Fully Conservative Characteristic Methods for Flow and Transport: - PowerPoint PPT Presentation

Fully Conservative Characteristic Methods for Flow and Transport: Part III, Nonlinear Two-Phase Flow Todd Arbogast Department of Mathematics and Center for Subsurface Modeling, Institute for Computational Engineering and Sciences (ICES) The University of Texas at Austin Chieh-Sen (Jason) Huang, National Sun Yat-sen University (Taiwan) Thomas F. Russell, U.S. National Science Foundation This work was supported by • U.S. National Science Foundation • KAUST through the Academic Excellence Alliance Center for Subsurface Modeling Institute for Computational Engineering and Sciences 1 The University of Texas at Austin, USA

Outline 1. Scalar Conservation Laws and Two-Phase Flow 2. Background on Scalar Conservation Laws 3. Some Standard Numerical Methods 4. A Fully Conservative Streamline Method 5. Some Numerical Results 6. Summary and Conclusions Center for Subsurface Modeling Institute for Computational Engineering and Sciences 2 The University of Texas at Austin, USA

Scalar Conservation Laws and Two-Phase Flow Center for Subsurface Modeling Institute for Computational Engineering and Sciences 3 The University of Texas at Austin, USA

The Scalar Conservation Law Recall the general conservation equation φc t + ∇ · ( c u ) = q c u ( x , t ) is the Darcy velocity of the fluid; φ ( x , t ) is the porosity of the medium (pore volume/bulk volume); c ( x , t ) is the concentration of tracer (mass/pore volume); q c ( x , t ) is an external source or sink of fluid. This equation is a first order hyperbolic partial differential equation for c . It is called a scalar balance law. In the absence of sources and sinks (i.e., q c = 0), we have the scalar conservation law for the transport of tracer particles. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 4 The University of Texas at Austin, USA

Immiscible, Incompressible, Two-phase Flow We consider the general immiscible, incompressible, water-oil system. For each pure phase α , where α = w or o, we have c �→ 1 (by immiscibility) φ �→ φs α (effective pore space) The general conservation equation gives the water-oil system φs t + ∇ · u w = q w and − φs t + ∇ · u o = q o , s = s w is the saturation of water (wetting fluid); 1 − s = s o is the saturation of oil (nonwetting fluid); u α is the Darcy phase velocity ( α = w or o); q α represent wells. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 5 The University of Texas at Austin, USA

Darcy’s Law and Capillary Pressure Darcy’s law governs the phase velocity and pressure (ignoring gravity) u α = − λ α k ∇ p α , α = w , o , p α is the phase pressure; k ( x ) is the absolute permeability; λ α ( s ) is the phase mobility , i.e., relative permeability divided by phase viscosity. The capillary pressure relation p c ( s ) = p o − p w completes the description of the system. This is a system of two nearly hyperbolic flow equations. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 6 The University of Texas at Austin, USA

Rearrangement into Flow and Transport Equations By a rearrangement, we can separate into two systems: • Elliptic flow or pressure equation; • (Nearly) hyperbolic transport or saturation equation. Pressure Equation: The sum of the two equations is ∇ · u = q ≡ q w + q o , u = − k [ λ w ∇ p w + λ o ∇ p o ] = − k [ λ ∇ p w + λ o ∇ p c ] = − k [ λ ∇ p o − λ w ∇ p c ] u = u w + u o is the total velocity; λ = λ w + λ o is the total mobility; Saturation Equation: The water conservation equation φs t + ∇ · f ( s ) = q w where the wetting flux function is f ( s, x ) = u w = λ w λ u + kλ w λ o ∇ p c λ Center for Subsurface Modeling Institute for Computational Engineering and Sciences 7 The University of Texas at Austin, USA

Alternate Saturation Equation However, we could equivalently choose the oil conservation equation φ (1 − s ) t + ∇ · F (1 − s ) = q o with F (1 − s, x ) = u o = λ o λ u − kλ w λ o ∇ p c λ Key point: Our numerics should not be biased toward either fluid! Center for Subsurface Modeling Institute for Computational Engineering and Sciences 8 The University of Texas at Austin, USA

A Simplified Model System—1 Assume: • one space dimension, domain (0 , 1); • appropriate boundary conditions so q = q w = q o = 0; • φ ≡ 1 for simplicity; • flux functions depend only on s and not on x (or t ); • z = 1 − s = s o . Remark: The pressure equation is u x = 0, so u is a known constant. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 9 The University of Texas at Austin, USA

A Simplified Model System—2 Model system: Writing both transport equations, we have s t + f ( s ) x = 0 (scalar conservation law for water) z t + F ( z ) x = 0 (scalar conservation law for oil) s + z = 1 f ( s ) + F ( z ) = u (for constant u ) Since u is known, this system is redundant. Boundary and initial conditions: For simplicity, assume the phases travel to the right, i.e., u , f , f ′ , F , and F ′ are nonnegative. s (0 , t ) = s B ( t ) and z (0 , t ) = z B ( t ) = 1 − s B ( t ) s ( x, 0) = s 0 ( x ) z ( x, 0) = z 0 ( x ) = 1 − s 0 ( x ) and where 0 ≤ s B ( t ) ≤ 1 and 0 ≤ s 0 ( x ) ≤ 1. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 10 The University of Texas at Austin, USA

Background on Scalar Conservation Laws Center for Subsurface Modeling Institute for Computational Engineering and Sciences 11 The University of Texas at Austin, USA

The Linear Case General scalar conservation equation in 1-D: � � c t + f ( c ) x = 0 Linearity: Asume f is linear, meaning there is some v independent of c , such that f ( c ) = cv Then we have c t + ( cv ) x = 0 In general, v = v ( x, t ). Center for Subsurface Modeling Institute for Computational Engineering and Sciences 12 The University of Texas at Austin, USA

The Case of Constant Velocity If v is constant, with an IC, we have c t + vc x = 0 , −∞ < x < ∞ , t > 0 , c ( x, 0) = c 0 ( x ) , which is solved by c ( x, t ) = c 0 ( x − vt ) . If v > 0, we have a wave traveling to the right, of fixed shape. c ✻ ✲ t ✲ t c 0 c ( x, t ) ✲ t ✲ x Particles simply translate to the right with velocity v . We could have a jump in the IC, which propagates as a contact discontinuity. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 13 The University of Texas at Austin, USA

Inflow Boundary Conditions On a domain 0 < x < ℓ , we have only one BC. If v > 0 is constant, we need to specify c on the inflow side x = 0, but not on the outflow side x = ℓ . That is, c t + vc x = 0 , 0 < x < ℓ, t > 0 , c (0 , t ) = c I ( t ) , c ( x, 0) = c 0 ( x ) , which is solved by  c 0 ( x − vt ) , x − vt ≥ 0 ,  c ( x, t ) = c I ( t − x/v ) , x − vt < 0 .  c ✻ Particle enters “ c I ” ✲ domain at x = 0 ① c ( x, t ) ✲ ① and some t > 0 ✲ × ① and then Particles leave c 0 ✲ × transports in ① the domain. x ✲ time. 0 ℓ Center for Subsurface Modeling Institute for Computational Engineering and Sciences 14 The University of Texas at Austin, USA

Streamlines Fluid particles travel along paths called streamlines. Given a starting position x , the path would be ( ξ ( t ) , t ), where Given a starting position x , the Time streamline is ( ξ ( t ) , t ): ✻ t dξ � � dt = v ξ ( t ) , t , t > 0 , ξ (0) = x. ✲ Space x ξ ( t ) Given v ( x, t ), this is easily solved. In the linear case and when v x = 0, c is constant along streamlines, since d � � ξ ( t ) , t = c t + c x ξ t = c t + c x v = c t + ( cv ) x = 0 . dtc If we know where the particle starts, we can simply follow it in time along the streamlines. Center for Subsurface Modeling Institute for Computational Engineering and Sciences 15 The University of Texas at Austin, USA

Streamlines for Constant Velocity If v is constant, then  dξ  dt = v, t > 0 ,  = ⇒ ξ = x + vt,  ξ (0) = x.  which are straight lines. Time ✻ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ t ✡ ✡ t ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ c I ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✢ ✡ ✡ ✡ ✡ ✡ ✡ ✲ Space t x x − vt c 0 Conclusion. If you know c 0 and c I , you know the solution for all time. We saw that in fact  c 0 ( x − vt ) , x − vt ≥ 0 ,  c ( x, t ) = c I ( t − x/v ) , x − vt < 0 .  Center for Subsurface Modeling Institute for Computational Engineering and Sciences 16 The University of Texas at Austin, USA