Hyperbolic-elliptic models for well-reservoir flow
Steinar Evje Kenneth H. Karlsen
Networks & Heterogeneous Media 2006, 1(4): 639-673 doi: 10.3934/nhm.2006.1.639
We formulate a hierarchy of models relevant for studying coupled well-reservoir flows. The starting point is an integral equation representing unsteady single-phase 3-D porous media flow and the 1-D isothermal Euler equations representing unsteady well flow. This $2 \times 2$ system of conservation laws is coupled to the integral equation through natural coupling conditions accounting for the flow between well and surrounding reservoir. By imposing simplifying assumptions we obtain various hyperbolic-parabolic and hyperbolic-elliptic systems. In particular, by assuming that the fluid is incompressible we obtain a hyperbolic-elliptic system for which we present existence and uniqueness results. Numerical examples demonstrate formation of steep gradients resulting from a balance between a local nonlinear convective term and a non-local diffusive term. This balance is governed by various well, reservoir, and fluid parameters involved in the non-local diffusion term, and reflects the interaction between well and reservoir.
keywords: hyperbolic-elliptic model existence advanced well entropy weak solution uniqueness. coupled well-reservoir flow Non-local conservation law
Modelling of wettability alteration processes in carbonate oil reservoirs
Liping Yu Hans Kleppe Terje Kaarstad Svein M. Skjaeveland Steinar Evje Ingebret Fjelde
Networks & Heterogeneous Media 2008, 3(1): 149-183 doi: 10.3934/nhm.2008.3.149
Previous studies have shown that seawater may alter the wettability in the direction of more water-wet conditions in carbonate reservoirs. The reason for this is that ions from the salt (sulphat, magnesium, calsium, etc) can create a wettability alteration toward more water-wet conditions as salt is absorbed on the rock.
   In order to initiate a more systematic study of this phenomenon a 1-D mathematical model relevant for spontaneous imbibition is formulated. The model represents a core plug on laboratory scale where a general wettability alteration (WA) agent is included. Relative permeability and capillary pressure curves are obtained via interpolation between two sets of curves corresponding to oil-wet and water-wet conditions. This interpolation depends on the adsorption isotherm in such a way that when no adsorption of the WA agent has taken place, oil-wet conditions prevail. However, as the adsorption of this agent takes place, gradually there is a shift towards more water-wet conditions. Hence, the basic mechanism that adsorption of the WA agent is responsible for the wettability alteration, is naturally captured by the model.
   Conservation of mass of oil, water, and the WA agent, combined with Darcy's law, yield a 2x2 system of coupled parabolic convection-diffusion equations, one equation for the water phase and another for the concentration of the WA agent. The model describes the interactions between gravity and capillarity when initial oil-wet core experiences a wettability alteration towards more water-wet conditions due to the spreading of the WA agent by molecular diffusion. Basic properties of the model are studied by considering a discrete version. Numerical computations are performed to explore the role of molecular diffusion of the WA agent into the core plug, the balance between gravity and capillary forces, and dynamic wettability alteration versus permanent wetting states. In particular, a new and characteristic oil-bank is observed. This is due to incorporation of dynamic wettability alteration and cannot be seen for case with permanent wetting characteristics. More precisely, the phenomenon is caused by a cross-diffusion term appearing in capillary diffusion term.
keywords: cross-diffusion. capillary diffusion convection-diffusion equations physico-chemical processes wettability alteration spontaneous imbibition porous media flow molecular diffusion adsorption
Global weak solutions for a viscous liquid-gas model with singular pressure law
Steinar Evje Kenneth Hvistendahl Karlsen
Communications on Pure & Applied Analysis 2009, 8(6): 1867-1894 doi: 10.3934/cpaa.2009.8.1867
We study a viscous two-phase liquid-gas model relevant for well and pipe flow modelling. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid leading to a pressure law which becomes singular when transition to single-phase liquid flow occurs. In order to handle this difficulty we reformulate the model in terms of Lagrangian variables and study the model in a free-boundary setting where the gas and liquid mass are of compact support initially and discontinuous at the boundaries. Then, by applying an appropriate variable transformation, point-wise control on masses can be obtained which guarantees that no single-phase regions will occur when the initial state represents a true mixture of both phases. This paves the way for deriving a global existence result for a class of weak solutions. The result requires that the viscous coefficient depends on the volume fraction in an appropriate manner. By assuming more regularity of the initial fluid velocity a uniqueness result is obtained for an appropriate (smaller) class of weak solutions.
keywords: Two-phase flow Lagrangian coordinates free boundary problem. weak solutions
Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operations
Steinar Evje Huanyao Wen
Discrete & Continuous Dynamical Systems - A 2013, 33(10): 4497-4530 doi: 10.3934/dcds.2013.33.4497
In this work we study a compressible gas-liquid models highly relevant for wellbore operations like drilling. The model is a drift-flux model and is composed of two continuity equations together with a mixture momentum equation. The model allows unequal gas and liquid velocities, dictated by a so-called slip law, which is important for modeling of flow scenarios involving for example counter-current flow. The model is considered in Lagrangian coordinates. The difference in fluid velocities gives rise to new terms in the mixture momentum equation that are challenging to deal with. First, a local (in time) existence result is obtained under suitable assumptions on initial data for a general slip relation. Second, a global in time existence result is proved for small initial data subject to a more specialized slip relation.
keywords: two-phase flow slip law free boundary problem local solutions global solutions. weak solutions Drift-flux model Lagrangian coordinates
A mathematical model for dynamic wettability alteration controlled by water-rock chemistry
Steinar Evje Aksel Hiorth
Networks & Heterogeneous Media 2010, 5(2): 217-256 doi: 10.3934/nhm.2010.5.217
Previous experimental studies of spontaneous imbibition on chalk core plugs have shown that seawater may change the wettability in the direction of more water-wet conditions in chalk reservoirs. One possible explanation for this wettability alteration is that various ions in the water phase (sulphate, calcium, magnesium, etc.) enter the formation water due to molecular diffusion. This creates a non-equilibrium state in the pore space that results in chemical reactions in the aqueous phase as well as possible water-rock interaction in terms of dissolution/precipitation of minerals and/or changes in surface charge. In turn, this paves the way for changes in the wetting state of the porous media in question. The purpose of this paper is to put together a novel mathematical model that allows for systematic investigations, relevant for laboratory experiments, of the interplay between (i) two-phase water-oil flow (pressure driven and/or capillary driven); (ii) aqueous chemistry and water-rock interaction; (iii) dynamic wettability alteration due to water-rock interaction.
   In particular, we explore in detail a 1D version of the model relevant for spontaneous imbibition experiments where wettability alteration has been linked to dissolution of calcite. Dynamic wettability alteration is built into the model by defining relative permeability and capillary pressure curves as an interpolation of two sets of end point curves corresponding to mixed-wet and water-wet conditions. This interpolation depends on the dissolution of calcite in such a way that when no dissolution has taken place, mixed-wet conditions prevail. However, gradually there is a shift towards more water-wet conditions at the places in the core where dissolution of calcite takes place. A striking feature reflected by the experimental data found in the literature is that the steady state level of oil recovery, for a fixed temperature, depends directly on the brine composition. We demonstrate that the proposed model naturally can explain this behavior by relating the wettability change to changes in the mineral composition due to dissolution/precipitation. Special attention is paid to the effect of varying, respectively, the concentration of $\text{SO}_4^{2-}$ ions and $\text{Mg}^{2+}$ ions in seawater like brines. The effect of changing the temperature is also demonstrated and evaluated in view of observed experimental behavior.
keywords: aqueous chemistry precipitation water-rock interaction countercurrent flow wettability alteration. spontaneous imbibition dissolution
A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions
Steinar Evje Aksel Hiorth Merete V. Madland Reidar I. Korsnes
Networks & Heterogeneous Media 2009, 4(4): 755-788 doi: 10.3934/nhm.2009.4.755
In this work a mathematical model is proposed for modeling of coupled dissolution/precipitation and transport processes relevant for the study of chalk weakening effects in carbonate reservoirs. The model is composed of a number of convection-diffusion-reaction equations, representing various ions in the water phase, coupled to some stiff ordinary differential equations (ODEs) representing species in the solid phase. More precisely, the model includes the three minerals $\text{CaCO}_3$ (calcite), $\text{CaSO}_4$ (anhydrite), and $\text{MgCO}_3$ (magnesite) in the solid phase (i.e., the rock) together with a number of ions contained in the water phase and essential for describing the dissolution/precipitation processes. Modeling of kinetics is included for the dissolution/precipitation processes, whereas thermodynamical equilibrium is assumed for the aqueous chemistry. A numerical discretization of the full model is presented. An operator splitting approach is employed where the transport effects (convection and diffusion) and chemical reactions (dissolution/precipitation) are solved in separate steps. This amounts to switching between solving a system of convection-diffusion equations and a system of ODEs. Characteristic features of the model is then explored. In particular, a first evaluation of the model is included where comparison with experimental behavior is made. For that purpose we consider a simplified system where a mixture of water and $\text{MgCl}_2$ (magnesium chloride) is injected with a constant rate in a core plug that initially is filled with pure water at a temperature of $T=130^{\circ}$ Celsius. The main characteristics of the resulting process, as predicted by the model, is precipitation of $\text{MgCO}_3$ and a corresponding dissolution of $\text{CaCO}_3$. The injection rate and the molecular diffusion coefficients are chosen in good agreement with the experimental setup, whereas the reaction rate constants are treated as parameters. In particular, by a suitable choice of reaction rate constants, the model produces results that agree well with experimental profiles for measured ion concentrations at the outlet. Thus, the model seems to offer a sound basis for further systematic investigations of more complicated precipitation/dissolution processes relevant for increased insight into chalk weakening effects in carbonate reservoirs.
keywords: dissolution porous media flow upscaling. precipitation physico-chemical processes convection-diffusion-reaction equations damage mechanics water-rock interaction
Global solutions to a one-dimensional non-conservative two-phase model
Steinar Evje Huanyao Wen Lei Yao
Discrete & Continuous Dynamical Systems - A 2016, 36(4): 1927-1955 doi: 10.3934/dcds.2016.36.1927
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.
keywords: uniqueness long-time behavior. Navier-Stokes Two-fluid model existence

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