Numerical modelling of three-phase immiscible flow in heterogeneous porous media with gravitational effects

Abreu, Eduardo

doi:10.1016/j.matcom.2013.09.010

Cited by 31 publications

(119 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…5 and 6 were carried out using an accurate and locally conservative computational method described in detail in [1][2][3][4]. This numerical simulator is able to handle non-classical shock waves in three-phase problems for one-dimensional [3,4] and for two-dimensional heterogeneous flow problems in porous media with gravity [1,3,4], see also [33]. The operator splitting technique employed allows the use of distinct time steps for the solution of advection (hyperbolic) and diffusion (parabolic) problems defined by the procedure.…”

Section: A Computational Methods For Three-phase Flow In Porous Mediamentioning

confidence: 99%

On a universal structure for immiscible three-phase flow in virgin reservoirs

et al. 2016

Self Cite

View full text Add to dashboard Cite

We discuss the solution for commonly used models of the flow resulting from the injection of any proportion of three immiscible fluids such as water, oil, and gas in a reservoir initially containing oil and residual water. The solutions supported in the universal structure generically belong to two classes, characterized by the location of the injection state in the saturation triangle. Each class of solutions occurs for injection states in one of the two regions, separated by a curve of states for most of which the interstitial speeds of water and gas are equal. This is a separatrix curve because on one side water appears at breakthrough, while gas appears for injection states on the other side. In other words, the behavior near breakthrough is flow of oil and of the dominant phase, either water or gas; the nondominant phase is left behind. Our arguments are rigorous for the class of Corey models with convex relative permeability functions. They also hold for Stone's interpolation I model [5]. This description of the universal structure of solutions for the injection problems is valid for any values of phase viscosities. The inevitable presence of an umbilic point (or of an elliptic region for the Stone model) seems to be the cause of this universal solution structure. This universal structure was perceived recently in the particular case of quadratic Corey relative permeability models and with the injected state consisting of a mixture of water and gas but no oil [5]. However, the results of the present paper are more general in two ways. First, they are valid for a set of permeability functions that is stable under perturbations, the set of convex permeabilities. Second, they are valid for the injection of any proportion of three rather than only two phases that were the scope of [5].

show abstract

Section: A Computational Methods For Three-phase Flow In Porous Mediamentioning

confidence: 99%

On a universal structure for immiscible three-phase flow in virgin reservoirs

et al. 2016

Self Cite

View full text Add to dashboard Cite

show abstract

“…A motivation is the occurrence of models such as (1) in numerous engineering problems. There are a number of prototype relevant models with discontinuous nonlinearities in oil trapping phenomenon [9,10,23,50,70,78], a Whitham model of car traffic flow on a highway [49,67] and a model of continuous sedimentation in ideal clarifier-thickener units [21], see also [17,52].…”

Section: Introductionmentioning

confidence: 99%

Weak asymptotic methods for scalar equations and systems

Abreu

Colombeau

Panov

2016

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

Please cite this article in press as: E. Abreu et al., Weak asymptotic methods for scalar equations and systems, J. Math. Anal. Appl. (2016), http://dx. AbstractIn this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslov asymptotic analysis. We obtain a sequence of functions which tend to satisfy the equation(s) in the weak sense in the space variable and in the strong sense in the time variable. To this end we reduce the partial differential equations to a family of ordinary differential equations in a classical Banach space. For scalar equations we prove that the initial value problem is well posed in the L 1 sense for the approximate solutions we construct. Then we prove that this method gives back the widely accepted solutions when they are known. For systems we obtain existence in the general case and uniqueness in the analytic case using an abstract Cauchy-Kovalevska theorem.

show abstract

“…In the approximation of the source term Q(U (x , t)), we use the interpolants linear in the score term, yielding the predictor-corrector central differencing scheme (for further details see [1,2]):…”

Section: An Unsplitting Approximate Algorithm For Relaxation Balance mentioning

confidence: 99%

“…, along with a rexalation time factor > 0 (1) where G, F and Q are smooth vector functions of U = U (x , t) : × + → n . System (1) appears in real world problems in biology, chemistry, physics, geophysics, gas dynamics and recovery of petroleum, just to name a few.…”

Section: Introductionmentioning

confidence: 99%

“…The rest of the manuscript is organized as follows. In the Section 2 we present a description of our unsplitting approximate algorithm for system (1). In the Section 3 we present and discuss numerical experiments for three balance law problems: injection of steam and nitrogen in a porous medium under and without thermodynamical equilibrium, the Euler equations with gravity and friction, and a shallow-water flow model with friction.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A unsplitting finite volume method for models with stiff relaxation source terms

Abreu

Bustos²,

Lambert

2016

Bull Braz Math Soc, New Series

Self Cite

View full text Add to dashboard Cite

We developed an unsplitting finite volume scheme to account the delicate nonlinear balance between numerical approximations of the hyperbolic flux function and the source linked to balance laws. The method is Riemann-solver-free and no upwinding technique is used. By means of this new approach, we conducted an analysis for two new models of balance laws linked to compositional and thermal flow in porous media problems, under and without a thermodynamic equilibrium hypothesis. For concreteness, we adopt the nitrogen and steam injection models in a porous media. To this model we found an interesting behavior linked to the relaxation term, which is the existence of a non-monotonic traveling wave. We applied this numerical technique to others well-known differential models with relaxation terms available in the literature. Qualitatively we were able to reproduce the expected results.

show abstract

Numerical modelling of three-phase immiscible flow in heterogeneous porous media with gravitational effects

Cited by 31 publications

References 49 publications

On a universal structure for immiscible three-phase flow in virgin reservoirs

On a universal structure for immiscible three-phase flow in virgin reservoirs

Weak asymptotic methods for scalar equations and systems

A unsplitting finite volume method for models with stiff relaxation source terms

Contact Info

Product

Resources

About