New Solutions for Well-Test-Analysis Problems: Part 2 Computational Considerations and Applications

Abstract

New Solutions for Well-Test-Analysis Problems: Part 2 Computational Problems: Part 2 Computational Considerations and Applications Summary. Computational considerations in obtaining well responses andpressure distributions for several problems presented in Part 1 are pressure distributions for several problems presented in Part 1 are discussed. In addition, new asymptotic expressions for pressuredistributions in closed drainage volumes applicable during theboundary-dominated flow period are derived. Interestingly, theseexpressions, which are much simpler than those available in the literature, can be used to derive shape factors for a variety of completion conditions(vertical, horizontal,) and vertically fractured wells). Application ofconstant-rate solutions to more complex conditions is also presented. Introduction In Ref. 1, we derived point-source solutions in the Laplace-transformdomain for a wide variety of unsteady-state flow problems ofinterest to well-test analysis. These solutions can also be used to computepressure responses in infinite and bounded reservoirs for a wide pressure responses in infinite and bounded reservoirs for a wide range of wellbore configurations: partially penetrating vertical wells, horizontal wells, fractured wells, etc. An extensive library ofsolutions is documented in Ref. 1. The primary purpose of this paper is to document procedures toovercome some of the practical problems in computing well responses for constant rate production noted in Ref. 1. We examine the following problems of interest.A vertically fractured well in an in finite reservoir that isnaturally fractured and produced at a constant rate. This problemexamines the computation of integrals involving the modified Besselfunction, K0 (x), for small values of its argument. Problemsinvolved in such computations are discussed in Refs. 2 and 3.Computation of horizontal well responses in homogeneous ornaturally fractured reservoirs. The primary issue we deal withinvolves recasting series that behave as though they are a divergentseries over certain time ranges.Computation of dimensionless pressure distributions forfractured wells in bounded systems (circles and rectangles). Considerationof this problem allows us to discuss procedures to compute responses forall times. Thus, "patching" of infinite reservoir andbounded reservoir solutions is avoided.Some solutions from Ref. 1 involve computation of doubleinfinite fourier series. These computations can be exceedinglydifficult. These computational problems are addressed by examininghorizontal well performance in a closed rectangular drainage region. This discussion should enable the reader to obtain solutionsfor other boundary conditions. In the second section of this paper, we derive asymptoticapproximations for some of the problems noted above and use thelong-time asymptotic expressions to compute shape factors for wellsproducing in bounded systems. These shape factors can be used producing in bounded systems. These shape factors can be used to compute inflow relationships for a wide variety of conditions. The principal advantage of the solutions given in Ref. 1 is thatthey permit us to solve more complex problems, particularlyvariable-rate problems. In the third section of this paper, wedemonstrate this aspect by considering two variable-rate problems ofinterest for vertically fractured and horizontal wells: the influenceof wellbore storage and skin and constant-pressure production. All results presented here examine the flow of a slightlycompressible fluid in a homogeneous of heterogeneous porousmediim subject to the idealization of Warren and Root. It is wellknown that responses for wells producing naturally fracturedreservoirs are governed by the matrix storativity, w, and the transfercoefficient. These variables are defined by ...............................................(1) and ...............................................(2) ...............................................(3) where s = Laplace-transform variable with respect to dimensionlesstime td, td=0.00002637 (nt/L2),........................(4) where ...............................................(5) Because f (s) appears in our solutions multiplied by s, forconvenience, we define u=sf (s) .....................................(6) When we discuss responses in homogeneous reservoirs, we setf (s) = 1 (i.e., u=s) and replace n given by E q. 5 with n = K(oc, u). We present all solutions in dimensionless form for generality. Thedefinitions of dimensionless pressure, pd, dimensionless distancerd, dimensionless horizontal distance, ld, and the dimensionlessvertical distance, zd, are given by ...............................................(7) ...............................................(8) ...............................................(9) ZD=Z/h.........................................(10) The dimensionless drainage area is AD = A/L2,.....................................(11) where A = pi r2e or A = xeye, and tAD = tD/AD....................................(12) We establish the following notation for convenience f(a+b+c..)=f(a+b+c...)+f(a-b+c...)+...,........(13) i.e. f (a+b+c...) represents the sum of the function f for allpossible combinations of signs in its argument. possible combinations of signs in its argument. ...............................................(14) SPEFE P. 369