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Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 26–29, 2020

Paper Number: SPE-201294-MS

... in the lower unit, thus to promote the potential of infill drill to enhance ultimate gas recovery. reservoir surveillance drilling operation production monitoring directional drilling pressure transient analysis drillstem testing pressure transient testing

**dimensionless****time**channel...
Abstract

This paper presents three sets of new integrative production decline type curves for the analysis of horizontal well production data in order to systematically estimate reservoir flowing capacity ( kh ), effective horizontal wellbore length and original oil/gas in place. In addition, diagnostics of dynamic flow regimes possible for the entire flow duration from early time transient to late time boundary dominated flow gives the proposed type curves unique value in field data analysis through using different idealized flow regimes as the reference systems. Comparing to the traditional classic type curves, these three sets of type curves offer more practical implemental strategy for well performance evaluation and reservoir formation characterization, resulting in less ambiguity in the outcome of data interpretation. The effects of horizontal wellbore length and reservoir geometry on the appearances and the characteristics of these three sets of new type curves are studied and documented. A detailed curve matching process has also been comprehensively developed to achieve systematically proper matching with field data using all of the three sets of type curves synchronously. To showcase this work in field application, the production dynamics of a gas field containing two vertical non-commingled units, each being produced by a horizontal well independently, is analyzed using the three sets of new type curves. With successful estimation of the kh and the effective horizontal wellbore length, it helps geologists and geophysicists locate the "sweet spot" with more confidence and offer the potential to extend the seismic response pattern to support further exploration activity in this geology unit. The outcome of this work helped identify the existence of an untapped gas reserve in the lower unit, thus to promote the potential of infill drill to enhance ultimate gas recovery.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 26–29, 2020

Paper Number: SPE-201448-MS

... form. By subtracting the derivative form from the pressure form, we remove the "dominant" wellbore storage term from the asymptotic solution. We then need to normalize that difference by the square root of time (or

**dimensionless****time**) to obtain the final formulation of the DVPF which leaves a single...
Abstract

This paper proposes the difference-value plotting function (DVPF) for the diagnostic analysis and interpretation of pressure transient test data in low-permeability reservoirs. Specifically, this work uses the approximation of the analytical solution for the performance of a vertical well with a single finite conductivity vertical fracture, where a Taylor Series expansion is used to obtain an asymptotic solution for early-time flow, which includes terms for wellbore storage and fracture conductivity. The well-testing derivative of this result is then obtained and is of a similar form. By subtracting the derivative form from the pressure form, we remove the "dominant" wellbore storage term from the asymptotic solution. We then need to normalize that difference by the square root of time (or dimensionless time) to obtain the final formulation of the DVPF which leaves a single constant parameter multiplied by time on the right-hand-side. Our contention is that this formulation leaves us with a diagnostic plotting function which provides a unique and contrasting behavior compared to using the pressure drop and/or pressure drop derivative functions alone for diagnostics and interpretations. As is typical of pressure transient or well testing data at early times, the observed pressures often exhibit random data noise. As such, we have adapted a noise reduction algorithm that was originally used for signal processing to smooth both the pressure and derivative functions. Lastly, we demonstrate the difference-value plotting function (DVPF) on several cases of synthetic and field-derived data to illustrate the utility of this methodology. Specifically, we have applied this method to cases in which it is difficult to determine unique interpretations using traditional methods (e.g., insufficient duration tests, lengthy WBS distortion, and effects of ultra-low permeability). The proposed DVPF allows us to observe underlying characteristics that are obscured at early times in traditional pressure and derivative analysis, and for the demonstration examples provided in this work, the DVPF does provide a strong auxiliary means of interpretation.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 27–29, 2014

Paper Number: SPE-170969-MS

.... enhanced recovery pore network shale gas Brine complex reservoir

**dimensionless****time**wettability index porosity imbibition MT sample wettability imbibition experiment contact angle affinity spontaneous imbibition Reservoir Characterization Upstream Oil & Gas pyrobitumen IMBIBITION data...
Abstract

This paper aims at understanding the factors controlling the wettability of unconventional rocks. In the first part, we report the results of comparative imbibition experiments on several binary core plugs from the Montney tight gas formation, which is an enormous tight gas fairway in the Western Canadian Sedimentary Basin. Both contact angle and imbibition data indicate that the formation is strongly oil-wet. However, the ratio between oil and water imbibition rate of these samples is higher than what capillary-driven imbibition models predict. This discrepancy can be explained by the strong adsorption of oil on the surface of a well-connected organic pore network that is partly coated by pyrobitumen. We also define a wettability index by using the equilibrium imbibed volume of oil and water in binary plugs. Oil wettability index is in general positively correlated to the total organic carbon (TOC), measured by the Rock-Eval technique. In the second part, we report similar imbibition experiments on several binary core samples collected from the cores drilled in the shale members of the Horn River Basin. In contrast to the Montney (MT) samples, the Horn River (HR) samples imbibe significantly more water than oil. This observation contradicts the contact angle results which suggest that the HR samples are strongly oil-wet. Clay hydration, imbibition-induced microfractures, depositional lamination, and osmotic potential are collectively responsible for the excess water uptake. We also measure and compare spontaneous imbibition of oil and water into the crushed packs of the similar HR samples. Interestingly, in contrast to the intact samples, the crushed samples consistently imbibe more oil than water. The comparative study suggests that the connected pore network of the intact HR samples is water-wet while the majority of rock including poorly connected pores is oil-wet. Overall, the results suggest that the well-connected pore network of the MT samples is dominantly hydrophobic and is very likely to be coated by pyrobitumen. This is the main reason why these samples imbibe more oil than water. On the other hand, the well-connected pore network of the HR samples is strongly hydrophilic primarily due to the presence of clay minerals and precipitated salt crystals coating the rock grains.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 8–10, 2012

Paper Number: SPE-159379-MS

... minutes on common PC platforms. data mining fracture pattern Simulation Upstream Oil & Gas

**dimensionless****time**Artificial Intelligence modeling Mayerhofer boundary condition fracture uniform flux dimensionless pressure productivity index Economide reservoir permeability equation...
Abstract

Newly-developed, generalized analytic solutions to the heat equation for arbitrary 3D well trajectory in anisotropic media are demonstrated to solve benchmark horizontal and slanted well productivity problems with unprecedented speed and accuracy. Arbitrary well trajectory is constructed as an assemblage of spatially integrated, linear well segments, as opposed to a distribution of numerically integrated point sources, to provide advantages in both computational speed and accuracy in singularity handling. Production from each arbitrarily-oriented segment is reduced to a combination of purely analytic expressions and rapidly-convergent, exponentially-damped infinite sum approximations. With offered flexibility in cell boundary conditions, the expressions can yield stand-alone well productivity estimates for complex wells or serve as the basis for advanced well equations, if integrated within a numerical reservoir simulator. Transients are also computed with analytical integrations in time, thus requiring no time marching. The breakthrough speed and accuracy in productivity assessment opens possibilities for advanced well testing and reservoir characterization methods. We further demonstrate the usefulness of analytic methodology with several time-dependent, discrete fracture problems for shale gas production with typical Barnett conditions, allowing direct use of complex fracture patterns, such as those interpreted from microseismic. In addition to uniform flux and uniform pressure modeling options, a new analytic model is introduced that is capable of modeling both time-dependent material transport between matrix and a stimulated zone and the interplay between a well and fracture. We illustrate our solution method with fractured Barnett well examples from the literature. With optional effects such as gas desorption and stress-dependent fracture conductivity as easy add-ons, we can produce full operational life production forecasts for shale or tight gas reservoirs from discrete, complex fracture patterns along with reservoir pressure mappings in a matter of minutes on common PC platforms.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 8–10, 2012

Paper Number: SPE-160316-MS

... Solvent SAGD process. This analysis can also be applied for a mixture of solvents based on available experimental data. thermal method steam-assisted gravity drainage concentration viscosity enhanced recovery upstream oil & gas

**dimensionless****time**steam chamber artificial intelligence...
Abstract

With increasing world demand for energy, more attention has been given to the exploitation of the huge resources present in the form of heavy oil and bitumen. Although thermal methods such as steam assisted gravity drainage (SAGD) are very successful in recovering heavy oil and bitumen, the low thermal efficiency of the process and the high level of greenhouse gas emissions and water usage remain major concerns. Co-injection of solvent with steam has shown to be promising in enhancing oil rates as well as in reduction of energy and water consumption with lower environmental impacts. In hybrid steam-solvent methods, there is a balance between the solubility of the solvent and its ability to reduce bitumen viscosity, and the viscosity reduction due to temperature increase. Therefore, proper selection of the solvent for the operating conditions is key to improving the overall efficiency of the steam- solvent process over the steam-only method. In this study, enhancement of the oil flow rate in the hybrid steam-solvent process is investigated using steady state temperature and unsteady state concentration distribution ahead of the interface with different operational parameters. The Integral Method is employed for prediction of solvent distributions, and the viscosity variation in the mobile zone is determined using the Shu (1984) correlation. The results show that the fractional increase in oil flow rate depends on type and mole fraction of the solvent in the steam chamber. It is also observed that large values of mechanical dispersivity factor enhances oil rate significantly; and drainage rate increases linearly with operating pressure. The results can be used to find the optimum solvent candidate and the injection strategy to maximize the flow rate of the Expanding Solvent SAGD process. This analysis can also be applied for a mixture of solvents based on available experimental data.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 8–10, 2012

Paper Number: SPE-160219-MS

....

**dimensionless****time**imbibition surfactant enhanced recovery capillary force imbibition cell test wettability alteration upstream oil & gas scaling group ift reduction capillary pressure matrix block size permeability experiment gravity imbibition experiment oil recovery reservoir...
Abstract

Oil recovery from fractured carbonate reservoirs by water flooding is often inefficient due to the commonly oil-wet nature of these rocks and the lack of sufficient spontaneous capillary imbibition driving force to push oil out from the matrix to the fracture network. Chemical processes such as surfactant/alkali-induced wettability alteration and interfacial tension (IFT) reduction have shown great potential to reduce the residual oil saturation in matrix blocks, leading to significant incremental oil recovery (IOR). However, the IOR response time is the most crucial decision factor in field projects. The magnitude and time efficiency of recovery depend on the degree of wettability alteration and IFT reduction, the nature and density of fracture network, and the dimensions of matrix blocks. Oil recovery experiments were performed for the same matrix rock and chemical formulation, but for different sized cores to gain a better understanding of the time dependence of the recovery process. The measured oil recoveries were history-matched. The simulation models were then used to predict the recovery response times for larger cores. The controlled and systematic laboratory measurements for several core sizes helped in developing dimensionless scaling groups to aid in understanding the time dependence and upscaling of laboratory results to field-scale applications. This finding is significant as it illustrates the extent of wettability alteration and IFT reduction needed in fractured reservoirs. Laboratory measurements and simulation work substantiate the validity and the range of applicability of upscaled procedures, and indicate the importance of viscous and buoyancy forces in larger field cases. The results of this work will be useful for the design of future field projects.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 19–22, 2010

Paper Number: SPE-135804-MS

... production monitoring

**dimensionless****time**pseudo time reservoir area drillstem/well testing equation real time Estimation of oil-in-place, gas-in-place, and reserves is an integral part of reservoir development and management. This requires knowledge of initial reservoir pressure and average...
Abstract

A knowledge of the average reservoir pressure ( p ¯ ) and its changes as a function of time or cumulative production is essential to determine the oil-in-place (OIP) or original gas-in-place (OGIP), to estimate reserves and to track and optimize reservoir performance. The common practice of determining p ¯ in moderate permeability reservoirs has been to run pressure buildup tests. In the current economic environment, buildup tests are almost non-existent except for very expensive exploratory wells. Moreover, time required for a pressure buildup test to reach p ¯ in low permeability reservoirs is prohibitively long. Fortunately, flowing pressures and rate data are continually collected from oil and gas wells. Data quality and quantity is usually good especially from wells installed with permanent pressure gauges. Such data for gas wells is currently being analyzed by assuming OGIP and estimating p ¯ required for calculating pseudo time. This is done in an iterative manner for using advanced decline curve analysis methods. The purpose of this paper is to discuss a new finding that will enable direct estimation of p ¯ using flowing pressures and rate data obtained from oil and gas wells during the pseudo steady-state flow period. In theory, pseudo steady-state flow requires that a well is produced at a constant rate. However, this limitation can be easily removed based on the work published in the SPE literature by this author and others whereby variable rate data can be converted to constant rate production data. The significance of the subject paper is that it will permit: a) direct determination of p ¯ using flowing wellbore pressures and rate data thus facilitating estimates of OGIP and OIP, b) estimation and/or validation of the value of the initial reservoir pressure ( p i ), which is normally suspect, and finally, c) enhancement or possible elimination of the current iterative process used for determining OGIP by advanced decline curve analysis methods.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 24–27, 2006

Paper Number: SPE-102641-MS

... Artificial Intelligence enhanced recovery Upstream Oil & Gas correlation displacement Fluid Dynamics boundary condition IMBIBITION data flow in porous media

**dimensionless****time**imbibition radial flow spontaneous imbibition recovery Morrow viscosity ratio permeability frontal position...
Abstract

Abstract Spontaneous imbibition data for Berea sandstone cores, that are very strongly wetted by the aqueous phase and initially 100% saturated with mineral oil, are reported for linear, radial, and all-faces-open boundary conditions. Oil viscosities were 4, 63 and 173 cP and aqueous phase viscosities ranged from 1 to 495 cP. Oil/aqueous phase viscosity ratios were varied by over four orders of magnitude (0.01 to 173.1). Near-linear relationships, with slopes close to one half, between the frontal position and imbibition time on a log-log scale were obtained for both linear and radial counter-current flow. Behavior is consistent with near piston-like displacement by the imbibing aqueous phase. The results are analyzed by a new mathematical model that accounts for counter-current spontaneous imbibition with symmetrical flow patterns. The model assumes that saturation and permeabilities to counter-flowing phases behind the front are constant and that any effect of local change in interfacial curvature with distance is negligible. The results from the model are used to extend scaling to include the measured effect of viscosity ratio for linear and radial flow. For the all-faces-open boundary condition, commonly used in core analysis studies, oil recovery vs. imbibition time is estimated by a combination of spherical and radial flow. Consistently close agreement was obtained between experiments and behavior predicted by the model. Introduction Laboratory spontaneous imbibition experiments are commonly used to investigate the mechanism of oil recovery from fractured reservoirs. The rate of oil transfer from the rock matrix into the fractures determines oil production. Although capillary force is the dominant driving mechanism for spontaneous imbibition, the rate of oil recovery depends on many factors, including fluid viscosities, sample size, shape and surfaces open to imbibition. 1, 2 A model of spontaneous imbibition is needed that can be verified by laboratory experiments and has predictive capability. Differential equations of mass balance and extension of Darcy's equation to two-phase flow in porous media 3 often serve as the basis for modeling countercurrent spontaneous imbibition. For counter-current flow, the rate of water imbibition is assumed to be equal and opposite to the rate of oil production. If boundary conditions, relative permeability, and capillary pressure functions are specified, the progress of saturation and pressure profiles can be calculated. The effect of relative permeability and capillary pressure functions may be lumped together as a single saturation function 4–7 in order to reduce the number of function parameters. In practice, saturation functions have to be tuned to either match the measured saturation profiles or the imbibition rate calculated by integration of the saturation profiles. When cores are very strongly water-wet, it can be difficult to tune the saturation functions to match the sharp imbibition front. More significantly, because of lack of experimental data, the traditional approach has not been fully tested against imbibition experiments under different flow patterns and a wide range of viscosity ratios. Another approach to prediction of the rate of imbibition involves development of dimensionless scaling groups that compensate for the effects of sample size, shape, boundary condition and rock and fluid properties. A scaling group proposed by Mattax and Kyte 8 was later modified by Ma et al. 1 to ............................................ (1) where k is the rock permeability, f? is rock porosity, s is the water/oil interfacial tension, µ w and µ o are the water and oil viscosities, and L c is the characteristic length, which depends on the sample size, shape, and boundary conditions. The scaling group (Eq. 1) correlated available oil/water imbibition data satisfactorily. 1, 9

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 24–27, 2006

Paper Number: SPE-102834-MS

... (these responses were computed by using a single-block with ten horizontal-well segments to simulate infinite conductivity). The

**dimensionless****time**used in Fig. 12 is based on the reference permeability of 10 md. Qualitatively, the order of the pressure and derivative responses shown in Fig. 12 are in...
Abstract

Abstract This paper presents a semianalytical model for the pressure-transient analysis of horizontal wells in composite, layered, and compartmentalized reservoirs. The model divides the reservoir into blocks that represent locally homogeneous substructures of the reservoir and couples the analytical, pressure-transient solutions at the block boundaries. This approach is consistent with the averaging effect of pressure transients and provides an alternative to full numerical modeling of horizontal-well pressure-transient responses in heterogeneous formations. The model can also be generalized for multiple wells of different geometry including multiple laterals. Introduction Conventional pressure-transient models assume that the entire length of a horizontal or multilateral well remains in the same formation with uniform properties. In reality, most horizontal and multilateral wells extend through subsections of the reservoir consisting of layers, sections, or pockets of reservoir with different properties. Furthermore, the reservoir subsections may be commingled or in partial or full communication. This paper presents a semi-analytical, pressure-transient model for the analysis of horizontal- and multilateral-well responses in composite, layered, and compartmentalized reservoirs. The model presented in this study divides the reservoir into blocks that represent the subsections of the reservoir characterized by uniform average properties. An analytical pressure-transient solution is developed for each block with the Neumann condition at the outer block boundaries together with the inner boundary condition appropriate for the particular well type and production condition. The analytical solution is discretized at the inner and outer block boundaries and coupled by the solution for the adjacent blocks by using the continuity of pressure and flux. Adding the production constraint for the well(s) creates a matrix problem to be solved for the wellbore pressure together with the pressures and fluxes at the block boundaries. Knowing the flux and pressure conditions at the block boundaries, pressure distributions within the blocks can also be obtained from the analytical solutions. This solution approach is similar to the boundary element method. Although this paper focuses on horizontal wells, the solution procedure can be generalized for other types of wells and it is also valid for multiple wells of different geometry. The accuracy and the efficiency of the method depend on the grid system chosen to discretize the domain boundaries that consist of the well and block boundary surfaces. As has been known for the boundary element solution of the Laplace's equation [Gaul and Fischer (2006)], even crude discretizations of the boundaries lead to reasonable approximations of the pressures (Dirichlet data) while the accuracy of the flux (Neumann data) may require finer discretizations. The semi-analytical approach presented here provides an alternative to full numerical modeling of horizontal- and multilateral-well pressure-transient responses in heterogeneous formations. Although modeling finer details of heterogeneity may remain to be a numerical task for some applications, such as tracking saturation distributions, dividing a heterogeneous reservoir into substructures with relatively uniform properties is consistent with the averaging effect of pressure transients and is supported by the known characteristics of many heterogeneous fields. The basic solution and methodology presented in this work may also be extended to other applications. For example, the fundamental solution developed in this work can be used to replace the simpler well indices in the numerical reservoir simulators to eliminate the requirement of very fine grid to capture flow convergence around wells. Our objective in the present paper, however, is to lay down the mathematical fundamentals of the proposed method, verify its results, and present some examples to demonstrate its utility. Mathematical Model First, we will present the Green's function formulation of the pressure-transient solution for a locally homogeneous reservoir substructure with arbitrary flux conditions on the boundaries. Then, we will explain the coupling of the substructures of the reservoir to obtain the pressure-transient solution for the combined system.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 24–27, 2006

Paper Number: SPE-103229-MS

... enhanced recovery compressibility regime Reservoir Characterization Fluid Dynamics

**dimensionless****time**gas well pressure drop Upstream Oil & Gas displacement water saturation different**dimensionless****time**evaporation regime permeability boundary condition capillarity equation...
Abstract

Abstract Gas expansion near the wellbore during production causes the evaporation of connate water. When the reservoir permeability is low, capillarity is controlling, causing liquid movement to the near-wellbore region where drying rates are higher. In tight gas sands or in shale gas formations, where capillarity is high, the gas production itself can cause depletion of the water saturation below residual values, due to such evaporation. In this work we present a study of the fundamental processes involved during the flow of a gas in a liquid- saturated porous medium. We have modeled evaporation by accounting for the capillary driven film flow or ‘wicking’ of saline liquid to the wellbore or the near-fracture region and the effect of gas expansion. It is shown that, for gas reservoirs with connate water saturation, large pressure drawdowns lead to a drying front that develops at the formation face and propagates into the reservoir. When pressure drops are lower, water rapidly redistributes due to capillarity-induced movement of liquid from high-to low-saturation regions. This phase redistribution causes higher drying rates near the wellbore. The results show, for the first time, the effect of both capillarity- induced film flow and gas compressibility on the rate of drying in gas wells. The model can be used to help maximize gas production under conditions such as waterblocking by optimizing the operating conditions. Additionally it can be used to obtain a better understanding of the impact of capillarity on evaporation and consequent processes, such as salt precipitation. Introduction Problems involving gas flow past trapped liquids in porous media are encountered in a variety of contexts, such as waterblock removal in gas wells, evaporation of volatile oils, and recovery of residual oil. In the case of a binary system, such as gas and water, the thermodynamic phase equilibrium can be represented by a simple linear law and gas injection reduces to a drying problem where the remaining liquid is evaporated by the flowing gas. Drying of wetting liquids in porous media has been studied by several authors. These studies mainly focused on pass-over drying where gas is passed over a porous medium saturated with the wetting liquid. This form of drying is controlled by the gas flow rate. However when the liquid recedes into the porous medium, drying is controlled by the rate of diffusion of the components in the liquid phase in the pore spaces. Early in 1949, Allerton et al. 1 studied through-drying of packed beds of crushed quartz and other porous materials by convection of dry gas. The study however did not consider the effect of gas compressibility or capillarity. Whitaker 2 developed a diffusion theory of drying using volume averaging methods with constant pressure in the gas phase. This eliminated the effect of compressibility of gas on the drying rates and hence is useful only in a pass-over drying context. Experimental and simulation studies of gas injection 3,4,5 showed that trapped water is first removed by a viscous displacement followed by a long period of evaporation. These studies showed that higher pressure drop, permeability and temperatures caused greater rates of evaporation and faster progression of saturation drying fronts in both fractured and unfractured wells.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 9–12, 2005

Paper Number: SPE-96812-MS

...). upstream oil & gas fluid property core analysis imbibition data oil recovery recovery boundary condition linear imbibition final oil recovery

**dimensionless****time**viscosity waterflooding afo imbibition ooip imbibition time viscosity ratio spontaneous imbibition characteristic length...
Abstract

Abstract Laboratory results of oil recovery through spontaneous imbibition are commonly scaled-up to forecast oil recovery from fractured reservoirs. This study addresses oil recovery from cylindrical sandstone cores by spontaneous imbibition at very strongly water-wet conditions for viscosity ratios of unity (a special case of the Mattax and Kyte scaling condition that viscosity ratios be matched). Combinations of mineral oils and aqueous solutions of glycerol were used to obtain matched viscosities ranging from 4 to 172 cp. In all, 25 imbibition data sets are reported for boundary conditions of all core faces open, two ends closed (radial imbibition), and one end open (linear imbibition). The data sets for individual boundary conditions were satisfactorily correlated by the Mattax and Kyte scaling group. Overall correlation of results was obtained after compensating for different boundary conditions by means of a characteristic length. An obvious limitation of correlations based on linear scaling of time is that differences in the shapes of imbibition curves cannot be eliminated. The correlated data provides clear distinction between the overall shape of recovery curves for linear versus radial imbibition. Final oil recoveries for radially dominated imbibition were independent of viscosity whereas recoveries for linear imbibition were consistently lower and decreased by up to 2.5 % PV with increase in viscosity. Shortly after the onset of imbibition, oil recovery for linear flow is generally close to linearly proportional to the square root of time until the imbibition front reaches the end of the core. Introduction In naturally fractured reservoirs, spontaneous imbibition of brine can be the dominant mechanism of oil production from the rock matrix. Rock properties, fracture geometry, fluid properties, and rock fluid interactions govern the rate and extent of recovery.[1–2] Oil recovery from laboratory measured spontaneous imbibition is commonly scaled to obtain mass transfer functions that are used to predict oil production from fractured reservoirs. Many questions remain unanswered in regard to the mechanism of spontaneous imbibition and valid scaling of laboratory imbibition data. Mattax and Kyte[3] introduced a scaling group based on the theoretical analysis of Rapoport and Leas.[4–5] (1) where µw is the aqueous phase viscosity and µo is the oil phase viscosity. Identical viscosity ratio was one of six listed conditions for valid scaling. The complexities of spontaneous imbibition in porous media are such that perfect correlation by linear scaling of time should not be expected. This study tests aspects of scaling for matched oil and aqueous phase viscosities. Results are presented for cores with all faces open, one end open (linear imbibition) and two ends closed (radial imbibition) for nearly two orders of magnitude variation in viscous forces but only small change in capillary forces (caused by small differences in interfacial tension).

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 9–12, 2005

Paper Number: SPE-96785-MS

... equation drillstem testing injection storage coefficient flow rate fracture fracture half length pressure transient testing

**dimensionless****time**storage drillstem/well testing Copyright 2005, Society of Petroleum Engineers This paper was prepared for presentation at the 2005 SPE Annual Technical...
Abstract

Abstract A new refracture-candidate diagnostic test is presented that requires a brief injection at a pressure exceeding the fracture initiation and propagation pressure followed by an extended shut-in period with the pressure falloff recorded. Provided the time of injection is short relative to the reservoir response, the pressure falloff can be analyzed as a slug test by transforming and plotting the falloff data on a variable-storage, constant-rate drawdown type curve for a well producing from single or multiple finite- or infinite-conductivity vertical fractures in an infinite-acting reservoir. Characteristic variable storage behavior is used to diagnose a pre-existing fracture retaining residual width and to determine if a pre-existing fracture is damaged. Using the proposed model and analysis methodology, a quantitative type-curve/model match can be used to estimate reservoir properties. In addidtion to the new fracture diagnostic solution, a new single-phase fracture-injection/falloff dimensionless pressure solution is also provided along with new pressure-transient solutions for a well producing from multiple arbitrarily-oriented finite- or infinite-conductivity fractures. Introduction Oil and gas wells often contain potentially productive layers bypassed either intentionally or inadvertently during an original completion.Subsequent refracturing programs designed to identify underperforming wells and recomplete bypassed layers have sometimes been unsuccessful in part because the programs tend to focus on commingled well performance and well restimulation potential without thoroughly investigating individual layer properties and the refracturing potential of individual layers. Perhaps the most significant impediment to investigating layer properties is a lack of representative and cost-effective diagnostics that can be used to determine layer permeability, reservoir pressure, and to quantify the effectiveness of previous stimulation treatments. Post-frac production logs,[1–2] near-wellbore hydraulic fracture imaging with radioactive tracers,[3–4] and far-field microseismic fracture imaging[5] all suggest 10% to 40% of the layers targeted for completion during primary fracturing operations using limited-entry fracture treatment designs are bypassed or ineffectively stimulated. Quantifying bypassed layers has proved difficult because imaged wells represent a very small percentage of all wells completed. Consequently, bypassed or ineffectively stimulated layers may not be easily identified, and must be inferred from analysis of the commingled well stream, production logs, or conventional pressure-transient tests of individual layers. A refracture-candidate diagnostic used prior to a refracture treatment should complete the following objectives.[6] To determine if: A pre-existing fracture exists. A pre-existing fracture is damaged. To estimate: Effective fracture half-length of a pre-existing fracture. Fracture conductivity of a pre-existing fracture. Reservoir transmissibility. Average reservoir pressure. When the diagnostic test objectives are achieved, the benefits of refracturing can be easily evaluated, and the incremental production from a refracture treatment can be predicted. Quantitative conventional pressure-transient testing, which includes drawdown, drawdown/buildup, or injection/falloff tests at a pressure less than the fracture propagation pressure, can be used to achieve the objectives of a refracture-candidate diagnostic test. However, conventional pressure-transient tests are best suited for evaluating a single layer. For wells producing from multiple layers, multilayer pressure-transient tests have been published,[7] but in practice, determining layer flow rates for test interpretation from multiple layers is problematic—especially with upwards of 20 layers producing.[8] In general, a cost-effective quantitative diagnostic test does not exist for wells producing from multiple layers. Diagnostic testing in low permeability multilayer wells has been attempted, and Hopkins et al.[9] describe several diagnostic techniques used in a Devonian shale well to diagnose the existence of a pre-existing fracture(s) in multiple targeted layers over a 727 ft interval. The diagnostic tests included isolation flow tests, wellbore communication tests, nitrogen injection/falloff tests, and conventional drawdown/buildup tests. The post-frac diagnostic program described by Hopkins et al.[9] is thorough and addresses the objectives of a refracture-candidate diagnostic. However, the diagnostic program is also expensive and time consuming - even for a relatively simple four layer case. Many refracture candidates in low permeability gas wells contain stacked lenticular sands with between 20 to 40 layers which need to be evaluated in a timely and cost effective manner.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 26–29, 2004

Paper Number: SPE-90132-MS

... waterflood recovery water saturation upstream oil & gas initial water saturation permeability recovery imbibition

**dimensionless****time**water relative permeability relative permeability oil recovery pore experiment saturation wettability index Copyright 2004, Society of Petroleum Engineers...
Abstract

Abstract Counter-current imbibition is an important recovery mechanism during waterflooding in fractured reservoirs. While this may be a rapid and efficient recovery process in strongly water-wet systems, the vast majority of reservoirs show some mixed-wet or oil-wet characteristics. If the reservoir is mixed-wet, it is possible for some water to imbibe spontaneously, but the ultimate recovery is lower and the imbibition rate may be several orders of magnitude slower than for strongly water-wet rock. We use quasi-static pore-scale network modeling as a tool to study the behavior of mixed-wet rocks and to predict relative permeability and capillary pressure. The model uses a topologically disordered network that represents the pore space of Berea sandstone. We adjust the distribution of contact angles at the pore scale to match previously publised experimental co-current waterflood recoveries and wettability indices on Berea. We then input the relative permeabilities and capillary pressures into a conventional grid-based code and simulate counter-current imbibition in one dimension. We make predictions, with no matching parameters, of the recovery as a function of time and compare the results with the experimental measurements. We are able to reproduce the observed dramatic increase in imbibition time as the system changes from being water-wet to mixed-wet. In a mixed-wet system spontaneous imbibition, where the capillary pressure is positive, is limited to a narrow saturation range where the water saturation is small. At these low saturations the water is poorly connected through the network in wetting layers and the water relative permeability is extremely low, leading to recovery rates tens to thousands of times slower than for water-wet media. We present a semi-empirical equation to correlate imbibition recovery in mixed-wet rocks with different wettability states and for a wide range of viscosity ratios. We show that the recovery rate is proportional to the water mobility at the end of imbibition. Introduction Fractured reservoirs are important oil and gas resources. The fracture network contains a small amount of oil in place compared to the lower permeability matrix that it is connected to. High well productivity and relatively low ultimate recovery are typical characteristics of these reservoirs. Waterflooding is one of the most important mechanisms of oil production from fractured reservoirs. Imbibition is the displacement of non-wetting phase by wetting phase. In a strongly water-wet rock water rapidly imbibes into the rock and displaces the non-wetting phase, oil. However, the majority of reservoir rocks are not strongly water-wet. 1 Salathiel 2 introduced the term mixed wettability for cases where the rock contains both water-wet and oil-wet fractions. After primary oil flooding those larger pores occupied by oil may change their wettability, while smaller water-filled regions of the pore space remain water-wet. The adsorption of surface-active agents in the oil, such as asphaltenes, to the pore surface in direct contact with the oil causes wettability alteration. 3–6 One of the important characteristics of mixed-wet rock is its ability to imbibe both water and oil. 6–9 Zhou et al . 10 performed 23 spontaneous imbibition and 27 waterflood experiments on Berea cores with different wettabilities and initial water saturations, corresponding to different wettabilities. To establish different initial water saturations after primary drainage, brine was displaced by Prudhoe Bay crude oil at different injection pressures. Then the samples were aged at a temperature of 88°C for between 0 and 240 hours to alter the wettability of the samples from water-wet towards mixed wettability. The oil that was in the core during aging was displaced with fresh crude oil prior to imbibition and waterflooding tests. Counter-current imbibition recovery was measured by the change in weight of the aged samples that were hung in a degassed brine solution. In flooding tests, the samples at initial water saturation were flooded at slow rates using a constant injection pressure. The waterfloods were stopped after 4 to 15 pore volumes of brine injection when the water-oil ratio, r wo , was greater than 99. The recovery at this point was used as an operational definition for the final waterflood recovery R wf .

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 29–October 2, 2002

Paper Number: SPE-77455-MS

... & gas flow rate hmax permeability anisotropy society of petroleum engineers interpretation reservoir permeability ellipse interference point drillstem/well testing drillstem testing interference minimum permeability hmin permeability interference timing

**dimensionless****time**spe 77455...
Abstract

Abstract This work provides a technique for determining the time and the point at which pressure disturbances originating from two adjacent wells begin to interfere significantly. Reservoir continuity and flow anisotropy due to natural fractures are common issues in tight-gas reservoir. Lacking and accuracy of pressure data makes the interpretation highly uncertain. The developed technique will provide a method to assist the interpretation of well-interference issue and the optimization of well spacing/pattern in tight-gas reservoirs. The technique is based on analytical model and is easy to apply. Introduction Well interference is important in determining the reservoir continuity and in optimizing well spacing/pattern of tight-gas reservoirs. Field data interpretation, however, is highly uncertain. Primary reasons may include: difficulty in achieving confident reservoir characterization, especially the trend of natural fracture or permeability anisotropy, the general lacking of pressure measurements due to both technical and economical reasons, and lack of practical techniques to assist interpretation. The objective of this study is to develop techniques for determining the time and the point at which pressure disturbances originating from two adjacent wells begin to interact, or interfere, significantly. This type of problem had been addressed by Stevens and Thodos 1 and Warren and Hartsock. 2 Their work, however, did not consider permeability anisotropy. This study extends their techniques to include the effect of permeability anisotropy. The developed technique provides a method to assist the interpretation of well-interference issue and the optimization of well spacing/pattern in tight-gas reservoirs.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 29–October 2, 2002

Paper Number: SPE-77544-MS

... cocurrent and countercurrent cases. We defined a

**dimensionless****time**with almost all the parameters considered. These include porosity, permeability, size, shape, boundary conditions, wetting and nonwetting phase relative permeabilities, interfacial tension, wettability, and gravity. The definition of the...
Abstract

Abstract Scaling the experimental data of spontaneous imbibition without serious limitations has been difficult. To this end, a general approach was developed to scale the experimental data of spontaneous imbibition for most systems (gas-liquid-rock and oil-water-rock systems) in both cocurrent and countercurrent cases. We defined a dimensionless time with almost all the parameters considered. These include porosity, permeability, size, shape, boundary conditions, wetting and nonwetting phase relative permeabilities, interfacial tension, wettability, and gravity. The definition of the dimensionless time was not empirical, instead, was based on theoretical analysis of the fluid flow mechanisms that govern spontaneous imbibition. The general scaling method was confirmed against the experimental data from spontaneous water imbibition conducted at different interfacial tensions in oil-saturated rocks with different sizes and permeabilities. A general analytical solution to the relationship between recovery and imbibition time for linear spontaneous imbibition was derived. The analytical solution predicts a linear correlation between the imbibition rate and the reciprocal of the recovery by spontaneous imbibition in most fluid-fluid-rock systems. Introduction An important fluid flow phenomenon during water injection or aquifer invasion into reservoirs is spontaneous water imbibition. Scaling the experimental data of spontaneous water imbibition in different fluid-fluid-rock systems is of essential importance to designing the water injection projects and predicting the reservoir production performances. Ignoring the effects of relative permeability, capillary pressure, and gravity in the dimensionless time might be the reason that the existing scaling methods do not always function successfully. It is known that these parameters influence the spontaneous imbibition in porous media significantly. For that reason, these parameters should be honored properly in the scaling. Many papers have been published to characterize and scale spontaneous water imbibition in both oil-water-rock systems 1–20 and gas-liquid-rock systems 21–24 . However few have included the effects of capillary pressure, relative permeability (both wetting and nonwetting phases), wettability, and gravity simultaneously. This is important because all the parameters may play an important role in many cases and may not be ignored. For example, a lot of enhanced/improved oil recovery processes relate to low interfacial tension (IFT). In these cases, capillary pressure as a driving force may be small and gravity may not be neglected. In some cases, gravity may also be a driving force as pointed out by Schechter et al . 25 . One of the frequently-used dimensionless time groups in the past to scale spontaneous imbibition data is defined as follows (Ma et al . 15 ): Equation 1 where t D is the dimensionless time, k is the rock permeability, f is the porosity, s is the interfacial tension between the wetting and nonwetting phases, t is the imbibition time, µ m is the geometric mean of the viscosities of the two phases and L a the characteristic length. The dimensionless time defined in Eq. 1 is suitable for oil-water-rock systems under specific conditions. These include: wettability must be the same, relative permeability functions must be identical, capillary pressure functions must be identically proportional to interfacial tension, initial fluid distributions must be duplicated, and gravity must be neglected. On the other hand, Eq. 1 implies that higher IFT systems have higher imbibition rate, which is not true in many cases. For example, Schechter et al . 25 observed experimentally that the imbibition rate in oil-water-rock systems with low IFT was greater than with high IFT in high permeability core samples. Al-Lawati and Saley 18 reported a similar phenomenon.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 29–October 2, 2002

Paper Number: SPE-77689-MS

... characteristics of a carbonate rock have been recognized by several works. dimensionless pressure matrix fracture interaction drillstem/well testing permeability drillstem testing decline curve behavior approximation flow in porous media

**dimensionless****time**flow period spe 77689 expression...
Abstract

Abstract This study presents a new way to model high secondary porosity, mainly vuggy porosity, in naturally fractured reservoirs. New solutions are presented for two cases, where there is no primary flow through the vugs, which is an extension of the Warren and Root model, and where dissolution process of pore throats has created an interconnected system of vugs and caves. In both cases there is an interaction between matrix, vugs, and fracture systems. New insights are provided. Both pressure and production responses during transient and boundary dominated flow periods are explored. In transient well tests, for the case where there is no primary flow through the vugs, a change of slope could be present during the transition period. Thus, this study shows that slope ratios of 2:1 of early or late-time segment versus transition segment do not necessarily imply transient interaction between matrix and fractures. It is also shown that the presence of vugs and caves may have a definitive influence on decline curve and cumulative production behaviors; therefore it is necessary to incorporate vuggy porosity in the process of type curve match. Finally, the use of the methodology obtained in this work is illustrated with synthetic and field examples. Introduction Most of the world's giant fields produce from naturally fractured and vuggy carbonate reservoirs that have complex pore systems, mainly because carbonate rocks are particularly sensitive to post-depositional diagenesis, including dissolution, dolomitization and fracturing processes. Complete leaching of grains by meteoric pore fluids can lead to textural inversion which may enhance reservoir quality through dissolution or occlude reservoir quality through cementation 1 . Some works have classified carbonates based on fabric selective and non fabric selective pore types. The non-fabric selective are vugs and channels, caverns, and fractures 1 . For the purpose of this work no distintion is made on vugs, caverns and channels, and they will be denoted by the term vugs. Thus, vugs may vary in size from millimeters to meters in diameter. Vuggs are the result of carbonate and/or sulfate dissolution. From cores observations, the matrix porosity types adjacent to the vuggy zones are moldic, solution-enlarged microfractures, and solution-enlarged intercrystalline. Thus, it is possible to have a permeability enhancement adjacent to the vuggy zones. Three porosity types, matrix, fractures, and vugs, are usually present in naturally fractured, vuggy carbonate reservoirs. The determination of permeability and porosity in vuggy zones from core measurements are likely to be pessimistic because of sampling problems. In areas lacking cores, open-hole wireline logs may be used to help identify vuggy zones; however, vugs are not always recognized by conventional logs because of their limited vertical resolution 2 . Vuggy porosity is common in many carbonate reservoirs and its importance in the petrophysical and productive characteristics of a carbonate rock have been recognized by several works.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, September 30–October 3, 2001

Paper Number: SPE-71586-MS

... boundary injection rate pressure transient analysis injection plane pressure derivative plane spe 71586 segment plane society of petroleum engineers

**dimensionless****time**upstream oil & gas main plane channel reservoir SPE 71586 Transient Pressure Response of Fluvial Reservoir With...
Abstract

Abstract Fluvial reservoirs with branching channels and splays comprise a commonly encountered depositional system, however, characterization of this type of system using transient pressure analysis has not yet been fully explored. This article presents a semi-analytical method to compute the transient pressure and pressure response for this system type. It simplifies the computation by decoupling the complex-geometry system into a discrete set of simple-geometry systems that are in hydraulic contact with each other, and exchange fluids at their hydraulic contacts [1,2] . The computed pressure and pressure derivatives were compared with those of other simple well/reservoir systems to gain insight into the information contained in the responses. The source and sink method [3] was used to compute the pressure response in the Laplace domain and the results were inverted numerically using the Stehfest Inversion algorithm [4] . The discussion in this paper is focused on four cases, consisting of a main channel and a side branch that connects with the main branch at angles of 30, 45, 60 and 90 degrees, respectively. In each of these cases, the set of "image" wells that create no-flow boundaries is easy to generate, and an efficient computational algorithm is developed. Excellent pressure and pressure derivative responses have been obtained; detailed examination of these responses provides insight into methods that may aid in the identification and characterization of this type of system. Introduction In fluvial and deltaic depositional environments, channel reservoirs with connected slanted branching channels and splays are frequently encountered. There is a need to understand the pressure transient behavior in this type of system, however, to our knowledge, this topic has not been addressed until now. Pressure transient data in this kind of reservoir system contains valuable information regarding the dimensions and properties of the reservoir. To the best of our knowledge, there are not many publications in this area. Some pertinent papers are listed in the references [1,2,5,6] . Larsen presented work on a network of interconnected linear reservoirs [5] and dealt with long-time productivity, with no special consideration given to the geometry at the regions of intersection. Chen's study solved the transient pressure analysis problem for a wedged reservoir with an arbitrary intersection angle [6] . As in our previous work [1,2] , we approach the computation of the pressure transient response for the complex-geometry reservoir by first decomposing the system into a set of simple-geometry systems, then writing down analytical solutions for each of the simple reservoir components in terms of the unknown pressures and fluxes at their boundaries. The coupled systems are then solved to compute the desired complex-geometry pressure response. This paper presents an efficient algorithm to compute the semi-analytical transient pressure response of fluvial reservoirs with connected branching channels at branch angles of 30, 45, 60 and 90 degrees. The proposed methodology can be extended to any system where the Laplace-space solution can be easily written in terms of integrals of real-space source/sink functions, including production at constant bottom-hole pressure, wellbore storage effects or naturally fractured systems. The method of sources and sinks [3] is used to compute the pressure response in the Laplace domain and the results are inverted numerically using the Stehfest Inversion algorithm [4] . We have applied fast, accurate methods of taking numerical Laplace transforms of the source/sink solutions that make the computations reasonably fast and efficient [2] . Methodology The physical model considered in Fig. 1 consists of two channel reservoirs in communication with each other. One has flow properties of the channel, and the other has properties of the branching channel. The coordinate system is defined in Fig. 2 . The following steps were applied to solve the problem [2] .

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 1–4, 2000

Paper Number: SPE-62921-MS

...

**dimensionless****time**permeability spe 62921 drillstem testing downhole water sink completion vertical permeability pressure response dual completion equation Copyright 2000, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2000 SPE Annual Technical Conference and...
Abstract

Abstract Downhole Water Sink (DWS) well completions segregate production in the wellbore by producing water from the water leg in the reservoir and oil from the upper oil bearing portion of the reservoir. A downhole pump drains water from the bottom completion and creates pressuredrawdown that inhibits water coning up to the upper producing zone. The pump injects the water to a disposal zone lower in the wellbore. Successful application of DWS technology in wells with water coning problem requires effective zonal isolation between the top and bottom completions of the well and good knowledge of permeability anisotropy. As these wells are dual completed, they are uniquely configured for vertical interference testing. The problem is that such test involves two fluids, water and oil, and is typically performed in a reservoir rock with no isolating layer between the top and bottom completion. The paper presents a new mathematical model and analysis method for vertical interference testing using top completion (in the oil leg) for production, and bottom completion (in the water leg) for observation. The model is analytical and accommodates partial penetration and permeability anisotropy. The analysis method employs a family of type curves. Also shown in the paper are examples of possible applications of this new testing method. Introduction Vertical permeability is an important parameter of a reservoir with water coning because it controls the well's critical rate and water breakthrough time. Conventional techniques for estimating vertical permeability include analysis of water production history to discern vertical permeability from the breakthrough time correlations, analysis of the early spherical flow patterns in pressure transient tests, or vertical interference testing. Known correlations for the water breakthrough time were developed on the scaled models or using numerical simulators. Even if the correlations had universal applicability to the real field data and relatively high accuracy, use of these correlations for estimation of the vertical permeability would require very detailed and accurate records of the oil and water production. This is usually not the case in routine production operations. Analyzing of the spherical flow pattern during pressure build-up in a partially-penetrating well requires substantial duration of the spherical flow period. Otherwise, the pattern may be completely covered by the wellbore storage effects. Thus this method is applicable in thick reservoirs having a very short completion. Vertical interference test is an expensive and complicated procedure that would require creation of two perforated intervals completed at a distance that can accommodate pressure gage, packer, and producing interval. 1–4 After the test, the bottom completion should often be cemented to prevent early water breakthrough into the well. However, in spite of the inconvenience and cost vertical interference tests should have better resolution and quality of the obtained results comparing to a single-well test. Recently, a new method of well completion, downhole water sink (DWS), was proposed and demonstrated in several field trials. 5 A schematic of the DWS completion is shown in Fig. 1a. DWS is a dual completion that has one completion in the oil and another completion in the water column of the reservoir. A packer separates the completion inside the well. The main concept of DWS is to create a hydro-dynamic effect at the oil-water contact (OWC) interface. The pressure drawdown created by oil production through the top completion is counter-balanced by the drainage of water from the bottom completion. This completion configuration gives an operator a tool to reduce or eliminate water-coning effects.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 1–4, 2000

Paper Number: SPE-63239-MS

.... This is especially true for those cases where the reservoir flow contribution to total leak-off is the controlling factor, as can be the case for fracpacking operations 5,9,10 . nonlinear leak-off spe 63239

**dimensionless****time**permeability upstream oil & gas nonlinear fluid leak-off...
Abstract

Abstract Fluid leak-off in hydraulic fracturing is conventionally described using the well-known linear (Carter) model. Although this works well in low-permeability formations, the linear leak-off assumption may lead to -sometimes significant- overestimation of fracture dimensions in medium to high permeability formations. At present, no methodology is available that allows an easy estimation of the impact of non-linear fluid leak-off on fracture dimensions and on pressure decline (mini-fracture) analysis. The current paper aims at resolving that deficiency. An exact numerical solution is presented to the fully transient elliptical fluid flow equation around a propagating hydraulic fracture for arbitrary pump rate(s). In addition, a simple analytical formula for leak-off rate is presented that is shown to yield an excellent approximation of the numerical results, both during fracture growth and after shut-in. This formula can be easily incorporated into any existing hydraulic fracture model, and is applicable over the entire range of fluid leak-off rates, i.e. from low-permeability fracture stimulation on one hand to high-permeability waterflood fracturing on the other hand. The above result is applied to a variety of hydraulic fracturing field examples to explore the limits of the linear (Carter) leak-off assumption, both in pressure-decline analysis of mini-fractures and in fracture design. It is, amongst others, shown that in frac-packing and high-perm CRI, the linear leak-off assumption may lead up to a tenfold overestimation of fracture dimensions. This result helps to explain the common field observation that CRI in unconsolidated sandstones results in fractures that appear to be significantly smaller than predicted by conventional hydraulic fracture models. 1.Introduction Fluid leak-off from hydraulic fractures is normally described by a one-dimensional (Carter) fluid flow model. In its simplest form, the leak-off rate within this model is, for a propagating fracture of constant height h, given by the equation Equation 1 where Q l is the leakoff rate at time t, h and L are fracture height and length, respectively, C T is the total leakoff coefficient, and t(x) is the first time of exposure of x to injection fluid. It is well-known that Eq. (1) only works properly if the fracture propagation rate is large compared to the leak-off diffusion rate. If this is not the case, the use of Eq. (1) can lead to overestimation of fracture length. For example, in waterflooding under fracturing conditions, this overestimation may be up to two orders of magnitude 1,2 . In this case, Eq. (1) needs to be replaced by a proper description of the reservoir fluid flow around the fracture 1–4 . Also, for hydraulic fracture stimulation (frac-packing) 5–11 and cuttings re-injection (CRI) 12–14 in high-permeability reservoirs, leak-off rate may be high enough compared to fracture propagation rate to the extent that using the 1D Carter model Eq. (1) is not justified anymore. This is especially true for those cases where the reservoir flow contribution to total leak-off is the controlling factor, as can be the case for fracpacking operations 5,9,10 .

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Annual Technical Conference and Exhibition, October 1–4, 2000

Paper Number: SPE-63174-MS

... near unity. injection rate fracture pressure slope analysis equation hydraulic fracturing fluid loss eff 50 tso closure

**dimensionless****time**efficiency pressure slope analysis fracture efficiency tso efficiency eff 10 mfrac net pressure bottomhole pressure compliance factor...
Abstract

Abstract The relative popularity and success of the frac & pack technique in hydraulic fracturing has resulted in some misconceptions regarding the objective, procedure and pressure analysis after a screen-out. This paper addresses frac & pack procedures and the pressure response after a tip screen-out (TSO). An analytical method has been developed for analyzing pressure slope behavior after a TSO in high permeability formations. These equations incorporate the first order parameters affecting the fracture pressure, rate of change of pressure (derivative) and pressure slope behaviors after a screen-out. The fundamental equations for pressure slope analysis are similar to those originally developed by Nolte for pressure decline analysis. The major difference is that after the fracture stops propagating (i.e., after a TSO) the injection rate is not zero. Consequently, if the injection rate is greater than the leakoff rate, the fracture volume and net pressure (constant compliance) must increase. If the injection rate falls below the leakoff rate, the fracture net pressure must decrease. Although analytical equations will not replace three dimensional fracturing simulators normally used for design and real-time history matching, they do provide insight into the major parameters affecting pressure behavior after a TSO without running a numerical simulator. The analytical equations presented in this paper demonstrate why pressure slopes after a screen-out are typically much greater than unity for low efficiency fractures. A generalized set of equations is presented for analyzing the pressure slope behavior after a screen-out. Numerous graphs are provided which illustrate the parametric effects of fracture efficiency, spurt loss and fracture net pressure at the time of a screen-out on the pressure, derivative and slope behaviors after a TSO. Comparisons of the analytical pressure slope equations with a three dimensional fracturing simulator are presented to show the application of the analysis. A new methodology of frac & pack post analysis is presented using the pressure slope technique. This methodology utilizes the pressure slope during a screen-out as a check on the minifrac and fracture efficiency. Two frac & pack cases with bottomhole data are analyzed using a three dimensional hydraulic fracturing simulator to illustrate the pressure slope analysis for low effi-ciency fractures. Introduction Godbey and Hodges 1 recognized the importance of analyzing fracture pressure data during a hydraulic fracturing treatment in 1957. However, it wasn't until 1979 that Nolte 2 developed a classic method of pressure decline analysis (fracture calibration or minifrac analysis) for estimating closure pressure, efficiency, leakoff coefficient, and fracture geometry. Nolte and Smith 3 in 1981 presented a technique for interpreting fracturing treating pressures based on the "Mode" of the pressure slope. The unit slope or Mode III behavior in log-log space implied "that the incremental pressure change is proportional to the incremental injected-fluid volume. " The interpretation is "that a unit slope implies that a significant flow restriction has formed in the fracture (e.g., proppant screenout). " Although Nolte and Smith pointed out that the net pressure increases at a rate proportional to the net injection rate, their analysis to find the screen-out location was based on a fracture efficiency of unity. Smith 4 in 1984 first presented a paper identifying the method of "a controlled screenout to achieve enough propped fracture width to ensure lasting facture conductivity. " The controlled terminology distinguishes itself as a designed screen-out rather than one which occurs inadvertently. Nolte and Economides 5 in 1988 (1 st Edition 1987) presented a chapter on "Fracturing Diagnostics Using Pressure Analysis" where they identified that "a log-log slope approaching one indicates restricted fracture extension at the fracture's extremeities and the requirement for a larger pad; whereas a slope greater than one indicates a restriction within the fracture . . . " This statement, however, was based on the understanding that the fracture efficiency was near unity.