Introduction Reliable relative permeability data are important to the overall planning for prospective thermal recovery projects. Empirically derived relative permeability functions presented here have characteristics quite different from those determined experimentally. The empirical functions emerged from simulations of numerous ongoing thermal projects in unconsolidated heavy-oil reservoirs from Trinidad, West Germany, Canada, and the U.S.The cyclic steam process involves strongly changing saturation and pressure conditions during alternating imbibition and drainage cycles. Three-phase flow effects are to be expected in this process, where both steam and mobile water are found in the presence of oil due to steam condensation. Experimental procedures for determining relative permeability on a microscopic scale often ignore dependency on saturation history (hysteresis), changing pressure levels, and three-phase effects. Hence, relative permeability in this process (and in other thermal processes) often is considered a weakly determined parameter. The thermal simulator, which has been available since 1973, provides insight into the nature of relative permeability on a reservoir scale. Thermal Reservoir Simulator A volatile-oil steamflood simulator developed by Todd, Dietrich & Chase Inc. was used during analysis of the field thermal projects. This model is a fully implicit, finite-difference-based program that can treat up to four mass conservation equations and an energy balance. There are three phases: aqueous, liquid hydrocarbon, and vapor. The four mass species are water, a nonvolatile heavy oil, and two volatile components. Although only water and a dead-oil component were used in the majority of field projects, all four mass species were needed in some projects. The volatile components have been used to model (1) noncondensable-gas and CO2 additives to steam, (2) solution-gas and free-gas effects, and (3) distillation of hydrocarbon pseudocomponents.The model was employed in two-dimensional areal and vertical (cross section) simulations and in three dimensions, using Cartesian (x, y, z) and cylindrical (r, o, z) coordinates. Relative permeability functions were changed between the injection and production stages of the cyclic steam process to model the previously noted imbibition and drainage hysteresis effects. Empirical Relative Permeability Functions Relative permeability relationships shown in Fig. 1 are typical of those developed empirically (through simulation) to reproduce observed producing water/oil ratios in cyclic steam projects worldwide. In this work, two-phase data similar to those shown in Figs. 1a and 1b were used with a modified form of Stone's expression to generate three-phase oil relative permeability functions. These functions were used with adjustments for temperature dependency (Table 1) to match drainage production behavior. In each field application, very low water relative permeability curves (0.0025 is lesser than krwro is lesser than 0.25) were required to reproduce field performance. These empirically developed curves are much lower than those routinely measured (0.05 is lesser than krwo is lesser than 0.25) during imbibition water/oil relative permeability tests using unconsolidated core material.Experimentally determined imbibition water relative permeability curves often are used successfully to match observed steam injection pressures and rates. Physical Basis Several complementary conjectures are offered regarding a physical basis for the empirically derived relative permeability functions. Hysteresis of drainage and imbibition nonwetting-phase relative permeability curves is an established phenomenon. Only recently are experimental results appearing in the literature that indicate wetting-phase hysteresis. Jones and Roszelle showed that water relative permeability exhibits pronounced hysteretic behavior in water-wet sandstone, and Schneider and Owens demonstrated oil relative permeability hysteresis in oil-wet carbonate. These results support the speculation that water relative permeability (in water-wet sands) is much lower on the production cycle (drainage) than on the injection (imbibition) cycle during cyclic steam stimulation.A second complementary conjecture is related to mechanical stress. JPT P. 1987^

Summary Diatoms and radiolarians are microorganisms that precipitate Opal-A to form siliceous tests that accumulate on the seafloor to form siliceous oozes. Progressive diagenesis of these deposits during burial results in thick, highly compressible reservoirs of exceptionally high porosity and low permeability, not unlike the chalk reservoirs of the North Sea. During burial and over time, the amorphous silica phase (Opal-A) becomes unstable and gradually changes in its structure to more stable, ordered Opal-A' and crystalline forms or phases of silica, namely Opal-CT and quartz. The Opal-A ? Opal-A' ? Opal-CT ? quartz transformation results in a naturally occurring densification and compaction process that is accelerated by an application of heat. Reservoir compaction and surface subsidence can usually be controlled by injecting fluid to control the effective stress. However, in heavy-oil diatomite reservoirs undergoing steam injection, the injected fluid causes competing effects: it controls effective stress to some degree, yet at the same time it accelerates compaction and subsidence. This paper describes selected results of a diatomite laboratory testing program and features of a unique thermal reservoir simulator formulated to handle the effects on compaction caused by stress, temperature, and time-dependent strain (creep). Elevated temperature in amorphous Opal-A diatomite is shown to be capable of causing a sample compression of 25% or more and a severe reduction in permeability. The effects of thermally induced compaction are expected to accelerate surface subsidence as diatomite steam projects mature. Introduction There is a class of problems involving reservoir compaction of cohesive rocks (e.g. chalk, shale, and diatomite) in which the effects of stress are of a second-order importance compared to those of temperature. The injection of cold seawater in North Sea chalk reservoirs under conditions of invariant effective stress has led to continued compaction and subsidence (Cook et al. 2001; Sylte et al. 1999). The North Sea chalks are nearly pure calcium carbonate, and it is well known that the solubility of calcium carbonate increases as the water temperature decreases. Thus, even under conditions of unchanging effective stress, one would expect gradually increasing dissolution of calcium carbonate and compaction as the reservoir temperature of the chalk (~ 270°F) is gradually lowered by cold seawater injection (Dietrich 2001). In the giant Wilmington field of California, the shaly siltstones that are interbedded with the unconsolidated sands have recently been shown to be much more susceptible to thermally induced compaction than to stress-induced compaction (Dietrich and Norman 2003). And finally, diatomite is known to undergo a silica-phase transformation as temperature is raised, whereby amorphous Opal-A is converted to a more dense, crystalline Opal-CT. The injection of steam into California diatomite reservoirs is expected to accelerate this naturally occurring process and lead to rapid densification and compaction. In each case, for chalk, shaly rocks, and diatomite, there is both a laboratory and field basis that demonstrates the dominant role played by temperature.

Compaction is incorporated into a field-scale finite-difference thermal simulator to allow practical engineering analysis of reservoir compaction caused by fluid withdrawal. Capabilities new to petroleum applications include hysteresis in the form of limited rebound during fluid injection and the concept of relaxation time (i.e. , creep).

A series of numerical experiments have been performed to compare horizontal well performance using coarse and fine grid reservoir models. The objective of this work has been to determine the effect of using a standard well model with coarse (i.e., field-scale) grid systems to approximate performance computed with high resolution grid systems in which a well model was not employed. For the systems studied, the theoretical transmissibility connection factor (i.e., well model) calculated using the Peaceman formulation must be reduced to provide reasonable rate forecasts. In homogeneous systems, the calculated value for the well model may need to be reduced by about 1/2. In heterogeneous systems, the magnitude of the required multiplicative factor (λ) was shown to vary over a wide range (0.3 < λ < 0.8), depending upon the single value of the effective vertical permeability which was assumed for the calculation of the connection factor. Owing to this uncertainty and that associated with grid dependency, the use of field-scale grid systems for horizontal well rate projections does not, on its own, seem feasible in the absence of an opportunity to tune the model against observed field response. Introduction The problem of predicting well productivity in the absence of analog field performance is an important one. Although it can be addressed through the use of either analytical or numerical approximations of real systems, there are well known shortcomings to each method of analysis. There are few real situations which meet the limiting assumptions required by the analytical approach and many numerical analyses are limited by both the well model and spatial truncation error. In many engineering studies it is the productivity of a horizontal well which must be predicted and then compared to known performance of conventional producing wells. The objective of the work described here was to determine whether or not a conventional well model developed for horizontal well applications and typical field-scale grid systems can be used to numerically approximate accurate horizontal well performance. To satisfy this objective it was first necessary to establish accurate performance for a series of different reservoir and well placement conditions; this was done by developing an explicit wellbore modelling approach and validating its results with those derived from analytical solutions. Explicit Modelling Technique By an explicit modelling approach we mean that a finite-difference grid system is used to explicitly represent the horizontal wellbore itself and the surrounding reservoir volume. In this approach the wellbore is represented by a series of very small gridblocks scaled to represent the length and diameter of the horizontal well. The permeability of the wellbore gridblocks is set at a very high level (7.5E + 09 md) to establish a negligible wellborepressure gradient (< 0.02 kPa/m) and, as described later, the well model is effectively inactivated in the single completion grid block at the heel of the well by using a very high transmissibility connection factor.

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract Two variations of an indirect technique for predicting three-phase oil relative permeability are described in this paper. permeability are described in this paper. Relative permeabilities and the dependence of waterflood residual oil saturation on trapped gas saturations predicted by these two models are compared with the three-phase experimental data available in the literature. Both models provide acceptable schemes for interpolating in the three-phase region, and qualitatively match the curvature of isoperms generated in the more carefully conducted (and previously published) threephase experiments. Limitations of each model are described, including the need for establishing derivative constraints to prevent calculation of relative permeabilities greater than unity. Guidelines are also presented for determining which model should be used in specific applications. Introduction Thorough analysis of a displacement process in which three-phase flow predominates process in which three-phase flow predominates (viz. steam drive and alternate gas/water) requires three-phase relative permeability data. The effort involved in determining these data experimentally suggests that indirect methods should be investigated. The difficulty in direct experimental work lies mainly in determination of saturation distributions along the length of the core. Electrical resistance circuits and either gamma-ray or x-ray absorption techniques are required to measure two of the three saturations in a three-phase system. Only nine experiments concerning three-phase relative permeability have been found in the literature. As an alternative to the direct methods, Stone has suggested that more easily measured two-phase data can be used to predict relative permeability to both the predict relative permeability to both the wetting and nonwetting fluids in three-phase flow. Then an indirect approach involving a technique which uses two sets of two-phase data-water/oil and gas/liquid - can be used to predict the relative permeability of the predict the relative permeability of the intermediate wettability phase. THE TWO RELATIVE PERMEABILITY MODELS Two variations of the indirect technique are described here. The first is a three-phase oil relative permeability expression developed by Hirasaki. This is a modification of a model originally proposed by Stone in 1970. The second model was presented in a paper by Stone during 1973, as a revision paper by Stone during 1973, as a revision of his first technique. Model I It has been observed experimentally in water-wet systems that the water relative permeability is approximately only a function of permeability is approximately only a function of the water saturation, and that the gas relative permeability is approximately only a function of permeability is approximately only a function of the gas saturation.