A new mathematical model for the miscible displacement in fractured-porous reservoirs is developed. A model is obtained by the upscaling of the traditional miscible displacement equations from the scale, which is lower than the fracture opening, up to the scale, which is much larger than the block size.
The model is based on simultaneous use of different mathematical methods: areal averaging of fluid fluxes, asymptotical averaging in systems with periodical heterogeneity, analytical solutions of the problem of interaction between the flux via fractures and the single block. The model takes into account diffusive, gravitational and advective mechanisms of the mass exchange between blocks and fractures and also hydraulic interaction between fluxes via systems of blocks and fractures.
The formula for the modified fractional flow functions which depends on the geometric tortuosity of the system of fractures are proposed. The tortuosity coefficient can be found from the 1-D laboratory displacement data.
The model developed has been validated by comparison with the number of laboratory studies of the miscible displacement in fractured-porous media.
Miscible gas injection into fractured-porous oil and gas-condensate reservoirs is the effective enhanced hydrocarbon recovery method. Nevertheless the efficiency of this IOR method is strongly dependent on parameters of fractured-porous system and on properties of reservoirs and of injected fluids.
The major oil reserves are located in the block system, so the recovery efficiency during the miscible gas injection is determined by mechanisms of the 'fracture-block' mass exchange and by the resulting displacement from blocks.
The displacement efficiency in fractured-porous media with the large opening and the high permeability of fractures and the low permeability of blocks is poor due to the fast breakthrough of the injected gas through the fractured system and low recovery from blocks. For the fractured-porous media with the less contrast between the fracture and the block conductivity the recovery is high.
Nevertheless, the recovery is determined not by the heterogeneity of the fractured-porous system only. If the fractured system conductivity is significantly higher then the one for the block system, but the blocks are small, the recovery could be still high.
The diffusive, gravitational and advective (convective) mechanisms of the fracture-block mass exchange during the two-phase partly miscible flow have been distinguished. The intensity of these three mechanisms are affected differently by the variation of the displacement velocity and also by the viscosity and the density of the injected fluid. Therefore, the comprehensive mathematical model which takes into account all the above mentioned recovery mechanisms is required for the optimization of the miscible gas injection into fractured-porous reservoirs.
Mathematical description of the flow in fractured-porous media are based on the image of the mass transfer in the double-porosity system (two fluxes via systems of fractures and of blocks interact with each other), by G. Barenblatt, Y. Zheltov and I. Kochina. In some other models the flux in blocks is neglected and blocks are treated as the source-sink terms with the given law of the mass exchange.
Another approach is based on the consideration of the fractured-porous media as a periodically heterogeneous media, and the upscaled model is obtained by the asymptotic averaging in periodical systems. The method allows to derive more complex formulae for the fracture-block mass exchange and to obtain the explicit formulae for effective permeabilities for systems of fractures and of blocks.
Diffusive mechanism of the fracture-block mass exchange corresponds to the linear term proportional to the concentration difference in blocks and fractures.
An analytical model for the coreflood honouring the capillary pressure is developed. The analytical solution for the problem of the displacement of oil by water taking into account the capillary pressure is obtained using the method of matched asymptotic expansions.
The model is based on the matching of the capillarity-free Buckley-Leverett solution with two asymptotic expansions in regions of strong capillary pressure effects: with the continuous stabilized-zone solution in the neighbourhood of the displacement front, and with the end-effect solution near to the core outlet. Numerical simulation shows the high accuracy of the analytical solution proposed.
The analytical solution is used for regularization of the inverse problem for determination the relative permeabilities from the coreflood data. Analytical solution allows to transform the inverse problem, which is the ill-posed one in the general formulation, to the system of two integral equations. The system can be solved asymptotically using the regular small parameter method.
Relative phase permeabilities are the basic functions which determine the efficiency of the waterflooding So the accurate laboratory determination of these functions from the routine coreflood procedure is an important problem of applied reservoir engineering.
The most widely spread procedure is the non-steady-state method which consists on the displacement of oil with connate water by water. Measurements of the water cut during the displacement allow to calculate the fractional flow function (Welge's method), while the pressure drop on the core after the breakthrough allows to determine the total mobility function. The fractional flow and the total mobility functions determine relative permeabilities for both phases.
The above mentioned methods are derived from the Buckley-Leverett equation and are based on the assumption that the capillary forces are negligible if compared with the viscous ones. Actually, they do not use the solution of the direct problem by itself, being based only on the self-similarity of the solution.
In short low permeable cores or for low displacement velocities the capillary pressure is of the same order of magnitude as the pressure drop on the core during the displacement. So the above mentioned assumption which allows to neglect the capillary pressure in the governing equation is not valid any more.
To overcome this difficulty, numerous papers have reported simplified approximate solutions of the inverse problem of determination of relative permeabilities honouring capillary pressure from the coreflood tests.
Another possible approach is to couple optimization methods with the numerical solution of flow equations including the capillary effects, where the solution of the inverse problem is found from minimization of the difference between the data measured and the ones predicted by the numerical model. This inverse problem is an ill-posed one. Therefore, this approach can give stable results only for fixed analytical expressions for relative permeability. Nevertheless, very different shapes of relative permeability have been reported in the literature, thus the more-or-less general formulae are to contain a lot of adjusting parameters, which reduces the stability of the optimization method.
Another possibility for the solution of the inverse problem could be the use of an analytical solution for the direct problem. The self-similar solution of the 1-D displacement problem without the capillary pressure is well known.
To describe two-phase displacement with hysteresis we use the Buckley-Leverett model with the imbibition, drainage and scanning fractional flows. Mathematical theory for the initial-boundary non-self-similar problems is developed. Structure of solutions is presented together with the physical interpretation of phenomena.
Analytical solutions for the injection of the water slug with the gas drive and for the sequential injection of water and gas slugs with the water drive are obtained. The solutions show that the hysteresis decreases the gas flux in the case where the drainage relative permeability is lower than the imbibition one, which is a positive effect for the WAG injection.
Hysteretic behaviour of relative permeability curves has long been recognised. Numerous laboratory studies have reported hysteresis of relative permeabilities for one-dimensional flow in cores and physical explanation of the phenomena have been presented.
Non-monotonic change of phase saturations takes place in almost every process of oil recovery (waterflood in heterogeneous reservoirs, WAG injection, water coning, gravitational separation, steam and polymer huf-n-puf stimulations, etc.). Therefore, numerous researches were dedicated to the mathematical modelling of the two-phase history-dependent flow.
Formulae for the drainage, imbibition and scanning relative permeability curves have been developed by C. S. Land and J. E. Killough. In these works, the hysteresis between the primary drainage and imbibition curves have been considered, however, all the imbibition curves have been assumed to be reversible. This assumption is relevant with the simulation of creation of the initial saturation and further floods in the laboratory cores. More complex formulae for hysteretic relative permeability have been derived in.
The mathematical model for two-phase displacement honouring hysteresis consists on the basic mass balance equations and the modified Darcy's law. This model is the same as in the hysteresis- free model, the only difference is the hysteretic parameter which shows the direction of the saturation process.
Introduction of the hysteresis into the Buckley-Leverett two-phase flow model changes the mathematical type of the governing equation as well as initial and boundary conditions. Therefore, a more detailed mathematical investigation which provides with the existence of the global solution, is required ( A monograph is an example of such a study). It is important for the development of numerical methods for reservoir simulators with hysteresis.
A mathematical model for the two-phase flow with hysteresis was developed by D. Marchesin, H. Medeiros and P. J. Paes-Leme. The model contains two unknowns: saturation and the number of previous drainages and imbibitions in each point of the reservoir. The complete solution of the Riemarin problem was obtained. The existence of the global solution have been established. This model considers two extreme fractional flow curves and the scanning curves are not incorporated.
A mathematical model for the fractional flow with scanning curves was proposed by Kh. Furati. The system contains two continuous unknowns: saturation and the hysteretic parameter, which is the maximum value of saturation which have been reached in each point of the reservoir. The author introduced immobile saturation shocks which allowed to obtain solution of the self-similar problem of the decay of the initial discontinuity. The Riemann problem was solved and the classification of self-similar configurations was presented. Self-sharpening solutions have been found for the waterflooding (one hyperbolic equation) and for the polymer flooding (system of two hyperbolic equations).
A new analytical model for the tertiary miscible CO2-WAG is developed.
The model is based on analytical solutions of non-self-similar and self-similar problems for a system of hyperbolic equations of mass conservation laws. Explicit formulae allow to analyze the propagation of displacement and phase transition fronts, mechanisms of trapping of oil with the sequential injection of water and gas slugs, mobility ratios on shock fronts and the dynamics of water and gas slugs.
Six different regimes for gas-water injection, after waterflooding, have been distinguished depending on the water-gas ratio, They differ from each other by the structure of the mixture zone and by the mechanisms of displacement caused by two-phase displacement and phase transitions phenomena.
The analytical model presented shows that the higher the WGR the lower the recovery, but the more favourable is the mobility ratio on the displacement front, which suggest the existence of an optimal water-gas ratio (WGR) for the tertiary miscible WAG.
As it follows from the analytical model there does exist a minimum slug size which prevents gas breakthrough via all the water slugs. With the injection of thinner slugs a connected gas network appears in the reservoir and it catches the front of water creating an unstable gas-oil front at the presence of the connate water only. So, simultaneous injection of gas and water, which corresponds to the reduction of slug size to the zero limit, is not an optimal WAG regime as it was suggested in the literature. On the other hand, the thinner the slugs the higher the displacement efficiency, which suggest the existence of an optimal slug size with the tertiary miscible WAG.
Tertiary miscible gas injection after waterflooding is an effective method of improved oil recovery. The mechanism of an additional recovery is the dissolution of the low mobility residual oil in the gas injected.
Nevertheless, the injected gas has a high mobility, compared to the water one, and this leads to unstable displacement because the injected gas moves mainly in highly permeable zones resulting in low areal sweep efficiency. Due to the lower mobility of water, when compared with oil mobility, the injected gas cannot displace oil from the low permeable zones which have not been swept during the waterflooding.
The displaced oil forms a high viscosity bank in front of the injected gas improving sweep, but not significantly.
Also, some residual oil in the gas swept zone remains unrecovered due to the blocking by water during the primary flood, this water is not removable by the gas injected.
Injection of water during the tertiary gas flood decreases the mobility of the injected fluid. So, the displacement from the water-swept zones is occurs with a higher sweep coefficient when compared with the tertiary gas flooding. At some water-gas ratio the mobility of the gas-water system is even higher than the water mobility, so the injected fluid enters in some zones which that were not swept during the waterflooding.
The local redistribution of the reservoir pressure near to water and gas slugs, which happen due to the different viscosities for water and gas, also results in some improvement of the sweep efficiency.
The effect of the increased sweep efficiency caused by the use of water during the gas flooding has been observed in a number of laboratory simulation studies and in pilot tests.
The physical mechanisms of the incremental recovery using miscible WAG can be captured by a 1-D model that takes into account interaction between water and gas slugs during the sequential injection, phase transitions and effects of phase compositions on relative permeabilities and phase viscosities. This model, however, does not take into account areal heterogeneity and viscous fingering.
A new analytical model for the miscible CO2-WAG is developed. The model is based on analytical solutions of non-self-similar and self-similar problems for system of hyperbolic equations of mass conservation laws. Explicit formulae allow to analyse propagation of displacement and phase transition fronts, mechanisms of trapping of oil with the sequential injection of water and gas slugs, mobility ratios on shock fronts and the dynamics of water and gas slugs.
Five different regimes for gas-water injection have been distinguished depending on the water-gas ratio. They differ from each other by different structure of the mixture zone and displacement mechanisms caused by phenomena of two-phase displacement and phase transitions.
The analytical model presented shows that the higher the WGR the lower the recovery, but the more favourable is the mobility ratio on the displacement front. These suggest the existence of an optimal water-gas ratio (WGR).
As it follows from the analytical model there does exist a minimum slug size which prevents gas breakthrough via all the water slugs. With the injection of thinner slugs a connected gas network appears in the reservoir which will catch up the front of water and will create an unstable gas-oil front at the presence of the connate water only. So, simultaneous injection of gas and water, which accords to the reduction of slug size to the zero limit, is not an optimal WAG regime, as it was suggested in the literature. On the other hand the thinner the slugs the higher the displacement efficiency. These speculations suggest the existence of an optimal slug size with the miscible WAG.
Disadvantages of the traditional waterflooding of oil reservoirs are: - unfavourable mobility ratio on the displacement front for highly viscous oil leading to low sweep efficiency
- strong capillary forces resulting in high residual oil saturation. However, capillary forces with waterflooding cause capillary imbibition of low permeable patterns allowing some incremental oil recovery during the late stage of production.
Miscible gas injection when compared with waterflood causes a significant reduction of the residual oil due to phase transitions and inter phase mass transfer between oil and gas. Nevertheless, the mobility ratio on the gas-oil displacement front is even less favourable than with the waterflood resulting in unstable displacement and viscous fingering. Diffusive mechanism of the oil sweep from low permeable patterns by gas is less intensive than imbibition with the waterflooding.
Sequential injection of water and miscible gas slugs in an oil reservoir (Water-Alternate-Gas Injection, WAG) seems to gain advantages of both, waterflood and miscible gas flooding: mobility of the displacing water-gas system is even lower than water mobility at the same total saturation, so sweep due to heterogeneity and well placing geometry increases with WAG viscous fingering and the early breakthrough are also prevented; miscible gas reduces saturation of oil trapped by water and capillary imbibition of low permeable parts by water works with WAG also.
The importance of all the above mentioned mechanisms was shown by analysis of a number of pilot tests.
The physical mechanisms of the incremental recovery using WAG, mentioned above, can be captured by 1-D model which takes into account interaction between water and gas slugs during the sequential injection, phase transitions and effects of phase compositions on relative permeabilities and phase viscosities. This model, however, does not take into account viscous fingering.
The purpose of this work is to develop an analytical model that produces results close to numerical 3D model for miscible displacement using horizontal wells. An analytical model has been developed for three-dimensional flow in system of horizontal wells. Different systems of horizontal injection and producing wells are discussed. The analytical model includes 3D picture of flow between injection and producing wells, bottom and top of the a reservoir. different viscosities of displaced and injected fluids. The model automatically calculates location of displacement front, breakthrough time and composition of fluid produced. It allows to compare sweep efficiency with different locations of injection and producing wells.
A computer floppy disk for 386 PC is available that contains software with PASCAL programs of the model. and sample data files. Files listing the results with graphical representation are also included.
Miscible displacement by gas injection has been proven to be an effective mean of recovering conventional and volatile oil and gas condensate. Miscible gas or solvent drive, or miscible gas flooding, is a process in which oil or gas condensate is displaced from injection wells toward production ones. Usually viscosity of gas or solvent injected is lower than viscosity of displaced oil or condensate. it leads to the poor areal sweep in a system of vertical wells. Thus application of horizontal wells which improve mainly areal sweep coefficient is particularly effective for the above mentioned cases of miscible displacement. Predictions of reservoir response to the application of gas or solvent are necessary before starting a miscible drive project. Numerical 3D models are available to provide forecasts. However. these models are expensive, require large computers and consume a great deal of computer time. Accurate simulation of reservoir performance with horizontal wells requires complex curvilinear numerical grid with decreasing size in neighborhoods of wells. These difficulties don't enable multivariant comparative study of different systems of horizontal wells in order to optimize plan of the reservoir development plan.
The use of an analytical model is an alternative to the numerical modeling. Analytical models are fast, but do not provide the flexibility and generality that can be obtained from a finite-difference model. The assumptions necessary to generate analytical models may lead to poor results. However, analytical models allow to compare a large number of project options and to perform multivariant sensitivity study during the design and optimization of the reservoir performance.
Explicit analytical formulae for 3D miscible displacement via horizontal wells allow quick estimation of sweep coefficient and other indicators of effective recovery with different locations of horizontal injection and producing wells at different viscosity ratios.
Bedrikovetsky, Pavel (Moscow State Oil & Gas Academy)
In this paper we discuss the problem of the reservoir characterisation from the tracer test data. Oil-water rates and tracer concentration on the productional wells are available. The problem is to determine the permeability profile.
The new pseudorised equations for the displacement of oil by water with tracer in heterogeneous stratified reservoir are derived. Analyses of the tracer wave propagation into the layer cake reservoir during the waterflooding was given on the basis of analytical solution of the direct displacement problem Exact analytical solution of the inverse reservoir characterisation problem provides explicit formulae for the profile of permeability.
The method developed was applied for the interpretation of pilot test on gas injection in the Vuktyl field (Russia, TimanPechora region). The results obtained from the tracer analyses data fits well with the data of logging and core analyses.
The heterogeneity of oil reservoirs is the main factor that determines the recovery during the waterflooding. The profile of heterogeneity could be obtained from the waterflood data . Use of tracers for the control on waterflood increases the quality of reservoir characterisation. But the inverse problem of the permeability profile determination from the waterflood data as well as from the tracer analyses data does not have the unique solution. Tuning of the reservoir model from a history matching can lead to different pictures of heterogeneity.
The inverse problem of permeability profile determination can be solved in frames of models with the low dimensions. Pseudofunctional flow approach is widely used to reduce the dimension of the reservoir model. It reduces the number of degrees of freedom for the inverse problem as well.
Many oil reservoirs operate under the conditions of vertical equilibrium (VE). The VE assumption (gravity and capillary forces are equal) permits to describe 2D displacement process in layer cake reservoir by one quasilinear hyperbolic equation [2-6].
Viscous dominated case assumes that the lateral pressure gradient is independent of the vertical coordinate. The main assumption is that the displacement of oil by water in different layers is going on in order of decreasing of their permeabilities. This model consists on the one hyperbolic equation as well [7 - 9].
Application of the Welge's method  to each of the mentioned above models allows us to determine the pseudofractional flow curve and profile of permeabilities from the waterflooding data.
The range of validity of different pseudorised models was obtained in [11 - 13] by approaching to zero different dimensionless parameters in frames - of 2D model. 5 pseudorised models were obtained in  as an asymptotical limits of 2D model: two mentioned above models; averaged model of commingled layers, gravity dominated model [14,15] and model for homogeneous reservoir.
All these models permit solution of the inverse problem by the Welge's method. It is necessary to highlight, that actually we do not need the solution of the direct displacement problem to obtain solution of the inverse reservoir characterisation problem. The self-similarity of the displacement problem is enough for existence of the explicit solution of the inverse problem [17,18].
In this work we derive a pseudorised equations for the two-phase flow with tracer In stratified heterogeneous reservoir. The problem of oil displacement by water with tracer in layer cake reservoir is solved. We analyze the propagation of the tracer front in regions with the different permeability. Inverse problem is solved as well. Extension to Welge's method developed allows us to determine more precisely profile of permeabilities from the tracer analyses data.
TRACER MOVEMENT IN THE OIL-WATER FLUX
Discuss the 1D tracer flow with the two-phase oil-water flux.
It is obvious from the Fig. 1 that the tracer front lags behind the front of water, Dc < Df. There are three reasons for the delay between the tracer front and water front:
sorption of tracer on the matrix surface and on the interphase water-oil surface;
solubility of tracer in the oil phase;
presence of connate water in the reservoir before the injection.