Secondary Water Recovery by Air Injection: 2. The Implicit Simultaneous Solution Method

This paper develops a general method for solving reservoir simulation equations, which is based upon an implicit finite difference formulation of the partial differential equations of compositional thermal simulation, permitting the simulation of nearly all enhanced oil recovery techniques. The general method is used to derive 10 solution schemes, which include the well-known IMPES, Simultaneous Solution (SS), and Sequential Solution (SEQ) methods as sub-cases. The novelty of the general method lies in showing the interrelations of these methods, and also affording the possibility of deriving still other methods. The theoretical development of the general method is given in detail, and the special methods are derived in reference to the reservoir problems for which they were employed. The starting point is a system of nonlinear finite difference equations, typical of compositional thermal simulation. Newtonian iteration then leads to a system of equations. The partitioning of the left-hand side (containing the unknown vector) in a particular manner permits the derivation of 10 solution approaches, which are identified in detail. The proposed general method bridges the gap in the development of the reported solution methods, and also points to several new approaches. The procedure described can be utilized for a variety of simulation problems, with fully implicit source-sink terms (including heat loss, thermal cracking, etc.), for both the standard and multipoint difference schemes in multi dimensions.

Reservoirs quality assessment and application of capillary pressure models in understanding saturation relationship in ‘Keyz’ field, Niger Delta Basin, Nigeria

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract The simultaneous solution method in reservoir simulation gives rise to a system of coupled linear equations. This paper shows that the system can be efficiently solved by a block successive over-relaxation method even for large 3-D, 3-phase models. Examples are given which include a large 3-D problem and a radial coning problem. problem and a radial coning problem Introduction In reservoir simulation problems involving flow in the vertical direction, it is desirable to formulate a semiimplicit or fully implicit finite difference model and simultaneously solve for as many unknowns as practical. This approach yields a very stable solution, and is most useful when the reservoir is highly heterogeneous and when the throughput across the grid blocks is high. Simultaneous solution also allows a convenient implementation of strongly coupled production and reservoir flow terms, which further contribute to stability. However, for easy reservoir simulation problems, the implicit pressure explicit saturation method is often more economical. The technique of solving for the unknowns simultaneously is known as the simultaneous solution method. There are, however, disadvantages in solving for several unknowns by this solution method. The number of linear equations to be solved is equal to the number of mesh points multiplied by the number of unknowns. For example, if a model has 1500 mesh points and 3 unknowns (oil pressure, points and 3 unknowns (oil pressure, gas saturation, and water saturation), there will be 4500 linear equations. A large system of this size precludes the use of direct methods because of excessive storage requirement, unless a special algorithm such as the storage management algorithm is available. For 3-D problems, direct methods also become rapidly uneconomical, as the number of mesh points increases. For example, the computational work for a cubic mesh with Nc3 mesh points is proportional to Nc6. Indeed, it is usually proportional to Nc6. Indeed, it is usually uneconomical to apply direct methods to simultaneous solution models, if the mesh exceeds a few hundred points. Iterative methods require less storage than direct methods and also less computational work if they converge rapidly.

The spray impulse and injection pressure of each hole of the 4-hole injector of F186 engine were tested by using the transient test system for flow coefficient and injection rate of each hole of the injector, and the injection rate, fuel injection quantity and flow coefficient of each hole were obtained. According to the measured injection rate of each hole and cycle fuel injection quantity, a 3D model of the diesel engine was established with the FIRE software, and the uniform injection rate and uneven injection rate of each hole were made as the injection setting for the combustion simulation to comparatively analyze the influence of the uneven injection rate of each hole on the combustion process of diesel engine. As the injection of each hole is uneven, some holes with high injection rate have long fuel spray penetration distance and the fuel spray is fully mixed with the air, and the ignition starting point is moved ahead. When compared with the case of uniform injection rate, the phase corresponding to the heat release rate in the case of uneven injection rate is moved forwards, the maximum cylinder pressure and temperature rise fast, the maximum cylinder pressure and maximum average temperature in cylinder are higher, the NO generation is larger, and the Soot generation is slightly larger. In the case of uneven injection rate from four holes, as the utilization of air in the hole-located cylinder space by the fuel spray is uneven, the hole with a large quantity of injection makes the oxygen in the hole-located cylinder space relatively low, the wall collision of the hole with high injection rate is more, and the diffusion combustion is incomplete and insufficient.

Forced and spontaneous imbibition of water is performed to displace oil from strongly water-wet Gray Berea (~130 mD) and Bentheimer (~1900 mD) sandstone core plugs. Two nonpolar oils (n-heptane and Marcol-82) were used as a non-wetting phase, with viscosities between 0.4 and 32 cP and brine (1 M NaCl) for the wetting phase with viscosity 1.1 cP. Recovery was measured for both imbibition modes, and pressure drop was measured during forced imbibition. Five forced imbibition tests were performed using low or high injection rates, using low or high oil viscosity. Seventeen spontaneous imbibition experiments were performed at four different oil viscosities. By varying the oil viscosity, the injection rate and imbibition modes, capillary and advective forces were allowed to dominate, giving trends that could be captured with modeling. Full numerical simulations matched the experimental observations consistently. The findings of this study provide better understanding of pressure and recovery behavior in strongly water-wet systems. A strong positive capillary pressure and a favorable mobility ratio resulting from low water relative permeability were main features explaining the observations. Complete oil recovery was achieved at water breakthrough during forced imbibition for low and high oil viscosity and the recovery curves were identical when plotted against the injected volume. Analytical solutions for forced imbibition indicate that the pressure drop changes linearly with time when capillary pressure is negligible. Positive capillary forces assist water imbibition, reducing the pressure drop needed to inject water, but yielding a jump in pressure drop when the front reaches the outlet. At a high injection rate, capillary forces are repressed and the linear trend between the end points was clearer than at a low rate for the experimental data. Increasing the oil viscosity by a factor of 80 only increased the spontaneous imbibition time scale by five, consistent with low water mobility constraining the imbibition rate. The time scale was predicted to be more sensitive to changes in water viscosity. At a higher oil-to-water mobility ratio, a higher part of the total recovery follows the square root of time. Our findings indicate that piston-like displacement of oil by water is a reasonable approximation for forced and spontaneous imbibition, unless the oil has a much higher viscosity than the water.

Chapter 11 - Introduction to modeling multiphase flow in petroleum reservoirs

Summary We simulate transient behavior of viscous- and capillary-dominated water invasion at mixed-wet conditions directly in scanning-electron-microscope (SEM) images of Bentheim sandstone by treating the pore spaces as cross sections of straight tubes. Initial conditions are established by drainage and wettability alteration. Constant rate or differential pressure is imposed along the tube bundle. The phase pressures vary with positions along the tube length but remain unique in each cross section, consistent with 1D core-scale models. This leads to a nonlinear system of equations that are solved for the interface positions as a function of time. The cross-sectional fluid configurations are computed accurately at any capillary pressure and wetting condition by a semianalytical model that is based on free-energy minimization. The fluid conductances are estimated by newly derived explicit expressions that are shown to be in agreement with numerical computations performed directly on the cross-sectional fluid configurations. An SEM image of Bentheim sandstone is taken as input to the developed model for simulating the evolution of saturation profiles during waterfloods for different flow rates and several mixed-wet conditions, which are established with various initial water saturations and contact angles. It is demonstrated that the simulated saturation profiles depend strongly on initial water saturation at mixed-wet conditions. The saturation profiles exhibit increasingly gradual behavior in time as the contact angle, defined on the oil-wet solid surfaces, increases or the initial water saturation decreases. Front menisci associated with positive capillary pressures promote oil displacement by water, whereas for large and negative capillary pressures at small flow rates, oil displaces water because the associated front menisci retract. This results in the development of pronounced gradual saturation fronts at mixed-wet conditions. The waterfloods simulated at conditions established with a large initial water saturation and small contact angle on the oil-wet solid surfaces exhibit sharp Buckley-Leverett saturation profiles for high flow rates because the capillary pressure is small and less important. The shape of the saturation profiles is interpreted on the basis of the simulated capillary pressure curves and the corresponding fluid configurations occurring in the rock image.

This paper describes the field operations at the Fry in situ combustion project in Crawford County, Ill. Field operations may be divided into three phases preliminary engineering studies of 1960 and 1961, ignition, and, finally, operation of the test. The Fry combustion project was a pilot test of cocurrent in situ combustion on a 3.3-acre inverted five-spot. In the summer of 1960, air injection tests were conducted using old wells in the area and a new air injection well drilled especially for the test. In the summer of 1961, four new oil wells were drilled on a 3.3 acre pattern for the pilot test. All old wells in the near vicinity of the pilot test were abandoned to preclude any possibility of gas escaping to adjacent formations. A compressor plant was also installed at the test site in 1961. Air injection tests and other engineering studies were made. At the conclusion of these tests, ignition was accomplished with a 40 kw electrical igniter. Operation of the Fry combustion test was little different from the usual oilfield operation with the exception of the high rate of air injection. No difficulty was found with the openhole completion. There were no unusual corrosion or emulsion problems until bottom-hole temperatures exceeded 150F. Four cores were drilled into the burned-out portion of the reservoir with no mechanical difficulties. Introduction A cocurrent or conventional in situ combustion test was conducted on the Fry unit of the Marathon Oil Co., in Crawford County, Ill. Besides testing combustion in the Fry type of reservoir, the test was intended to evaluate various operating procedures. This paper presents a discussion of the Fry combustion test field operations. The geology of the reservoir and the test performance are discussed in other papers. HISTORY OF THE FRY UNIT The Fry unit consists of 11 leases totaling 337 acres in an area in which oil was originally discovered in 1906. Production from this reservoir, as from all Robinson sand reservoirs in the area, was very high during the first few years. As the dissolved-gas energy was dissipated, early attempts to retard the decline consisted of vacuum, air, and gas repressuring, and recently, most of the area has been subjected to water flooding. Following early air repressuring, the Fry waterflood unit was formed and water flooding initiated in July, 1952. In the area in which the Fry combustion test was conducted, a peripheral water flood of a 160-acre area was developed as shown in Fig. 1. Between 1956 and 1960, it became apparent that the water flood was unsuccessful. In looking for a suitable site for an in situ combustion test, the Marathon Oil Co. selected the Fry unit as possessing those features most desirable for such a research project. Field operations associated with the Fry combustion test began in the spring of 1960. JPT P. 343ˆ

Relative permeability functions are developed for both two- and three-phase systems with the saturation changes in the imbibition direction. An empirical relation between residual nonwetting-phase saturation after water imbibition and initial nonwetting-phase saturations is found from published data. From this empirical relation, expressions are obtained for trapped and mobile nonwetting-phase saturations which are used in connection with established theory relating relative permeability to pore-size distribution. The resulting equations yield relative permeability as a function of saturation having characteristics believed to be representative of real systems. The relative permeability of water-wet rocks for both two- and three-phase systems, with the saturation change in the imbibition direction, may be obtained by this method after properly selecting two rock properties: the residual nonwetting-phase saturation after the complete imbibition cycle, and the capillary pressure curve. Introduction Relative permeability is a function of saturation history as well as of saturation. This fact was first pointed out for two-phase flow by Geffen et al. and by Osaba et al. Hysteresis in the relative permeability-saturation relation also has been reported for three-phase systems. Since saturations may change simultaneously in two directions in a three-phase system, four possible relationships arise between relative permeability and saturation for a water-wet system. The four saturation histories of this system were given by Snell as II, ID, DI and DD. I refer to the direction of saturation change (imbibition and drainage), with the first letter of the symbol indicating the direction of change of the water phase. As used in this paper, the second letter of the symbol refers to the direction of saturation change of the gas phase, i.e., D and I indicate an increase and decrease, respectively, in gas saturation. Only a few three-phase relative permeability curves have been published. Leverett and Lewis published three-phase curves for unconsolidated sand, and Snell reported results of several English authors for both drainage and imbibition three-phase relative permeability of unconsolidated sands. Three-phase relative permeability curves for a consolidated sand were published by Caudle et al. for increasing water and gas saturations (ID). Corey et al. reported drainage (DD) three-phase relative permeability for consolidated sands. Recently, Donaldson and Dean and Sarem calculated three- phase relative permeability curves from displacement data on consolidated sands, also for saturation changes in the drainage direction. The only published three - phase relative permeability curves for consolidated sands with saturation changes in the imbibition direction (II) are those of Naar and Wygal. These curves are based on at theoretical study of the model of Wyllie and Gardner as modified by Naar and Henderson. Interest in three-phase relative permeability has increased recently due to the introduction of new recovery methods and refinements in calculation procedures brought about by the use of large-scale digital computers. The scarcity of empirical relations for three-phase flow, and the experimental difficulty encountered in obtaining such data, have made the theoretical approach to this problem attractive.

Immiscible two‐phase flow through porous materials exhibits different invasion patterns controlled by dynamic conditions, competition between the viscous and capillary forces, and the contrast between the fluids viscosities. Two distinct invasion patterns are viscous and capillary fingering. While the first one happens under unfavorable viscosity ratios at high injection rates, the second one happens when the viscous forces are very small compared to the capillary forces. Depending on whether the invasion is under the capillary fingering or viscous fingering regime, the remaining oil saturation and the effective permeability of the fluids can significantly change. The contribution of the present work has two key aspects: (a) It addresses how the remaining saturation changes at different flow rates (i.e., capillary numbers) for different unfavorable viscosity ratios in a three‐dimensional system; (b) it presents a new dynamic pore network model using the fully implicit scheme which has been enhanced by the graphic processing unit (GPU) parallel computing. Additionally, the model has been carefully validated against micromodel experiments in both time and space, which to our best knowledge has not been reported in such detail in the literature. The results of the validated 3‐D dynamic pore network model demonstrate the remaining saturation at the breakthrough time as a nonmonotonic trend with the imposed capillary number.

A study of the impact of instability on relative permeability and capillary pressure

Summary This paper shows that mud-filtrate invasion prior to drillstem testing in low-permeability gas formations can cause significant reduction of gas flow rate observed during the test. In addition, apparent gas permeabilities determined from test data can be much lower than the true values. These conclusions are supported by simulator results and field data. Introduction Interpretation of drillstem tests (DST's) in low-permeability gas formations is difficult. and this difficulty leads to uncertainty in identifying formations that, when stimulated, could become commercial completions. Operators have tong suspected that mud-filtrate invasion into these low-permeability formations could contribute to these uncertainties because of significant alterations in relative permeability and capillary pressure. These uncertainties led us to a study of the effects of mud-filtrate invasion on response in a DST; this paper reports the results of that study. In this investigation, we used a two-phase, two-dimensional, fully implicit reservoir simulator to model DST's. The input data for the simulator were obtained from special core analyses. openhole tops, and DST's from the Falher sandstone in western Canada. Model Description The two-dimensional, two-phase model used in this study has been described previously in the literature. The model simulates the flow of gas and water in the formation using a simultaneous determination of saturations and pressures from finite-difference approximations of the gas and water flow equations. Capillary pressure and relative permeability data are input into the model; the model is therefore capable of simulating imbibition, water blocking, and cleanup effects. These effects are potentially important in DST performance. To simulate a DST, uniform initial pressures and saturations were established in the model for all reservoir cells. To model the imbibition of filtrate into the formation prior to the DST, the pressure in the wellbore was set equal to (and maintained at) the hydrostatic mud pressure, and the water (filtrate) saturation was maintained at 100%. Water saturations and pressures around the wellbore increased during the imbibition period as filtrate moved into the reservoir. To model a mud filter cake, the permeabilities in the formation adjacent to the wellbore were reduced. Following the imbibition period, a flow period was simulated. During the flow period, the pressure in the wellbore was maintained at a fixed pressure and the wellbore volume was set equal to the value of a typical drillpipe open to flow. Following the flow period, a pressure buildup test was simulated, with the wellbore volume set equal to a typical volume below the shut-in valve in a DST tool assembly . JPT P. 299^

The water and air flow in soil pores in response to rainfall infiltration is an important factor affecting the stability of rainfall-induced landslides. The objective of this paper is to investigate the effect of water-air flow on the deformation and stability of rainfall-induced landslides based on a coupled infiltration and hydromechanical model. The model is formulated by the water and air flow equation, the mechanical equilibrium, and porosity equations. The rainfall infiltration is introduced as a flux boundary, which is determined according to the comparison of the infiltration capacity and rainfall intensity. The numerical model and solution approach are validated by simulating the Liakopoulos drainage test and a rainfall infiltration experiment with satisfactory results. Taking the Tanjiawan landslide as a case study, the water and air flow in response to rainfall infiltration and its effect on deformation and stability are examined. The results show that the pore air is gradually trapped and compressed due to rainwater infiltration. The entrapped air has a slowing effect on the rainfall infiltration and a pushing-out effect on the front sliding body, which is an important driving force for the evolution of the slope deformation and stability due to rainfall infiltration.

Improved oil recovery from fractured oil-wet carbonate reservoirs is a great challenge. The water-flooding efficiency will be low because of higher permeability in fractures than in matrix, and negative capillary pressure retains oil inside the matrix blocks. Studies of oil-wet chalk have shown that sulphate ions in the seawater may alter the wettability towards increased water-wetness. One-dimensional spontaneous imbibition tests of seawater into preferentially oil-wet chalk cores are performed. To get a better understanding, a numerical model has been developed which includes effects of wettability alteration. The experiments are carried out on cylindrical, sealed core plugs with only top open or with both ends open. Only countercurrent imbibition takes place for cores with top end open. For cores with both ends open, both countercurrent and cocurrent imbibition take place, and oil recovery rate is obviously accelerated. Taking formation water as the base case, higher oil recovery is observed with seawater imbibition. To simulate the wettability alteration process caused by seawater, a model is developed which includes molecular diffusion, adsorption of wettability alteration (WA) agent, gravity and capillary pressure. The WA agent diffuses into the formation water initially present in the core, adsorb onto the rock surface and induce wettability alteration. Consequently, the capillary pressure curve is shifted to higher values. In particular, the capillary pressure at the initial water saturation changes from negative to positive values and seawater is imbibed into the core. The shapes of relative permeability curves also depend on the wettability. The simulation results can fairly well match the experimental data. With the experimental and modeling work we explore the interplay between capillarity and gravity, and especially the importance to consider wettability alteration process is again confirmed.

Measurements of capillary pressure and relative permeability curves for foam transport in porous media-A capillary bundle approach