A Numerical Simulation Framework for the Design, Management and Optimization of CO 2 Sequestration in Subsurface Formations

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1 A Numerical Simulation Framework for the Design, Management and Optimization of CO 2 Sequestration in Subsurface Formations Investigators Hamdi Tchelepi, Associate Professor of Petroleum Engineering; Lou Durlofsky, Professor of Petroleum Engineering; and Khalid Aziz, Professor of Petroleum Engineering. Introduction The objective is to develop a numerical simulation framework for modeling CO 2 sequestration operations in large-scale heterogeneous formations, such as deep saline aquifers and depleted oil and gas reservoirs. This framework is constructed based on in depth understanding of the physics in the parameter space of interest for geologic sequestration of CO 2, accurate and computationally efficient numerical algorithms for modeling of highly detailed descriptions of large-scale porous formations, and a flexible extensible computational framework for the optimization of real CO 2 projects during both the injection and long term storage periods. The work described in our proposal was divided into several interrelated items: (1) detailed understanding of the physics, (2) general AIM (adaptive implicit method) formulation, including the development and implementation of new physical models, (3) multiscale finite-volume simulation, and (4) optimization under uncertainty. This annual progress report is divided into six projects: (1) stability of two-phase vertical flow in homogeneous porous media, (2) gravity currents in horizontal porous layers, (3) construction of high order adaptive implicit methods, (4) algebraic multiscale modeling of compressible flow, (5) a stochastic particle framework for nonlinear two-phase flow, and (6) development of basic CO 2 sequestration modeling capabilities in Stanford s general purpose research simulator. These projects relate to different aspects of the numerical framework we are building. The first two projects are aimed at improved understanding of the physical mechanisms that dictate the behavior of CO 2 plumes in porous media. The third and fourth projects are concerned with the development of accurate and scalable (efficient for large problems) numerical methods for modeling the complex behavior of CO 2 in multi-component, multiphase porous media flow settings. The fifth project relates to a new stochastic framework for modeling nonlinear multiphase flow across multiple scales (e.g., pore, Darcy, upscaled Darcy), accounting for the possibility that the physics at the different scales may be described by different equation sets. Building up the basic CO 2 modeling capabilities of our General Purpose Research Simulator (GPRS) is the subject of the sixth sub-project. In the first project, the evolution of a CO 2 plume that is immiscible with the resident brine is studied. The differences in density and viscosity between the CO 2 and GCEP Technical Report

2 the brine can lead to flow instabilities. This complex problem is analyzed using linear stability analysis and high accuracy nonlinear simulations. In the second project, the long term fate of gravity currents of a CO 2 plume in the vicinity of the injection well that is ponded against the top boundary of a two-dimensional horizontal aquifer is investigated. The similarity analysis yields three time scales of interest in the design and monitoring of CO 2 sequestration projects in deep saline aquifers of large extent: a self similar early time regime with the gravity current spreading horizontally with a speed proportional to the square root of time, and a late-time self similar propagation regime with slower (proportional to t 1/3 ) asymptotic propagation speeds. In between the two self similar regimes lies a transition period. These time scales can vary over a wide range depending on the initial plume size and the viscosity ratio. The adaptive implicit method (AIM) is central to our GPRS-based simulation framework. AIM is a valuable tool for enabling accurate and computationally efficient modeling of CO 2 sequestration processes for both the injection and post-injection (long term) phases. It is believed that accurate simulation of the physics governing subsurface CO 2 sequestration requires solving coupled nonlinear conservation equations for highly detailed descriptions of the properties of the target porous formation. For such models, AIM helps to allocate the computational resources when and where necessary. However, standard AIM is a low-order discrete approximation. Specifically, it is first order in both space and time. We are interested in high fidelity numerical solutions of CO 2 related processes in large-scale domains of practical interest. As a result, we are developing an AIM method that is high order (at least second order) in both space and time. We have encountered several challenges. In this report, we provide a detailed analysis of AIM schemes, and we discuss the progress made to date. In the fourth project, we describe the extension of the multiscale finite-volume method for more complex flow processes. The focus so far has been on handling the compressible multiphase flow problem in strongly heterogenous domains. The new algebraic framework accommodates the presence of strong compressibility effects in a natural manner. Moreover, this operator base multiscale method (OBMM) is a promising platform for including additional flow mechanisms (e.g., capillarity, buoyancy) as well as for solving problems on unstructured grids. In collaboration with researchers at ETH, Zurich, we are developing a new particle-based framework for nonlinear multi-phase flow in porous media. This development is consistent with our long term objective of developing accurate multiscale, multi-physics techniques that accurately represent the complex flow behaviors associated with subsurface injection and storage of CO 2. Finally, in the sixth project we provide an update on our effort to build up the basic CO 2 modeling capabilities in GPRS. Specifically, we report on the latest implementation of hysteresis models as well as fast phase behavior computations for simple two-phase systems. GCEP Technical Report

3 Stability of two-phase vertical flow in homogeneous porous media Investigators Hamdi Tchelepi (Associate Professor of Petroleum Engineering), Amir Riaz (Post-Doctoral Researcher) Introduction Two phase flow in porous media with gravity oriented in a direction parallel to the mean flow is of fundamental importance for CO 2 sequestration in saline aquifers. The physical nature of viscous and buoyancy related instabilities of two-phase, immiscible, vertical flows in porous media is not completely understood. The linear stability theory for immiscible flows in porous media is well known for neutrally buoyant flows [33, 63, 51]. Nonlinear characteristics of such flows have recently been investigated by Riaz and Tchelepi [52]. The behavior of the viscous instability for immiscible flow in the presence of density differences, when gravity is perpendicular to the mean flow direction, has also been analyzed by Riaz and Tchelepi [52]. What is required in terms of CO 2 sequestration is a thorough understanding of the fate of the CO 2 plume during the injection and the post injection periods. The case of the late time miscible, gravity driven flow has recently been analyzed [49]. Figure 1 shows a CO 2 plume accumulated above the horizontal injection well. The top region of the plume is subjected to an unstable density contrast because of higher brine density. The viscosity of brine is also higher, therefore, the plume is also viscously unstable. In this report we focus on the characterization of instability at the plume-brine interface in the top region. For this purpose we employ the well known characterization of vertical flows with shocks located at each end of an expansion wave [48, 22]. We do not take into account the dynamics of the lower section of the plume where trapping may occur. The development of shocks in the top region of the flow is due to the fractional flow function, which depends on both gravity and viscosity differences [58]. The shocks at both ends of the spreading wave are regularized by capillary dispersion [62, 60]. A sketch of the saturation distribution in Fig. 2 shows the case of injection from the top, which is exactly equivalent to that of injection from the bottom when density values are reversed. Two shock fronts develop that move in opposite directions. It is important to note that the rear saturation front will move in the upward direction even in the case of downward injection. This phenomenon, which is related to the fractional flow function discussed in section 2, is due to the macroscopic representation of microscopic pore scale flow; it allows the rarefaction wave to spread in the upward direction against the mean flow even for a 1D problem [36]. The saturation gradients are relaxed around the shock due to capillary dispersion. Instability can occur at both shock locations with distinct characteristics. Unlike single phase miscible flows [40], the two-phase immiscible flow can be unstable both at the front and back ends. Contrary to the usual behavior observed for neutrally buoyant displacements, the GCEP Technical Report

4 Figure 1: Sketch of the CO 2 plume during the initial injection period into a brine aquifer. At the top boundary, the plume is unstably stratified because the brine is heavier than injected CO 2. The plume is also viscously unstable due to its lesser viscosity. growth rate of instability at the front end decreases with an increase in the mobility ratio. We also report an unexpected increase in the preferred mode of instability, at the back end, with an increase in the mobility ratio. A thorough understanding of these anomalous behaviors is clearly necessary and has not been developed previously. A high accuracy numerical method is employed to test the predictions of the linear stability analysis as well as to understand the late time fully developed nonlinear profiles. Numerical simulations support the validity of analyzing the linear stability behavior independently at each shock location, and they confirm the preferred modes and growth rates given by the linear analysis. Governing equations The dimensional equations governing two-phase flow in porous media can be expressed [45] as φ S i t u i = k k ri µ i ( P i ρ i g z), (1) = u i, (2) where, subscript i represents the wetting and the non-wetting phase, as w and n respectively. u is the Darcy velocity. All lengths are scaled with the domain width W, shown in Fig. 2. The wetting phase saturation, 0 < S w < 1, is scaled with (1 s rn s rw ), where s rw and s rn are the residual saturation values of the wetting and the non-wetting phase, respectively. Relative permeability functions, k ri, are scaled with respective end point values, k rw (1) = k rw (S w = 1) and k rn (0) = k rn (S w = 0) Absolute permeability, scaled with its average value, is k. Velocity is scaled with injection velocity U. Time is scaled with W φ (1 s rn s rw ) /U. Pressure is scaled with k/µ i U. By using the total velocity, u = u w + u n, and capillary pressure, P c = P n P w, we can express the dimensionless governing equations in terms of GCEP Technical Report

5 w z g,u injection velocity U s=1 ρ 1 µ 1 v sb z=0 s sf 1-s sb rarefaction region H v sf ρ 2 µ 2 s=0 dispersed shock regions s x Figure 2: Sketch of a 1-D saturation profile is shown in the left plot, where we take ρ 1 > ρ 2. The right plot shows the 2-D configuration in x-z plane. Both the injection velocity, U, and gravity points in the positive z direction. Shocks, produced at both ends of the rarefaction wave, move with speeds v f in the downward and v b in the upward, directions. The shocks are relaxed due to capillary dispersion. The saturation value at the front shock is s sf and that at the back end shock is s sb. Shock saturations and the shock speeds are determined by the fractional flow function. wetting phase saturation and pressure as u = 0, (3) ( S w Mkrw = u + G k rwk rn z + k ) rwk rn dp c S w, (4) t λ T λ T Caλ T ds w u = λ T M P w + G k rn M z k rn dp c S w, (5) M Ca ds w where the total mobility function is λ T = Mk rw + k rn. (6) Three dimensionless parameters appear in the above equations, the mobility ratio M, the capillary number Ca and the gravity number G, which are defined as M = µ n k (1) rw µ w k (0) rn Ca =, (7) µ w UW γ nw (1 s rn s rw ) G = k (ρ w ρ n ) g µ w U φ k, (8). (9) Based on previous work, [63, 51] we choose dp c /ds w = 1/k rw k rn. Relative permeability functions are specified as [29, 55] k rw = S 4 w, (10) k rn = erf (1 2S w ), (11) GCEP Technical Report

6 (a) 2 (b) 2 rarefaction region s sb s sf vsb λ T k r λ T k rn k rw f w v sf G=20 G= S w S w Figure 3: (a) Relative permeability functions for the wetting (k rw ) and the non-wetting (k rn ) phase, obtained respectively from Eqs. (10) and (11), and the total mobility function, λ T, given by Eq. (6) for M = 2. (b) The fractional flow functions according to Eq. (13) for M = 2 at G = 10 and G = 20. Shock speeds are given by the slopes v sf and v sb, which also determine the shock saturations s sf and s sb. An expansion wave is formed for s sf < S w < s sb. Variation in G changes the shock speeds as well as the shock saturation. The latter alters the λ T profile across the shock. where erf() is the error function. k rn will be normalized to vary between 1 and 0 for 0 < S w < 1. A one-dimensional form of Eq. (4), with constant velocity in the z-direction, w = 1, is given by S w t = z ( Mkrw λ T + G k rwk rn 1 ) S w λ T Caλ T z. (12) In the absence of capillary dispersion (Ca ) one can define a fractional flow function as f w = Mk ( rw 1 + G k ) rn. (13) λ T M The solution to the Riemann problem of Eq. (12) can be obtained by the method of characteristics [16, 36, 11, 57]. Both the shock saturation s s and the shock speed v s, for both the front and the rear shocks, respectively, can be obtained from v sf = df w ds w = 1 {f w (s sf ) f w (0)}, (14) Sw=ssf s sf v sb = df w 1 ds w = {f w (1) f w (s sb )}. (15) Sw =s sb 1 s sb Equation (14) gives the positive speed of the front shock v sf, while Eqn. (15) gives the negative speed of the rear shock v sb. Instability is governed by the ratio of λ T values across the shock, rather than the end-point value M. The shock mobility ratio, GCEP Technical Report

7 λ Ts, can be defined for the front shock as λ T (s sf )/λ T (0) and for the rear shock as λ T (s sb )/λ T (1). Figure 3(a) plots the relative permeability curves according to Eqs. (10) and (11), as well as the total mobility function, λ T, given by Eq. (6) for M = 2. The corresponding fractional flow functions at G = 10 and G = 20, are plotted in Fig. 3(b). Tangents drawn on the f w profile, give the shock speeds at the front and back ends, respectively, as the slopes v sf and v sb. The intersection of the tangents with f w gives the shock saturations s sf and s sb [59]. A one-dimensional flow profile, based on the fractional flow plotted in Fig. 3(b), is sketched in Fig. 2 for a finite value of Ca. Linear stability analysis We will use Eq. 12 as the base state to compute the growth rate of linearized perturbations in the transverse x-direction. Riaz and Tchelepi [51] show that instability occurs only at the shock location while the rarefaction region is stable. Moreover, the saturation gradient due to capillary dispersion is steady at the shock front [60]. Hence, stability of the base state can be analyzed individually for both shocks, with saturation values 0 < S w < s sf and s sb < S w < 1, respectively, in a frame of reference, ξ = z v s t, moving with respective shock speeds, v sf and v sb. The steady state base profiles are given by ds o dξ ds o dξ = λ T Ca (f w v sf s o ) s o {0, s sf }, (16) = λ T Ca (f w + v sb s o ) s o {s sb, 1}, (17) where ξ in each case is the reference frame moving at the corresponding shock speed. Equation (1), in terms of the wetting phase velocity, along with Eqs. (3) and (5) are used for the linear stability analysis. Since stability properties scale linearly with the capillary number [51], all results for the stability analysis will be presented for Ca = 1. Saturation and pressure variables are expanded in terms of base state and perturbation components. The latter are further decomposed into streamwise eigenfunctions and normal modes in the transverse direction (S w, P ) (x, ξ, t) = (s o, P o ) (ξ) + ( ŝ, ˆP ) (ξ) exp (i n x + σ t), (18) where n is the wavenumber and σ the growth rate. Equations for linearized perturbations [51] can be expressed as ( d d k ˆP ) rwo dξ dξ + dp o dξ k r wo ŝ + v s ŝ ( d d λ ˆP To dξ dξ + λ dp o T o dξ ŝ 1 Ca k r wo ds o dξ ŝ 1 Ca k rwo n 2 k rwo ˆP = σŝ,(19) ) dŝ dξ G k r wo ) +n 2 ( 1 Ca k rwo ŝ λ To ˆP = 0, (20) where v s is the speed of the shock at which the problem is being solved. Primes denote derivatives with respect to the base saturation s o and subscript o represents base state GCEP Technical Report

8 (a) 2 M= 1/ (b) 3 M= 1/ x 10-1 σ 1 σ x n x n x 10-1 Figure 4: Growth rate vs. wavenumber curves for different value of M at G = 20, (a) front end, (b) back end. At both ends, the maximum growth rate decreases with an increase in M while the maximum wavenumber increases. variables. Perturbations are prescribed to decay far from the shock location. Details on the numerical solution of the above algebraic eigenvalue problem can be found in [50] and [51]. Figure 4 plots the unstable growth rate σ as a function of the wavenumber n for different values of M at G = 20. Instability related to the front and back ends, is shown respectively in Fig 4(a) and 4(b). At both ends, the maximum growth rate decreases with an increase in M. While this behavior is expected at the back end, where increased stabilization should occur for larger values of M, it is surprising to find σ max decreasing with M at the front end since larger values of M are generally associated with higher levels of instability. Similarly, the most dangerous wavenumber, n max, is observed to increase with M at the front end as expected but again, contrary to expectation, increases at the back end. The magnitude of the maximum growth rate as a function of M at various values of G is plotted for the front end in Fig. 5(a) and for the back end in Fig. 5(b). As M is increased at the front end from 1/100, there is first an increase in σ max, followed by a decrease. Finally, σ max increases again for large values of the mobility ratio. This non-monotonic behavior as a function of the mobility ratio, occurs only for gravitationally unstable flows and becomes more pronounced for larger values of G, as shown in Fig. 5(a). The influence of G is observed to diminish for larger mobility ratios and the maximum growth rate approaches the value of the G = 0 case. The maximum growth rate at the back end, shown in Fig. 5(b), also displays a nonuniform behavior as a function of the mobility ratio for all values of G. It increases for small values of M and decreases, as expected, for larger values of M. Contrary to the front end, σ max at the back end increases uniformly as a function of G for all values of M. GCEP Technical Report

9 (a) (b) G= σ max σ max G= M M Figure 5: Maximum growth rate, σ max, as a function of M for different values of G at (a) the front end, (b) the back end. 3 M=50, G=0 M=50, G=10 λ T ρ 2 1 M=10, G=0 M=10, G=10 M=5, G=10 density ξ Figure 6: Total mobility at the front end (solid lines), for different values of M and G. Dashed line shows the saturation distribution, since density varies linearly with saturation, it gives the density profile across the front. λ T profile is monotonic for large M and non-monotonic for small M. The minimum value of λ T for non-monotonic profiles decreases with M. GCEP Technical Report

10 Physical mechanisms Based on the behavior of the G = 0 case shown in Fig. 5(a), one expects σ max at the front end to increase uniformly as the mobility ratio increases. However, when G > 0, this occurs only at large values of M and small values of G. At intermediate values of M, we observe a relatively flat profile of σ max for small G, while a decrease in σ max occurs for large G as M is increased in this range, as shown in Fig. 5(a). These unexpected behaviors can be understood by considering the total mobility profiles across the front. Figure 6 shows in solid lines, the total mobility profiles at the front end for several values of M and G. The saturation profile is shown with the dashed line. Since the density at a given location is the sum of densities of the wetting and the non-wetting fluids, it varies linearly with s and therefore takes the same profile across the front as the saturation. The saturation is normalized to vary between 0 and 1 for all cases. At M = 50 and G = 0, λ T increases uniformly from 1, at s = 0, to its maximum value, at s = 1. On the other hand, a non-monotonic profile of λ T is obtained when M = 10. Note that an increase in G lowers the value of λ T at s = 1 for both M = 10 and M = 50. The minimum value of the total mobility, λ Tmin, for the non-monotonic profile at M = 10, is however, not affected by a change in G. λ Tmin for non-monotonic profiles is only a function of M and is further reduced for the M = 5 case. Total mobility across the shock is a function of both M and the shock saturation. Since the shock saturation depends on G, as given by Eq. (13), λ T profiles in Fig. 6 are different for different G. In order to understand the stability characteristics in light of the above observations, we consider the perturbation vorticity eigenfunctions, where the vorticity, ω, is given by ω = w x u ξ. (21) The stability equations, Eqs. (19) and (20), are solved for saturation and pressure eigenfunctions. Velocity eigenfunctions can be obtained from ( ) 1 u = i n λ To ˆP Ca λ w o P c o ŝ e i n x, (22) { d w = λ ˆP To dξ 1 Ca λ w o P c dŝ o dξ + dp o dξ λ T o ŝ 1 ( ) λwo P c ds o Ca o dξ ŝ + Gλ w o ŝ } e i n x, (23) where primes denote derivative with respect to base-state saturation s o, subscript o denotes base-state variables and i is the imaginary number 1. The real part of ω, with u and w given by Eqs. (22) and (23) respectively, is the two-dimensional disturbance eigenfunction. Contours of real values of ω are plotted in Fig. 7. Negative values of vorticity are shown with dashed lines. Figure 7(a) plots the case with monotonic total mobility GCEP Technical Report

11 (a) -2 (b) -2 (dλ T /dξ) min (dλ T /dξ) min -1-1 ξ ξ 0 0 (dρ/dξ) min (dλ T /dξ) max (dρ/dξ) min 1 x 1 x (c) (d) ξ ξ x 1 x Figure 7: Two-dimensional vorticity eigenfunctions at the front end given by Eq. (21) for different values of M and G. Negative values are shown in dashed lines. (a) M = 50, G = 0, (b) M = 10, G = 0, (c) M = 10, G = 10 and (d) M = 5, G = 10. Horizontal lines give the location of maximum spatial gradients of total mobility and density. GCEP Technical Report

12 profile, as shown in Fig. 6 for M = 50 and G = 0. Vortices with alternating signs appear in a single row along the front that push the more mobile fluid into the less mobile fluid, thereby creating an instability. The location of the maximum value of the eigenfunctions coincides with the location of the minimum gradient of λ T as shown by the horizontal line in Fig. 7(a). When G is increased to 10 at M = 50, the vorticity structure remains more or less unchanged but the location of the minimum gradient of density, shown by the horizontal line towards the bottom in Fig. 7(a), does not coincide with the location of the eigenfunction maximum. Therefore, the density contrast is unable to substantially influence the overall instability. As a result, Fig. 5(a) shows that the growth rate increases by a relatively small amount when G is increased from 0 to 10 at M = 50, compared to smaller values of M. When the total mobility profile is non-monotonic, e.g. at M = 10 and G = 0 as shown in Fig. 6, the vorticity eigenfunction shows two rows of vortices in Fig. 7(b). Vortices in the top row again have an elongated structure, while the bottom row vortices are symmetrically distributed around their maximum value. The location of the eigenfunction maximum coincides with the locations of (dλ T /dξ) min and (dλ T /dξ) max, shown by the two horizontal lines in Fig. 7(b). The top row of vortices is destabilizing, pushing high mobility fluid into the less mobile fluid, while the bottom row, which transports low mobility fluid to high mobility region, is stabilizing. Hence we note, that the non-monotonic mobility profile results in a vorticity structure, that simultaneously creates as well as counters instability, similar to that for miscible flows [39, 44]. When the vorticity at the top is stronger than that at the bottom, the flow is unstable, otherwise it is stable. Figure 7(c) once again shows the eigenfunction structure for M = 10, but now with a density contrast at G = 10. The corresponding λ T profile in Fig. 6 shows that λ Tmin remains constant, compared to the G = 0 case, while λ T (s = 1) is about the same as λ T (s = 0). Hence, viscous destabilization is neutralized because a non-monotonic mobility profile with equal end-point values gives rise to two rows of vortices which have approximately equal magnitudes (Fig. 7(c)). Both rows of vortices, however, produce buoyancy related instability by pushing the heavier fluid down into the lighter fluid. Moreover, the location of (dρ/dξ) min now coincides with the location of the maximum value of the bottom row of vortices, shown in Fig. 7(c). Hence, these vortices, although stabilizing in terms of the mobility variation, are more effectively destabilizing in terms of the density contrast. The top row of vortices, while also being gravitationally unstable, has a smaller influence by being located away from the minimum gradient of density. Therefore, although the increase in G results in lowering the value of λ T (s = 1) by roughly the same factor for both M = 10 and M = 50, as shown in Fig. 6, a much larger increase in the growth rate is produced at M = 10 (Fig. 5(a)) because the eigenfunction maximum coincides with (dρ/dξ) min. Vorticity contours related to the case M = 5 and G = 10 are plotted in Fig. 7(d). Corresponding λ T profile in Fig. 6 shows that λ Tmin is smaller as compared to the M = 10 case at G = 10. Also, λ T (s = 1) < λ T (s = 0), which denotes a stable flow in terms of the mobility contrast. Consequently, the bottom row of vortices is dominant in Fig. 7(d), which reflects a stronger stabilizing influence due to the more favorable mobility contrast, compared to the M = 10 case. However, an increase in the vorticity GCEP Technical Report

13 (a) M=10 M=50 M=1 (b) ω 10 0 Σ t DNS linear stability t Figure 8: (a) Evolution of the vorticity norm, ˆω, and (b) the vorticity growth rate, Σ, obtained from the numerical simulation for various values of M at G = 20 and Ca = 100. Dashed lines show corresponding linear stability results. A good match is observed between the two. strength at this location, which coincides with the position of (dρ/dξ) min, results in a stronger gravitational instability. Compared with the M = 10 case in Fig. 7(c), the destabilizing influence due to density is reduced at the top row but increases at the more effective, bottom row of vortices. Therefore, although the mobility contrast decreases, compared to the M = 10 case, the higher influence of buoyancy driven instability at M = 5, results in a relatively small decrease in the growth rate, as shown in Fig. 5(a). Numerical simulations The linear stability analysis is expected to be valid for practical displacements only in the limit of short times. The most dangerous mode will grow to large amplitude unstable fingers at a later time where the dynamics are governed by nonlinear mechanisms. We therefore employ high accuracy numerical simulations both as a check on the accuracy of predictions of the linear stability theory as well as to understand nonlinear mechanisms. Details of the numerical method are given in [52]. In order to determine the rate of growth of initial disturbances imposed on the vorticity field, we need to measure the rate of increase of some quantity obtained from the numerical simulation. Based on our previous analysis[52], the norm of the vorticity magnitude gives a reasonable estimate of the growth rate. Figure 8(a) provides the vorticity norm in a semi-log plot. Results for three mobility ratios at G = 20 and Ca = 100 are shown where s rw = s rn = 0.2. The highest magnitude is attained for the M = 1 case. The corresponding, measured, growth rate, Σ, plotted in Fig. 8(b), shows that the rate of growth increases quickly to the value given by the linear stability analysis, shown by the dashed line. The agreement between the linear stability and the numerical simulations results is quite good. The sudden decay in the magnitude of Σ is related to nonlinear saturation which occurs when the unstable fingers begin to grow to a larger amplitude. Fig. 8(b) further shows that the growth rate for the M = 1 is the highest while M = 50 and M = 1 cases have lower growth rates, in accordance with GCEP Technical Report

14 Figure 9: Saturation contours for the full problem for (a) M = 0.6, G = 10, t = and (b) M = 1.57, G = 20, t = at Ca = 400 and A = 2. Unstable fingers are localized at both ends and evolve independently without any interaction from the fingers at the other end. The number of fingers is consistent with the linear stability analysis which justifies the approach of isolating the linear analysis at each end. the results of the linear stability analysis. Hence, the numerical simulations appear to support the existence of the nonuniform behavior of the growth rate as a function of M. Below we present saturation profiles obtained from nonlinear simulations. Figure 9(a) shows the saturation contours of the full problem at an early time of t = for G = 10 and M = 0.6, while Fig. 9(b) plots the contour for G = 20 and M = 1.57 at t = Both profiles are obtained at Ca = 400, A = 2, s rw = 0.3 and s rn = 0.3 with F1 relative permeability functions. Different values of M are selected for each G to obtain the same shock speed for both ends. Figure 9 shows that instability appears in the form of small fingers that are clearly localized at individual fronts. The number of fingers given by the linear stability analysis for the G = 10 case are 18 and 11 while for the G = 20 case they are 36 and 32, at the front and the back end, respectively. Figure 9 shows the number of fingers that develop in the numerical simulation are approximately consistent with that prediction. Hence, the assumption of instability localized at the two fronts, in order to carry out the linear stability analysis independently for each, appears to be justified in view of our numerical simulation of the full problem. The structure of the unstable fingers at a later time is plotted in Fig. 10 for the same parameters as in Fig. 9. The G = 10 case is shown at t = while the G = 20 case is shown at t = The front reaches the vertical boundary earlier for the G = 20 case because of a higher shock speed as well as larger growth rate. Notice that the number of fingers is reduced significantly at both ends, for both cases, as compared to an early time shown in Fig. 9. The unstable fingers have a structure quite similar to that of the neutrally buoyant case discussed in [52]. The mechanisms of finger interaction and collapse, discussed in detail by these authors, appear to be also applicable to the current problem. GCEP Technical Report

15 Figure 10: Saturation contours for the full problem for (a) M = 0.6, G = 10, t = 0.15 and (b) M = 1.57, G = 20, t = 0.08 at Ca = 400 and A = 2. Unstable fingers are have grown to relatively larger amplitudes by this late time and are also reduced in number. The G = 20 case shows more fingers with a larger amplitude then the G = 10 case. Conclusions Immiscible, two-phase, vertical displacements in porous media, for the case of unstable density stratification, give rise to shocks separated by a rarefaction wave. The dynamics of the top region of the CO 2 plume during the injection period can be modeled by such an approach. We have examined the instability characteristics at both the front and back ends. We have shown by numerical simulations that the results obtained from the linear analysis are valid for nonlinear displacements. The linear stability analysis, carried out independently at each shock location, makes the assumption that the presence of the neighboring shock, as well as the rarefaction wave, does not influence the results. This fundamental assumption is validated by numerical simulations. Our linear stability results show that both the maximum growth rate and the most dangerous mode tend to become independent of the gravity number, for large values of the mobility ratio at the front end and for small mobility ratios at the back end. The nonuniform dependence of the maximum growth rate as a function of M at the front end occurs for all values of G. The total mobility profile is always monotonic across the back end while across the front end, M determines whether it is monotonic or non-monotonic and sets the minimum value of λ T, in the latter case. At the front end, in the case of monotonic λ T, a single row of vortices is concentrated at the location of (dλ T /dξ) min for all G. Therefore, gravitational instability is less severe compared to the case of non-monotonic λ T, where it is more effective through the row of vortices located at (dρ/dξ) min. We therefore observe that increasing values of G increase the growth rate more efficiently for smaller values of M. GCEP Technical Report

16 Interesting dynamics occur at the front due to the non-monotonic mobility profile, where we find that the strength of stabilizing vortices, positioned at (dρ/dξ) min, increases for smaller values of M and λ Tmin. However, stronger, stabilizing vortices at smaller values of M, are more unstable gravitationally, for the same density contrast. Due to this competing influence, an increase in M at intermediate values, although producing a smaller stabilizing effect, weakens the gravitational instability. Therefore, only a small increase in the growth rate is observed for small G, and a decrease in the growth rate occurs for large G, when M is increased. The increasing effect of gravitational instability with decreasing values of M, however, reverses when M is sufficiently small. Our analysis shows that during CO 2 injection, moderate values of M and G that are expected will not lead to a substantial instability. However, the level of instability will definitely be higher then that of the neutrally buoyant case. Given a density difference, even a small decrease in the viscosity contrast, for example as a function of temperature, will substantially increase instability. Moreover, a decrease in density due to the variation in temperature as the plume moves away from the injection site, will also enhance instability. Future Plans The stability analysis and numerical simulations presented here describe the first order response to disturbances in a homogeneous porous media. The influence of spatially varying permeability in real systems can be explored based on this analysis. Another direction in which this study can be extended is by investigating three dimensional displacement processes. Moreover, the stability characteristics related to a stationary interface and zero injection velocity are of significant importance during the post injection period of CO 2 sequestration. Modeling of residual trapping at the trailing end of the rising plume also represents a challenging problem. These effects will be explored in future investigations. GCEP Technical Report

17 Gravity currents in horizontal porous layers: Transition from early to late self-similarity Investigators Hamdi Tchelepi (Associate Professor of Petroleum Engineering), Franklin M. Orr, Jr (Professor of Petroleum Engineering), Brian Cantwell (Professor of Aeronautics and Astronautics), Marc Hesse (Graduate Research Assistant) Introduction When CO 2 is injected into an aquifer, typically at a depth greater than 800 m, it forms an immiscible CO 2 -rich vapor phase. In the temperature and pressure range encountered in geological CO 2 storage, the density of the CO 2 -vapor is less than the density of the brine [5]. The buoyant CO 2 -vapor will spread underneath the seal and along the top of the aquifer as a gravity current. An estimate of the area invaded by CO 2 -vapor is of great interest during site selection and subsequent monitoring efforts. The distribution of the fluids at the end of the injection period provides an initial condition for the long term evolution. Therefore we are interested in the evolution of a finite release of fluid into a saturated porous medium, with a particular initial distribution. For simplicity we study the release of fluid into a two-dimensional horizontal porous medium saturated by an ambient fluid. The detailed discussion of the work summarized below can be found in [27]. Problem statement Governing equation We consider the flow of fluid 1 with density ρ 1 = ρ and viscosity µ 1 and of fluid 2 with density ρ 2 = ρ + ρ and viscosity µ 2 in a horizontal porous layer of thickness H and infinite lateral extent. We assume that the porous medium is homogeneous and isotropic with permeability k and porosity φ, and is bounded on the top and the bottom by an impermeable boundary. The fluids are separated by a sharp interface, we denote the thickness of each fluid by h p (x, t) where p {1, 2}, so that h 1 (x, t) + h 2 (x, t) = H. We consider an aquifer with a large aspect ratio (length/height 1), so that we can assume hydrostatic pressure in both fluids, and we neglect the vertical velocity. An asymptotic analysis of this simplification has been presented in [61]. In this case the volume flux per unit width q p of phase p is given by Darcy s law q p = kλ p p/ x, where we introduce the mobility of phase p defined as λ p = 1/µ p (p {1, 2}). The flow rate per unit width of phase p is given by Q p = h p q p. ( pi Q 2 = h 2 λ 2 x g (ρ + ρ) h 2 x ( pi Q 1 = h 1 λ 1 x + gρ h 1 x In the absence of a source term the global conservation of volume is given by Q 2 + Q 1 = 0. Using this constraint we can eliminate p I / x from the expressions for ). ) GCEP Technical Report

18 a) plane of symmetry x f ~ t 1/2 early similarity solution early time b) interface detatches back-propagation time c) non-self-similar transition intermediate time d) x f ~ t 1/3 late similarity solution late time Figure 11: The evolution of buoyant CO 2 -vapor (gray) released into a horizontal porous layer saturated by brine (white). the flow rates and obtain Q 2 = Q 1 = g ρ h 2λ 2 h 1 λ 1 h 2 λ 2 + h 1 λ 1 h 2 x. To obtain an equation for the evolution of the interface, we consider the conservation of the volume of fluid p over region x and time t. The change in volume V p is given by V p = h p xφ = ( ) Q p x Q p x+ x t. Introducing two parameters, the diffusivity of the released fluid p given by κ p = kg ρλ p /φ and the mobility ratio M p = λ p /λ q, we obtain h p t = κ p x ( hp (H h p ) h p (M p 1) + H ) h p. (24) x This choice of parameters allows a simple reduction of (24) in the limits max(h p [x, t]) H or M p 1, as discussed in. Initial and boundary conditions We consider the evolution of the interface after injection has stopped (t t 0 ). Figure 12 illustrates the initial condition with CO 2 injection into a saline aquifer, but GCEP Technical Report

19 0 L d x o,0 x plane of symmetry H fluid 1 fluid 2 h 0 h * 0 x i,0 h 1 L f Figure 12: The geometry of the initial condition and the three associated length scales: the height of the layer H, the distance of displacement L d, and the width of the front L f. A particular initial condition h 0 [x] is shown as a solid line, where the inward propagation tip position is initially x i,0, and the outward propagating tip position is x o,0. The idealized step function initial condition h 0 [x] is shown as dashed line, in this case L d = x i,0 = x o,0. all arguments also hold for injection of a dense fluid. Near the injection site the gas has completely displaced the water over an average distance of L d. The gas-water interface transitions from h 0 [x i,0 ] = H to h 0 [x o,0 ] = 0 over a frontal region of width L f = x o,0 x i,0. The length scale L d is chosen so that an idealized step function initial profile located at x = L d has the same gas volume as the particular initial condition (2φL d H = V ). The idealized initial condition is { H, for x h Ld ; 0[x] = 0, for x > L d. The two boundary conditions for (24) require that h[x, t] 0 for x. Self-similar solution at early times For L f < L d the fronts are initially separated, and will evolve independently until their inward propagating tips collide (figure 11b). Each front can be analyzed in isolation, and it is convenient to shift it to the origin (ˆx = x L d ). The problem defined above has three dimensions: the length L, the height H, and the time T. The dimensions of the variables and parameters appearing in (24) and the initial condition are [ĥ] = [H] = H, [ˆx] = [L f ] = L, [t] = T, [κ] = L 2 H 1 T 1, [M] = 1. We choose ˆx, L f, and M as dependent parameters. The independent parameters give the length scale l = (κht) 1 2, which we use to obtain the following dimensionless parameters Π = ĥ H, Π 1 = ζ = ˆx (κht) 1 2, Π 2 = ζ f = L f, Π (κht) 1 3 = M. (25) 2 The non-dimensional interface height Π can be written as a dimensionless function ψ of the dimensionless variables Π = ψ [Π 1, Π 2, Π 3 ]. We seek a similarity solution for GCEP Technical Report

20 θ(ζ) 0.4 M = 10 0 M = 10 2 M = M = 10 3 M = ζ Figure 13: Similarity solution is shown for different values of M. Note that θ is the nondimensional thickness of the released fluid, measured from the top of the domain for a buoyant current. times after the details of the particular initial condition have disappeared. Following the procedure given by [9] we assume complete similarity in the parameter Π 2, and we seek a solution of the form Π = ψ [Π 1, 0, Π 3 ] = θ [Π 1, Π 2 ]. The expression for the h and x in these variables are ĥ = Hθ [ζ, M], ˆx = ζ [M] (κht) 1 2. (26) Dimensional analysis shows that the tip propagation is proportional to t 1/2 when this scaling analysis is valid. The inner tip position is given by ˆx i = ζ i [M] (κht) 1/2, and the outer tip position by ˆx o = ζ o [M] (κht) 1/2, where ζ i and ζ o are dimensionless quantities that depend only on the mobility ratio M. Substituting relationships (26) into (24), we obtain a nonlinear ordinary differential equation for θ: ζ dθ 2 dζ = d ( ) θ(1 θ) dθ. (27) dζ θ(m 1) + 1 dζ The mobility ratio M is the only parameter determining the shape of the similarity solution at early times. The inner and outer boundaries of integration ζ i and ζ o are unknown, and must be determined as part of the solution. The boundary conditions are: dθ θ (ζ i ) = 1, dζ = ζ im ζi 2 ; dθ θ (ζ o ) = 0, dζ = ζ o ζo 2. The conditions on dθ/dζ come from inserting the conditions on θ into (27). The evolution of the interface at early times has been reduced to a nonlinear eigenvalue problem for a second order ordinary differential equation, with two unknown eigenvalues and four boundary conditions. The two additional boundary conditions GCEP Technical Report

21 allow the unique determination of the eigenvalues as a function of the mobility ratio M. The nonlinear eigenvalue problem has been solved numerically for the shape of the interface and the tip positions as a function of M. The resulting interface shapes are shown in figure 13. As the mobility ratio increases a gravity tongue develops along one of the horizontal boundaries. For large values of M the tip positions follow scaling laws given by ζ i = e M M, M > 10; (28) ζ o = e M , M > 200, (29) M 0.42 So that the position of the outward propagating tip of the interface at early times is given by { x e L d + ζ o [M] (κht) 1 2, M 1; o = L d ζ i [M 1 ] ( ) 1 (30) κht 2, M < 1; M where the superscript e is used to indicate the scaling for the early similarity solution. The position of the inward propagation tip is given by { L d + ζ i [M] (κht) 1 2, M 1; x i = L d ζ o [M 1 ] ( ) 1 (31) κht 2, M < 1. M The similarity solutions described above were obtained under the assumption of complete similarity in Π 2, which corresponds to a step function initial profile (L f = 0). For other initial conditions the similarity solution will be valid after the details of the initial conditions have dissipated, because Π 2 = L f / (κht) 1/2 approaches zero for L f (κht) 1/2. Hence every particular initial condition will be asymptotic to the similarity solution for t t e = L2 f κh. (32) For t t e the similarity solution is valid until the inward propagating tip reaches the origin x i (t b ) = 0, where t b is the back-propagation time (figure 11b). Solving (31) for t b we obtain t b = L 2 d, κhζ i [M] 2 M 1; L 2 d M, κhζ o[m 1 ] 2 M < 1. Hence the early self-similar solution is valid for t e t t b. Self-similar solution at late times Reduction to the porous medium equation At the back-propagation time t b the interface detaches from one of the horizontal boundaries, and the thickness of the released fluid h decreases monotonically as a (33) GCEP Technical Report

22 function of time (figure 11c, d). At late times h H, and we expect the solution for a finite layer to be similar to the solution in a half-space. The equation for the half-space can be obtained from (24) by taking the limit for H, for finite h and M. Consider the limit of the nonlinear diffusion coefficient in (24) for small h, keeping M and H constant, lim h 0 h (H h) h(m 1) + H = h. (34) We will refer to this limit as the half-space approximation. In this limit (24) reduces to h x = κ ( h h ). (35) x x We expect that the limit (34) becomes a good approximation even if h remains finite but h H, so that (35) becomes a good approximation for (24) after some finite time. Barenblatt s solution We briefly summarize the work of [8] (for an English version, see [10]), and present it in a notation consistent with the developments above. Consider a finite initial distribution of gas given by h[x, t = 0] = { h[x], for x < LD ; 0, for x L D. (36) in a half-space otherwise saturated by water. The evolution of the gas-water interface h[x, t] is governed by (35). 2L d is the width of the aquifer that is initially saturated by gas. The volume of current is given by V = Ld L d h(x)dx = 2Ld H. (37) The dimensions of the problem are still the height H, the length L, and time T. The dimensions of the five governing parameters in (35) and (37) are: [x] = [L b ] = L, [t] = T, [κ] = L 2 H 1 T 1 [V ] = LH, where L b is the initial width of the current at back-propagation time t b. L b is an appropriate choice of the initial length scale for M 0, where (35) becomes valid very soon. We choose t, V, and κ as independent parameters, the length scale (κv t) 1/3, and the scale (V 2 /κt) 1/3 for the height. We obtain the following non-dimensional parameters Π = h(κt) 1 3, Π1 = ξ = V 2 3 x (κv t) 1 3, Π2 = L b. (38) (κv t) 1 3 Hence we can write the problem in non-dimensional form as Π = ψ( Π 1, Π 2 ). For finite L b the parameter Π 2 0 as t increases, and [8] assumes complete similarity in the parameter Π 2. This corresponds to assuming an idealized initial condition where L b = 0, corresponding to a delta function given by h 0 = δ(x). This assumption allows GCEP Technical Report

23 the reduction of the problem to the following expressions for the interface h and position x, ( ) 1 V h = ϕ(ξ), x = ξ(κv t) 3. (39) κt The dimensional expression for the evolution of the fluid interface becomes ( ) ( ) 1 1 V 2 3 ξ h(x, t) = 2 6 κt o x2, for x x (κv t) 2 o ; 3 0, for x > x o. The non-dimensional tip position ξ o is determined by the volume of the released fluid, and can be obtained by inserting (40) into (37). The front position at late time is given by (40) x l o = ξ f (κv t) 1 3 = (9κLd Ht) 1 3. (41) The superscript l identifies the tip scaling for the late similarity solution. The late similarity solution depends on the volume of the released fluid, or the displacement distance L d, but it is independent M. Non-self-similar transition We have obtained a description of the front propagation speed at early times from the similarity solution of the lock exchange problem ( ). At late times the governing equations simplify to (35), and the similarity solution of [8] gives the propagation speed at late times. The transition from the early to the late similarity solution will not be self-similar, and must be investigated numerically. For the numerical solution of (24) we have chosen the following non-dimensional variables: η = h/h, χ = x/l d, and τ = t/t with the characteristic time t = L 2 d κ 1 H 1. Substituting these definitions into (24) we obtain the following dimensionless equation, η τ = ( η(1 η) χ η(m 1) + 1 η χ ). (42) The dimensionless mobility ratio M is the only governing parameter. We consider initial distributions that are symmetric with respect to the origin, so that we only need to consider the spatial domain [0 a] where a > 0 is chosen larger than the maximum propagation distance estimated from (41). The initial condition in all simulations is the following step function { 1, χ 1; η (χ, 0) = (43) 0, χ > 1. The problem is symmetric with respect to the origin, so that the boundary condition at the origin is η(0, τ)/ χ = 0, and the outer boundary condition is η(a, τ) = 0. GCEP Technical Report

24 Transition of the scaling for the tip position Figure 14(a-d) shows the numerical results for the non-dimensional position of the outward propagating tip χ o of the released fluid as a function of non-dimensional time τ. The figure shows the effect of increasing the mobility ratio M on the tip propagation and the timing of the transition. The scaling laws for the tip position obtained from the early and late similarity solutions are also shown. In non-dimensional coordinates these scaling laws (30, 41) simplify to χ e o = { 1 + ζo [M] τ 1 2, M 1, 1 ζ i [M 1 ] τ 1 2 M 1 2, M < 1, (44) χ l o = (9τ) 1 3, (45) respectively. Note, the shifted tip position χ o 1 is plotted as a function of time in logarithmic axes, so that the early scaling law (44) plots as a straight line with slope 1/2. In these variables the late scaling law (45) does not plot as a straight line, but it approaches a straight line with slope 1/3 for large times, where it becomes valid. The late scaling law is independent of M and therefore the same curve in all four figures, while the straight line corresponding to the early similarity solution is shifting downward as M increases. Comparison of the numerical results with the scaling laws from the early and late similarity solutions leads to the following four observations: 1. The numerical tip position initially follows the prediction given by the early scaling law χ o τ 1/2 (44), and then the scaling law for late times χ o τ 1/3 (45). 2. The transition time τ t increases monotonically with increasing M. Comparison of figure 14(a) and 14(b) shows that this increase is very small for M < Figures 14(c) and 14(d), on the other hand, show a rapid increase of the transition time for M > The tip position follows the early scaling law χ o τ 1/2 even after the early similarity solution has become invalid at τ b = t b /t (33). In figures 14(a-c) the early scaling law continues to be valid almost up to τ t. 4. In general the transition occurs over a period of time. This transition period is very short for M 10 1 (figure 14b), and increases rapidly for M > 10 1 (figure 14d). Although the early and the late scaling behavior are separated by a transition period, it is useful to define a dimensionless transition time τ t that falls within this transition period. This transition time defines a lower bound for the validity of the late similarity solution. The difference between the old and the new scaling law in figure 14 is given by f(τ; M) = ( χ l o 1 ) χ e o, (46) where χ l o is given by (45), and χ e o by (44). Using the substitution τ t = y 6, (46) reduces to a cubic in y. Let M t denote the value of M, where the early scaling law is tangent to the late scaling law such that f(τ; M t ) = 0. For M M t the non-dimensional GCEP Technical Report

25 (a) M = τ b χ o τ t (b) M = τ b χ o τ t (c) M = τ b χ o τ t (d) M = τ b χ o τ t τ Figure 14: The numerical results for the non-dimensional tip position χ o are shown as a function of non-dimensional time τ, for different mobility ratios M. In all figures the numerical solution is given by dots ( ), the tip scaling from the early similarity solution by a dashed line (- - -), and the tip scaling from the late similarity solution as a full line ( ). (a) M = 10 2, τ t = 2.0, τ b = 0.1; (b) M = 10 1, τ t = 2.5, τ b = 0.24; (c) M = 10 0, τ t = 29.3, τ b = 1; (d) M = 10 1, τ t = 811.3, τ b = 7.1. GCEP Technical Report

26 transition time τ t can be defined as the intersection of the early and late time scaling laws (figures 14a and 14b), and is therefore given by the largest real root of f(τ; M 1) = 0. For M < M t the two scaling laws do not intersect, but the transition time can be defined as the point of minimal vertical distance between the two scaling laws (figures 14c and 14d) given by the local minimum of (46). Solving for the appropriate root and the minimum we obtain the following expression for the non-dimensional transition time ( [ 1 9ζ cos π ]) θ 6 o[m] 6 3 3, M 1; ( [ M τ t = 3 9ζ i [M 1 ] cos π ]) θ , Mt M 1; (47) 64M 3, M M 9ζ i [M 1 ] 6 t, where θ is the principle argument of the following complex numbers [ Arg 2 + 3ζ o [M] 2 + iζ o [M] ] 12 9ζ o [M], M 1; θ = [ Arg 2M [ Mζ 1 ] 2 [ i M 1 ] ( [ iζi 3M 4M 3ζ 1 ] ) ] 2 M i, M M t M 1. Due to the change in the definition of the transition time at M t the graph is not smooth at this point. For M < M t the transition time increases very slowly with M, while it increases strongly for M M t. Discussion We have analyzed the evolution of a finite release of fluid into a horizontal porous medium saturated with an ambient fluid. The released fluid can be miscible or immiscible with the ambient fluid. The density difference between the fluids is the only driving force that has been considered. We first summarize our results in a regime diagram. Then we illustrate the implications of this study for CO 2 storage in saline aquifers. Figure 15 combines all time scales into a M-τ regime diagram, that determines the evolution of a finite release of fluid. The only parameter in this problem is the mobility ratio M = M p = λ p /λ q = µ q /µ p between the released fluid p and the ambient fluid q. The magnitude of a particular dimensional time scale is given by the characteristic time t = L 2 d κ 1 H 1. For all finite values of M the evolution can be divided into three dynamic stages: an early self-similar regime, a transition period, and a late self-similar regime. After the details of the initial condition have dissipated, the interface is asymptotic to an early similarity solution that corresponds to a tilting interface, generated by a lock-exchange flow. The early similarity variable is ζ = x(κht) 1/2, so that the non-dimensional tip position is given by χ o τ 1/2. The early similarity solution ends at the back-propagation time τ b, because inward propagating tips of the two initially separated fronts start to interact at the origin (figure 11b). Figure 15 shows that τ b increases monotonically with time, and follows simple scaling laws for big and small M. Physically we can explain the increase of τ b with increasing M by the increasing viscosity of the ambient fluid that slows down the inward propagating tip of the tilting GCEP Technical Report

27 10 10? τ M 5/2 t τ τ l 1.96 τ t lower bound late soln.?? transition time back-propagation time τ M b τ M 1/6 b M Figure 15: Regime diagram for a finite release of fluid into a horizontal porous slab, showing the non-dimensional time scales obtained in this study, and the shapes of the gravity current as a function of the mobility ratio M. The characteristic time to dimensionalize all results is t = L 2 d κ 1 H 1 GCEP Technical Report

28 interface. Figure 14 shows that the scaling law for the non-dimensional tip position χ o τ 1/2 is valid for a significant time after the early similarity solution itself has become invalid at τ b. The initial similarity solution is followed by a period where the solution is not self-similar and must be obtained numerically. The numerical results in figure 14(b-d) show that the transition period increases with increasing M. From the transition of the scaling laws for the tip position we have defined a transition time τ t, that provides a lower bound on the onset of the late similarity solution. Equation 47 shows a rapid increase of τ t with increasing M for M > M t. After the transition period the late similarity solution becomes valid, because the released fluid occupies only a small fraction of the height of the aquifer, and (24) reduces to (35). Equation 35 admits a similarity transformation in the variable ξ = x/(κv t) 1/3, and the analytic solution has been obtained by [8]. When the late similarity solution is valid, the non-dimensional tip position is given by χ o τ 1/3. Again the scaling for the tip position becomes valid before the solution is fully self-similar. During CO 2 storage in saline aquifers a highly mobile supercritical CO 2 -rich vapor phase is released into a storage aquifer saturated by a less mobile aqueous brine. The mobility ratio is M > 1, and we expect an extended early period where the tip position is given by x t 1/2. The theory developed above allows us to estimate the duration of the this period t b, and after which time we can expect the late scaling law to hold. Consider the example of the Sleipner injection site, on the Norwegian continental shelf [41]. The physical properties of the Utsira-formation used for CO 2 storage at Sleipner are, height H 200 m, permeability k m 2, porosity φ Picking intermediate values from the range of physical properties for fluids in a shallow cold aquifer, given by [42], we set ρ c = 710 kg m 3, ρ b = 1100 kg m 3, µ c = Pa s, and µ b = Pa s, so that M c = 20. For this case the dimensionless time scales are given by τ b 14 and τ t The characteristic time is t = L 2 d κ 1 c H 1, and we see that both time scales will increase quadratically with the injection distance L d, while they decrease with both increasing H and κ c = kg ρµ 1 c φ 1 = m s 1. Consider the effect of increasing L d from 1000 to 2000 m, the back-propagation time t b = τ b t increases from 3.4 to 13.5 yrs, and the transition time t t = τ t t increases from 680 to 2700 yrs. It is commonly assumed that CO 2 needs to be stored for several thousand years to contribute to the reduction of greenhouse gasses in the atmosphere. Assuming a storage period of approximately ten thousand years in the example above, the regime transition occurs relatively early in the storage period, but the transition time may still be much larger than the time over which the plume is actively monitored. Suppose that monitoring data, collected in the first decade after the end of injection, is used to estimate the subsequent spreading of the CO 2 plume, if the change in the propagation regime is not anticipated the extent of the plume will be severely overestimated after a few hundred years. GCEP Technical Report

29 Conclusions The dynamics of a finite release of fluid into a two-dimensional horizontal porous medium are governed by a nonlinear parabolic partial differential equation. The evolution of the released fluid is divided into three regimes. The mobility ratio M is the parameter that determines the magnitude of the non-dimensional time scales separating these regimes. We have obtained new similarity solutions in the variable ζ = x(κht) 1/2 that are valid at early times when the interface is tilting due to a horizontal exchange flow. In this regime the tip position is given by x t 1/2. In the limits h 0 and M 0 the governing equation simplifies to the porous medium equation. This equation admits a similarity solution in the variable ξ = x/(κv t) 1/3 that has been obtained by [8], where the tip position is given by x t 1/3. We have obtained an expression for the transition time t t from the t 1/2 to the t 1/3 scaling. The transition time t t increases monotonically with M, but it is a weak function of M for M < 0.18, and increases rapidly for M > The two self-similar regimes are separated by a transition period that is roughly centered on t t. Numerical solutions to the governing partial differential equation are used to obtain solutions during the non-self-similar transition period. Numerical results show good agreement with the early and the late similarity solutions. During CO 2 storage in saline aquifers, M 20 and the transition from the early to the late scaling is likely to occur within the first thousand years. It is therefore important to anticipate this change in the propagation regime, when estimating the extent of the CO 2 -plume. Future Plans We plan to extend this work to sloping aquifers, and to incorporate a simple model of residual trapping of CO 2 in the wake of the plume. Residual trapping will lead to a new length scale - the maximum migration distance of the injected CO 2. We will also investigate the effect of formation dip on this length scale. The maximum migration distance will be particularly important in large regional aquifers without structural closure. GCEP Technical Report

30 Construction of High Order Adaptive Implicit Methods Investigators Hamdi Tchelepi (Associate Professor of Petroleum Engineering), Amir Riaz (Post-Doctoral Researcher), Romain de Loubens (Graduate Research Assistant) Introduction The Adaptive Implicit Method (AIM) has been widely used in Reservoir Simulation for the last two decades. The success of this approach over the traditional IMPES or FIM formulations can be attributed to its significant reduction of the computational cost. However the standard AIM formulation is only first order accurate, so we propose to investigate high order extensions of this method. High order AIM schemes could be used to produce more accurate results for the same computational cost, or to reduce the cost of a simulation while maintaining the same level of accuracy. The general requirements for a high order AIM scheme are the following : at least second order accuracy both in time and space, unconditional stability and non-oscillatory behavior in the implicit regions, and local mass conservation. First we present an analysis of the standard AIM formulation that reveals an inconsistency in the discretization at the transition between implicit and explicit blocks. Although the discretization errors resulting from this inconsistency do not amplify with time, they may affect the overall accuracy of the method. The propagation of these errors is examined through a linear error analysis. In a particular case, we are able to show the convergence of the AIM scheme in spite of its inconsistency. Numerical experiments confirm the results of our analysis. They can also be used to reveal the presence of undesirable kinks in the solution profile. A general approach to fix this inconsistency is suggested and analyzed for a particular example. The consistency of the AIM formulation at the implicit-explicit boundaries is of primary importance when considering high order extensions of this approach. Clearly, the main difficulty is to combine high order explicit and high order implicit time integrations in a consistent manner. We present a second order AIM formulation that meets all of our requirements, except that the non-oscillatory condition in the implicit regions holds only for CFL numbers lower than two. Analysis of the standard AIM formulation Local inconsistency The one-dimensional Buckley-Leverett problem without gravity and capillarity is described by the following PDE in dimensionless form: s t + f(s) x = 0, (48) GCEP Technical Report

31 where s is the water saturation and f(s) the fractional flow of water. For simplicity we assume that f (s) > 0. The initial and boundary conditions are given by s(x, 0) = s wc, s(0, t) = s wi, (49) where s wc is the connate water saturation and s wi the inlet saturation. Let us consider the standard AIM discretization of (48) on a uniform grid. Although the implicit levels usually depend on the local CFL restrictions, we can assume for clarity that all the blocks to the left of block i 0 are implicit, whereas all the blocks to the right of it are explicit (as shown in figure 16). Figure 16: Schematic of the implicit and explicit blocks For this configuration, the standard AIM scheme reads : 0 = sn+1 i s n i t 0 = sn+1 i t s n i + 1 [ f(s n+1 i ) f(s n+1 i 1 x )], i < i 0, (50) + 1 x + 1 [ f(s n i ) f(s n+1 i 1 )], i = i 0, (51) 0 = sn+1 i s n [ i f(s n t x i ) f(s n i 1) ], i > i 0. (52) The numerical schemes in the implicit and explicit regions are respectively the Backward Euler and Forward Euler two points Upwind schemes. They both have a leading truncation error in O( x) + O( t). But a Taylor series expansion for block i 0 shows that the truncation error is Ei n 0 = t x [f( s) t] n+1 i 0 + O( x) + O( t) + O( t 2 / x), (53) where s is the exact solution to the IBVP (48)-(49). The above equation clearly indicates that the standard AIM discretization is inconsistent at the boundary between implicit and explicit blocks. This inconsistency arises from the fact that the fluxes across block i 0 are computed at different time levels. Similarly, the scheme is inconsistent at the transition between explicit and implicit blocks. So in general we can say that the standard AIM scheme is inconsistent at any implicit-explicit boundary (which will be referred to as an I/E boundary). At this point it is important to recall that by construction the standard AIM scheme is stable, so that we don t expect the discretization errors at the I/E boundary to be amplified. On the other hand, the convergence is not guaranteed a priori (since Lax equivalence theorem cannot be applied) and the accuracy of the scheme may be affected by these errors. In the following section, we present a linear error analysis that will allow us to show convergence in a particular case and more generally to understand how these discretization errors propagate. GCEP Technical Report

32 Linear error analysis To conduct this analysis we assume that f(s) = s and that the location of the I/E boundary remains fixed in time (i.e. i 0 is fixed). Let e n i = s n i s(x i, t n ) denote the numerical error. Substituting s n i = s n i + e n i in (50), (51), (52), and neglecting the terms of order two or higher, it yields: e n+1 i e n+1 i e n+1 i e n i t e n i t e n i t + 1 ( ) e n+1 i e n+1 i 1 = 0, i < i0, (54) x + 1 x + 1 x ( ) e n i e n+1 t i 1 = x ( s t) n+1 i, i = i 0, (55) ( ) e n i e n i 1 = 0, i > i0. (56) Due to the boundary condition, we use e n+1 0 = 0 in the first equation. Multiplying through by t and denoting the CFL number by λ = t, we obtain x (1 + λ)e n+1 i λe n+1 i 1 = en i, i < i 0, (57) e n+1 i λe n+1 i 1 = (1 λ)en i + λ t( s t ) n+1 i, i = i 0, (58) e n+1 i = (1 λ)e n i + λe n i 1, i > i 0, (59) which can be written under the matrix form AE (n+1) = BE (n) + S (n+1), (60) where E is the error vector, A and B are matrices that have non zero entries only on their main diagonal and first off-diagonal, and S is a source term due to the discretization error at the I/E boundary: S (n+1) i = { 0, i i 0 λ t( s t ) n+1 i, i = i 0., (61) Let T f denote the final time and N the total number of time steps (i.e. N t = T f ). Since E (0) = 0, we can compute the error at the final time as follows: E (1) = S (1), E (2) = C S (1) + S (2), E (3) = C 2 S (1) + C S (2) + S (3),. E (N) = N C N k S (k), (62) k=1 where S (n) = A 1 S (n), and C = A 1 B. The matrix C represents the standard AIM operator. In this simple case, it is easy to show that C has only two distinct eigenvalues, namely µ 1 = 1 λ, µ 2 = λ. (63) Due to the stability condition we have 0 < µ 1 < 1, and similarly 0 < µ 2 < 1, so ρ(c) < 1. This last result shows that as expected the standard AIM operator is stable. However, this is not sufficient to show the convergence. GCEP Technical Report

33 In fact, the matrix-vector products in (62) are easily obtained by computing the i 0 -th column of C n (note that only the i 0 -th component of S (k) is not zero). Denoting by (g 1,..., g p ) the canonical basis in R p, where p is the number of blocks, we can show that C n g i0 = l n k=0 C k n λ k (1 λ) n k g i0 +k. (64) where l n = min(n, p i 0 ). The above equation indicates that each individual error source term arising at an I/E boundary propagates with a unit speed to the right and that it diffuses with time (see figure 17). Figure 17: Propagation of an error source term arising at the I/E boundary From (64) we have E (N) = = It follows that N 1 n=0 l N 1 k=0 C n S (N n) = ( N 1 n=k N 1 n=0 ( ln ) Cn k λ k (1 λ) n k g i0 +k k=0 C k n λ k+1 (1 λ) n k t( s t ) N n i 0 ) g i0 +k. e N i0 +k = (E (N), g i0 +k) λ k+1 K t N 1 n=k λ t( s t ) N n i 0 C k n (1 λ) n k, (65) where K is an upper bound for the time derivative of s over the time interval [0, T f ]. Since 0 < λ < 1 we can write N 1 Cn k (1 λ) n k 1 [n(n 1)... (n k + 1)] (1 λ) n k. (66) k! n=k n=k The series expansion on the right-hand side of (66) corresponds to the k-th derivative of f(z) = z n = 1 1 z, (67) n=0 GCEP Technical Report

34 which is C on the open interval ] 1, 1[. Clearly f (k) (z) = [n(n 1)... (n k + 1)] z n k = n=k k!, (68) (1 z) k+1 so e N i0 +k λ k+1 K t 1 k! = K t. (69) k! λk+1 This upper bound is independent of k which shows that the error is uniformly bounded. Since t 0 as N, the error converges to zero for the infinite norm. In figure 18 we show the final error profiles obtained with different mesh sizes for an initial saturation profile s(x, 0) = e x2. As expected, when the grid is refined by a factor of 2, the total amplitude of the error is also divided by 2. Figure 18: Final error profiles for different mesh sizes Numerical tests In figure 19 below we show the solution and error profiles computed in the linear case for the same initial condition s(x, 0) = e x2 and the inlet condition s(0, t) = 1. The mesh size is 50 and like in the case studied previously, the I/E boundary is fixed. The standard AIM scheme is compared against the implicit and explicit two points Upwind Schemes. On the solution profile we can see that there is a small kink just after the I/E boundary. On the error profile we observe that the amplitude of this kink is about the same as the amplitude of the dissipative error terms, as expected from our previous analysis. GCEP Technical Report

35 Figure 19: Solution and error profiles with a fixed I/E boundary In the general case where the I/E boundary moves with time, an even larger kink may appear. We can actually move the I/E boundary with a unit speed, so that the newly created error source terms are added at the location where the error is supposed to be maximal. This way, we clearly maximize the total amplitude of the error, and we might expect to have an almost first order error. But as we saw previously, due to numerical dissipation, the error source terms are diffused with time so the amplitude of the error eventually dies away when the grid is refined. However, as shown in figure 20, when the mesh is refined by a factor of ten, the maximum error is only divided by a factor 2 or 3. Hence the convergence rate is very slow (clearly less than one). Figure 20: Solution profiles (50 and 500 blocks) with a moving I/E boundary Consistency fix In order to avoid an inconsistency at the I/E boundary, one could evaluate the flux at the old time level when updating the explicit block, and at the new time level for the GCEP Technical Report

36 implicit block. But that would obviously lead to a mass balance error. Another way to preserve the consistency (without any mass balance error) is to evaluate the numerical flux everywhere as a weighted average of the old and new time levels (or any combination of intermediate time levels). Using time levels n and n + 1 for instance, we would have s n+1 i = s n i t { } θ(f ñ+1 i f ñ+1 i 1 x ) + (1 θ)(f i n fi 1) n, (70) where n + 1 refers to the new time level or some predictor of it, and θ is a parameter between 0 and 1. In the implicit region we apply s n+1 i = s n i t { θ(f n+1 i f n+1 i 1 x ) + (1 θ)(f i n fi 1) } n, (71) whereas in the explicit region we have s n+1 i = s n i t { θ(f x i fi 1) + (1 θ)(fi n fi 1) } n, (72) where fi is evaluated using some explicit predictor s i of s n+1 i. For an explicit block that has an implicit block to its left, the scheme would be: s n+1 i = s n i t { θ(f x i f n+1 i 1 ) + (1 θ)(f i n fi 1) } n, (73) and a similar hybrid scheme would be applied for an implicit block that has an explicit block to its left. As a predictor, we could use s i = s n i t x (f i n fi 1) n. (74) The resulting scheme is clearly conservative and consistent, including at the I/E boundaries. The leading truncation error in the explicit and implicit regions is the same and it is given by ] (s xx ) n i. (75) [ E = ( 1 x θ) t 2 2 whereas for an explicit block that has an implicit block to its left we have E n i = [ ( 1 x θ) t 2 2 θ t ] ( t x) (s xx ) n i. (76) x A detailed study of this scheme (e.g. using Von Neumann analysis) shows that it is stable in the implicit regions under the condition (1 2θ)λ < 1 and that it is TVD (Total Variation Diminishing) for λ < 1/(1 θ). So by fixing the inconsistency we introduced a more restrictive CFL condition (note that the standard AIM scheme is unconditionally TVD in the implicit regions because it corresponds to the case where θ = 1). In the explicit regions, we can show that the maximum principle is satisfied if and only if λ min (1/(2θ), 1). So for θ > 1/2, the CFL condition becomes more restrictive than for the explicit Upwind scheme. If for instance we choose θ = 0.9, then we can use CFL numbers up to 10 in the implicit region and CFL numbers smaller than 5/9 in the explicit region. So in the end, the inconsistency can be fixed, but under more restrictive CFL conditions. Note however that a different choice for the predictor may give better results. GCEP Technical Report

37 Construction of High Order AIM formulations Basic framework High order spatial discretizations such as ENO, WENO or Central Schemes can serve as a framework for the construction of high order AIM formulations. As a preliminary step, we just consider a high order in space AIM scheme. In generic form, we can write it as s n+1 i = s n i t ( ˆf ñ i+ x ˆf ) ñ, ñ = n or n + 1, (77) 1 i where ˆf i+ 1 is a high order numerical flux. For instance, we may use the second order 2 WENO flux (written for f > 0): ˆf i+ 1 2 = (ω 0 ) i+ 1 2 { 1 2 f i f i } + (ω 1 ) i+ 1 2 { 1 2 f i + 1 } 2 f i+1, (78) where ω 0, ω 1 are the weights associated with each stencil. Hence (77) combined with (78) gives a scheme that is second order in space but only first order in time because the time integration is Forward Euler or Backward Euler. Figure 21 below shows a numerical simulation of a 1D Buckley-Leverett problem using Honarpour relative permeability curves with a unit mobility ratio. We observe, as expected, that the second order in space AIM scheme is less dissipative than the standard AIM scheme. But both schemes lead to a relatively large error near the first I/E boundary, whereas the first order explicit Upwind Scheme is much closer to the MOC solution. Clearly, the loss of accuracy at the I/E boundary due to the inconsistency of the AIM formulation is a major issue for the construction of high order AIM schemes. As a matter of fact we need to incorporate a consistent, high order time integration in order to achieve high accuracy everywhere. Figure 21: Simulation with a high order in space AIM scheme GCEP Technical Report

38 Second order AIM scheme To achieve second order time accuracy, we can apply a trapezoidal time integration (second order Runge Kutta method) in the explicit regions and a Crank Nicolson scheme in the implicit regions. Both of these time integration schemes use the information at the old and new time levels (more exactly, the trapezoidal method uses the information from time level n and from a predictor of time level n + 1). So the consistency at the I/E boundary is easily maintained. For a 1D problem, we can write this scheme as s n+1 i = s n i t ( ˆf ñ+1 2 x i+ 1 2 ) ñ+1 ˆf t ( i ˆf n 1 i+ 2 2 x ˆf ) n, (79) 1 i where n + 1 refers again to the new time level or some predictor of it. More precisely, if there is an explicit node within the numerical stencil of an implicit block, then the implicit part of the flux is evaluated with the predicted saturation value of the explicit block. Conversely, if there is an implicit node within the stencil of an explicit block, we use the implicit value at this node instead of a predicted value. The resulting system of equations can be solved in three steps: 1) predict the saturation values at the new time level in the explicit blocks. 2) solve for the implicit blocks using Newton-Raphson method. 3) update the explicit blocks. It is important to note that this scheme is adaptive implicit in the sense that we can solve for the implicit blocks independently of the explicit ones. Besides, it is conserving mass, it is consistent at the I/E boundaries and globally second order accurate. However, the Crank-Nicolson scheme is non-oscillatory only for CFL numbers less than 2. Figure 22 shows 1D simulation results for a Buckley-Leverett problem. The solution computed with a CFL number lower than 2 appears to be very accurate even near the discontinuity. But as expected, the saturation profile becomes non monotonic for a CFL number greater than 2. Figure 22: 1D saturation profiles for CFL<2 (left) and CFL>2 (right) GCEP Technical Report

39 Extension of this scheme to 2D and 3D is straightforward. In figure 23 we show a simulation of a 2D Buckley-Leverett problem with gravity in the y-direction. Initially, the saturation is equal to unity inside a circle and zero outside. The total velocity is constant with time and aligned with the direction θ = π/4. Since there is no analytical solution to this problem, we give a reference solution computed with the explicit WENO2 scheme. There is a good match between this reference solution and the solution of our second order AIM scheme. Note that the simulation was run with a CFL number lower than 2. Figure 23: Contour plots calculated with explicit WENO2 (left) and second order AIM (right) The use of this second order AIM scheme is limited in practice due to the CFL restriction for non-oscillatory behavior. However, it demonstrates that the standard AIM formulation can be extended to a high order method both in space and time. Conclusions and future work Our analysis revealed that the discrete form of the standard AIM scheme is locally inconsistent at the I/E boundaries. But in practice, this is not an issue because the saturation fronts are most of the time treated fully implicitly, or the error source terms arising at the I/E boundaries do not accumulate repeatedly at the same location. From our linear error analysis and our numerical experiments, it appears that the convergence depends on the dissipative nature of the scheme. But this analysis would also need to be extended to the nonlinear case. For the construction of high order AIM formulations, high order spatial discretizations and time integration schemes are available in the literature and can be used as a framework. But the challenge for the AIM formulation is to obtain a high order time integration procedure that is consistent at the I/E boundaries and also unconditionally non-oscillatory in the implicit regions. We plan to investigate further time integration schemes such as Backward differentiation Formula (BDF), or explicit and implicit Runge-Kutta schemes. Another option to consider would be to introduce artificial viscosity in order to remove oscillations. GCEP Technical Report

40 Algebraic Multiscale Modeling of Compressible Flow Investigators Hamdi Tchelepi (Associate Professor of Petroleum Engineering), Hui Zhou (Graduate Research Assistant) Introduction The accuracy of subsurface flow simulation relies on having detailed descriptions of the porous formation. Properties such as porosity and permeability typically vary over many scales. High-resolution reservoir models represent permeability as a discrete, highly discontinuous, full tensor property, and that presents difficult challenges to numerical solution methods. Detailed reservoir description models may require O(10 8 ) grid cells. That is too expensive since state-of-the-art numerical flow simulators can deal effectively with O(10 6 ) cells. Traditionally, this difficulty is tackled by upscaling techniques [21]. Upscaling is used to coarsen the fine scale geological model to computationally manageable sizes that can be used to make performance predictions. The upscaled coarse models are usually constructed using simple flow scenarios (e.g., single-phase flow) and strong localization assumptions. As a result, it is difficult to quantify the errors associated with using these upscaled models to make performance predictions of more complex multiphase flow processes. In a multiscale formulation, we do not solve equations for the global fine-scale problem. Instead, we solve for coarse-scale unknowns. The fine-scale information is usually integrated into basis functions obtained by solving local problems with special boundary conditions. Thus, the computational cost can be much less than that for solving the global fine-scale problem. Different from upscaling techniques, multiscale methods allow for the reconstruction of the fine-scale information from the coarse-scale solution using the basis functions. Moreover, multiscale methods operate at the formulation level. That is, the fine-scale problem statement of the complex flow process of interest is replaced by a discrete set of multiscale equations and operators that represent the physical processes under investigation, which is then solved for the domain of interest. In order to describe flow and transport of CO 2 in geologic formations, we need to be able to model compressible multiphase flow in large-scale heterogeneous porous media. Existing multiscale methods ([31], [30], [15], [3], [4]), however, deal only with the incompressible (elliptic) flow problem. Here, we extend the multiscale finite-volume (MSFV) method to handle compressible systems accurately and efficiently. Moreover, we cast the multiscale treatment into a general algebraic framework that is easy to implement for both structured and unstructured grids. Moreover, this algebraic framework can be easily extended to include more complicated physical mechanisms. GCEP Technical Report

41 Background The multiscale character of the properties of natural porous media makes the problem of predicting flow and transport in such systems a natural target for multiscale methods. Hou and Wu [30] proposed a multiscale finite element method (MSFEM) that captures the fine scale information by constructing special finite element basis functions within each element. However, the reconstructed fine-scale velocity of the MSFEM is not conservative. Later, Chen and Hou[15] proposed a conservative mixed finite-element multiscale method. Another multiscale mixed finite element method has been presented by Arbogast [3] and Arbogast and Bryant [4]. Numerical Green functions were used to capture the fine scale information. The multiscale finite-volume method (MSFVM) was first proposed by Jenny, Lee and Tchelepi[31] to deal with the flow problem (pressure and total-velocity) in detailed descriptions of strongly heterogeneous formations. They used two sets of local basis functions - dual and primal - in order to obtain conservative coarse-scale and fine-scale velocity fields. The finite-volume method is widely used in numerical modeling of multiphase flow and transport in heterogeneous porous media; as a result, the MSFVM is a natural framework for high resolution simulations of the complex flow processes associated with subsurface CO 2 sequestration in large-scale natural porous formations. Note that all the existing methods considered incompressible (elliptic problem) flow in porous media only. Very recently, Lunati and Jenny studied compressible multiphase flow[38], where the flow equation is parabolic. They propose three models to deal with the compressibility in the MSFVM framework. We are interested in extending the MSFVM to model the physical processes associated with subsurface CO 2 sequestration processes for both the injection and post-injection periods. This includes compressible multiphase flow with gravity and capillarity for large-scale highly heterogenous formations with special emphasis on saline aquifers. Results We present an Operator Based Multiscale Method (OBMM) for compressible multiphase flow in heterogeneous porous media. Construction of the two multiscale operators, namely, prolongation and restriction, as well as the assembly and solution of the coarse-scale system are described. Finally, the ability of the OBMM approach to resolve the fine-scale structures of compressible heterogeneous systems is demonstrated using several numerical test cases. The model equation is (λ p) = c p (80) t Note that in the context of compressible flow, λ and c are both functions of pressure, GCEP Technical Report

42 Figure 24: Two-dimensional multiscale grid with a typical dual control volume. The enlarged dual control volume shows the underlying fine grid. i.e., λ = kb µ c = φ b p where k is the absolute permeability, b is the reciprocal of the formation volume factor (FVF), µ is viscosity and φ is porosity. For simplicity, we take φ, µ as constants. b is a function of pressure that accounts for compressibility. Note that Eq. 80 is for single phase flow, but the pressure equation of multiphase flow can also be cast into this parabolic form with corresponding interpretations of λ and c for a multiphase system. The OBMM algorithm The OBMM algorithm can be summarized in the following steps: 1. Model the field with multiscale grid. A typical two-scale grid is shown in Figure (24). The properties of the system are defined on the underlying fine grid. The dual grid is constructed by connecting the cell centers of coarse blocks. A typical dual control-volume is shown in Figure (24) together with underlying fine grid. 2. Construct the fine-scale discretization equations. For example, we can use backward Euler time and central space difference to discretize Equation (80), which can be written in matrix form as (81) A f p f = r f (82) 3. Assemble the multiscale prolongation operator P from the basis functions. The prolongation operator is defined by p f = Pp c (83) 4. Construct the multiscale restriction operator R based on the coarse scale numerical scheme. The restriction operator projects a large number of fine scale equations into a small number of coarse-scale equations. In matrix form, we have R(A f p f ) = Rr f (84) GCEP Technical Report

43 5. Construct the coarse-scale equations. From the definition of the two multiscale operators, the coarse-scale system of equations can be written as where A c p c = r c (85) A c = RT f P r c = Rr f (86) 6. Solve coarse scale equation 85 to get the coarse-scale pressure p c, then Eq.83 can be used to reconstruct the fine-scale pressure field p f. 7. Solve local fine-scale problems (with boundary conditions obtained from the fine scale pressure) in each primal coarse block to get the fine-scale velocity field. The velocity is conservative and can be used to calculate the saturation field. The OBMM algorithm can be interpreted as the algebraic form of the multiscale finite-volume method. This algebraic form is compact, but is quite powerful. It offers several advantages: 1. The algorithm is expressed in algebraic form and thus is readily extendable to unstructured grid. 2. The algorithm is flexible and allows for including complex physical mechanisms (e.g., compressibility, capillarity) that may change the type of the model equations. 3. The method can make use of the fine scale discretization equations and only simple algebraic operations (as in Equation (86)) are need to convert fine scale equations to coarse scale equations. Therefore, it may be possible to build a multiscale reservoir simulator from existing fine-scale simulators. Compared with building a multiscale simulator from scratch, that will save a great deal of time and effort. 4. The algorithm is easy to implement and is computationally efficient (see following subsection). Construction of the multiscale operators From Eq.83, we know the prolongation operator is nothing but the interpolation from coarse-scale variables to fine-scale variables. Thus, the basis functions of the MSFVM[31] serve as the prolongation operator, P. Although those basis functions were designed for incompressible (elliptic) problems, we use them for the compressible (parabolic) multiphase flow problem. So that P a,a = φ A (x a ) (a = 1,..., n f ; A = 1,..., n c ), (87) where a denotes a fine node, A denotes a coarse node, φ A the basis function of coarse node A, n c is the total number of coarse nodes and n f the total number of fine nodes. Numerical examples show that this yields results that are in excellent agreement with reference solutions for a wide range of problems. GCEP Technical Report

44 For the multiscale finite-volume method, the restriction operator is R A,a = { 1 if Ωa Ω A 0 otherwise (a = 1,..., n c ; A = 1,..., n f ), (88) where Ω a is the fine block centered in fine node a, Ω A is the coarse block centered in coarse node A. Numerical Experiments We first test the simplest one-dimensional single-phase problem. Note that for 1D elliptic problems, the basis functions of the MSFVM are exact. So we can study the numerical performance of the OBMM when the elliptic basis functions are used to solve parabolic problems. In this and the following 1D cases, the initial and boundary conditions are the same. All the variables are in dimensionless form. The characteristic time is τ = cl 2 / k ( c, k are average values of c and k, respectively). The initial pressure is 100. Constant pressure boundary conditions of 100 and 1 are imposed on the left and right boundaries, respectively. The fluid is taken as an ideal gas, i.e., b = b 0 p 0 p. (89) The first case has a homogeneous permeability (k = 1). We compare the multiscale pressure solutions with reference fine-scale solutions for several time steps. Also shown is the pressure profile for steady state ( incompressible) flow. As can be seen from Figure (25(a)), the multiscale solution shows noticeable error in the beginning but the error decreases very quickly. At later times, very good agreement between the multiscale and reference solutions are observed. The early-time error results from taking the elliptic basis function for a parabolic problem. In elliptic problems, the signal propagates at infinite speed, while in parabolic problem the velocity is finite and depends on the compressibility. However, the error is both small and local and decreases rapidly with time. As a result, the overall performance of the multiscale method is quite good. Now, we consider a more challenging test case. The permeability is taken to be random. To make the compressibility effect stronger, we assume the dimensionless coefficient c to be highly oscillatory and 4 orders larger than k, which is given by k = (1 + sin(10 3 πx))ξ c = 10 2 [ sin(10 3 πx) ] ξ (90) where ξ is a random number in [0, 1]. Figure (25(b)) shows that the multiscale solutions have the same performance as in the homogeneous case. Again, the overall accuracy is very good. Now, we use the OBMM to study 2D single-phase flow. To challenge the proposed method, we use the permeability model from the 10th SPE Comparative Solution Project (SPE 10 [17]). We take the top layer of the model as shown in Figure (26(a)). The permeability varies by six orders of magnitude and the variance of the logarithm permeability is 5.5. This is a difficult problem due to the high contrasts in the GCEP Technical Report

45 A Numerical Simulation Framework for the Design t = 10 τ 2 t = 10 τ 1 t = 10 τ t = τ Steady state Pressure t = 10 τ 2 t = 10 τ 1 t = 10 τ t = τ Steady state 20 ms fine fine x (a) homogeneous case (b) heterogeneous case Figure 25: 1D single phase test case permeability field. We impose no flow boundary condition on the upper and lower boundaries. The left boundary condition is at a constant pressure, which is the same as the initial pressure (100), and the right boundary is kept at a constant pressure of one. The fluid is assumed to be an ideal gas. Figure (26(b)) shows a comparison of the multiscale pressure profiles with those obtained from reference fine-scale solutions for three time steps. 8 Fine pressure Multiscale pressure t = 0.01τ t = 0.1τ (a) Top layer of SPE 10 permeability model t = τ (b) fine scale and multiscale pressure profiles at different time steps Figure 26: 2D single phase test case The last test case is for 2D multiphase flow. We consider a depletion process in a liquid-gas reservoir. The reservoir is initially filled with 50% liquid and 50% gas. The initial pressure is 100. The boundary condition is the same as the above 2D single-phase case. The PVT properties of the two fluids are: b l = p/p 0, b g = 1 + p/p 0. The gas compressibility is three orders of magnitude higher than that of the liquid. The problem is purely compressibility driven, meaning that if the two fluids have the same (91) GCEP Technical Report

46 A Numerical Simulation Framework for the Design... compressibility the saturation will remain the same. The multiscale results are compared with fine scale results in Fig.27. Figure (27(a)) indicates that the multiscale pressure solution in this multiphase case is also in good agreement with the fine scale solution. As for the saturation field, the multiscale solution captures the saturation distribution accurately. There is some noticeable error in the saturation field near the right boundary. We are looking into this problem. Fine pressure Multiscale pressure Fine saturation Multiscale saturation 100 t = τ t = 0.05 τ t = 0.05 τ (a) Top layer of SPE 10 permeability model (b) fine scale and multiscale pressure profiles at different time steps Figure 27: 2D single phase test case From the above examples, we see that the OBMM is able to model the fine scale details of compressible single and multiphase flow in heterogeneous porous media accurately. The basis functions are determined only by the elliptic component. This avoids any time derivatives in the calculation of the basis functions and results in small early-time errors within a limited spatial region. Progress We developed an operator based multiscale method that serves as a general algebraic framework. We studied compressible flow within this framework and demonstrated good numerical performance without special treatment for the basis functions. This flexible framework allows for incorporating additional physical mechanisms; moreover, it is directly applicable to unstructured grids. Future Plans We will investigate adding buoyancy, capillarity, and well models into this algebraic multiscale finite-volume framework. GCEP Technical Report

47 A Stochastic Lagrangian Framework to Study Physical Processes Relevant for CO 2 Storage in Geological Formations Investigators Patrick Jenny (Assistant Professor, Institute of Fluid Dynamics, ETH Zurich), Hamdi A. Tchelepi (Associate Professor, Petroleum Engineering Department, Stanford), Ivan Lunati (Post Doctoral Fellow, Institute of Fluid Dynamics, ETH Zurich), Manav Tyagi (Ph.D Student, Institute of Fluid Dynamics, ETH Zurich) Introduction A novel Lagrangian simulation framework specifically designed for the study of the multiscale, multi-physics processes relevant to storage of CO 2 in geological formations has been developed. The basic motivation is due to the fact that the statistics of fluid states, properties, and composition can be modeled more naturally, if one keeps track of the histories of individual, infinitesimal small fluid volumes. In turbulence modeling of reactive flows a similar methodology called probability density function (PDF) modeling proved to have significant advantages, e.g. turbulent dispersion and chemical reaction appear in closed form. Moreover, the statistical information contained in the joint PDFs allows for improved understanding of the complex multiscale physical interactions that describe the physics, and that in turn allows for the development of more reliable models and computational algorithms. The major challenge of the initial phase of this project was the development of a particle method for nonlinear multiphase transport in porous media. Particle methods have been investigated extensively for single-phase tracer transport, but conventional particle approaches are unable to describe the correct nonlinear macroscopic flux observed in immiscible multi-phase flow. The new methodology is based on transporting the particles according to statistical rules. We demonstrate that local mass balance is satisfied, and that in the limit of having many particles, the results is consistent with the standard immiscible, Darcy-scale, two-phase flow equations. There are no inherent limitations in the methodology, provided the required Lagrangian statistics are available from physical experiments or pore network simulations, for example. The most attractive aspect of this new particle approach is the ability to bridge the small (e.g., pore) and large (e.g., Darcy) scales in a consistent framework. With such a consistent multi-scale multi-physics framework one can gain more insight into the physics relevant for CO 2 storage in geological formations. Eventually, this framework may be used to derive macroscopic models that incorporate sub-scale information appropriately, which can be used to describe the behavior of CO 2 processes in large-scale systems. In this preliminary report, however, we present the new particle method for the simple Buckley-Leverett two-phase flow problem. Background Particle tracking methods have been employed successfully for subsurface flow simulation. Several Eulerian-Lagrangian schemes have been introduced for linear tracer transport (see, e.g., [46, 25, 14, 34, 56]) and were extended to nonlinear GCEP Technical Report

48 problems such as solving the saturation equation for two-phase immiscible flow (see, e.g., [18, 19, 28]). From the pioneering works of [2] and [47], fully Lagrangian schemes based on random walk have been widely employed for tracer transport. Here, in contrast to Eulerian-Lagrangian methods, each particle represent a physical mass of a tracer (phases) and do not simply transport a concentration (saturation) value. Fully Lagrangian methods have also been applied to reactive-tracer transport with nonlinear accumulation terms (see, e.g., [26, 1]; or [20] for a comprehensive review), which requires the calculation of concentration at the node of a superimposed grid [6]. Motivation To our knowledge, fully Lagrangian methods have not yet been applied to subsurface flow problems with nonlinear fluxes. Here, we introduce a scheme to solve saturation transport equations for two-phase immiscible flow. Two types of particles are used, one for each phase, such that each particle represents the mass of a particular phase. Our motivation is to develop a methodology which can incorporate pore-scale physical processes in the form of Lagrangian statistics obtained from experiments or pore-scale simulations. The particle velocity, for instance, can be described by a given probability-density function (pdf) and spatial correlation tensor. Such a framework can naturally deal with compositions, chemical reactions and non-equilibrium effects. Mathematical Formulation In this section, we describe the new particle modeling framework for the study of multi-phase transport in porous media. Therefore, in order to explain the basic ideas, we consider the transport equations and Φ S 1 t (λ 1 p 1 ) = 0 (92) Φ S 2 (λ 2 p 2 ) = 0 (93) t for incompressible two-phase flow without gravity effects, where S 1,2 and p 1,2 are phase saturations and pressures, respectively, λ 1,2 = k r1,2 k/µ 1,2 mobilities and Φ the porosity. The relative permeabilities, k r1,2, are functions of saturation, the rock permeability, k, a function of space and the viscosities, µ 1,2, are constant. The difference between the phase pressures due to capillarity effects is given by the algebraic expression p 1 p 2 = p c = c/s 2. (94) With relation (94) Eq. (93) can be rearranged and one obtains Φ S ( 2 (λ 2 p 1 ) = (λ 2 p c ) = c λ ) 2 S t S2 2 2 with a diffusion term on the right-hand side. Finally, the sum of Eqs. (92) and (95) results in the pressure equation ( ((λ 1 + λ 2 ) p 1 ) = c λ ) 2 S S2 2 2 (96) (95) GCEP Technical Report

49 for p 1 and with the definition of the total flux and the fractional flow functions Φ S 1 t u tot = (λ 1 + λ 2 ) p 1 c λ 2 S S2 2 2 (97) f 1,2 = the saturation equations can be written in the form + (f 1 u tot) ( = and Φ S 2 t λ 1,2 λ 1 + λ 2 (98) ) cλ 2 f 1 S S2 2 1 (99) + (f 2 u tot) ( ) cλ 2 = f 1 S S (100) Note that u tot 0 and f 1 + f 2 1. Next, we describe how Eqs. (99) and (100) can be solved with a particle method. Stochastic Particle Method The motivation for a Lagrangian simulation framework is to model complex phenomena, which are difficult to treat with an Eulerian approach. At this point, however, the goal is to show that the method presented in this section satisfies local mass balance and converges to the correct solution. Consider a large number of computational particles, which represent small fluid volumes belonging either to phase one or phase two. An expression for the evolution of a particle that is consistent with Eqs. (99) and (100) is given by dx = ( f 1 u tot ΦS 1 ( f 2 u tot ΦS 2 + ( )) cλ f 2 1 ΦS2 2 ( )) cλ + f 2 1 ΦS2 2 dt + dt + ( f 1 cλ 2 2ΦS 2 2 ( f 1 cλ 2 2ΦS 2 2 dt) 1/2 ξ, if s = 0 dt) 1/2 ξ, if s = 1, (101) where x is the particle position. The saturation S 2 is the average of the s, which denotes the phase that the particle belongs to (s = 0 for phase one, and s = 1 for phase two). Note that a random walk (the terms with the normally distributed random variable ξ) is employed to partly account for the terms on the right-hand side of Eqs. (99) and (100). Together with the term (f 1 cλ 2 /(ΦS 2 2)dt in Eq. (101) diffusion is treated correctly. The expectation of the total particle flux is Φ(S 1 dx s = 0 + S 2 dx s = 1 )/dt = u tot. (102) The operators and denote expectation and conditional expectation, respectively. Numerically, however, S 1 has to be approximated by an estimate S 1 = 1 S 2 and the numerical expectation of the total particle flux becomes Φ(S 1 dx s = 0 + S 2 dx s = 1 )/dt = u tot (f 1 ( S 1 ) S 1 S 1 + f 2 ( S 2 ) S 2 S 2 ), (103) GCEP Technical Report

50 which is equal to u tot, only if S 2 = S 2, where S 2 is the exact expectation of s (the discretization error of the operator is neglected here). In general, this numerical mass balance can lead to large errors. Next, it is shown how this can be avoided using a simple correction scheme. Local Mass Balance We consider the particle evolution given by Eq. (101) together with the numerical approximation S 1,2 S 1,2. The resulting particle displacement is denoted by dx u. The goal is to come up with a correction dx c, such that local mass balance is achieved. We propose dx c = u tot Φ u tot Φ ( ) 1 f 1( S 1 ) S 1 ( ) 1 f 2( S 2 ) S 2 dt, if a = 0 dt, if a = 1, where a {0, 1} is an independent random variable with a = S 2. Note that the numerical expectation of the total particle flux based on the velocity v = (dx u + dx c)/dt is (104) Φ(S 1 v s = 0 + S 2 v s = 1 ) = u tot (f 1 ( S 1 ) S 1 S 1 + f 2 ( S 2 ) S 2 S 2 ) (105) + u tot (1 f 1 ( S 1 ) 1 a S 1 + f 2 ( S 2 ) a S 2 ), which becomes equal to u tot provided the relation a = S 2 = s is valid. Another requirement for a is independence of s. Honoring these two constraints for a is crucial, but not straightforward. Here, we explain how this can be achieved. We first consider the hypothetical case with an infinite number of particles. Since S 2 (x) = s (x), it is natural to set a equal to the property s of a randomly chosen particle located at x. In practical cases with finite particle numbers, this procedure can be replaced by randomly choosing a particle from the n nearest neighbors. Provided the particle number is large enough, the corresponding value of s can be regarded as a quasi-random number, which approximately fulfills the requirements for a. For all our test cases this method to determine a proved very successful. Next, it is shown that the particle displacement dx u + dx c is consistent with the saturation equations (99) and (100). Consistency If one neglects the discretization error of the operator, the expected flux of particles with s = 0 can be expressed as ΦS 1 v s = 0 = u tot f 1 ( S 1 ) S 1 S 1 (106) + u tot S 1 (1 f 1 ( S 1 ) S 1 + f 2 ( S S 2 ) S 2 ) 1 S ( 2 + S 1 f 1 ( S 1 ) cλ 2( S ) ( 2 ) f S ( S 1 ) cλ 2( S ) 2 ) S S GCEP Technical Report

51 With the following analysis, we show that this flux is consistent with the phase-one flux u 1 = u tot cλ 2 f 1 f 1 S S (107) Assuming S 1,2 = S 1,2 + O(ε) one can easily show that for small ε ΦS 1 v s = 0 = u tot f 1 (S 1 ) (108) + u tot S 1 (1 f 1 (S 1 ) + f 2 (S 2 )) ( + S 1 f 1 (S 1 ) cλ ) 2(S 2 ) S2 ( 2 + f 1 (S 1 ) cλ ) 2(S 2 ) S S O(ε) = u tot f 1 (S 1 ) f 1 (S 1 ) cλ 2(S 2 ) S S O(ε), (109) which is equal to u 1. Analogously, consistency for the phase two particle flux can be shown. Note that in general S 1,2 is approximated by kernel function estimates at grid nodes and subsequent interpolation to the particle positions. Therefore, O(ε) can be expressed as a function of grid spacing h as h n, where n is some positive number. Next, the time integration scheme used for particle tracking and the overall solution algorithm are described. Particle Tracking and Solution Algorithm The coupled flow and transport problems are solved sequentially, i.e. the pressure equation (96) using the estimate S 1,2 S 1,2 is solved with a classical finite-volume scheme to obtain u tot (from Eq. (97)), which is then used to compute the particle displacements v dt = dx u + dx c. Here we do not describe the finite-volume scheme used to calculate p 1. We assume that u tot is always zero, except where sources are applied. For particle tracking, we use a high-order (2nd or 4th order) Runge-Kutta scheme. Below we describe the second order method: The first half-step x n+1/2 = x n + ( ( cf1 ( S 1 )λ 2 ( S 2 ) Φ S 2 2 ) if a = 0 s = 0 ( ( cf1 ( S 1 )λ 2 ( S 2 ) Φ S 2 2 ) if a = 1 s = 0 ( ( cf1 ( S 1 )λ 2 ( S 2 ) Φ S 2 2 ) if a = 0 s = 1 ( ( cf1 ( S 1 )λ 2 ( S 2 ) Φ S 2 2 ) if a = 1 s = 1, + utot Φ + utot Φ + utot Φ + utot Φ ( )) 1 + f 1( S 1 ) S 1 f 1( S 1 ) dt S, 1 2 ( )) 1 + f 1( S 1 ) S 1 f 2( S 2 ) dt S, 2 2 ( )) 1 + f 2( S 2 ) S 2 f 1( S 1 ) dt S, 1 2 ( )) 1 + f 2( S 2 ) S 2 f 2( S 2 ) dt S, 2 2 (110) GCEP Technical Report

52 is to estimate the particle position x n+1/2 at the time t n + dt/2. Therefore, u tot and S 1,2 at x n and t n are used. Finally, the new particle position at t n+1 is computed by the second step x n+1 = x n + u dt, (111) where u tot and S 1,2 at x n+1/2 and t n are used to determine u. Note that it is important that a has the same value in both steps (Eqs. (110) and (111)). The overall solution algorithm is outlined below: pseudo code of solution algorithm: distribute particles consistently with initial conditions set time t = 0 and time step n = 0 do { estimate S 1,2 and compute u tot with finite-volume flow solver determine time step size dt, such that CFL condition is fulfilled dt = min(dt, t final t) particles: { determine a at location x n = x (t) determine u tot, S 1,2 and S 1,2 at location x n compute new position x n+1 } t = t + dt and n = n + 1 collect particles in inflow and outflow boundary regions distribute new particles consistently in inflow boundary regions } until (t = t final ) Initially, the particles are distributed consistently. During each time step iteration, first u tot is computed with a finite-volume scheme, and then each particle is moved according to Eqs. (110) and (111) as described above. Finally, the time step loop terminates when the final time t = t final is reached. Numerical Results In this section, we present numerical results for one-dimensional (1D) and two-dimensional (2D) immiscible, nonlinear, two-phase flow problems. The 1D test case is similar to the classical Buckley-Leverett problem except that we consider capillary pressure effect. For the 2D test case, we choose a quarter of five spot configuration. The reservoir is initially saturated with phase one, while phase two is injected at one of the corners. In both test cases, quadratic relative permeabilities were used for the two phases, with constant and equal viscosities. The numerical values of permeability, porosity and viscosity are equal to unity in all the results presented below. Figure 28 depicts results from the stochastic particle and finite difference method (SPM and FDM, respectively) for the 1D test case after 0.5s with u tot =1ms 1. GCEP Technical Report

53 Different capillary pressure were used, i.e. c=0.01pa for Fig 1a and c=0.02pa for Fig 1b. The reference solution was computed with the FDM using a very fine grid with 2500 nodes (dotted lines). The solid and dashed lines represent the SPM and FDM results, respectively, both based on a grid with 100 nodes. For all SPM simulations the time step size was 0.001s and in order to obtain smooth results 100,000 particles per cell were used. One can easily observe that the SPM gives more accurate results than the FDM for the same grid. While computational efficiency considerations are important, they are not discussed further here. Figure 29 depicts the particle number density of the results corresponding to Fig. 1a, where the dotted and solid lines denote results with and without correction, respectively. It can be seen that the correction reduces the local mass balance error; however, for the case presented, the results are quite good without applying the correction term. Figures 30 and 31 depict SPM results of the 2D test after 0.25 pore volume injected (PVI). The grid size is 100x100 and the wells are treated as constant source terms uniformly distributed over regions consisting of 10x10 cells at opposite corners. The source strength per unit area is 100s 1 and c=0.04pa. For this study, a frozen, but non-uniform velocity field, was employed, i.e. the flow field was not updated. Note, however, that this poses no limitations FDM (fine grid) FDM (same grid) SPM 0.8 FDM (fine grid) FDM (same grid) SPM S S x(m) (a) x(m) (b) Figure 28: Simulation results of the 1D test case; (a) c=0.01pa and (b) c=0.02pa Conclusions and Future Work A stochastic particle framework to model multi-phase transport in porous media is presented. This report provides a proof of concept for the new Lagrangian approach capable of modeling nonlinear immiscible transport in porous media. We are working on a careful multi-dimensional validation study. After that, gravity effects will be included. Then, residual trapping models will be incorporated and analyzed. The objective is to arrive at a better understanding of the link between the physical phenomena that govern multiphase flow at the microscopic (pore) and macroscopic GCEP Technical Report

54 1.04 with correction without correction 1.02 n x(m) Figure 29: Simulation results of the 1D test case; particle number density with and without correction y y x (a) x (b) Figure 30: Simulation results of the 2D test case after 0.25 PVI; (a) saturation contours of the injected phase; (b) particle distribution of the injected phase GCEP Technical Report

55 1 S y x Figure 31: Simulation results of the 2D test case after 0.25 PVI; saturation surface plot of the injected phase (Darcy) scales. Eventually, we plan to develop this particle-based framework so that highly resolved multi-scale multi-physics studies of subsurface CO 2 sequestration processes are achieved. GCEP Technical Report

56 Development of CO 2 Sequestration Modeling Capabilities in Stanford s General Purpose Research Simulator Investigators Lou Durlofsky (Professor of Petroleum Engineering), Hamdi Tchelepi (Associate Professor of Petroleum Engineering), Huanquan Pan (Research Associate), Yaqing Fan (Graduate Research Assistant) Introduction The sequestration of carbon dioxide in deep saline aquifers requires accurate and reliable modeling capabilities. Engineering models will eventually be used in all phases of the sequestration operation; i.e., for site selection, management and optimization during CO 2 injection, and long-term monitoring. To address these needs, and to provide a platform for the incorporation of research results (such as those described elsewhere in this report), we are developing a comprehensive CO 2 modeling capability within Stanford s General Purpose Research Simulator (GPRS). The initial version of GPRS was developed by Cao [12] in Over the last four years, GPRS has become the main research platform for the Reservoir Simulation Research Group at Stanford (SUPRI-B) and many extensions and enhancements have been introduced. The code is managed using modern software management tools and has a full-time research associate (H. Pan) responsible for (among other things) long-term code stability and maintaining compatibility between the many new algorithms developed by our research group. The GPRS capabilities and development directions are illustrated schematically in Figure 32. The code includes a generalized compositional formulation, which is well-suited for modeling CO 2 injection, and the ability to simulate advanced wells, which may be useful in CO 2 sequestration operations (advanced wells include multilateral or smart wells; e.g., wells with downhole sensors and flow control valves). In subsequent work we plan to assess the applicability of smart wells for CO 2 sequestration operations. GPRS is well-suited for this type of study as it is already linked to an adjoint-based optimization capability. With further development, the adaptive implicit method may become the method of choice for complex sequestration simulations. Finally, the ability to handle generally unstructured grids renders GPRS suitable for the resolution of geological complexity as well as for the geometric complications that may result from the use of advanced wells. Prior to incorporating advanced CO 2 sequestration modeling and optimization capabilities into GPRS, it is essential that we introduce some more basic but nonetheless necessary enhancements. In this initial stage of our work in this topic area, we have therefore extended GPRS to include additional capabilities in the areas of three-phase relative permeability modeling and relative permeability hysteresis modeling. These, as we will show later, are essential for practical modeling of later stages of sequestration operations (e.g., residual trapping). In addition, we have also implemented a fast flash calculation for CO 2 -H 2 O systems. Because in this case we GCEP Technical Report

57 have only two components in the system, the compositions of the gaseous and aqueous phases as functions of pressure can be computed and stored in tabular form before running the simulation. Using this treatment, approximately one third of the total GPRS simulation time is eliminated compared to using an equation of state during the simulations. In the following sections of this report, we describe our three-phase relative permeability and hysteresis models. We then apply GPRS to the simulation of a representative saline aquifer to demonstrate the impact of relative permeability hysteresis and residual trapping in a practical setting. This example also demonstrates that residual trapping (which is a desirable trapping mechanism) can be enhanced through CO 2 injection using a horizontal well. Figure 32: Schematic illustrating Stanford s General Purpose Research Simulator (GPRS) Three-phase relative permeability and hysteresis models The three-phase relative permeability and hysteresis modeling capabilities of GPRS were extended in order to enable more realistic modeling of CO 2 sequestration processes, as we now describe. GPRS now includes two options for three-phase relative permeability modeling, specifically the Baker [7] and Stone 1 [54] models. In Baker s model, also referred to as the segregation model, the gas and water relative permeabilities are first calculated by interpolation from the input rock-fluid properties and the oil phase relative permeability is calculated by [24]: k ro = S gk rog + (S w S wo )k row S g + S w S wco, (112) where S g is gas saturation, S w is water saturation, S wco is connate water saturation, GCEP Technical Report