 0 α b )( T h dt = G 4 ( T h (12) h , p h , t ) , T h ( 0 ) , p h ( 0 ) given .
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal with phase change Two-phase mixture model problems (C.Y. Wang, 2007) 1 The mass conservation of the two phase is given ∂φρ + ∇ · ( ρ u ) = Q f . (13) ∂ t Here Q f is the source of liquid and vapor. 2 The momentum conservation and is given by u = − K µ [ ∇ p − ρ k ( s ) g ] , (14) 3 The model is obtained by adding the equations of mass conservation of liquid phase and vapor phase, ρ u = ρ l u l + ρ v u v , ρ = ρ l s + ( 1 − s ) ρ v , ρ k = ρ l λ l + ρ v λ v , µ = ρυ , υ = 1 / ( kr l /υ l + kr v /υ v ) with u i = − K kr i [ ∇ p − ρ i g ] , i = { l , v } . (15) µ i
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal with phase change Two-phase mixture model problems 1 Monotone transformation of the thermodynamic state variables H = ρ ( h − 2 h vsat ) , ρ h = ρ l sh l + ρ v ( 1 − s ) h v � f ( s ) K ∆ ρ h fg � Ω ∂ H ∂ t + ∇ · ( γ h u H ) = ∇ · (Γ h ∇ H ) + ∇ · g (16) ν v 2 The temperature T and liquid saturation s can be calculated as H + 2 ρ l h vsat H ≤ − ρ l ( 2 h vsat − h lsat ) ρ l c pl T = T sat − ρ l ( 2 h vsat − h lsat ) < H ≤ − ρ v h vsat T sat + H + ρ v h vsat − ρ v h vsat < H ρ v c pv 1 H ≤ − ρ l ( 2 h vsat − h lsat ) H + ρ v h vsat = − − ρ l ( 2 h vsat − h lsat ) < H ≤ − ρ v h vsat s ρ l h fg + ( ρ l − ρ v ) h vsat 0 − ρ v h vsat < H .
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal with phase change Model problem (C.Y. Wang and al.) 1 Two-phase mixture model problem (C.Y. Wang and al.) ∂φρ ∂ t + ∇ · ( ρ u ) = Q f � f ( s ) K ∆ ρ h fg � Ω ∂ H ∂ t + ∇ · ( γ h u H ) = ∇ · (Γ h ∇ H ) + ∇ · g ν v Ω = φ + ρ s c ps ( 1 − φ ) dT dH 2 Wang model were recently tested with great success for steady state mass conservation by different authors 3 For geothemal, steady state mass conservation is less realistic
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal with phase change Our adapted model problem Decomposition ∂ ( φρ ) φ∂ρ ∂ t + ρ∂φ = (17) ∂ t ∂ t φ∂ρ φ ∂ρ ∂ p ∂ t + φ ∂ρ ∂ H ∂ p ∂ t + φ ∂ρ ∂ H = ∂ p | H ∂ H | p ∂ t = φρβ H ∂ H | p ∂ t . ∂ t Here, β H is called the pseudo fluid compressibility at constant mixture pseudo enthalpy H ∂ρ ∂ V β H = 1 ∂ p | H = − 1 ∂ p | H (18) ρ V We assume that the rock is weakly compressibility ∂φ ∂ p φ = φ 0 ( 1 + α b ( p − p 0 )) ∂ t = φ 0 α b ∂ t . (19)
Rosenbrock-type methods for geothermal reservoirs simulation Geothermal with phase change Our adapted model problem Note that in one phase region, by simplication we have: � ∂ρ βρ c p + α ( 1 − α T ) � α β H = ρ ( c p − α ( h − 2 h vsat )) , χ := = (20) ∂ H α ( h − 2 h vsat ) − c p p As we are dealing with two phase flow with phase change we compute the coefficients by ∂ v � ∂ v ∂ h | p � χ = − 1 | p = − 1 (21) v 2 v 2 ∂ H ∂ H ∂ h | p ∂ v ∂ H ∂ h | p − ∂ v ∂ H ∂ p | h ∂ h | p ∂ p | h β H = − ρ (22) ∂ H ∂ h | p
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