Chapter 4. Upscaling One-Phase Flow
4.4. Designing an Upscaled Model from the Two-Scale limit
It is hard to see how the fully homogenized equation (4.3.2), which we derived from the two-scale limit using Tartar’s method, can be used to design a numerical method. We will derive an efficient numerical method starting from the two-scale limit by using the method of averaging. We will also demonstrate that this equation is very close to the homogenization result of the previous section. We will use the higher moments of the saturation and the velocity to model the macrodispersion. In the context of two-phase flow this idea was introduced by Efendiev, Durlofsky, and
Lee [20, 21] and is referred to as Volume Averaging. In these papers they were facing a much more difficult problem since they were averaging along and across the flow.
In our case the velocity field is already upscaled in the direction along the flow. In addition we are selecting the shape of the cells so that the velocity varies less within each cell. The resulting algorithm is also simpler to implement in our case because of the decoupling of the full two-dimensional equation to one-dimensional transport along streamlines.
4.4.1. Physical Interpretation of the Subgrid Forcing. We would like a physical interpretation of the forcing term that appears in the equation for the mean saturation in (4.2.2). We will integrate the equation for the fluctuations along the characteristics to eventually form that forcing term. The characteristics are defined by
dP
dt = ˜v0, with P(p,0) =p.
The equation forS0 is S0 =−
Z t 0
v˜00(P(p, τ), ψ)Sp(P(p, τ), ψ, τ) + ˜v00Sp0) dτ.
We take the derivative of both sides with respect to p, multiply by ˜v00, and average over ψ. The second term will be third- order in fluctuating quantities and therefore small compared to the first. The macrodispersion becomes
˜
v00Sp0 =− Z t
0
˜ v00 ∂
∂p v˜00(P(p, τ), ψ)Sp(P(p, τ), ψ, τ) dτ.
We take ∂p∂ outside the integral to find
˜
v00Sp0 = − ∂
∂p Z t
0
˜
v00v˜00(P(p, τ), ψ)Sp(P(p, τ), ψ, τ)dτ +
Z t 0
∂v˜00
∂p v˜00(P(p, τ), ψ)Sp(P(p, τ), ψ, τ)dτ.
Looking at the last equation we can understand the influence of the macrodisper- sion in the upscaled equation (4.2.2). The first term corresponds to diffusion whose magnitude depends on the two point correlation of the velocity field, and the second corresponds to a convection. Similar results have been obtained in a probabilistic
framework (see for example [43]). When the velocity field does not vary a lot along the streamlines, which is realistic since we have already averaged along streamlines, we expect the convection term of the macrodispersion to be negligible. Convection by simple arithmetic average of the velocity across streamlines will then give the exact amount of fluid that is transported through the domain over large times. It will give the correct integral of the fractional flow curve over all time or the total amount of oil that can be obtained. If we want to retain some information about the break- through time or the precise form of the fractional flow rate at every time we have to retain some information about the fast and slow channels within the coarse cells.
This information would be contained in the macrodispersion.
4.4.2. Numerical Averaging across Streamlines for Linear Flux. The derivation in the previous section contained no approximations, and the average sat- uration equation cannot be solved on the coarse grid. In this section we follow the same idea as in that derivation to solve the equation for the fluctuations along the characteristics, but with the purpose of deriving an equation on the coarse grid. To achieve our purpose we will not do analytical upscaling in the sense that we are not interested in deriving a continuous upscaled equation as in the previous section. We will first discretize the equation with a FV method in space and then upscale the re- sulting equation. Our upscaled equation is then dependent on the numerical scheme.
We note that also the definitions of the average saturation and its fluctuations (4.2.1) are easier to understand in the context of a numerical scheme.
We use the same definition for the average saturation and the fluctuations as in (4.2.1) and follow the same steps until equation (4.2.2). We discretize the macrodis- persion term in the equation for the average saturation
˜
v00Sp0 = v˜00S0i+1−v˜00S0i
∆p +O(∆p).
A superscript ·i refers to a discrete quantity defined at the center of the conservation cell. Instead of solving the equation for the fluctuations on the fine characteristics as
before, which would lead to a fine grid algorithm, we solve it on the coarse character- istics defined by
dP
dt = ˜v0, with P(p,0) =p.
Compared to the equation that we obtained in the previous section forS0, this equa- tion for S0 has an extra term, which appears second
S0 =− Z t
0
˜
v00(P(p, τ), ψ)Sp(P(p, τ), ψ, τ) + ˜v00(P(p, τ), ψ)Sp0(P(p, τ), ψ, τ) + ˜v00Sp0) dτ.
The second term is second-order in fluctuating quantities, and we expect it to be smaller than the first term so we neglect it. As before, we multiply by ˜v00 and average overψ to find
˜
v00S0 =− Z t
0
˜
v00v˜0(P(p, τ), ψ)Sp(P(p, τ), ψ, τ)dτ.
In this form at time t it is necessary to know information about the past satu- ration in (0, t) to compute the future saturation. In the appendix we demonstrate that Sp(P(p, τ) depends weakly on time, in the sense that the difference between Sp(P(p, τ) andSp(P(p, t) is of third-order in fluctuating quantities. Therefore we can take Sp(P(p, τ) out of the time integral to find
˜
v00S0 =− Z t
0
˜
v00v˜00(P(p, τ), ψ)dτ Sp.
The term inside the time integral is the covariance of the velocity field along each streamline. The macrodispersion in this form can be computed independent of the past saturation.
4.4.3. Numerical Averaging across Streamlines for Nonlinear Flux. The nonlinearity of the flux function, the fact that the sum of the fine saturation fluxes is not equal to the flux of the sum of the fine saturations, introduces an extra source of error in the approximation. We Taylor expand f( ˜S) near S and keep only the first
term
(4.4.1)
S˜ = S(p, ψ, t) +S0(p, ψ, ζ, t)
˜
v0 = v˜0(p, ψ, t) + ˜v00(p, ψ, η, t) f( ˜S) = f(S) +fS(S)S0+O(S02) f(S)p = fS(S)Sp +f(S)S0+. . .
A reasonable objection is that this approximation will be inadequate near the shock since S0 is not small there. The region near the shock is important because there the macrodispersion is largest. Due to the dependence of the jump in the satu- ration on the mobility we expect this approximation to be better for lower mobilities.
Nevertheless this approximation works well in practice. For more accuracy it is also possible to retain more terms in the Taylor expansion, as was done in [21]. We will show that in realistic examples these higher-order terms are not important in our setting.
Using these definitions we derive the following equations for the average saturation and the fluctuations
St+ ˜v0f(S)p+ ˜v00(fS(S)S0)p = 0 (4.4.2)
St0 + ˜v00fS(S)Sp+ ˜v0fS(S)Sp0 −v˜00Sp0 = 0.
The macrodispersion is discretized as
˜
v00(fS(S)S0)p = v˜00fS(S)S0i+1−v˜00fS(S)S0i
∆p +O(∆p).
We solve the second equation on the coarse characteristics defined by dP
dt = ˜v0fS(S), withP(p,0) =p and form the terms that appear in the macrodispersion
˜
v00fS(S)S0 =− Z t
0
˜
v00fS(S) ˜v00(P(p, τ), ψ)fS(S(P(p, τ), ψ, τ))Sp(P(p, τ), ψ, τ)dτ.
As before we have dropped terms that are second-order in fluctuating quantities. We use an argument in the appendix to show that fS(S(P(p, τ), ψ, τ))Sp(P(p, τ), ψ, τ)
does not vary much along the streamlines and take it out of the integration in time to find
(4.4.3) v˜00fS(S)S0 =− Z t
0
˜
v00v˜00(P(p, τ), ψ)dτ fS(S)2Sp.
This expression is very similar to the one obtained in the linear case; however here the macrodispersion depends on the past saturation through the equation for the coarse characteristics.