Paper Number – 33

Assisted History Matching for Surface Coupled Conventional and Unconventional Gas Reservoirs

Deepankar Biswas, President SiteLark LLC 14001 N Dallas Pkwy Suite 1200 Dallas TX 75240 email: dbiswas@sitelark.com Abstract For a gas reservoir, the intake pressures are available at a delivery point and the operator is

obligated to supply gas honoring such downstream pressure requirements. As a result, it is

imperative that the reservoir deliverability prediction be coupled with the pressure drop in the

surface network. Additionally, this coupled system should not only honor the imposed delivery

point pressure constraints but also the historical attributes of pressures and rates. In this paper, a

modified Gauss-Newton method is utilized in conjunction with a non-linear parameter estimation

algorithm to history match a surface-reservoir coupled gas reservoir simulation. Examples are

shown for dry gas, shale gas and coal-bed methane reservoirs.

There are three significant contributions of this paper, namely (a) an assisted history matching

algorithm proves efficient in comparison to arbitrary perturbation of decision variables (b) a

coupled reservoir-surface model renders a more accurate pressure prediction of the total system

and finally, (c) an unstructured grid simulator provides the platform for accurate and cost-

effective reservoir modeling alternative.

Keywords History matching, conventional gas reservoirs, unconventional gas reservoirs, shale gas, coal-bed methane, CVFE formulation, Weymouth equation, perturbation, Gauss-Newton, decision variables

Introduction Literature is rife with papers (Emanuel and Ranney1, Schiozer and Aziz2, Hepguler et al.3 Litvak

et al.4 , Byer et al.5 etc) that investigate and apply various numerical experiments of coupling of

reservoir and surface models. Coupling techniques of explicit vs implicit, full system solve vs

domain decomposition and applications to large fields to optimize production and maximize

surface facility utilization have been extensively examined. However, one critical component of

any modeling effort, time-consuming calibration to known (observed) data points or history

matching process, is not emphasized. The proposed research is motivated by the fact that vast

amount of time and effort is spent to history match and condition reservoir models to dynamic

1

data. An additional motivation for this research is provided by the observation that the track

record of history matched predictions is erratic. Anecdotal evidence suggests that no more than

one in three history-matched predictions have proven to be accurate. Not surprisingly, some

practitioners even say no more than one in twenty. A good example of the disparity between

history matching and predicting is shown in Figure 1.

Figure 1 shows that there is generally a poor correlation between the "goodness" of the history

match and the goodness of the resulting prediction. Indeed, in one of the cases in Figure 1, the

best history match resulted in the worst prediction. Several reasons exist for this poor track record

of history matched reservoir models:

1. In brute force history matching procedures, the reservoir model is adjusted in an ad hoc

fashion until the available production history data is matched. In the process of these

adjustments, the model may cease to be consistent with the prior geological model.

Predictions made with such a model can therefore be in error.

2. History matching algorithms presume that the deviation of the predicted response from

the observations is attributable to only a small set of reservoir variables such as

permeability and porosity. In actuality, the production response is affected by a much

larger variable set for example relative permeabilities, capillary pressures, fractures etc.

3. The uncertainty in reservoir parameter distribution and their influence on the production

response is severely under-represented. The trend has been to start with a single

reservoir model and perform history matching to obtain a single perturbed reservoir

model. The history matched model is non-unique and a suite of models are possible,

each satisfying the available production data. In order to realistically represent the

uncertainty in predicted response, the predictions obtained from several history matched

realizations have to be aggregated.

In order to address some of the issues identified above, a new method is proposed where the

underlying heterogeneity is conditioned to production response. Specifically, the proposed

approach will:

• Focus predominantly on gas reservoirs where the number of decision variables

(parameters) is significantly less whereas the benefits of cost-effective history matching

are still immense. History matching using this reduced variable set will reduce

inaccuracies in the predictions for future performance.

2

• Utilize an inverse algorithm that is sufficiently optimized to converge to solutions faster

and more accurately. The inverse algorithm will utilize a reduced set of optimization

parameters to accomplish the history match by first performing a Gauss-Newton

convergence procedure followed by a rigorous non-linear regression of parameter

estimation.

• A flow simulator based on a CVFE will be utilized which is suitable to incorporate

permeability tensors. In the context of inverse problem this is significant because during

the iterative inverse procedure, the depiction of heterogeneity in the reservoir model

alters. As a result, the simulation grid has to be frequently re-oriented in order to

constantly align with the principal directions of permeability. Since in our approach, the

flow simulator can handle a tensor representation of permeability, the simulation grid

remains invariant albeit providing extra degrees of freedom for the history matching

algorithm to perturb. The resultant history matched models are thus likely to be accurate.

Approach In order to facilitate a seamless framework to test and validate the above methodology, following

three components are needed, namely (1) reservoir simulator, (2) network model and (3) inverse

algorithm. The reservoir simulator for the dry gas reservoir is based on control-volume finite

element (CVFE) scheme where mass conservation equations are solved coupled with well bore

equations. The reservoir models for the shale gas and coal-bed methane reservoirs solve the

material balance and deliverability equations for water and gas phases simultaneously. These

zero dimensional model parameters can be systematically perturbed to match historical

production. Finally, the network simulator is based on equations describing steady-state gas flow

(Weymouth Equation) through pipes. Mass flow rates are conserved at every node of the surface

configuration.

History Matching Algorithm Despite their apparent utility, inverse models are used much less compared to calibrations

conducted using only trial-and-error methods. This is partly because of the difficulties inherent in

inverse modeling, which are related to the mathematics used to represent the processes, the

complexity of the simulated systems, and the sparsity of data in most situations; and partly due to

a lack of effective, versatile inverse models. On the other hand, inverse algorithms, though an

evolving tool, provide modelers to leverage the insight available from their models and data. For

the current research, a publicly available computer program, UCODE13, is used. It has two

3

attributes that are not jointly available in other inverse models: (1) the ability to work with any

mathematically based model or pre- or post-processor with ASCII or text only input and output

files, and (2) the inclusion of more informative statistics.

UCODE is designed to allow inversion using existing application models that use numerical

(ASCII or text only) input, produce numerical output, and can be executed in batch mode.

Specifically, the code was developed to:

1. manipulate application model input files and read values from application model output

files

2. compare user-provided observations with equivalent simulated values derived from the

values read from the application model output files using a weighted least-squares

objective function

3. use a modified Gauss-Newton method to adjust the value of user selected input

parameters in an iterative procedure to minimize the value of the weighted least-

squares objective function

4. report the estimated parameter values

5. calculate and print statistics to be used to

a. diagnose inadequate data or identify parameters that probably cannot be

estimated

b. evaluate estimated parameter values

c. evaluate how accurately the model represents the actual processes

d. quantify the uncertainty of model simulated values.

UCODE offers flexibility to the user in terms of defining observed values. For instance, in case

the observed value does not intersect with a computational node then the computer program can

be instructed to interpolate between neighboring values. Moreover, if calibration is needed for a

variable which is not directly observable yet is a function of the observed values, UCODE can be

requested to read in a function that defines this relationship and perform the calibration

accordingly.

Implementation Details The reservoir simulator and the network model are strongly coupled at the end of each time step.

In other words, for every timestep given the reservoir parameters and production rates, the

reservoir simulator solves for the wellbore flowing variables. Thereafter, the network simulator

solves for the pressures at every node of the network configuration while honoring the

downstream (trunkline) and upstream (wellbore variables) constraints. The same process is

repeated for every timestep until the final time of simulation is reached.

4

The history matching algorithm, on the other hand, acts as the outer loop of the composite

algorithm. In particular, the algorithm progresses with performing forward simulations, perturbing

the parameters one at a time. A forward simulation is a combination of reservoir and network

simulation. For instance, if there are three matching parameters then the history matching

algorithm will perform three forward simulations in a given iteration where each run incorporates

perturbed value of one and only one matching parameter. The values of the other matching

parameters are kept the same for that particular iteration. The updated magnitude of perturbation

for each parameter for the next iteration is computed based on sensitivity analysis. This process

proceeds until convergence is achieved i.e. the perturbations are less than a set tolerance value.

Results and Discussions The sample problem consists of a gas reservoir with three wells. The three wells start production

at different times. Their schedule and production rates are shown in Table 1. The purpose of the

reservoir simulation part is to demonstrate the effects of interference between wells because of

the fact that their drainage radii overlap. Therefore, when the last well was drilled the reservoir

pressure encountered at the drilling location was less than initial reservoir pressure. Figure 2

depicts the surface network configuration of the sample problem. As can be seen in the figure,

the pressures at points A, B and C are the unknowns of the system. The goal of the surface

network simulator is to compute these pressures while honoring the downstream pressure

(pressure specified at point D) and the pressures and rates suggested at the wells by the

reservoir simulator. Table 2 enlists the observations that are used as the basis of perturbing and

matching the history matching parameters. These are shut-in pressures of the wells at different

times. Notice that Well 1 has more observations than the rest, mainly because it was the first well

drilled and monitored better. For the other two wells there is only one observation point each

which is the shut-in pressure at the end of the third year. Finally, for the history matching

algorithm, permeabilities in x and y directions and the porosity are selected as the matching

variables. However, several different cases have been run spanning from a case where the bulk

properties of the reservoir are altered to a case where properties of each simulation grid block is

treated as a matching parameter.

Figures 4-7 show the gradually improving solution behavior during the evolution of the history

matching algorithm. If the surface algorithm was not coupled with the reservoir flow equations,

the history matching algorithm would have terminated in the situation shown in Figure 6 where

the surface model prediction shows significant departure from observed values.

5

Multi-Parameters Substitution To test the robustness of the history matching algorithm a case where all the grid block variables

i.e. 3 times 40 times 5 (600 variables) are selected as perturbation candidates. Figure 16

presents the match with the starting values. Figures 16 and 17 show the progressive

improvement of the reservoir simulation match after the 1st and 10th iteration, respectively. Given

the complexity of the problem, the surface network was de-coupled from the reservoir simulator

for these runs. Clearly, comparison of Figures 17 and 18 indicate that the history matching

algorithm is proceeding towards the correct solution, although the convergence is slow.

Furthermore, the standard deviation is increasing with iterations indicating that the history

matching algorithm is perturbing localized areas around the wellbore and keeping variables away

from them intact. This fact further confirms the observation that the localized parameter

perturbation is preferred over global substitution as exhibited by the faster convergence of the

partial substitution experiments performed elsewhere.

Unconventional Gas Example Finally, an unconventional gas reservoir example is included. The main difference in terms of gas

storage between conventional and unconventional gas reservoirs is that for the latter surface

adsorbed gas could be significant part of the in-place volumes thereby producing them would

require pressures to deplete below the adsorption pressure. As is shown in Figure 10, as the

water is produced from a coal-bed methane reservoir, gas production steadily increases until it

reached its maximum beyond which it starts to decline. Similarly for Barnett Shale example

(Figure 11), although there is no water production, the gas production exhibits distinct behavior

where early production is governed by fracture flow whereas the mid term and late production

gets increasing contribution from the matrix. Clearly, the unconventional reservoirs have a

different set of history matching parameters. The model developed provides a means to

systematically alter such parameters to obtain a satisfactory match.

Conclusions A generalized framework is developed where a coupled model of reservoir and surface

simulators is driven by a parameter estimation algorithm to obtain calibrated solutions and

ultimately, a history matched output. It is shown that not only this is feasible but also it results in

better accuracy as compared to using one model (reservoir simulator) in a standalone fashion.

Examples for altering bulk reservoir properties exhibited gradual convergence towards observed

values. A case where parameters for all the reservoir simulation blocks are treated as calibration

6

parameters is also considered. Although the results showed gradual improvements with

increasing iterations, the convergence is slow as compared to the former cases. This clearly

indicates that there is a limit with respect to the size of the parameter set that can be handled by

this algorithm. Unconventional gas reservoir example is also included to demonstrate the

different set of parameters that govern the flow in such fractured reservoirs.

Finally, it is also emphasized that mathematically consistent yet non-physical solutions may arise

while applying this technique. Caution must be exercised in accepting the solutions of the

coupled method. Judicious discussions with geologists and prior knowledge of the reservoir and

its analogs should be utilized to bridge this shortcoming.

References

1. Emanuel, A.S. and J.C. Ranney: “Studies of Offshore Reservoir with an Interfaced Reservoir/Piping Network Simulator,” JPT, March 1981.

2. Schiozer, D.J. and K. Aziz: “Effect of Chokes on Simultaneous Simulation of Reservoir and Surface Facilities,” SPE 26308, May 1993, unsolicited paper.

3. Hepguler, G., S. Barua, and W. Bard: “Integration of a Field Surface and Production Network with a Reservoir Simulator,” SPE Computer Applications, June 1997.

4. Litvak, M.L. et al.: “Integration of Prudhoe Bay Surface Pipeline Network and Full Field Reservoir Models,” paper SPE 38885 prepared for presentation at the 1997 SPE Annual Technical Conference and Exhibition held in San Antonio, TX, 6-8 Oct.

5. Byer, T.J., M.G. Edwards and K. Aziz: “Preconditioned Newton Methods for Fully Coupled Reservoir and Surface Facility,” paper SPE 49001 prepared for presentation at the 1998 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, Sept 27-30.

6. Sorensen, F. and L.C. Little: “A 3-Dimensional Approach to the Modeling of the Hoadley-Westrose Gas Reservoir-Surface Pipeline Network,” paper SPE 26143 prepared for presentation at the 1993 SPE Gas Technology Symposium held in Calgary, Canada, June 28-30.

7. Holst, R. and L. Moalowany: “Computer Optimization of Large Gas Reservoir with Complex Gathering Systems,” paper SPE 56548 prepared for presentation at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, TX, Oct 3-6.

8. Yee, D. and R. Poitevien: “A Cost Effective Approach to Modeling and Managing Large Gas Fields,” paper SPE 84222 prepared for presentation at the 2003 SPE Annual Technical Conference and Exhibition held in Denver, CO, Oct 5-8.

9. Aziz, K., and S. Verma: “A Control Volume Scheme for Flexible Grids in Reservoir Simulation,” SPE 37999 presented at the 1997 Reservoir Simulation Symposium, Dallas, TX, June 8-11.

10. Gunasekera, D., P. Childs, J. Herring, and J. Cox: “A Multi-Point Flux Discretization Scheme for General Polyhedral Grids,” paper SPE 48855 prepared for presentation at the 1998 SPE International Oil & Gas Conference and Exhibition held in Beijing, China, Nov 2-6.

11. Naji, H.S.A.: Finite difference and control-volume finite element simulation of naturally fractured reservoirs, Ph.D. Dissertation, Colorado School of Mines (Oct. 1993).

12. Biswas, D. et al.: “An Improved Model to Predict Reservoir Characteristics During Underbalanced Drilling,” paper SPE 84176 prepared for presentation at the 2003 SPE Annual Technical Conference and Exhibition held in Denver, Colorado, U.S.A., Oct. 5 – 8.

7

13. Poeter, E.P. and M.C. Hill: Documentation of UCODE – a Computer Code for Universal Inverse Modeling, US Geological Survey, Water-Resources Investigations Report 98-4080.

14. Hill, M.C.: “Methods and Guidelines for Effective Model Calibrations,” US Geological Survey, Water-Resources Investigations Report 98-4005.

15. Bos, C.F.M., "Optimizing the Value Chain of the Upstream Oil & Gas Industry - Why we need a new approach to training," Journal of Petroleum Sciences and Engineering, pp. 1-10, 2003.

.

Time (Days)

Well-1 (SCF/D)

Well-2 (SCF/D)

Well-3 (SCF/D)

0 40,000 0 0 365 60,000 40,000 0 730 100,000 60,000 60,000

Table 1. Well recurrent data for the sample problem

OBS-Name

Shut-in press (psia)

Time (days)

1A 6000 0

1B 5907 73

1C 5856 170

1D 5779 365

1E 5605 535

1F 5446 730

1G 5233 842

1H 4857 1095

2A 4972 1095

3A 5036 1095

Table 2. Observed data points for the sample problem. In OBS-Name, first digit indicates well number and the second letter denotes the observation point

8

Figure 1. Relationship between quality of history match and prediction accuracy [adapted

from (Bos et al.,15)]. The vertical axis shows the "goodness" of agreement between prediction and actual performance. The plot shows that the agreement with actual

performance is always poorer than with the history match.

Figure 2. Network configuration for the sample problem

Figure 3. Numerical and well locations for the sample problem

9

Figure 4. History matching results for experiment 1 with start values of the matching

parameters

Figure 5. History matching results for experiment 1 with 1st iteration values of the matching parameters

Figure 6. History matching results for experiment 1 with 4th iteration values of the matching parameters

10

Figure 7. History matching results for experiment 1 with 10th iteration values of the matching parameters

Figure 8. History matching results for experiment 4 with 1st iteration values of the

matching parameters

Figure 9. History matching results for experiment 4 with 10th iteration

11

1.0

10.0

100.0

1000.0

0.10 1000.10 2000.10 3000.10 4000.10 5000.10 6000.10 7000.100.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

Qg (MSCF/D)Qw (STB/D)Qg-A (MSCF/D)Qw-A (STB/D)CumG (MM SCF)CumW (M STB)

Time (Days)

Rate

Cum

ulat

ive

Pro

duct

ion

Time (Days)

Rate

Cum

ulat

ive

Pro

duct

ion

Figure 10. Model prediction and history match for a coal-bed methane example

1.0

10.0

100.0

1000.0

10000.0

0.10 1000.10 2000.10 3000.10 4000.10 5000.10 6000.10 7000.10 8000.10-500.0

0.0

500.0

1000.0

1500.0

2000.0

2500.0

Qg (MSCF/D)Qw (STB/D)Qg-A (MSCF/D)Qw-A (STB/D)CumG (MM SCF)CumW (M STB)

Figure 11. Model prediction and history match for Barnett Shale example

12