Performance of a horizontal well in a bounded anisotropic reservoir: Part II: Performance analysis of well length and reservoir geometry

: Evaluation of the performance of horizontal wells is an important aspect in the enhancement of their productivity. This study provides mathematical computations, and analysis for theoretical well and reservoir considerations. The study investigates how well design and reservoirs geometry a ﬀ ect the overall performance of a horizontal well in a completely bounded reservoir throughout its productive life. A horizontal well in a rectangular reservoir with completely sealed boundaries is considered and the e ﬀ ect of dimensionless well length L D , dimensionless reservoir length x eD , and dimensionless reservoir width y eD on the pressure response over a given period of production using dimensionless time t D is studied. The mathematical model used was derived using source and Green ’ s functions presented in part I of this study. Appropriate well and reservoir parameters are considered and the respective dimensionless parameters are computed which are then used in computing dimensionless pressure P D and its dimensionless pressure derivative ′ P D . From the computations, the results obtained are analysed in diagnostic log – log plots with a discussion of the ﬂ ow periods. The results obtained indicate that an increase in dimensionless well length decreases pressure response during the in ﬁ nite-acting ﬂ ow at early times and during transition ﬂ ows at middle time but increases the pressure response during the pseudosteady state ﬂ ow at late times. The dimensionless reservoir width and length are observed not to in ﬂ uence dimensionless pressure response during the in ﬁ nite-acting ﬂ ow at early times and during the transition ﬂ ows at middle time, only a ﬀ ecting the prevalence time of the ﬂ ow periods. However it is observed that during the pseudo-steady state ﬂ ow at late times, dimensionless pressure response reduces with increased dimensionless reservoir length and width.


Introduction
The performance of any well will depend on its initial design and any improvements that can be made to enhance its productivity. Analysis of horizontal wells has continued to pose challenges due to the complexity when it comes to analysing the flow periods with several boundaries involved. This is due to the occurrence of transient and pseudosteady flows as different boundaries are affected. Previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] as discussed in part I [21] of this study have made major strides in studying pressure response in horizontal wells. However, the considerations of infinite-acting, separate sealed boundaries, and isotropy in calculations have continued posing challenges to the accuracy of results obtained. The consideration of completely sealed boundaries and a study of pressure response from inception to date have not been fully considered. In this study, by including individual directional permeabilities in computations, the effect of well length and reservoir geometry on the performance of a horizontal well from inception to date is investigated.

Reservoir description
The well and reservoir geometry is as described in Figure 1 and the well is considered to be centrally placed in the reservoir. The drainage volume considered is completely sealed in all axes.

Mathematical description
It is assumed that the vertical boundary will be felt first since the formation thickness is expected to be far much smaller compared to the width and length of a given reservoir. The mathematical model used considers two possible cases obtained from previous studies [22] and [23]. First, a case where the horizontal boundary parallel to the well (y-boundary) is felt first. In this case, the dimensionless pressure derived in part I [21] of this study is given by equation (1) and its equivalent dimensionless derivative is given by equation (2).
Second, a case where the horizontal boundary perpendicular to the well (x-boundary) is felt first, the dimensionless pressure as derived from part I of this study is given by equation (3) and its equivalent dimensionless pressure derivative is given by equation (4).

Results and discussion
Theoretical well and reservoir parameters for a centrally located well in a single layer such that = = = m n l 1 are considered and their respective dimensionless equivalents are computed. A line source is considered such that = y y D w D and = − r z z wD D wD . A monitoring point at the centre of the well such that = x x D w D approximates = β 2 during early time. Using Odeh and Babu Strategies discussed in part I [21] of this study, the approximate integration limits are identified. To compute dimensionless pressure and its dimensionless pressure derivative, all the considered and computed parameters are substituted into the models. The results are presented and analysed in diagnostic log-log plots.

Effect of dimensionless horizontal well length
To study how the dimensionless horizontal well length, L D affects the flow periods and dimensionless pressure, dimensionless horizontal well length is varied keeping the other parameters constant. Table 1 shows the theoretical dimensional values considered and the computed dimensionless variables. Table 2 shows the dimensionless flow period times as computed using Odeh and Babu strategies. From Table 2, it is observed that the y-boundary is felt first. Using equation (1), dimensionless pressure is computed, and dimensional pressure derivative is computed using equation (2).
The dimensionless pressure and dimensionless pressure derivative values computed are shown in Table 3. Figure 2 shows the plot of P D and ′ P D against t D on log-log axes where the solid lines represent P D against t D , while the dashed lines represent ′ P D against t D . From Figure 2, the infinite-acting flow is identified as the first flow period. This flow period starts shortly after the well is put into production. This flow period is evident with the flattening of the dimensionless pressure derivative plot at early times. This flow period is radial in the y-z plane and can be considered as the early radial flow period. It is noted that this flow period might end even before the vertical boundary is felt for the cases where the flow coming from the ends of the well starts influencing the pressure response. In such a case, a very short transition flow which is still radial continues until the flow reaches the vertical boundary. When the vertical boundary is felt, it is noted that as the dimensionless horizontal well length increases, an early linear flow period occurs in the y-z plane. This flow period is observed to prevail for a very short time and is evident when the dimensionless pressure derivative plot stops flattening and shows an upward trend. Where the early linear flow period does not occur, a transition flow occurs from the time the vertical boundary is felt until the time the flow starts coming from beyond the ends of the wellbore. This is evident for the first two cases of dimensionless horizontal well length considered. Where the early linear flow occurs as shown in the last three cases of the dimensionless horizontal well length considered, the period will end when the flow moves beyond the ends of the wellbore. At this point a transition flow occurs until the flow starts coming from beyond the ends of the wellbore. When the transition flow ends, a second radial flow is identified with the flattening of the graph of dimensionless pressure derivative. This flow period starts after the flow has started coming from beyond the ends of the wellbore and can be considered to be the late pseudoradial flow period with one boundary having been felt. This flow period will end when the y-boundary is felt which is followed by a transition flow.
The transition flow will prevail until the x-boundary is felt at a point where it can be considered that all boundaries have been felt and a pseudosteady state flow begins. On the plot, this is identified with the straight upward line at late time. This flow period will prevail to date. For the parameters considered, it is noted that as the dimensionless horizontal well length increases, the pressure response decreases during early time but increases during late time when a pseudosteady state behaviour is observed.

Effect of dimensionless reservoir length
To investigate the effect of dimensionless reservoir length, x eD on the flow periods and dimensionless pressure, constant values of well and reservoir parameters are considered and the dimensionless reservoir length is varied. Table 4 shows the dimensional values considered and the computed dimensionless parameters. Table 5 shows the dimensionless flow period times as computed using Odeh and Babu strategies. For the first two values of x eD considered, it is observed that the x-boundary is felt earlier than the y-boundary and thus the dimensionless pressure is computed using equation (3) and the dimensionless pressure derivative is computed using equation (4). For the other values of x eD considered, it is observed that the y-boundary is felt first and thus dimensionless pressure is computed using equation (1) and dimensionless pressure derivative is computed using      (2). Table 6 shows the computed values of dimensionless pressure and dimensionless pressure derivative. Figure 3 shows the plots of P D and ′ P D against t D on log-log axes where the solid lines represent P D against t D , while the dashed lines represent ′ P D against t D . From the parameters considered, for all the values of x eD , an infinite-acting flow is identified from the time the well is put into production. This flow is radial in the y-z plane and considered as the early radial flow period. This is evident in the flattening of the graph of dimensionless pressure derivative during early time. It is noted that the early radial flow period will end when the wellbore end effects start affecting the flow. This is followed by a transition flow until the vertical boundary is felt. When the vertical boundary is felt, a transition flow is identified which prevails until the flow starts coming from beyond the ends of the wellbore. This is indicated in the plot by an upward straight line. When the flow starts coming from beyond the ends of the wellbore, a radial flow in the x-y plane is    When all the boundaries have been felt, a pseudosteady state behaviour is identified with an upward straight line in the dimensionless pressure derivative graph. This flow will prevail for the rest of the productive life of the well. It is observed that there is no effect on the change in dimensionless reservoir length on the pressure response during early time. However, this is not the case as observed during middle and late times. During these times, we note that as the dimensionless reservoir length increases, the pressure response decreases with a large decrease during late times.

Effect of dimensionless reservoir width
Since it is expected that the dimensionless reservoir width, y eD will have an effect on the flow periods prevalence time, it is expected to have an effect on the pressure response.
To investigate this effect, constant well and reservoir parameters are considered and the dimensionless reservoir width is varied. Table 7 shows the dimensional values considered and the dimensionless parameters computed. Table 8 shows the dimensionless flow period times as computed using Odeh and Babu strategies. For the first three values of y eD considered, it is observed that the y-boundary is felt first and thus dimensionless pressure is computed using equation (1) and dimensionless pressure derivative is computed using equation (2). For the last two values of y eD considered, it is observed that the x-boundary is felt first and thus dimensionless pressure is computed using equation (3) and dimensionless pressure derivative is computed using equation (4). The computed values of dimensionless pressure and dimensionless pressure derivative are shown in Table 9. Figure 4 shows the plots of P D and ′ P D against t D on log-log axes where the solid lines represent P D against t D , while the dashed lines represent ′ P D against t D . From the parameters considered, an infinite-acting flow period is identified from the time the well is put into production. This flow period is radial in the y-z plane and considered to be the early radial flow period. This is evident as seen     from the flattening of the graph of dimensionless pressure derivative during early time. The identified early radial flow is observed to end when the wellbore end effects start affecting the flow. At this point a transition flow is identified which prevails until the vertical boundary is felt. When the vertical boundary is felt, a transition flow that prevails until the flow starts coming from beyond the ends of the wellbore is also identified. This is identified in the graph of dimensionless pressure derivative with an upward straight line. At the point when the flow starts coming from beyond the ends of the wellbore, a radial flow in the x-y plane considered to be the late pseudoradial flow period is identified. For the first two cases of y eD considered, it is observed that this flow will end when the y-boundary is felt. For the last three values of y eD considered, it is observed that this flow ends when the x-boundary starts having an effect on the flow. This flow is evident with the flattening of the dimensionless pressure derivative graph. Analysing the third value of y eD considered, a short transition flow is identified after the late pseudoradial flow period ends. This transition flow ends when the y-boundary is felt. For the first three cases of y eD considered, when the y-boundary is felt, a transition flow is identified and it prevails until the x-boundary is felt. This flow is identified from the straight upward line on the plot of dimensionless pressure derivative. For the last two cases of y eD considered, when the x-boundary is felt, a linear flow in the x-y plane which can be considered as the late linear flow period is also identified. This flow is evident in the plot of dimensionless pressure derivative by the upward straight line. This flow period will end when all the boundaries have been felt. At this point, a pseudosteady state behaviour is evident with a straight upward line. It is also noted that as much as the increase in dimensionless reservoir width has no effect on the pressure response during early time, it shows a decrease in pressure response during late time. Further, it is noted that an increase in the dimensionless reservoir width increases the prevalence time of the late pseudoradial flow period.

Conclusion
The study investigated how well design and reservoirs geometry affected the overall performance of a horizontal  well in a completely bounded reservoir throughout its productive life. A horizontal well in a rectangular reservoir with completely sealed boundaries was considered and the effect of dimensionless well length L D , dimensionless reservoir length x eD , and dimensionless reservoir width, y eD on pressure response over a given period of production using dimensionless time, t D was studied. Appropriate well and reservoir parameters were considered and the respective dimensionless parameters were computed which were then used in computing dimensionless pressure P D and its dimensionless pressure derivative ′ P D . From the computations, the results obtained were analysed in diagnostic log-log plots with a discussion of the flow periods.
From this study, the dimensionless horizontal well length and the reservoir geometry have an effect on flow periods and influence how a horizontal well performs from inception to date. It is observed that (1) From the time when the well is put into production, as many as seven flow periods can occur from the infiniteacting flow to full pseudosteady state behaviour. However, when considering the effect of boundaries separately, only five mathematical models are applicable. (2) Longer horizontal wells will experience low pressure response during early time but this will increase at late times. The results obtained indicate that an increase in dimensionless well length decreases pressure response during the infinite-acting flow at early times and during transition flows at middle time but increases the pressure response during the pseudosteady state flow at late times. (3) Large drainage volumes will have a reduced pressure response at late times. The dimensionless reservoir width and length are observed not to influence dimensionless pressure response during the infinite-acting flow at early times and during the transition flows at middle time, only affecting the prevalence time of the flow periods. However, it is observed that during the pseudosteady state flow at late times, dimensionless pressure response reduces with increased dimensionless reservoir length and width.
Nomenclature h reservoir thickness, ft h D dimensionless reservoir thickness k reservoir permeability, md k x directional permeability in the x-direction, md k y directional Permeability in the y-direction, md k z directional Permeability in the z-direction, md L well length, ft length in x-direction, ft x D dimensionless reservoir length in the x-direction x e reservoir length, ft x eD external dimensionless reservoir length x w source coordinate in the x-direction, ft x wD dimensionless source coordinate in the xdirection y width in y-direction, ft y D dimensionless reservoir width in the y-direction y e reservoir width, ft y eD external dimensionless reservoir width y w source coordinate in the y-direction, ft y wD dimensionless source coordinate in the y-direction z thickness in z-direction, ft z D dimensionless reservoir thickness z w source coordinate in the z-direction, ft z wD dimensionless source coordinate in the zdirection τ D dimensionless dummy variable for time Conflict of interest: Authors state no conflict of interest.