Showing papers by "Steinar Evje published in 2017"

On global solutions to the viscous liquid–gas model with unconstrained transition to single-phase flow

Abstract: In this paper, we consider the Dirichlet problem of a one-velocity viscous drift-flux model. One of the phases is compressible, the other one is weakly compressible. Under weak assumptions on the initial data, which can be discontinuous and large as well as involve transition to pure single-phase points or regions, we show existence of global bounded weak solutions. One main ingredient is that we employ a decomposition of the pressure term appearing in the mixture momentum equation into two components, one for each of the two phases. This paves the way for deriving a basic energy equality. In particular, upper bounds on the masses are extracted from the estimates provided by the energy equality. By relying on weak compactness tools we obtain an existence result within the class of weak solutions. An essential novel aspect of this analysis, compared to previous works on the same model, is that the solution space is large enough to allow for transition to single-phase flow without any constraints. In particular, one of the phases can vanish in a point while the other phase can persist. The key to achieve this result, which represents a major step forward compared to previous results for this model, is that we do not rely on any higher-order (i.e. derivatives in space) estimates on the masses or pressure, only low-order estimates provided by the energy equality and the uniform upper bounds on the liquid and gas mass.

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

Abstract: A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as $$R_{{i}}$$ (i = water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms $$\lambda _\mathrm{wo}$$ and $$\lambda _\mathrm{ow}$$ are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient $$I>0$$ which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio $$\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} $$ is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms $$R_{{i}}$$ and the fluid–fluid interaction term R.

Relaxation limit of a compressible gas–liquid model with well–reservoir interaction

Abstract: This paper deals with the relaxation limit of a two-phase compressible gas–liquid model which contains a pressure-dependent well–reservoir interaction term of the form \(q (P_r - P)\) where \(q>0\) is the rate of the pressure-dependent influx/efflux of gas, P is the (unknown) wellbore pressure, and \(P_r\) is the (known) surrounding reservoir pressure. The model can be used to study gas-kick flow scenarios relevant for various wellbore operations. One extreme case is when the wellbore pressure P is largely dictated by the surrounding reservoir pressure \(P_r\). Formally, this model is obtained by deriving the limiting system as the relaxation parameter q in the full model tends to infinity. The main purpose of this work is to understand to what extent this case can be represented by a well-defined mathematical model for a fixed global time \(T>0\). Well-posedness of the full model has been obtained in Evje (SIAM J Math Anal 45(2):518–546, 2013). However, as the estimates for the full model are dependent on the relaxation parameter q, new estimates must be obtained for the equilibrium model to ensure existence of solutions. By means of appropriate a priori assumptions and some restrictions on the model parameters, necessary estimates (low order and higher order) are obtained. These estimates that depend on the global time T together with smallness assumptions on the initial data are then used to obtain existence of solutions in suitable Sobolev spaces.

Improved Modeling of Gravity-Aided Spontaneous Imbibition Using Momentum-Equation-Based Relative Permeabilities

Abstract: It is well known that relative permeabilities (RPs) can vary depending on the flow configuration and are lower during counter-current flow as compared to co-current flow. In this paper we use a novel two-phase momentum-equation approach to generate effective RPs where this dependence (and others) is well captured whereby the fluids transfer momentum due to fluid-rock interaction and fluid-fluid interaction. During co-current flow the faster moving fluid accelerates the slow fluid, but is itself decelerated, while for counter-current flow they are both decelerated. We investigate recovery of oil from a matrix block surrounded by water due to a combination of gravity drainage (GD) and spontaneous imbibition (SI), relevant for fractured reservoirs. In capillary-dominated systems the flow is counter-current and viscous coupling can result in increased time scale of the recovery process. During gravity-dominated flow it is more co-current and applying co-currently measured relative permeabilities from the lab becomes a better assumption. Using one set of parameters the momentum-equation approach is thus able to model the behavior of blocks of different operating at different Bond numbers in the reservoir.