Fully-upwind Galerkin method FUG

3.2 Petrov–Galerkin method PG

In order to ensure convergence of the numerical solution, a modified Petrov–Galerkin method was devel- oped in which the test functions are up to two polynomial degrees higher than the shape function. 7,20,21 We consider the case of piecewise quadratic test functions in one dimension having the following representation for an ele- ment x i , x j W a , k ¼ N k þ ˜ a a , k F Q , k [ {i , j} and a [ {w , n} 20 where the quadratic term F Q is given by F Q ¼ 4 ˜x 2 ¹ ˜x ÿ ; ˜x ¼ x ¹ x i = x j ¹ x i . Let k ¼ i, then the corresponding upwind parameter ˜ a a , i for the a phase equation is defined by ˜ a a , i ¼ ¹ 1 if w a , j ¹ w a , i , 0 0 if w a , j ¹ w a , i ¼ , a [ {w , n} , 1 if w a , j ¹ w a , i . 0 8 : 21 where w a,i and w a,j are evaluated at the previous iteration step. The use of a two-point Gaussian quadrature rule with Gauss points at x i þ 1=2ð1 6 ð1=  3 p ÞÞ x j ¹ x i ÿ results in the following weighting of the nodal mobilities l PG a , ij ¼ 7 6 l a , i ¹ 1 6 l a , j 22 in the case that node i is the upstream node for phase a. Consequently, in this case the mobility term l PG a , ij is par- tially upstream weighted. The PG method coincides with the SG method in the case when ˜ a a ¼ 0. Analogously to the development of modified upwind weighting functions for linear one-dimensional elements, it is possible to derive polynomials for bi- and tri-linear elements in two and three dimensions, respectively. 7 In order to avoid artificial cross diffusion in multidimensional problems, the upwind para- meters ˜ a are calculated from the streamlines. Fig. 2 illus- trates this for the derivation of parameter ˜ a e ij between nodes i and j of a rectangular element e. The velocity is evaluated at the two nearest Gauss points with respect to an element side. The two velocity vectors are normalized and projected to this element side. Consequently, the value of higher magnitude is selected as the upwind parameter for this element side. This upwind parameter is used to modify the components of the weighting functions corres- ponding to nodes i and j in the direction of this element side. This method ensures that the residual of this ele- ment is weighted in the upstream direction. The extension to arbitrary quadrilateral elements is straightforward.

3.3 Fully-upwind Galerkin method FUG

Since W i ¼ N i holds for both methods, the Fully-Upwind Galerkin and the Standard Galerkin method, the respective transmissivity integrals are the same. For FUG, the mobility is fully upstream weighted, and it is equal to the value of the mobility at the upstream node, i.e. l FUG a , ij ¼ l a , i if g ij w a , j ¹ w a , i l a , j if g ij w a , j ¹ w a , i . 0 , a [ {w , n}: 23 It is also valid for the multidimensional case. In the one- dimensional case g ij ,i Þ j, is always positive. However, when discretizing a two-dimensional domain of interest, obtuse triangles and rectangles with a side length ratio of greater than  2 p should be avoided, since they lead to negative transmissibilities g ij which give rise to non- physical discrete fluxes and can cause an oscillatory behaviour in the Newton iteration Positive Transmissi- bility Condition 22 . A similar argument holds for the three-dimensional case. Because the Fully-Upwind Galerkin method uses a single-point quadrature rule to compute the integrals, it entails much less computational effort than the Petrov–Galerkin method which requires four Gauss points. Equation 17 in combination with a Standard Galerkin discretization represents a type of finite volume formulation. A fully upwind scheme utilizing the above formulation is called a Fully Upwind Control Volume Finite Element FU- CVFE method. 6 The Fully-Upwind Galerkin method FUG satisfies these conditions and subsequently belongs to this class of discretization methods see e.g. Helmig 23 . 4 DISCUSSION OF TWO-PHASE FLOW IN HETEROGENEOUS MEDIA Primary variables that show discontinuities at heterogene- ities are responsible for severe numerical difficulties. To illustrate this we will look at one-dimensional two-phase flow at the interfaces of two sands having different perme- abilities and porosities. The example considers flow in a porous medium structured into zones of coarse high perme- able sand sand I and fine low permeable sand sand II. The flow problem described by finite element formulation, eqn 8, requires two conditions for the transition from one sand to the other. The first condition is derived from the continuity equation and states that the flux from one side over an interface is equal to the influx over this inter- face into the other side, i.e. Q I a þ Q II a ¼ , a [ {w , n} 24 Fig. 2. Petrov–Galerkin method: Upwind coefficients in two dimensions for a rectangular element. Comparison of Galerkin-type discretization techniques 701 The second condition comes from the continuity of ‘inten- sive’ physical quantities, e.g. pressure. But this constraint is only valid for mobile phase pressures. Thus, capillary pressure is continuous across the interface of a hetero- geneity, unless one phase is immobile on one side. Only if both phases are mobile on each side of the heterogeneity interface does the following equality hold p I c ¼ p II c 25 Consider the one-dimensional displacement of a wetting phase water by a non-wetting phase DNAPL-dense non-aqueous phase liquid through a porous medium divided into three zones Fig. 3 where zones 1 and 3 con- sist of sand I and zone 2 of sand II. At the two interfaces of the configuration different processes take place. At the left interface the non-wetting phase passes a transition from coarse sand to fine sand which we refer to as case A. At the right interface the transi- tion occurs in the other direction from fine sand to coarse sand case B. It is possible to express capillary pressure as a function of permeability, porosity, and the J-Leverett function 24 in the following way p c ¼ j  f k r J S w 26 where j is the interfacial tension and k the scalar intrinsic permeability. The square root term reflects the mean pore diameter, and the J-Leverett function depends on the saturation and represents the lithology of the porous medium. Appropriate models of p c versus S w are the ones of Brooks and Corey BC 25 and van Genuchten VG. 26 The capillary pressure–saturation function of Brooks and Corey is given by p c ¼ P d S ¹ 1 l e 27 where S e ¼ S w ¹ S wr 1 ¹ S wr is the effective saturation, S wr the wetting phase residual saturation, and l the pore size distribution index of the corresponding sand. The model for the capillary pres- sure–saturation relationship of Brooks and Corey assigns a characteristic value, the so-called entry or bubbling pres- sure P d , to each sand. P d is the capillary pressure required for a non-wetting fluid phase to displace a wetting fluid phase inside the corresponding sand see the solid curves in Fig. 4. The material data of sand I and II are listed in Table 1. The lower curve corresponds to the coarse sand sand I, the upper curve to the fine sand sand II. Consequently for case A at the left interface the non- wetting phase is unable to enter the low permeability zone until the capillary pressure at the interface is greater than or equal to the specified entry pressure. This can only occur when the non-wetting phase saturation has increased up to a certain value S p n ¼ 1 ¹ S p w where the capillary pressure is Fig. 3. Displacement of water by DNAPL within a heterogeneous medium. Fig. 4. Capillary pressure–saturation curves. Table 1. Material properties and model parameters Unit Sand I Sand II Intrinsic permeability K m 2 5·04·10 –10 5·26·10 –11 Porosity f – 0·40 0·39 Wett. phase res. saturation S wr – 0·08 0·10 Non-wett. phase res. saturation S nr – 0·00 0·00 BC: entry pressure P d Pa 370 1324 BC: pore size distribution index l – 3·86 2·49 VG: a Pa -1 0·0023 0·0006 VG: n – 8·06 5·34 702 R. Helmig, R. Huber equal to the entry pressure Fig. 4, i.e. p c S p w ÿ ¼ P dII 28 As a consequence, the non-wetting phase is dammed up at the interface until the entry condition above is satisfied. At interfaces where case A occurs, discontinuities in the saturation appear. At the second interface case B the transition is from fine to coarse sand. In contrast to case A, no accumulation of the non-wetting phase takes place. This is because the capillary pressure at the interface is greater than the entry pressure of zone 3. As in case A, a jump in the saturation develops at the interface. This jump is caused by a ‘sucking’ effect from the zone of fine sand into the zone of coarse sand. The non-wetting phase saturation level within the fine sand zone is higher than in coarse sand. In contrast to the Brooks–Corey model, van Genuchten’s capillary pressure–saturation curves are continuous Fig. 4. They are given by the following equation p c ¼ 1 a S ¹ 1 m e ¹ 1 B 1 C A 1 n 29 where m ¼ 1 ¹ 1n. The parameters of the VG model are a and n. Eqn 29 shows that a is inversely proportional to P d , the entry pressure of BC. The pore size distribution is taken into account by parameter n. While low values of n i.e. n → 1 indicate soils with broad pore size distributions, higher n-values point to more uniform distributions. Remark: The main difference between the two models lies in the way they account for entry pressure effects. For water saturation values close to one, the van Genuchten model gives a very small capillary pressure which converges for increasing S w to zero. The model of Brooks and Corey with non-zero entry pressure P d yields, in contrast to the other model, values close to the entry pressure, which in some cases is of significant magnitude. In the van Genuchten approach, for any saturation on one side of the interface, there exists a corresponding saturation on the other side such that the capillary pressure is continous. If, however, for the Brooks–Corey approach the entry pressure is non- zero, in Fig. 4 solid line it can be seen that there is a specific saturation S p w such that continuity of capillary pressure can only be achieved if the wetting phase saturation on the side corresponding to the lower curve is below or equal to S p w . Lenhard et al. 27 investigated the relationship between these two formulations and derived a formula to convert the parameters of one model into parameters of the other model. The pore size distribution index of the Brooks– Corey model is derived in the following manner l ¼ n ¹ 1 1 ¹ ˜S 1 m e B 1 C A 30 where ˜S e is selected as the medium effective saturation within the range of S e . The entry pressure P d of the Brooks–Corey model is obtained by P d ¼ 1 a ˜S 1 l x ˜S ¹ 1 m x ¹ 1 B 1 C A 1 n 31 where ˜S x is a match-point effective wetting phase satura- tion, i.e. a value of S e for which the p c –S w curves of the two models are closest to each other for a selected norm. Since the relative permeability–saturation curves of the two models are essentially the same we will not consider them for our investigations. 5 COMPARISON OF THE DIFFERENT DISCRETIZATIONS The ability of the previously introduced Galerkin finite ele- ment discretizations to describe flow in heterogeneous media, especially at interfaces, will now be investigated. For the Brooks–Corey model, we define the total poten- tial of the non-wetting phase at node i by This formulation is compatible with the extended pressure condition see e.g. van Duijn 28 which states that capillary pressure must be continuous across an interface unless the non-wetting phase is immobile on the low permeability side. The non-wetting phase pressure is undefined if it is immo- bile. Hence p c ¼ p n ¹ p w is not meaningful and does not need to be continuous. Using for the BC model, eqn 32 instead of eqn 18 with eqn 4, we can define w n,j ¹ w n,i at an interface which constitutes a discontinuity in material properties within the porous medium. When at such a heterogeneity interface w n,j ¹ w n,i , i denoting the upstream node, is negative, the entry pressure has not yet been achieved see Fig. 5. However, if it becomes positive, the entry pressure has been achieved and an influx of the non- wetting phase into the low permeability zone takes place. w n , i : ¼ ˆp w , i þ ˆp c , i ¹ r n g ˆ D i if ˆS n , i . S nr non ¹ wetting phase is mobile ˆp w , i þ P d ¹ r n g ˆ D i if ˆS n , i S nr non ¹ wetting phase is immobile 32 Comparison of Galerkin-type discretization techniques 703 The discretizations for each method are graphically presented in Fig. 6, Fig. 7 and Fig. 8. The lower illustrations show the test and shape functions for each method. As mentioned in the previous section, the Standard Galerkin and Fully-Upwind Galerkin methods employ the same piecewise linear test and shape functions. In contrast, PG uses quadratic weighting functions eqn 20. In the upper illustrations, the discretized mobility of the non-wetting phase is presented for the situation when the non- wetting phase has reached the interface to the low perme- ability zone and the capillary pressure is smaller than the entry pressure. The following statements can be derived from Figs. 5–8: • At node i see Fig. 5 the gradients of the non- wetting phase total potential w n as defined by eqn 18 and eqn 32, respectively, are opposite for all three discretizations, i.e. w n , i ¹ w n , i ¹ 1 , 0 and w n , i þ 1 ¹ w n , i . 0 33 • Because physical properties are specified at nodes, an element having nodes which belong to different permeability zones can be considered as a transition element. For the Standard Galerkin method, the mobility of the non-wetting phase within this tran- sition element is non-zero. w n,iþ1 ¹ w n,i at the Fig. 5. Total potential of a non-wetting phase reaching an inter- face with a transition to a zone with a higher entry pressure. Fig. 6. Standard Galerkin discretization for a non-wetting phase at an interface constituting case A. Fig. 7. Petrov–Galerkin discretization for a non-wetting phase at an interface constituting case A. Fig. 8. Fully-Upwind Galerkin discretization for a non-wetting phase at an interface constituting case A. 704 R. Helmig, R. Huber interface within the transition element is positive Fig. 5. Since the discrete phase flow is propor- tional to the product of phase mobility l a and w n,iþ1 ¹ w n,i , the Standard Galerkin method produces a flow of the non-wetting phase from node i þ 1 to node i although at node i þ 1 the non-wetting phase is immobile or not present S n ¼ 0. This non-physical process produces a peak of the non-wetting phase saturation distribution at node i and probably a negative saturation at node i þ 1. The saturation level at the interface necessary for penetration of the non-wetting phase into the low permeability zone is achieved, at the same rate, or, sooner than it would be for the physical solution, because of an unphysical flow. This gives numerical results where the non-wetting phase reaches the entry pressure and enters a low permeability lens although physically this does not happen. The scheme implies a continuous distribu- tion of the saturation at the interface which yields in the case of the Brooks–Corey model a discontinu- ous approximation of the capillary pressure. • The test function of the Petrov–Galerkin method for the non-wetting phase equation at node i is simi- lar to the one for the Standard Galerkin method because the gradients of w n at this node are opposite Fig. 7. Thus the problems arising with the Petrov– Galerkin approach are comparable to the ones with the Standard Galerkin method. • Because the Fully-Upwind Galerkin method uses the fully upstream value of the mobility as defined in eqn 23, a flow of the non-wetting phase across the interface will not occur unless the pressure at the interface has reached the entry pressure. No unphysical flow occurs that could produce oscilla- tions in the numerical solution. At the interface, mobilities are approximated discontinuously, the capillary pressure continuously. To solve the system of non-linear equations eqn 17 for the corresponding discretization methods, a residual-based Newton–Raphson iterative concept is used. 6 NUMERICAL EXAMPLES 6.1 1-D problem in a homogeneous medium without capillary effects First we will show the performance of SG, PG, and FUG, with respect to flow in a homogeneous medium. The Buckley–Leverett problem 29 is a simple test problem without capillary pressure effects whose analytical solution is easily found and which is excellent for investigating the resolution of discontinuities for each method. A non-wetting phase displaces a wetting phase from left to right. The initial total velocity of the two-phase system, defined as the sum of the phase Darcy velocities, is 1·0 m s ¹ 1 , the ratio of the dynamic viscosities is one, residual saturations are zero and the Brooks–Corey function l ¼ 2·0 is used for the relative permeabilities. A space–time discretization of Dx ¼ 0·025 m and Dt ¼ 0·005 s is chosen. Fig. 9 shows the saturation profiles at time t ¼ 0·4 s for the three methods and the analytical solution. The SG method does not converge to the analytical solution because only numerical schemes using a form of upwind technique are able to capture correctly convection-dominated processes. 30

6.2 1-D problem in a heterogeneous medium with capillary effects