The dotted line is for a fine fractional flow curve on a 100 x 100 grid, the dotted line is for a coarse model with a non-uniform coarsening method on a non-uniform coarse 12 x 12 grid. The dotted line is for a fine fractional flow curve on a 100 x 100 grid, the dotted line is for a coarse model with a non-uniform coarsening method on a non-uniform coarse 10 x 10 grid. The dotted line is for a fine fractional flow curve on a 100 x 100 grid, the dotted line is for a coarse model with a non-uniform coarsening method on a non-uniform coarse 9 x 9 grid, a dotted line.
Chapter 1 Overview
In this part of the work we analyze the Ms-FEM for different possible situations both analytically and numerically. In this part of the work we use the techniques developed in the MsFEM analysis to improve the calculated absolute permeability values. The last part of the work is the application of MsFEM for the two-phase flow rate increase.
Chapter 2 Introduction
A key feature of MsFEM is that the construction of basis functions is a local operation within the elements. Resonance represents a fundamental difficulty due to the mismatch between the local construction of multiscale basis functions and the global nature of elliptic problems. This mismatch between the local solution and the global solution produces a boundary layer in the first-order rectifier of the local solution which causes the resonance.
Formulation and overview
- Homogenization for one periodic case and related estimates
- Overview of MsFEM convergence
For now, we assume that the basis functions are linear along the element boundaries, i.e. As we can see, the only difference between MsFEM and the traditional finite element method is the construction of basis functions. As mentioned in the introduction, the purpose of the multicriteria method is to capture a large-scale solution.
MsFEM for problems with many separated scales
- Summary
- Formulation In this section we investigate MsFEM for
- Estimates for first order correctors
- H1 estimates
- H1 estimates for two scale case
The calculations are extremely large and are done on parallel computers, eg, the Intel Paragon computer. But various numerical examples show that in some cases one of them may be dominant, while in other cases both may be important. Furthermore, similar to a two-scale case it can be shown that for smooth domains I;2.
- L2 estimates
- Numerical experiments
- Concluding remarks and generalizations
One would expect that when h is close to €1 the first term dominates and when h is close to €2 the second term dominates. As we see from the calculated l2 error of MsFEM (see Table 4.3), the method does not reveal the -1 order convergence for h = O(e2). This important problem and its numerical solution, e.g. the oversampling method, has been analyzed for small-scale problems (see non-conforming MsFEM).
Chapter 5 MsFEM for discontinuous case
- Summary
- Formulations
- fll estimates In this section we consider H 1 error estimate for MsFEM for
- Asymptotic expansion of the discrete solution
- Numerical experiments
We present the numerical results for the cases where the element boundaries are aligned with the discontinuity surfaces and when they are not. The analysis differs from the analysis of the case with smooth coefficients due to the fact that the generalized VX is no longer continuous. For the latter, we compare the discrete solution of (4.1) and that of the homogenized equation at the nodes.
For that reason, we assume that the point of intersection between dK and the interfaces of the discontinuity of the coefficients is a countable number of points. In this section, we study the convergence and accuracy of the multiscale method through numerical experiments. However, due to the change of the mesh size, the elements may include more discontinuities of the coefficients.
To calculate the basis functions, we use finite element discretization adapted to the discontinuity interfaces of the coefficients with Black-Box multigrid. We also test separately the cases where multiscale elements are aligned with the discontinuity interfaces of the coefficients and when they are not. The numerical tests show that the tb2 error estimate (5.49) is the case when multiscale elements are aligned with the discontinuity interfaces of the coefficients.
In Tables 5.1 and 5.2, numerical tests are performed when the elements at multiple scales are aligned with coefficient discontinuity interfaces.
Chapter 6 MsFEM for problems with weakly dependent random coefficients
- Discussion
- Formulations
In other words, to denote @(A) as the o-algebra generated by the parameters of the problem in the physical domain A, we assume that. The presence of the ratio in the error estimates of MsFEM was observed earlier and it is called the resonance. We observe that in the ergodic case there is the equality < f >= Ef; where E denotes the operator of the mean with respect to the measure P (mathematical expectation).
When aij (y, w ) is strictly stationary and ergodic, it was found that there exists aij (if j = 1,2,3) such that if uo(s) is the solution of the Dirichlet determinant problem. However, the extension of the result on periodic structures to homogeneous random structures encounters a difficulty related to the intractability of (6.12) in terms of homogeneity. For the calculations of the degree of convergence of MsFEM we require an assumption about the weak dependence of the field values of the coefficients of (6.13), k = (aij, gi) at distant points.
This estimate is obtained using non-trivial estimates for the correlations of the field h and by choosing the large T, T = E-P (P > 0). Here we state the main theorem and the inequalities used in the derivation of the convergence rate of the MsFEM. The proof of this theorem is based on the analysis of the paths of the diffusion processes corresponding to (6.13), and it can be found in [56].
These rates represent the relationship between the small scale and the macroscopic scale of the problem.
Lemma 6.6.3
Consequently, we only need to estimate the last three terms in (6.29), which we do in the next three lemmas. As we will see from the estimates, the main resonance error is caused by the first order corrector, I18vI)1,K than in periodic case. Furthermore, by equating the first and the last terms, we can find the optimum value for m.
Lemma 6.6.4
We can observe numerically that the resonance error is no longer c / h if the coefficients are random.
Chapter 7 Nonconforming MsFEM and its analysis
- Summary
- Description of nonconforming MsFEM
- Cell resonance and averaged over-sampling
- Some observations
- Averaged over-sampling method
7.2) The analysis of the behavior of qp is a complex problem, which has yet to be rigorously performed. Using the oversampling technique, we get rid of a large part of the error caused by 0. Due to the conformity of the linear basis functions, the second term on the right-hand side of (7.12) is zero.
We note that the proof of the theorem for the case of an arbitrary number of elements per sample can be performed following our proof with Remark 3. For the latter, we compare the discrete solution of (4.1) with that of the homogenized equation ( 3.9) at the nodal points . First step: Asymptotic expansion of the discrete operator Using the asymptotic expansion of the basis functions,.
We say two nodal points are neighbors if they are the vertices of the same element. We have ignored the sums of the boundary terms generated by the summation by parts (analogous to the boundary integrals produced by the integration by parts). In the following, we first show the subtle behavior of the resonance error through numerical tests.
The arithmetic mean of the above mentioned elements of the cell matrix with the same integrand h-2 JK [ ( x / e ) d x will indeed be averaged to SQ [(x/e)dx where diam (f2) is of order 1 .
Chapter 8 Upscaling of absolute permeability
- Formulations. Effective and grid block permeability
- Overview .1 Local Laplacian formulations
- Volume vs. surface averages
- Accuracy of upscaling
- Some estimates
- Estimates for upscaled permeability and solutions
- Remarks
- Over-sampling method
We show that the upscaling error appears as a resonance between the medium's small physical scales and the artificial mesh scale (size). The problem is examined in more detail in the first part of the work. In particular, K depends on the location and geometry of the grid block in which it is calculated.
Note that the source term is set to zero because of the second property of K* mentioned above. The velocity solution of the scaled-up equation thus approximates the volume average of the fine-scale velocity in the lattice blocks as E + 0. As we see, the cause of the resonance error is the discrepancy between volume V and the period of the problem.
Note that the discretization error in the test is fixed because NE is held constant; therefore, the error reduction is mainly due to the decrease in the resonance error (Eq. K thus depends solely on the geometry of the lattice block and the underlying fine-scale permeability. On the other hand, the use of the Dirichlet boundary condition (8.7) should always are associated with oversampling.
For the calculation of the effective coefficients, the fine resolution cell problem was solved numerically.
MEDIA
Chapter 9 Applications of MsFEM to upscaling of displacements in heterogeneous porous media
- Introduction
- Unit mobility case
- The flow features in reservoir and their modeling
- Non-uniform coarsening method
- Derivation of the coarse model equations for unit mobility case The coarse model we are going to derive in this section is known in literature [14, 571
- Rigorous derivation of upscaled equations for layered system
- The formulations of the diffusion in coarse models
- The use of MsFEM in coarse models
- Boundary conditions for coarse models
- Numerical results for unit mobility case
- Upscaling of two-phase flow
- Numerical results for two-phase flow
- Concluding remarks
In this case, as we noted at the beginning of the section, the fine curves of relative permeability are preserved. Our rough models as we see from the previous section require average flow characteristics. The numerical examples we present here compare the fractional flow curves of the coarse model (9.41) and the nonuniform coarsening method on the same coarse mesh.
We see from this figure that our method improves the result of the non-uniform coarsening method. We see that for this case too the coarse model (9.42) gives a better approximation of the fractional flow curve of the detailed flow than the non-uniform coarsening on a coarse grid of 10 x 10. If we increase the correlation length in the y-direction , the performance of the coarse model (9.42) deteriorates (see Figure 9.13).
The derivation of the coarse model is based on the perturbation of saturation and the velocity fields around their average. Furthermore, the coarse model with diffusion improves the predictions of the non-uniform magnification method. The implementation of the coarse model formulated in the previous section to the general two-phase flow is currently under investigation.
The main idea of the approximate models I am currently working with is to assign three velocity fields (for practical purposes) to each thick block rather than one as in the non-uniform thickness approach. In the non-uniform coarsening approach, we simply represent the mean flow while missing important flow features. As we see from these figures, the predictions on a coarse 2 x 2 grid are better than a coarse 5 x 5 approximation of the fractional flow curves using non-uniform coarsening.
Bibliography
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