Analytical solutions for forced and spontaneous imbibition accounting for viscous coupling

Fluid-fluid momentum transfer can cause higher flow resistance when fluids flow in opposite directions as compared to the same direction. Conventional modelling of flow in porous media using simple, saturation dependent relative permeabilities does not account for such variations. We consider a generalized theory for multiphase flow in porous media based on mixture theory, where fluid mobilities follow from water-rock, oil-rock and water-oil interaction terms defined in momentum equations. Under strictly co- or counter-current flow modes, the generalized model produces explicit relative permeability expressions dependent on the flow mode, saturations, viscosities and interaction parameters. New expressions for counter-current relative permeabilities are derived assuming zero net flux, representative of counter-current spontaneous imbibition. These functions are compared to previously derived co-current relative permeabilities (assuming equal phase pressure gradients). The functions are incorporated into analytical solutions for forced and spontaneous imbibition (FI and SI) using the theory by Buckley and Leverett (1942) and McWhorter and Sunada (1990), respectively. Our results show that when accounting for viscous coupling; Counter-current relative permeabilities are always lower than co-current ones, including the end points. Both phase curves are reduced by the same saturation dependent coefficient. Increased viscous coupling in the FI case led to a more effective displacement, seen as an increased front saturation and average water saturation behind the front. For counter-current SI, increased viscous coupling resulted in lower imbibition rate. Increased viscosities reduces both oil and water counter-current relative permeabilities, and predict greater reduction in imbibition rate than only modifying the viscosities. The analytical solutions for SI were in agreement with numerical solutions of both a conventional and generalized model. The solutions for SI could be scaled exactly to a square root of time curve for arbitrary input parameters in the generalized model, especially including the strength of viscous coupling.publishedVersio

Combination of Conventional and Optimisation Techniques for Performance Prediction in Large Waterflood Projects

Imperial Users onl

Simulation study of co-current spontaneous imbibition

Master's thesis in Petroleum engineeringSpontaneous imbibition is the main driving mechanism for obtaining high recovery from the naturally fractured reservoirs with low permeable matrix. The present thesis presents the results of a simulation study of one-dimensional, co-current spontaneous imbibition in a strongly water-wet sample. Experimental data used for this work was taken from Haugen et al. (2014, 2015). The circumstances of the experiments were characterized by one end face of the core to be open to brine (an inlet) and the other end face to be open to oil (an outlet). Under this Two-Ends-Open (TEO) boundary condition both co- and counter-current flow can take place at the same time, in other words, the inlet can be produced counter-currently and the outlet - co-currently. The simulation program IORCoreSim was used in this thesis to model the system. The water-oil flow was developed by using Corey relative permeability type and J-function capillary pressure correlation. The experiments were matched by establishing relative permeability and capillary pressure curves. After the match was obtained, the saturation functions were used to perform the sensitivity analysis. It was done by varying several parameters: mobility ratio by holding one of viscosities fixed while changing the other, then both viscosities at fixed mobility ratio, and furthermore capillary back pressure. The last two cases were performed at M=0.01 and M=11. With increased oil viscosity at fixed water viscosity, the imbibition rate was observed to be lower with decreasing co-current recovery, while counter-current recovery was increased. The breakthrough time was delayed. With increased water viscosity at fixed oil viscosity, the trends for inlet and outlet recovery were similar with increased imbibition time. The breakthrough time was also delayed. For fixed mobility ratio with varying both viscosities, the trend showed that increased viscosity ratio has no impact on total production and co-current recovery was reduced as M increased whereas counter-current increased. The capillary back pressure influenced essentially the system at M=11 when compared with M=0.01. Counter-current recovery decreased with increasing capillary back pressure at values beyond the threshold capillary pressure

An analytical model to predict the effects of suspended solids in injected water on the oil displacement efficiency during waterflooding

Suspended solids in the injection water cause impairment of water injectivity during waterflooding operations. Suspended solids affect reservoir properties and decrease the permeability of reservoir rocks causing an increase of injection pressure and a decrease in water injectivity. Removal of all suspended solids from injection water is an expensive and economically unfeasible process. To minimize the effects of suspended solids to the formation, it is necessary to determine an impairment mechanism of suspended solids on oil displacement and, therefore, optimize the water treatment process. In this paper, an analytical model that describes the relationship between injection water quality and impairment mechanisms on oil displacement is presented. A formation impairment was calculated, introducing the parameter called impairment ratio, which represents the ratio between suspended solids and pore size distribution of reservoir rock. Based on the impairment ratio, decreases in porosity and permeability were calculated with changes in capillary pressure, relative permeability, and displacement efficiency. The model was tested for three different types of injection water. Results indicated the presence of formation impairment even with the smallest particles. Suspended solids had the greatest influence on porosity and permeability impairment. The model could be used as input for reservoir modelling studies for monitoring and controlling displacement efficiency during waterflooding as well as for planning and modification of water treatment units

Traveling waves in a finite condensation rate model for steam injection

Steam drive recovery of oil is an economical way of producing oil even in times of low oil prices and is used worldwide. This paper focuses on the one-dimensional setting, where steam is injected into a core initially containing oil and connate water while oil and water are produced at the other end. A three-phase (oil, water, steam) hot zone develops, which is abruptly separated from the two-phase (oil + water) cold zone by the steam condensation front. The oil, water and energy balance equations (Rankine–Hugoniot conditions) cannot uniquely solve the system of equations at the steam condensation front. In a previous study, we showed that two additional constraints follow from an analysis of the traveling wave equation representing the shock; however, within the shock, we assumed local thermodynamic equilibrium. Here we extend the previous study and include finite condensation rates; using that appropriate scaling requires that the Peclet number and the Damkohler number are of the same order of magnitude. We give a numerical proof, using a color-coding technique, that, given the capillary diffusion behavior and the rate equation, a unique solution can be obtained. It is proven analytically that the solution for large condensation rates tends to the solution obtained assuming local thermodynamic equilibrium. Computations with realistic values to describe the viscous and capillary effects show that the condensation rate can have a significant effect on the global saturation profile, e.g. the oil saturation just upstream of the steam condensation front

Early and Late Time Analytical Solutions for Co-current Spontaneous Imbibition and Generalized Scaling

We propose an explicit analytical solution for 1D cocurrent (COC) spontaneous imbibition (SI) in which a core is exposed to water (inlet) and oil (outlet). The system is described using an advection-capillary diffusion transport equation combined with a pressure equation. By ignoring the capillary diffusion term in the transport equation, the analytical solution follows in terms of Buckley-Leverett (BL) saturation profiles. The capillary force appears in the pressure equation and determines the advective term of the transport equation. The time when the front reaches the outlet (critical time) is calculated and used for scaling. The solution is extended to after critical time (late time) by maintaining the BL profile inside the system, thus preserving continuity in recovery and spatial profiles. The solution is characterized by an effective total mobility and capillary pressure (incorporating the entire saturation functions), both constant at early time (before critical time). At late times, they change dynamically. The model states that the imbibition rate can increase, decrease, and stay constant with time based on a new mobility ratio being less than, more than, or equal to unity, respectively. The ratio also indicates effectiveness of oil displacement. The square root of time recovery is a special case only seen for a (very) favorable mobility ratio. The model predicts that COC imbibition scales with the square of length both at early and late times and that the solution can scale saturation functions. The analytical solution was compared against numerical simulations of the full system. The new mobility ratio reflected the evolution in COC recovery better than total recovery. The analytical solution showed a too-high imbibition rate at a favorable mobility ratio. The diffusion term is important then due to strong saturation gradients, and the resulting smoothened profile yields a lower imbibition rate from the pressure equation. The analytical solution showed a too-low imbibition rate at early times for unfavorable mobility ratio due to not accounting for rapid early countercurrent (COUC) production. The analytical solution predicted a too-high imbibition rate at late times because the BL profile does not capture the oil mobility restriction at the outlet at late times. The time of water reaching the outlet was underestimated by a factor ∼ 2 for strongly water-wet (SWW) simulations and ∼ 10 for mixed-wet (MW) simulations. Scaling recovery with length squared was exact for all times. Scaling recovery until water reaching the outlet demonstrated consistency across saturation functions and viscosities. The analytical solution could match literature experimental data and produce corresponding saturation functions. To our knowledge, previous analytical solutions have only considered infinite-acting systems (early time), assumed piston-like displacement (PLD) (uniform saturations on both sides of a saturation shock front) or are implicit, thus not providing more insight than numerical simulations.acceptedVersio

Benchmark of different CFL conditions for IMPES

The IMplicit Pressure Explicit Saturation (IMPES) method is a prevalent way to simulate multiphase flows in porous media. The numerical stability of this sequential method implies limitations on the time step, which may depend on the flow regime studied. In this note, three stability criteria related to the IMPES method, that differ in their construction on the observed variables, are compared on homogeneous and heterogeneous configurations for different two-phase flow regimes (viscous/capillary/gravitational). This highlights that there is no single optimal criterion always ensuring stability and efficiency. For capillary dominated flows, the Todd’s condition is the most efficient one, while the standard Coat condition should be preferred for viscous flows. When gravity effects are present, Coat’s condition must be restricted, but remains more efficient than the Todd’s condition

Pore-Scale Modeling: Stochastic Network Generation and Modeling of Rate Effects in Waterflooding

Pore scale network modeling has been used to predict transport flow properties for multiphase flow successfully. The prediction is based on having geologically realistic networks that are computationally expensive to generate and normally represent only a very small section of the rock sample. We present a new method to generate stochastic random networks representing the pore space of different rocks with given input pore and throat size distributions and connectivity – these distributions can be obtained from an analysis of pore-space images. The stochastic networks can be arbitrarily large and hence are not limited by the size of the original image. The basic assumption made in the prediction of transport flow properties using most pore-scale models is that the flow is capillary dominated. This implies that the viscous pressure drop is insignificant compared to the capillary pressure. However, at the field scale, gravity and viscous forces dominate displacement processes. We develop a rate-dependent network model that accounts for viscous forces by solving for the wetting and non-wetting phase pressure and which allows wetting layer swelling near an advancing flood front. We propose a new time-dependent algorithm by accounting for partial filling of elements. We use the model to study the effects of capillary number and mobility ratio on imbibition displacement patterns, saturation and velocity profiles. We also investigate the effects of capillary number and mobility ratio on the water fractional flow curve, cumulative oil production and residual oil saturation for water-wet and mixed-wet systems. By using large networks we reproduce Buckley-Leverett profiles directly from pore-scale modeling thereby providing a bridge between pore-scale and macroscale transport