Showing papers by "Steinar Evje published in 2015"

A Model for Wettability Alteration in Fractured Reservoirs

Abstract: We present a mathematical model for wettability alteration (WA) in fractured reservoirs. Flow in the reservoir is modeled by looking at a single fracture surrounded by matrix on both sides. Water is injected into the formation with a chemical component that enters the matrix and adsorbs onto the rock surface. These changes of the mineral surface are assumed to alter the wettability toward a more water-wet state, which leads to enhanced recovery by spontaneous imbibition. This can be viewed as a representation of “smart water” injection in which the ionic composition of injection brine affects recovery. The WA is described by shifting curves for relative permeability and capillary pressure from curves representing preferentially oil-wet (POW) conditions toward curves representing more-water-wet conditions. The numerical code was successfully compared with ECLIPSE for the specific case in which a fixed wetting state is assumed. Also, the relevance of the WA model was illustrated by modeling a spontaneous-imbibition experiment in which only a modification of the brine composition led to a change in oil recovery. The model can predict sensitivity to matrix properties such as wettability, permeability, and fracture spacing and to external parameters such as schedule of brine compositions and injection rate. Our model illustrates that one cannot use conventional reservoir modeling to capture accurately the behavior we observe. The rate of recovery and the level of recovery have a strong dependency on the component chemistry and its distribution. A significant feature of gradual WA by injecting a component is that the rate of fluid transfer is maintained between matrix and fracture. The resulting recovery profile after water breakthrough can behave close to linear as opposed to the square-root-of-time profile that is observed when the wetting state is fixed (Rangel-German and Kovscek 2002). The water will typically break through early as dictated by the initial POW state, but a higher final recovery will be obtained because higher saturations can imbibe. Improved understanding of the coupling between WA controlled by water/rock chemistry and fracture/matrix flow is highly relevant for gaining more insight into recovery from naturally fractured reservoirs.

Global solutions of a viscous gas-liquid model with unequal fluid velocities in a closed conduit ∗

Abstract: We consider a compressible gas-liquid drift-flux model with a general slip law commonly used to describe realistic two-phase flow scenarios. The slip law will introduce a difference in the magnitude of the two fluid velocities, and they possibly will also have different sign. This allows the model to describe the effect of buoyant forces, for example, in a vertical conduit, where heavy liquid will move downward due to gravity whereas light gas will be displaced upwardly. Combining the two mass equations and the mixture momentum equation with the slip law, the model can be expressed in terms of the gas-fluid velocity and a generalized pressure function that depends on the common pressure and three new terms that depend on the liquid mass and the gas velocity. This generalized pressure function introduces new nonlinear effects and coupling mechanisms between the two mass equations and the mixture momentum equation which require careful refinements of techniques previously used for the analysis of the classi...

On the large time behavior of the compressible gas–liquid drift-flux model with slip

Abstract: The main purpose of this work is to study the long-time behavior of a compressible gas–liquid model based on the drift-flux formulation. The model is composed of two mass conservation equations and one mixture momentum equation. The flow domain is closed at one end and involves a free gas–liquid interface at the other where both phases vanish. The model includes a slip between the gas and liquid phases, i.e. they move with different velocities. This is a main reason why the model is useful for many industrial applications. We introduce a reformulation of the model based on the gas velocity. This gives rise to new nonlinear terms in the mixture momentum equation which account for the difference in the gas and liquid velocity. New challenges in the mathematical analysis will then appear. In particular, under appropriate smallness conditions on initial data (initial energy) various time-independent estimates of gas and liquid masses, as well as fluid velocities, are obtained. Novel upper and lower bounds on masses are provided that contain precise information about the time-dependent decay rate. Hence, the long-time behavior can be directly extracted from these estimates.

Global solutions to a one-dimensional non-conservative two-phase model

Abstract: In this paper we investigate a basic one-dimensional viscous gas-liquid model based on the two-fluid model formulation. The gas is modeled as a polytropic gas whereas liquid is assumed to be incompressible. A main challenge with this model is the appearance of a non-conservative pressure term which possibly also blows up at transition to single-phase liquid flow (due to incompressible liquid). We investigate the model both in a finite domain (initial-boundary value problem) and in the whole space (Cauchy problem). We demonstrate that under appropriate smallness conditions on initial data we can obtain time-independent estimates which allow us to show existence and uniqueness of regular solutions as well as to gain insight into the long-time behavior of the model. These results rely strongly on the fact that we can derive appropriate upper and lower uniform bounds on the gas and liquid mass. In particular, the estimates guarantee that gas does not vanish at any point for any time when initial gas phase has a positive lower limit. The discussion of the Cauchy problem is general enough to take into account the possibility that the liquid phase may vanish at some points at initial time.

Analysis of a Compressible Two-Fluid Stokes System with Constant Viscosity

Abstract: Basic properties of a reduced viscous compressible gas–liquid two-fluid model are explored. The model is composed of two conservation laws representing mass balance for gas and liquid coupled to two elliptic equations (Stokes system) for the two fluid velocities and obtained by ignoring acceleration terms in the full momentum equations. First, we present a result that shows existence and uniqueness of regular solutions for a fixed time T 0 > 0 which depends on the initial data and the constant viscosity coefficients. Moreover, T 0 can be large when the viscosity coefficients are large. However, for a fixed set of viscosity coefficients, we conjecture that the smooth solution might blow up, at least, as time tends to infinity. This result is backed up by considering a numerical example for a fixed set of viscosity coefficients demonstrating that for smooth and small initial data with no single-phase regions, the solution may tend to produce both single-phase regions and blow-up of mass gradients as time becomes large.

A model for low salinity flooding experiments: dissolution and ion exchange

Abstract: Low salinity water as a means of improving oil recovery has recently generated interest both from petroleum science and industry. A number of experimental studies have highlighted the importance of carbonate solubility in the recovery process. In this paper, which is a continuation of the work in [28], we present a one dimensional mathematical model that couples multicomponent ion exchange, carbonate solubility and transport of the water and oil phases. The transport of the phases is linked to the desorption of the divalent ions from the clay surface in such a way that increased desorption of the divalent ions leads to improved flow function. We first compare the flow model against some published experimental data, demonstrating that the model is able to capture important trends in the experimental results. We then use the flow model to study the effect of multicomponent ion exchange and carbonate solubility on oil recovery during simulated floods with different low salinity brines and sea water like brines. We find that the recovery is very dependent on the individual brine composition and that dissolution of calcite tends to reduce desorption of calcium ions from the rock surface and hence the possibility to improve recovery by the multicomponent ion exchange mechanism. The study demonstrates how the calcite dissolution can change the brine composition of the injected brine deep in the reservoir and hence alter the intended chemistry of the brine-rock interaction.

An Analytical Model for Imbibition Experiments with Porous Plate

Abstract: Imbibition experiments with porous plate can be used to derive accurate capillary pressure curves for porous media flow. A setup is considered where a brine spontaneously imbibes cocurrently through a water-wet porous disc and into a mixed-wet core. The capillary pressure is reduced in steps to allow for imbibition of more brine and determine distinct points on the capillary pressure curve. The aim of this work is to present a mathematical model which can be used for the interpretation or design of such experiments. A numerical discretization of the proposed model approach is used to historymatch experimental data (Ahsan et al, 2012) and evaluate the role of mechanisms controlling the process. An analytical solution of the model is then derived. It is validated against the full numerical solution. The simplest analytical solution for the imbibition time profile is of an exponential form with a time scale $tau$. Such a model has been applied earlier in the literature, however we show that it is an appropriate formulation and provide an explicit expression for the time scale.