Because of the influence of dispersion on miscible-displacement processes, diffusion and dispersion phenomena in porous rocks are of current interest in the oil industry. This paper reviews and summarizes a great deal of pertinent information from the literature.Porous media (both unconsolidated packs and consolidated rocks) can be visualized as a network of flow chambers, having random size and flow conductivity, connected together by openings of smaller size. In such a porous medium, the apparent diffusion coefficient D is less than the molecular diffusion coefficient Do, as measured in the absence of a porous medium. For packs of unconsolidated granular material the ratio D/Do is about 0.6 to 0..7. For all porous rocks, both cemented and unconsolidated, the ratio of diffusion coefficients can also be represented as where F is the formation electrical resistivity factor and is the porosity.If fluids are flowing through the porous medium, dispersion may be greater than that due to diffusion alone. At moderate flow rates the porous medium will create a slightly asymmetrical mix zone (trailing edge stretched out), with the longitudinal dispersion coefficient approximately proportional to the first power of average fluid velocity (if composition is nearly equalized in pore spaces by diffusion). If the velocity in interstices is large enough, there will be insufficient time for diffusion to equalize concentration within pore spaces. In this region, longitudinal dispersion increases more rapidly than fluid velocity.At low velocities in interstices, transverse dispersion is characterized by a region in which transverse diffusion dominates. If the fluid velocity gets high enough, there will be a transition into a region where there is stream splitting with mass transfer but with insufficient residence time to completely damp-out concentration variations within pore spaces.There are several variables that must be controlled to get consistent longitudinal and transverse dispersion results, viz.,edge effect in packed tubes,particle size distribution,particle shape,packing or permeability heterogeneities,viscosity ratios,gravity forces,amount of turbulence, andeffect of an immobile phase. Introduction Diffusion and dispersion in porous rocks are of current interest to the oil industry. This interest arises because of the influence of dispersion on miscible-displacement processes.In a recovery process utilizing a zone of miscible fluid, there is the possibility of losing miscibility by dissipating the miscible fluid or by channeling or ‘fingering" through the miscible zone. Diffusion and dispersion are two of the mechanisms that may lead to mixing and dissipation of the slug. On the other hand, dispersion may tend to damp-out viscous fingers which may be channeling through the miscible slug. Hence, dispersion may be detrimental or beneficial (if it prevents fingering through the miscible zone). Therefore, it is doubly important that we understand these processes.In this paper we review, summarize and interpret a great deal of information from the literature. In particular, we will briefly discuss molecular diffusion in miscible fluids. Then we will discuss what differences to expect for diffusion in a porous rock. If there is movement of the fluid through the rock, then there may be an additional mixing or "dispersion". Furthermore, the dispersion longitudinally (in the direction of gross fluid movement) and transverse to the direction of fluid movement will not be equal. We will discuss both types of dispersion as well as several variables which can affect dispersion (viscosity differences, density differences, turbulence, heterogeneity of media, etc.). This group of variables has sometimes led to difficulty when comparing literature data. DIFFUSION OF MISCIBLE FLUIDS If two miscible fluids are in contact, with an initially sharp interface, they will slowly diffuse into one another. SPEJ P. 70^
Recent improvements in processes for recovering viscous reserves has renewed interest in the phenomenon of immiscible fingering. This paper describes studies of immiscible fingering in linearHele-Shaw and bead-packed models. Immiscible fingers were readily initiated in all models. The fingers, however, were damped out before traveling very far in the uniform bead packs that contained connate water. The damping mechanism is believed due to the movement of the two phases in a direction transverse to the direction of gross flow.To study the transverse flow phenomenon under controlled conditions, oil and water were injected simultaneously and side by side in linear models. Transition zones were formed that grew broader as the distance from the inlet increased. The saturation distribution in the transition zones could be described mathematically by an "immiscible dispersion coefficient" and the well known error function solution of the dispersion equation. The immiscible dispersion coefficients were found to be proportional to the interstitial velocity and proportional to the product of the bead diameter and packing inhomogeneity factor.
A simplified theory of viscous fingering in miscible systems has been developed. It predicts the correct functional relationship between pertinent variables and permits the calculation of the order of magnitude of fingering behavior for simple systems, such as linear or radial. The theory is based on the following four observations:cross-flow takes place only near the ends of fingers;the relative finger width is about 0.5;fingers can be suppressed by transverse dispersion, the suppression being quantitatively described by a critical ratio of dispersion times; andthe widths of extending fingers are close to the minimum size finger that can grow at any stage of displacement. Fingering is studied in two-dimensional linear and diverging radial systems, both theoretically and experimentally. For linear systems, the length of the fingered region is proportional to mean displacement; the finger width is proportional to square root of mean displacement; and there is a small initial region void of fingers because of suppression by longitudinal dispersion. For the radial system, two limiting conditions are recognized. If the mean displacement is small compared with the wellbore radius, the length of the fingered region is proportional to the mean displacement. The width is proportional to the square root of mean displacement. If the mean displacement is large compared with wellbore radius, length of the fingered region is proportional to mean displacement, but the number of fingers approaches a constant value. Also, in the radial case there is a small initial region void of fingers because of longitudinal dispersion. Introduction The behavior of viscous fingers in miscible systems has been of interest to the oil industry for many years. Previous studies have clearly shown the existence and nature of fingers in small models,1,2,3,4 Engineering techniques for extrapolating to reservoir situations have been proposed.5 Still, because of the lack of a real understanding of the mechanics of fingering, there remains uncertainty and disagreement as to the best way to scale or calculate fingering behavior in the reservoir. In this paper we discuss a study of the mechanics of fingering in miscible systems (i.e., why do fingers behave as they do?). A simplified theory is developed which we believe willpredict the correct functional relationship between pertinent variables,permit us to calculate the order of magnitude of fingering behavior for simple systems such as linear and radial, andgive further insight into the problem of scaling or otherwise extending the results to more complicated reservoir conditions. The paper includes the following sections:brief summary of the mixing behavior of miscible fluids in linear and radial systems;four fundamental observations of fluid flow (under fingering conditions);based on these four observations, a simplified theory of fingering in linear and radial systems is developed; andthe theoretical equations are compared with experimental fingering data measured in laboratory models. A Review of Diffusion and Dispersion As will be shown later, the behavior of viscous fingers in miscible systems is controlled in large degree by the mixing between the two fluids. A quantitative description of fingers will first require a quantitative description of mixing. Fortunately, much work has been done to clarify this subject; a fairly comprehensive review has been presented in a previous paper.7 For present purposes let usbriefly summarize the quantitative description of diffusion and dispersion (both longitudinal and transverse) within a differential element;present a simplification of the "width of mixed zone"; anddescribe the effect of geometry on width of mixed zone.
The application of a method for analyzing pressure build-up curves to determine the effective permeability in the Spraberry is presented. Sixteen Upper Spraberry wells of the Driver area are analyzed and show variations in the in-place effective permeability of 2 to 183 md. Since these values are much larger than the.5 md, or less, reported for the matrix rock, a large part of the vertical fractures noted in Spraberry cores must be considered native to the formation. The effectiveness of the well fracture treatment in connecting the well bore to the native fracture system is analyzed by comparison of the effective permeability as determined from the PI test to that obtained from a pressure build-up curve. These results show that the fracture treatment is sufficiently effective in connecting the well bore to the native fracture system to yield a flow capacity within about ?50 per cent of that dictated by the native fracture system. About two-thirds of the wells show some degree of local blockage. Introduction Fractured reservoirs have been known in the past: However, during the past two years the Spraberry formation has directed the attention of the petroleum industry to a study of the performance of a fractured reservoir. The large variation in the number of fractures noted in Spraberry cores, the large variation in the producing ability of offset wells, the dramatic effect of well treatment on well productivity, the unknown quantity of oil stored in the fractures and matrix rock, and the unknown oil to be recovered were part of the initial Spraberry puzzle. In this type reservoir where the well treatment is an important factor in determining the productivity, it is particularly desirable to determine the in-place flow capacity of the drainage area at distances far removed from the well bore. This value, when compared to the flow capacity obtained from a PI measurement, will show the effectiveness of the well treatment (completion, fracture treatment, acidization, etc.) in establishing a flow capacity in the vicinity of the well bore equal to that in the more remote region. The object of this paper is to show the application of a method of pressure build-up analysis for this purpose in studying the Upper Spraberry formation in the Driver area.
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