Summary. New analytic solutions are presented in real time and as Laplacetransforms for horizontal wells in reservoirs bounded at the top and bottom byhorizontal planes. Two types of boundary conditions are considered at theseplanes. and the Laplace-transform pressure solutions are used to includewellbore-storage and skin effects. Solutions are based on the uniform-flux, line-source solution, but differ from most existing solutions owing to the use of pressure averaging to approximate the infinite-conductivity wellborecondition and use of the correct equivalent wellbore radius for an anisotropicreservoir. New flow periods (regimes) are identified, and simple equations andexistence criteria are presented for the various flow periods that can occurduring a transient test. Introduction Determination of transient pressure behavior for horizontal wells hasaroused considerable interest over the past 10 years. An extensive literaturesurvey on horizontal wells can be found in Ref. 2. Most work dealing with thehorizontal-well problem uses the instantaneous Green's function techniquedeveloped by Gringarten and Ramey to solve the 3D isotropic diffusivityequation. Goode and Thambynayagam used finite Fourier transforms to solve theanisotropic problem for the line-source case. Because the infinite-conductivityinner-boundary condition (uniform pressure over the sand face) poses a verydifficult boundary-value problem. a uniform-flux condition on the innerboundary is commonly used. The infinite-conductivity solution is thenapproximated with either an equivalent-pressure-point or pressure-averagingtechnique. We prefer the pressure-averaging method because it requires no apriori information. is exact in the limit of a small wellbore radius, and ismore accurate at intermediate times than the equivalent-pressure-point method. These reasons are discussed further in Appendix A. Another feature of thesolutions presented in this paper is the use of the correct equivalent wellboreradius for an anisotropic formation, which guarantees that elliptical-floweffects near the well are treated correctly at late times. At early times it ispreferable to use the elliptical-cylinder solution. Solutions presented in thispaper. however, are sufficient for most practical problems. Using thesetechniques. we extend the work of Goode and Thambynayagam and Clonts and Rameyto obtain new analytic solutions for horizontal wells with and without theeffects of gas cap or aquifer. The wellbore-storage effect is accounted for, and new formulas are presented for the determination of reservoir parametersfrom the characteristics of different flow regimes. Solutions With and Without Gas Cap or Aquifer First we discuss the basic solutions for horizontal wells for theconstant-rate case without wellbore-storage and skin effects. These solutionswill then be combined with constant wellbore storage and/or measured downholeflow rate. The horizontal well shown in Fig. 1 is considered to be completed inan infinite anisotropic medium bounded above and below by horizontal planes. The boundaries of the reservoir in the horizontal directions are considered tobe so far away that they are not seen during the test. The permeabilities inthe principal directions are denoted by kx, ky, and kz. We develop thesolutions for the general case where the three permeabilities are all differentin Appendices A and B, but in the text we consider a transversely isotropicmedium and write k, kv - kH and kz -kv. The flow of a slightly compressiblefluid of constant compressibility and viscosity is assumed throughout themedium. Gravity effects are neglected. Two types of top and bottom boundaryconditions are considered. In the first case. both the top and the bottomboundaries have no-flow conditions. In the second case, one of the boundariesis at constant pressure, while the other is a no-flow boundary as before; thiscase can represent either a gas cap at the top boundary or an active aquifer(in which the water mobility is high compared with the mobility of thereservoir fluid) at the bottom. For convenience, we refer to the first model asthe no-flow-boundary model and to the second as the constant-pressure-boundarymodel. The notation of this paper assumes that in the latter model, theconstant-pressure boundary is at the top (the gas-cap case). but the formulasmay be readily adapted for the case of an aquifer at the bottom. During thelast few years, several solutions for horizontal wells have been presented. Most of these solutions are for the no-flow-boundary model, and apart from thework of Goode and Thambynayagam. none present solutions in the Laplace-transform domain. A solution for the constant-pressure condition atboth the top and bottom boundaries was presented by Daviau et al. This solutionis different from the constant-pressure-boundary solution presented here, inwhich one of the boundaries (top or bottom) is no-flow. This flexibility isimportant because if we have a constant-pressure boundary such as a gas cap, the well is usually drilled close to the other (no-flow) boundary. The solutionmethod is discussed in Appendix A, and the actual solutions are developed in Appendix B. Our solutions differ somewhat from other solutions given in theliterature because we approximate the infinite-conductivity condition byaveraging the pressure along the well length instead of using an equivalentpressure point. A discussion of the pressure-averaging technique is given in Appendix A, together with a derivation of the correct equivalent wellboreradius to be used for an anisotropic formation. We define dimensionless timeand pressure (in field units) by ................(1) ................(2) and other dimensionless parameters ............................(3a) ............................(3b) ............................(3c) In the time domain, the dimensionless pressure response, pD, forconstant-rate drawdown is most conveniently given as a time integral over theinstantaneous Green's function (see Appendix B): ...................(4) .......(5) .........................(6) SPEFE P. 86^

Fractures and faults are common features of many well-known reservoirs. They create traps, serve as conduits to oil and gas migration, and can behave as barriers or baffles to fluid flow. Naturally fractured reservoirs consist of fractures in igneous, metamorphic, sedimentary rocks (matrix), and formations. In most sedimentary formations both fractures and matrix contribute to flow and storage, but in igneous and metamorphic rocks only fractures contribute to flow and storage, and the matrix has almost zero permeability and porosity. In this study, we present a mesh-free semianalytical solution for pressure transient behavior in a 2D infinite reservoir containing a network of discrete and/or connected finite-and infinite-conductivity fractures. The proposed solution methodology is based on an analytical-element method and thus can be easily extended to incorporate other reservoir features such as sealing or leaky faults, domains with altered petrophysical properties (for example, fluid permeability or reservoir porosity), and complicated reservoir boundaries. It is shown that the pressure behavior of discretely fractured reservoirs is considerably different from the well-known Warren and Root dual-porosity reservoir model behavior. The pressure behavior of discretely fractured reservoirs shows many different flow regimes depending on fracture distribution, its intensity and conductivity. In some cases, they also exhibit a dual-porosity reservoir model behavior.

Fractures are common features of many well-known reservoirs. Naturally fractured reservoirs contain fractures in igneous, metamorphic, and sedimentary formations. Faults in many naturally fractured carbonate reservoirs often have high-permeability zones, and are connected to many fractures with varying conductivities. Furthermore, in many naturally fractured reservoirs, faults and fractures can be discrete (i.e., not a connected-network fracture system). New semianalytical solutions are used to understand the pressure behavior of naturally fractured reservoirs containing a network of discrete and/or connected (continuous) finite-and infinite-conductivity fractures. We present an extensive literature review of the pressure-transient behavior of fractured reservoirs. First, we show that the Warren and Root (1963) dual-porosity model is a fictitious homogeneous porous medium because it does not contain any fractures. Second, by use of the new solutions, we show that for most naturally fractured reservoirs, the Warren and Root (1963) dual-porosity model is inappropriate and fundamentally incomplete for the interpretation of pressure-transient well tests because it does not capture the behavior of these reservoirs.We examined many field well tests published in the literature. With few exceptions, none of them shows the behavior of the Warren and Root (1963) dual-porosity model. These examples exhibit very diverse pressure behaviors of discretely and continuously fractured reservoirs. Unlike the single derivative shape of the Warren and Root (1963) model, the derivatives of these examples exhibit many different flow regimes depending on fracture distribution and on their intensity and conductivity. We show these flow regimes with our new model for discretely and continuously fractured reservoirs. Most well tests published in the literature do not exhibit the Warren and Root (1963) dual-porosity reservoir-model behavior. If we interpret them by use of this dualporosity model, then the estimated permeability, skin factor, interporosity flow coefficient (k), and storativity ratio (x) will not represent the actual reservoir parameters.

Summary This paper presents a new general method for solving the pressure diffusion equation in laterally composite reservoirs, where rock and fluid properties may change laterally as a function of y in the x-y plane. Composite systems can be encountered as a result of many different types of depositional and tectonic processes. For example, meandering point bar reservoirs may be approximated as lateral composite systems. Reservoirs with edge water encroachment are another example of such systems. The new solution method presented is based on the reflection transmission concept of electromagnetics to solve fluid-flow problems in 3D nonhomogeneous reservoirs, where heterogeneity is in only one (y) direction. A general Green's function for a point source in 3D laterally composite systems is developed by using the reflection-transmission method. The solutions in the Laplace transform domain are then developed from the Green's function for the pressure behavior of specific composite reservoirs. The solution method can also be applied to many different types of wells, such as vertical, fractured, and horizontal in composite reservoirs. The pressure behavior of a few well-known laterally composite systems are investigated. It is shown that a network of partially communicating faults and fractures in porous medium can be modeled as composite systems. It is also shown that the existing solutions for a partially communicating fault are not valid when the fault permeability is substantially larger than the formation permeability. The derivative plots are presented for selected faulted, fractured, channel, and composite reservoirs as diagnostic tools for well-test interpretation. It is also shown that if the composite system's permeability varies moderately in the x or y direction, it exhibits a homogeneoj)s system behavior. However, it does not yield the system's average permeability. Furthermore, the composite systems with distributed low permeability zones behave as if the system has many two no-flow boundaries. Introduction Deltaic sandstone deposits are the world's most prolific reservoirs. In these reservoirs, rock properties vary laterally because of the sorting of sediments according to size and weight as they are transported. For example, the heavier and coarser materials, such as pebbles and gravel, settle closer to the shorelines, while lighter and finer ones are deposited in deeper water. This process often, but not always, creates regularly sorted channels. Degradation of rock properties, such as porosity and permeability, usually occurs in the direction of flow in these deltaic sediments. As noted by Levorsen,1 the boundaries between different zones are sharp planes, but more often they are gradational. Whether gradual or sharp, we assume that the lateral zonation is in the x-y plane, as shown in Fig. 1. This type of formation is called laterally composite reservoir, which may exhibit both sharp and gradual changes in the formation properties. The other flow problems, such as edgewater drive or encroachment, also may be treated as laterally composite systems. Braided and meandering stream reservoirs are also good examples of lateral composite systems. The lateral continuity between stream channels can be quite good or poor, depending on the shale content of each sand body. Since the early work of Hurst,2 many studies of radially composite reservoirs have been made in connection with secondary and tertiary oil recovery processes. Studies on laterally composite media, which are called linear discontinuities, are few; the first was conducted by Bixel, Larkin, and Poollen.3 They solved the pressure diffusion equation in a 2D infinite two-zone reservoir. Fractures and/or faults are common features of many well-known reservoirs and their effects on the reservoir behavior are well studied. The effect of a partially communicating fault on the pressure behavior of a well was investigated numerically by Stewart et al.4 and analytically by Yaxley.5 Although Yaxley's5 paper is a comprehensive study on the behavior of a partially communicating fault, it is not valid when the permeability of the fault is larger than the permeability of the formation. In other words, the flux parallel to the fault plane is neglected. In general, when the porosity of the formation is low, say less than 15%, it is possible that the permeability of the fault zone can be greater than the formation permeability. It is well known that many reservoirs can have a system of sealing or partially communicating (sealing) faults. For example, the Cormorant field in North Sea, as shown in Fig. 2, is extensively faulted, as discussed by Ruijtenberg, Buchanan, and Marke.6 For these types of faulted reservoirs, a single partially communicating fault model could be misleading. If the multiple faults are partially communicating, it is most likely that a well behavior will be affected by these faults. Analytical solutions are not available even for a well between two partially communicating faults. The behavior of a well in a system of sealing or partially communicating multiple faults will be investigated in this study. A similar problem but in a different context, the behavior of a well near an infinite-conductivity vertical fracture, was studied by Cinco, Samaniego, and Dominguez.7 As in faulted reservoirs, it is again unlikely that a reservoir will have a single fracture. Most reservoir studies indicate that a system of fractures may cut through the field according to the regional stress field and the tectonic past of the formation. If fractures form a network in a reservoir, it may then be modeled as a dual-porosity system. If the fractures are not densely connected, as shown in Fig. 3, they also may be modeled as laterally composite systems. This is particularly important if the conductivity is finite. If the fractures are filled and had gone through certain geochemical and tectonic changes, it is quite possible that the permeability of the fractured zone could be lower than the formation permeability. If the effects of the fault strikes are negligible, the behaviors of low-conductivity fractures and partially communicating faults also can be modeled as laterally composite systems. Another problem associated with composite systems is the application of radially composite models to laterally composite reservoirs. With the exception of unnaturally created conditions, such as water injection, it is difficult to imagine a natural sedimentary formation where its properties are symmetrical about the origin (radially composite) or about the axes (elliptically composite). It is almost impossible to drill a well and hit the center of a deltaic sedimentary formation, which could be approximated by either a circular or elliptical composite reservoir. The well will most likely strike one of the channels. Therefore, a laterally composite model is better suited for this case, given the geologic dimension of most sedimentary channels. In this study, we develop new transient solutions for general laterally composite reservoirs using the solution technique presented by Kuchuk and Habashy.8 The solutions are also presented for several practical well-test problems.

Summary Determination of the influence function of a well/reservoir system from thedeconvolution of wellbore flow rate and pressure is presented. Deconvolution isfundamental and is particularly applicable to system identification. A varietyof different deconvolution algorithms are presented. The simplest algorithm isa direct method that works well for presented. The simplest algorithm is adirect method that works well for data without measurement noise but that failsin the presence of even small amounts of noise. We show, however, that amodified algorithm that imposes constraints on the solution set works verywell, even with significant measurement errors. Introduction In reservoir testing, we generally know the characteristic features of thesystem from its constant-flowrate and constant-pressure behavior. Thus it isimportant to determine the constant-rate or -pressure behavior of the systemfor the identification of its characteristic features. For instance, identification of a one-half on a log-log plot of the pressure data mayindicate a vertically fractured well, as two parallel straight lines on a Homergraph may indicate a fractured reservoir. The presence of eitherwellbore-storage or flow-rate variations, however, usually masks characteristicsystem behavior, particularly at early times. For many systems, it is desirableto have a wellbore pressure that is free of wellbore-storage and/orvariable-flowrate effects to obtain information about the well/reservoirgeometry and its parameters. For example, the effects of partial penetration, hydraulic fractures, solution gas within the vicinity of the wellbore, gas cap, etc., on the wellbore pressure can be masked entirely by wellbore storage, flowrate variations, or both. Although deconvolution of pressure and flow rate has not been commonly usedfor reservoir engineering problems, one can still find a few works ondeconvolution (computing influence function) in the petroleum engineeringliterature. Hutchinson and Sikora, Jones et al., and Coats et al. presentedmethods for determining the influence function directly from field data. Jargon and van Poollen were perhaps first to use the deconvolution ofwellbore-flowrate and pressure data to compute the constant-rate behavior (theinfluence function) of the formation in well testing. Bostic et al. used adeconvolution technique to obtain a constant-rate solution from a variable-rateand -pressure history. They also extended the deconvolution technique tocombine production and buildup data as a single test. Pascals also usedproduction and buildup data as a single test. Pascals also used deconvolutiontechniques to obtain a constant-rate solution from variable-rate (measured atthe surface) and -pressure measurementof a drawdown test. Kucuk and Ayestaranpresented several deconvolution methods including the Laplace transform andcurve fit. Thompson et al. and Thompson and Reynolds presented a stableintegration procedure for deconvolution. This paper focuses on deconvolution methods. Mathematically, thedeconvolution operation can be defined as obtaining solutions forconvolution-type, linear, Volterra integral equations. In reservoir testing, itis defined as determining the pressure behavior (in-fluence function orunit-response behavior) of a system fro simultaneously measured downholepressure and flow rate. In other words, deconvolution computes the pressurebehavior of a well/reservoir system as if the well were producing at a constantrate. We call the computed pressure behavior of the system "deconvolvedpressure." Convolution Integral The convolution integral, which is a special case of the Volterra integralequations, is widely known for providing techniques for solving time-dependentboundary-value problems. It is also known as the superposition theorem (i.e., Duhamel's principle) and has played an important part in transient well-testanalysis. In recent played an important part in transient well-test analysis. In recent years, there has been more interest in the solution of theconvolution integral in connection with analysis of simultaneously measuredwellbore pressure and flow rate. Although we restrict our discussion mostly todetermining influence functions for the constant-rate case, we do treat theconvolution integral in a general manner. In other words, the influencefunction can also be the solution of the constant-pressure case. In a linear causal system (reservoir), the relationship between input (thetime-dependent boundary condition that can be either the flow rate or pressure)and output (the system response measured as either the flow rate or pressure)at the wellbore can be described as a convolution operation. We let thequantities measured at the wellbore, above the sandface, be p =wellborepressure, Q =cumulative wellbore production, and q =wellbore flow rate. The convolution integral is (1) where The functions delta p (t) and Q (t) or q (t) are the solutions of thediffusivity equation for the constant-flowrate or -pressure case with orwithout wellbore-storage and skin effects. Although it is usually small, thedeconvolved pressure will always be affected by the wellbore volume between themeasurement point and the sandface because the sandface flow rate is differentfrom the wellbore flow rate, q, when it is measured by the flowmeter at somewellbore location above the perforations. As shown by Coats et al., the general solutions of the diffusivity equationwith the first and second kind of internal boundary conditions and nonperiodicinitial and outer-boundary conditions, satisfy the constraints (2) (3) (4) and (5) when the real time is greater than 1 second and if the diffusivity constant, k/phi mu c, is not very small. Coats et al. used linear programming with theabove constraints to compute K(t) from measured g(t) programming with the aboveconstraints to compute K(t) from measured g(t) and f(t). Here we use theseconstraints to compute the system influence function. SPEFE P. 53