Previous workers have developed differential equations to describe oil displacement by water imbibition, but have not explicitly defined the relationship between recovery behavior for a single reservoir matrix block and its size. In the present work, imbibition theory is extended to show that the time required to recover a given fraction of the oil from a matrix block is proportional to the square of the distance between fractures. Using this relationship, recovery behavior for a large reservoir matrix block is predicted from an imbibition test on a small reservoir core sample. The prediction is then extended to analyze recovery behavior for fractured - matrix, water - drive reservoirs in which imbibition is the dominant oil-producing mechanism. Experimental data are presented to support the basic imbibition theory relating matrix block size, fluid viscosity level and permeability to recovery behavior. Introduction Imbibition has long been recognized as an important factor in recovering oil from water - wet, fractured-matrix reservoirs subjected to water flood or water drive. Recently, two approaches have been published which might be used to predict imbibition oil-recovery behavior for reservoir - sized matrix blocks. Graham and Richardson used a synthetic model to scale a single element of a fractured-matrix reservoir. Blair, on the other hand, used numerical techniques to solve the differential equations describing imbibition in linear and radial systems. This latter method requires auxiliary experimental data in the form of capillary pressure and relative permeability functions. These two approaches, i.e., synthetic models and numerical techniques, have been used to study a variety of reservoir fluid-flow problems. One purpose of this work is to present, with experimental verification, a third method for predicting imbibition oil recovery for large reservoir matrix blocks. This method uses scaled imbibition tests on small reservoir core samples to predict field performance. The imbibition tests are easier to perform than the capillary pressure and relative permeability tests required to apply the numerical method. Furthermore, when suitably preserved reservoir-rock samples are used, the properties of the laboratory system are the same as those of the field. This offers an important advantage over the use of synthetic models because there is usually some question as to how accurately reservoir-rock properties can be duplicated in such models. Based on the recovery behavior for a unit matrix block, an analysis is presented to predict oil recovery for a fractured, water-drive reservoir made up of many such unit blocks. In the analysis, it is assumed that the flow resistance and the volume of the reservoir fracture system are negligible compared with that of the porous matrix. These assumptions are generally consistent with observed characteristics of many fractured-matrix reservoirs, and have been employed in previous studies. It is further assumed that the effect of gravity on flow in the matrix blocks is negligible. On first thought, the latter assumption might appear to seriously limit application of the method. However, in a fractured reservoir, the effect of gravity on flow in a matrix block will be restricted by the height of the block. Furthermore, matrix permeabilities are often very low (10 md or less) in such reservoirs. This means that capillary or imbibition forces will be large, thus tending to minimize the relative importance of gravity. For these reasons, imbibition should be the dominant oil-recovery mechanism in many fractured-matrix, water-drive reservoirs. The predictive method presented in this report is applicable to such reservoirs. SPEJ P. 177^

Published in Petroleum Transactions, AIME, Vol. 216, 1959, pages 330–333.Paper presented at the AIChE-SPE Joint Symposium in Kansas City, Mo., May 17–20, 1959. Abstract The effects of crude oil components on the wettabilities of sandstone and limestone were investigated. Fractions containing components differing in molecular weight and molecular structure were obtained from crude oils by distillation, extraction and chromatography. Individual fractions were then tested for their effects on rock wettability. Tests indicate that sandstone wettability may be changed by a complex variety of surfactants varying both in molecular structure and molecular weight. Limestone appears to be particularly sensitive to basic, nitrogenous surfactants. Introduction Investigations in recent years have shown that petroleum reservoir rock wettability can exert a significant influence on the efficiency with which oil can be produced by water flooding. While most reservoirs are presumably water-wet, they may range in their degree of water-wettability from near-neutral to strongly water-wet. Reservoir wettabilities other than strongly water-wet are likely to be induced by adsorption of surface-active components from the crude oil on the pore walls of reservoir rock. Little is known, however, about the nature of the surface-active materials which are likely to be adsorbed by the reservoir rock. Due to the complexity of crude oils, attempts made in the past to isolate these surface-active components have met with only limited success. It is probable that many different types of surface-active materials are indigenous to crude oils and that many of these may be adsorbed to varying degrees by reservoir rock. This was explored in the studies discussed in this paper.

"... the precision with which variations In reservoir properties can be modeled Is determined by the number of blocks In the model." Introduction In the past 30 years, reservoir simulation has evolved from a research area into one of the most flexible and widely used tools in reservoir engineering. Use of reservoir simulation has grown because of its ability to predict the future performance of oil and gas reservoirs over a wide range of operating conditions. Reservoir simulators use numerical methods and high-speed computers to model multidimensional fluid flow in reservoir rock. Reliable simulators and adequate computing capacity are available to most reservoir engineers, so simulation is usually practical for all reservoir sizes and all types of reservoir performance studies. Although the use of simulation frequently is optional, it may be the only reliable way to predict the performance of a large, complex reservoir, especially if such external considerations as government regulations influence the production schedule. Even for small reservoirs where simple calculations or extrapolations may be adequate, simulation is often faster, cheaper, and more reliable than alternative methods for predicting performance. Modeling Concepts A reservoir simulator models a reservoir as if it were divided into a number of individual blocks (gridblocks). Each block corresponds to a designated location in the reservoir and is assigned properties-porosity, permeability, relative permeability, etc. -believed to be representative of the reservoir at that location. In the simulator, fluids can flow between neighboring blocks at a rate determined by pressure differences between blocks and flow properties assigned to the interfaces between blocks. In essence, the mathematical problem is reduced to a calculation of flow between adjacent blocks. For every block-to-block interface, a set of equations must be solved to calculate the flow of all mobile phases. The equations generally incorporate Darcy's law and the concept of material balance and contain terms describing the permeability "between" blocks, fluid mobilities (relative permeability and viscosity), and rock and fluid compressibilities. Fig. 1 illustrates the most common types of models used in simulation. Models range in complexity from a single block, useful only for classical material-balance calculations, to fully 3D models capable of modeling all major factors that influence reservoir performance. Each gridblock in a model has only one set of properties; there is no variation in any property within a block. For example, phase saturations in a model block will be volumetric averages of the saturations in that part of the reservoir represented by the block. In this respect, a model block can be visualized as a well-stiffed tank (i.e., its contents are homogeneous)connected to adjacent tanks with pipes whose flow capacities are determined by reservoir flow properties. This visualization, although simplistic, demonstrates that the precision with which variations in reservoir properties can be modeled is determined by the number of blocks in the model. A simulator also divides the life of a reservoir into discrete increments. Changes in a reservoir (pressure, saturation, etc.) are computed over each of many time increments, or timesteps. Conditions are defined only at the beginning and end of each timestep; nothing is defined at any intermediate time within a time interval. The accuracy with which reservoir behavior can be calculated generally will be influenced by the length of the timesteps as well as the number of gridblocks. The preceding discussion implies that for any size gridblock and any length of timestep, there will always be abrupt changes in reservoir conditions from one block to the next and from one timestep to the next. Fig. 2 illustrates this point. At the top of Fig. 2 are plan views of a hypothetical two-well reservoir being waterflooded and a four-gridblock simulator model of the reservoir. The two plots show-water-saturation distribution at a given time in both the reservoir and the model. JPT P. 692⁁

Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs. Introduction Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions". DESCRIPTION OF PSEUDO-RELATIVE PERMEABILITY FUNCTIONS PERMEABILITY FUNCTIONS Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities. SPEJ P. 175

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